LNTwww - User contributions [en]

LNTwww:Privacy policy

2023-02-24T14:55:32Z

Bene: Text replacement - "sysadmin@lnt.ei.tum.de" to "sysadmin@ice.cit.tum.de"

<h4>Datenschutzerklärung</h4>

Eine Nutzung der Internetseiten der TU München, Lehrstuhl für Nachrichtentechnik ist grundsätzlich ohne jede Angabe personenbezogener Daten möglich. Sofern eine betroffene Person besondere Services unseres Unternehmens über unsere Internetseite in Anspruch nehmen möchte, könnte jedoch eine Verarbeitung personenbezogener Daten erforderlich werden. Ist die Verarbeitung personenbezogener Daten erforderlich und besteht für eine solche Verarbeitung keine gesetzliche Grundlage, holen wir generell eine Einwilligung der betroffenen Person ein.

Die Verarbeitung personenbezogener Daten, beispielsweise des Namens, der Anschrift, E-Mail-Adresse oder Telefonnummer einer betroffenen Person, erfolgt stets im Einklang mit der Datenschutz-Grundverordnung und in Übereinstimmung mit den für die TU München, Lehrstuhl für Nachrichtentechnik geltenden landesspezifischen Datenschutzbestimmungen. Mittels dieser Datenschutzerklärung möchte unser Unternehmen die Öffentlichkeit über Art, Umfang und Zweck der von uns erhobenen, genutzten und verarbeiteten personenbezogenen Daten informieren. Ferner werden betroffene Personen mittels dieser Datenschutzerklärung über die ihnen zustehenden Rechte aufgeklärt.

Die TU München, Lehrstuhl für Nachrichtentechnik hat als für die Verarbeitung Verantwortlicher zahlreiche technische und organisatorische Maßnahmen umgesetzt, um einen möglichst lückenlosen Schutz der über diese Internetseite verarbeiteten personenbezogenen Daten sicherzustellen. Dennoch können Internetbasierte Datenübertragungen grundsätzlich Sicherheitslücken aufweisen, sodass ein absoluter Schutz nicht gewährleistet werden kann. Aus diesem Grund steht es jeder betroffenen Person frei, personenbezogene Daten auch auf alternativen Wegen, beispielsweise telefonisch, an uns zu übermitteln.

<h4>1. Begriffsbestimmungen</h4>
Die Datenschutzerklärung der TU München, Lehrstuhl für Nachrichtentechnik beruht auf den Begrifflichkeiten, die durch den Europäischen Richtlinien- und Verordnungsgeber beim Erlass der Datenschutz-Grundverordnung (DS-GVO) verwendet wurden. Unsere Datenschutzerklärung soll für die Öffentlichkeit einfach lesbar und verständlich sein. Um dies zu gewährleisten, möchten wir vorab die verwendeten Begrifflichkeiten erläutern.

Wir verwenden in dieser Datenschutzerklärung unter anderem die folgenden Begriffe:

<ul style="list-style: none">
<li><h4>a) personenbezogene Daten</h4>
Personenbezogene Daten sind alle Informationen, die sich auf eine identifizierte oder identifizierbare natürliche Person (im Folgenden „betroffene Person“) beziehen. Als identifizierbar wird eine natürliche Person angesehen, die direkt oder indirekt, insbesondere mittels Zuordnung zu einer Kennung wie einem Namen, zu einer Kennnummer, zu Standortdaten, zu einer Online-Kennung oder zu einem oder mehreren besonderen Merkmalen, die Ausdruck der physischen, physiologischen, genetischen, psychischen, wirtschaftlichen, kulturellen oder sozialen Identität dieser natürlichen Person sind, identifiziert werden kann.
</li>
<li><h4>b) betroffene Person</h4>
Betroffene Person ist jede identifizierte oder identifizierbare natürliche Person, deren personenbezogene Daten von dem für die Verarbeitung Verantwortlichen verarbeitet werden.
</li>
<li><h4>c) Verarbeitung</h4>
Verarbeitung ist jeder mit oder ohne Hilfe automatisierter Verfahren ausgeführte Vorgang oder jede solche Vorgangsreihe im Zusammenhang mit personenbezogenen Daten wie das Erheben, das Erfassen, die Organisation, das Ordnen, die Speicherung, die Anpassung oder Veränderung, das Auslesen, das Abfragen, die Verwendung, die Offenlegung durch Übermittlung, Verbreitung oder eine andere Form der Bereitstellung, den Abgleich oder die Verknüpfung, die Einschränkung, das Löschen oder die Vernichtung.
</li>
<li><h4>d) Einschränkung der Verarbeitung</h4>
Einschränkung der Verarbeitung ist die Markierung gespeicherter personenbezogener Daten mit dem Ziel, ihre künftige Verarbeitung einzuschränken.
</li>
<li><h4>e) Profiling</h4>
Profiling ist jede Art der automatisierten Verarbeitung personenbezogener Daten, die darin besteht, dass diese personenbezogenen Daten verwendet werden, um bestimmte persönliche Aspekte, die sich auf eine natürliche Person beziehen, zu bewerten, insbesondere, um Aspekte bezüglich Arbeitsleistung, wirtschaftlicher Lage, Gesundheit, persönlicher Vorlieben, Interessen, Zuverlässigkeit, Verhalten, Aufenthaltsort oder Ortswechsel dieser natürlichen Person zu analysieren oder vorherzusagen.
</li>
<li><h4>f) Pseudonymisierung</h4>
Pseudonymisierung ist die Verarbeitung personenbezogener Daten in einer Weise, auf welche die personenbezogenen Daten ohne Hinzuziehung zusätzlicher Informationen nicht mehr einer spezifischen betroffenen Person zugeordnet werden können, sofern diese zusätzlichen Informationen gesondert aufbewahrt werden und technischen und organisatorischen Maßnahmen unterliegen, die gewährleisten, dass die personenbezogenen Daten nicht einer identifizierten oder identifizierbaren natürlichen Person zugewiesen werden.
</li>
<li><h4>g) Verantwortlicher oder für die Verarbeitung Verantwortlicher</h4>
Verantwortlicher oder für die Verarbeitung Verantwortlicher ist die natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle, die allein oder gemeinsam mit anderen über die Zwecke und Mittel der Verarbeitung von personenbezogenen Daten entscheidet. Sind die Zwecke und Mittel dieser Verarbeitung durch das Unionsrecht oder das Recht der Mitgliedstaaten vorgegeben, so kann der Verantwortliche beziehungsweise können die bestimmten Kriterien seiner Benennung nach dem Unionsrecht oder dem Recht der Mitgliedstaaten vorgesehen werden.
</li>
<li><h4>h) Auftragsverarbeiter</h4>
Auftragsverarbeiter ist eine natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle, die personenbezogene Daten im Auftrag des Verantwortlichen verarbeitet.
</li>
<li><h4>i) Empfänger</h4>
Empfänger ist eine natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle, der personenbezogene Daten offengelegt werden, unabhängig davon, ob es sich bei ihr um einen Dritten handelt oder nicht. Behörden, die im Rahmen eines bestimmten Untersuchungsauftrags nach dem Unionsrecht oder dem Recht der Mitgliedstaaten möglicherweise personenbezogene Daten erhalten, gelten jedoch nicht als Empfänger.
</li>
<li><h4>j) Dritter</h4>
Dritter ist eine natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle außer der betroffenen Person, dem Verantwortlichen, dem Auftragsverarbeiter und den Personen, die unter der unmittelbaren Verantwortung des Verantwortlichen oder des Auftragsverarbeiters befugt sind, die personenbezogenen Daten zu verarbeiten.
</li>
<li><h4>k) Einwilligung</h4>
Einwilligung ist jede von der betroffenen Person freiwillig für den bestimmten Fall in informierter Weise und unmissverständlich abgegebene Willensbekundung in Form einer Erklärung oder einer sonstigen eindeutigen bestätigenden Handlung, mit der die betroffene Person zu verstehen gibt, dass sie mit der Verarbeitung der sie betreffenden personenbezogenen Daten einverstanden ist.
</li>
</ul>

<h4>2. Name und Anschrift des für die Verarbeitung Verantwortlichen</h4>
Verantwortlicher im Sinne der Datenschutz-Grundverordnung, sonstiger in den Mitgliedstaaten der Europäischen Union geltenden Datenschutzgesetze und anderer Bestimmungen mit datenschutzrechtlichem Charakter ist die:

TU München, Lehrstuhl für Nachrichtentechnik
Arcisstr. 21
80333 München
Deutschland
Tel.: +49 89 289-23466
E-Mail: sysadmin@ice.cit.tum.de
Website: www.lntwww.de

<h4>3. Name und Anschrift des Datenschutzbeauftragten</h4>
Der Datenschutzbeauftragte des für die Verarbeitung Verantwortlichen ist:

TU München
Arcisstr. 21
80333 München
Deutschland
Tel.: +4989289-17052
E-Mail: sekretariat@datenschutz.tum.de
Website: https://www.datenschutz.tum.de/der-datenschutzbeauftragte/

Jede betroffene Person kann sich jederzeit bei allen Fragen und Anregungen zum Datenschutz direkt an unseren Datenschutzbeauftragten wenden.

<h4>4. Cookies</h4>
Die Internetseiten der TU München, Lehrstuhl für Nachrichtentechnik verwenden Cookies. Cookies sind Textdateien, welche über einen Internetbrowser auf einem Computersystem abgelegt und gespeichert werden.

Zahlreiche Internetseiten und Server verwenden Cookies. Viele Cookies enthalten eine sogenannte Cookie-ID. Eine Cookie-ID ist eine eindeutige Kennung des Cookies. Sie besteht aus einer Zeichenfolge, durch welche Internetseiten und Server dem konkreten Internetbrowser zugeordnet werden können, in dem das Cookie gespeichert wurde. Dies ermöglicht es den besuchten Internetseiten und Servern, den individuellen Browser der betroffenen Person von anderen Internetbrowsern, die andere Cookies enthalten, zu unterscheiden. Ein bestimmter Internetbrowser kann über die eindeutige Cookie-ID wiedererkannt und identifiziert werden.

Durch den Einsatz von Cookies kann die TU München, Lehrstuhl für Nachrichtentechnik den Nutzern dieser Internetseite nutzerfreundlichere Services bereitstellen, die ohne die Cookie-Setzung nicht möglich wären.

Mittels eines Cookies können die Informationen und Angebote auf unserer Internetseite im Sinne des Benutzers optimiert werden. Cookies ermöglichen uns, wie bereits erwähnt, die Benutzer unserer Internetseite wiederzuerkennen. Zweck dieser Wiedererkennung ist es, den Nutzern die Verwendung unserer Internetseite zu erleichtern. Der Benutzer einer Internetseite, die Cookies verwendet, muss beispielsweise nicht bei jedem Besuch der Internetseite erneut seine Zugangsdaten eingeben, weil dies von der Internetseite und dem auf dem Computersystem des Benutzers abgelegten Cookie übernommen wird. 

Die betroffene Person kann die Setzung von Cookies durch unsere Internetseite jederzeit mittels einer entsprechenden Einstellung des genutzten Internetbrowsers verhindern und damit der Setzung von Cookies dauerhaft widersprechen. Ferner können bereits gesetzte Cookies jederzeit über einen Internetbrowser oder andere Softwareprogramme gelöscht werden. Dies ist in allen gängigen Internetbrowsern möglich. Deaktiviert die betroffene Person die Setzung von Cookies in dem genutzten Internetbrowser, sind unter Umständen nicht alle Funktionen unserer Internetseite vollumfänglich nutzbar.

<h4>5. Erfassung von allgemeinen Daten und Informationen</h4>
Die Internetseite der TU München, Lehrstuhl für Nachrichtentechnik erfasst mit jedem Aufruf der Internetseite durch eine betroffene Person oder ein automatisiertes System eine Reihe von allgemeinen Daten und Informationen. Diese allgemeinen Daten und Informationen werden in den Logfiles des Servers gespeichert. Erfasst werden können die (1) verwendeten Browsertypen und Versionen, (2) das vom zugreifenden System verwendete Betriebssystem, (3) die Internetseite, von welcher ein zugreifendes System auf unsere Internetseite gelangt (sogenannte Referrer), (4) die Unterwebseiten, welche über ein zugreifendes System auf unserer Internetseite angesteuert werden, (5) das Datum und die Uhrzeit eines Zugriffs auf die Internetseite, (6) eine Internet-Protokoll-Adresse (IP-Adresse), (7) der Internet-Service-Provider des zugreifenden Systems und (8) sonstige ähnliche Daten und Informationen, die der Gefahrenabwehr im Falle von Angriffen auf unsere informationstechnologischen Systeme dienen.

Bei der Nutzung dieser allgemeinen Daten und Informationen zieht die TU München, Lehrstuhl für Nachrichtentechnik keine Rückschlüsse auf die betroffene Person. Diese Informationen werden vielmehr benötigt, um (1) die Inhalte unserer Internetseite korrekt auszuliefern, (2) die Inhalte unserer Internetseite sowie die Werbung für diese zu optimieren, (3) die dauerhafte Funktionsfähigkeit unserer informationstechnologischen Systeme und der Technik unserer Internetseite zu gewährleisten sowie (4) um Strafverfolgungsbehörden im Falle eines Cyberangriffes die zur Strafverfolgung notwendigen Informationen bereitzustellen. Diese anonym erhobenen Daten und Informationen werden durch die TU München, Lehrstuhl für Nachrichtentechnik daher einerseits statistisch und ferner mit dem Ziel ausgewertet, den Datenschutz und die Datensicherheit in unserem Unternehmen zu erhöhen, um letztlich ein optimales Schutzniveau für die von uns verarbeiteten personenbezogenen Daten sicherzustellen. Die anonymen Daten der Server-Logfiles werden getrennt von allen durch eine betroffene Person angegebenen personenbezogenen Daten gespeichert.

<h4>6. Routinemäßige Löschung und Sperrung von personenbezogenen Daten</h4>
Der für die Verarbeitung Verantwortliche verarbeitet und speichert personenbezogene Daten der betroffenen Person nur für den Zeitraum, der zur Erreichung des Speicherungszwecks erforderlich ist oder sofern dies durch den Europäischen Richtlinien- und Verordnungsgeber oder einen anderen Gesetzgeber in Gesetzen oder Vorschriften, welchen der für die Verarbeitung Verantwortliche unterliegt, vorgesehen wurde.

Entfällt der Speicherungszweck oder läuft eine vom Europäischen Richtlinien- und Verordnungsgeber oder einem anderen zuständigen Gesetzgeber vorgeschriebene Speicherfrist ab, werden die personenbezogenen Daten routinemäßig und entsprechend den gesetzlichen Vorschriften gesperrt oder gelöscht.

<h4>7. Rechte der betroffenen Person</h4>
<ul style="list-style: none;">
<li><h4>a) Recht auf Bestätigung</h4>
Jede betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber eingeräumte Recht, von dem für die Verarbeitung Verantwortlichen eine Bestätigung darüber zu verlangen, ob sie betreffende personenbezogene Daten verarbeitet werden. Möchte eine betroffene Person dieses Bestätigungsrecht in Anspruch nehmen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.
</li>
<li><h4>b) Recht auf Auskunft</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, jederzeit von dem für die Verarbeitung Verantwortlichen unentgeltliche Auskunft über die zu seiner Person gespeicherten personenbezogenen Daten und eine Kopie dieser Auskunft zu erhalten. Ferner hat der Europäische Richtlinien- und Verordnungsgeber der betroffenen Person Auskunft über folgende Informationen zugestanden:

<ul style="list-style: none;">
<li>die Verarbeitungszwecke</li>
<li>die Kategorien personenbezogener Daten, die verarbeitet werden</li>
<li>die Empfänger oder Kategorien von Empfängern, gegenüber denen die personenbezogenen Daten offengelegt worden sind oder noch offengelegt werden, insbesondere bei Empfängern in Drittländern oder bei internationalen Organisationen</li>
<li>falls möglich die geplante Dauer, für die die personenbezogenen Daten gespeichert werden, oder, falls dies nicht möglich ist, die Kriterien für die Festlegung dieser Dauer</li>
<li>das Bestehen eines Rechts auf Berichtigung oder Löschung der sie betreffenden personenbezogenen Daten oder auf Einschränkung der Verarbeitung durch den Verantwortlichen oder eines Widerspruchsrechts gegen diese Verarbeitung</li>
<li>das Bestehen eines Beschwerderechts bei einer Aufsichtsbehörde</li>
<li>wenn die personenbezogenen Daten nicht bei der betroffenen Person erhoben werden: Alle verfügbaren Informationen über die Herkunft der Daten</li>
<li>das Bestehen einer automatisierten Entscheidungsfindung einschließlich Profiling gemäß Artikel 22 Abs.1 und 4 DS-GVO und — zumindest in diesen Fällen — aussagekräftige Informationen über die involvierte Logik sowie die Tragweite und die angestrebten Auswirkungen einer derartigen Verarbeitung für die betroffene Person</li>

</ul>
Ferner steht der betroffenen Person ein Auskunftsrecht darüber zu, ob personenbezogene Daten an ein Drittland oder an eine internationale Organisation übermittelt wurden. Sofern dies der Fall ist, so steht der betroffenen Person im Übrigen das Recht zu, Auskunft über die geeigneten Garantien im Zusammenhang mit der Übermittlung zu erhalten.

Möchte eine betroffene Person dieses Auskunftsrecht in Anspruch nehmen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.
</li>
<li><h4>c) Recht auf Berichtigung</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, die unverzügliche Berichtigung sie betreffender unrichtiger personenbezogener Daten zu verlangen. Ferner steht der betroffenen Person das Recht zu, unter Berücksichtigung der Zwecke der Verarbeitung, die Vervollständigung unvollständiger personenbezogener Daten — auch mittels einer ergänzenden Erklärung — zu verlangen.

Möchte eine betroffene Person dieses Berichtigungsrecht in Anspruch nehmen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.</li>
<li>
<h4>d) Recht auf Löschung (Recht auf Vergessen werden)</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, von dem Verantwortlichen zu verlangen, dass die sie betreffenden personenbezogenen Daten unverzüglich gelöscht werden, sofern einer der folgenden Gründe zutrifft und soweit die Verarbeitung nicht erforderlich ist:

<ul style="list-style: none;">
<li>Die personenbezogenen Daten wurden für solche Zwecke erhoben oder auf sonstige Weise verarbeitet, für welche sie nicht mehr notwendig sind.</li>
<li>Die betroffene Person widerruft ihre Einwilligung, auf die sich die Verarbeitung gemäß Art. 6 Abs. 1 Buchstabe a DS-GVO oder Art. 9 Abs. 2 Buchstabe a DS-GVO stützte, und es fehlt an einer anderweitigen Rechtsgrundlage für die Verarbeitung.</li>
<li>Die betroffene Person legt gemäß Art. 21 Abs. 1 DS-GVO Widerspruch gegen die Verarbeitung ein, und es liegen keine vorrangigen berechtigten Gründe für die Verarbeitung vor, oder die betroffene Person legt gemäß Art. 21 Abs. 2 DS-GVO Widerspruch gegen die Verarbeitung ein.</li>
<li>Die personenbezogenen Daten wurden unrechtmäßig verarbeitet.</li>
<li>Die Löschung der personenbezogenen Daten ist zur Erfüllung einer rechtlichen Verpflichtung nach dem Unionsrecht oder dem Recht der Mitgliedstaaten erforderlich, dem der Verantwortliche unterliegt.</li>
<li>Die personenbezogenen Daten wurden in Bezug auf angebotene Dienste der Informationsgesellschaft gemäß Art. 8 Abs. 1 DS-GVO erhoben.</li>

</ul>
Sofern einer der oben genannten Gründe zutrifft und eine betroffene Person die Löschung von personenbezogenen Daten, die bei der TU München, Lehrstuhl für Nachrichtentechnik gespeichert sind, veranlassen möchte, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden. Der Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wird veranlassen, dass dem Löschverlangen unverzüglich nachgekommen wird.

Wurden die personenbezogenen Daten von der TU München, Lehrstuhl für Nachrichtentechnik öffentlich gemacht und ist unser Unternehmen als Verantwortlicher gemäß Art. 17 Abs. 1 DS-GVO zur Löschung der personenbezogenen Daten verpflichtet, so trifft die TU München, Lehrstuhl für Nachrichtentechnik unter Berücksichtigung der verfügbaren Technologie und der Implementierungskosten angemessene Maßnahmen, auch technischer Art, um andere für die Datenverarbeitung Verantwortliche, welche die veröffentlichten personenbezogenen Daten verarbeiten, darüber in Kenntnis zu setzen, dass die betroffene Person von diesen anderen für die Datenverarbeitung Verantwortlichen die Löschung sämtlicher Links zu diesen personenbezogenen Daten oder von Kopien oder Replikationen dieser personenbezogenen Daten verlangt hat, soweit die Verarbeitung nicht erforderlich ist. Der Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wird im Einzelfall das Notwendige veranlassen.
</li>
<li><h4>e) Recht auf Einschränkung der Verarbeitung</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, von dem Verantwortlichen die Einschränkung der Verarbeitung zu verlangen, wenn eine der folgenden Voraussetzungen gegeben ist:

<ul style="list-style: none;">
<li>Die Richtigkeit der personenbezogenen Daten wird von der betroffenen Person bestritten, und zwar für eine Dauer, die es dem Verantwortlichen ermöglicht, die Richtigkeit der personenbezogenen Daten zu überprüfen.</li>
<li>Die Verarbeitung ist unrechtmäßig, die betroffene Person lehnt die Löschung der personenbezogenen Daten ab und verlangt stattdessen die Einschränkung der Nutzung der personenbezogenen Daten.</li>
<li>Der Verantwortliche benötigt die personenbezogenen Daten für die Zwecke der Verarbeitung nicht länger, die betroffene Person benötigt sie jedoch zur Geltendmachung, Ausübung oder Verteidigung von Rechtsansprüchen.</li>
<li>Die betroffene Person hat Widerspruch gegen die Verarbeitung gem. Art. 21 Abs. 1 DS-GVO eingelegt und es steht noch nicht fest, ob die berechtigten Gründe des Verantwortlichen gegenüber denen der betroffenen Person überwiegen.</li>

</ul>
Sofern eine der oben genannten Voraussetzungen gegeben ist und eine betroffene Person die Einschränkung von personenbezogenen Daten, die bei der TU München, Lehrstuhl für Nachrichtentechnik gespeichert sind, verlangen möchte, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden. Der Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wird die Einschränkung der Verarbeitung veranlassen.
</li>
<li><h4>f) Recht auf Datenübertragbarkeit</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, die sie betreffenden personenbezogenen Daten, welche durch die betroffene Person einem Verantwortlichen bereitgestellt wurden, in einem strukturierten, gängigen und maschinenlesbaren Format zu erhalten. Sie hat außerdem das Recht, diese Daten einem anderen Verantwortlichen ohne Behinderung durch den Verantwortlichen, dem die personenbezogenen Daten bereitgestellt wurden, zu übermitteln, sofern die Verarbeitung auf der Einwilligung gemäß Art. 6 Abs. 1 Buchstabe a DS-GVO oder Art. 9 Abs. 2 Buchstabe a DS-GVO oder auf einem Vertrag gemäß Art. 6 Abs. 1 Buchstabe b DS-GVO beruht und die Verarbeitung mithilfe automatisierter Verfahren erfolgt, sofern die Verarbeitung nicht für die Wahrnehmung einer Aufgabe erforderlich ist, die im öffentlichen Interesse liegt oder in Ausübung öffentlicher Gewalt erfolgt, welche dem Verantwortlichen übertragen wurde.

Ferner hat die betroffene Person bei der Ausübung ihres Rechts auf Datenübertragbarkeit gemäß Art. 20 Abs. 1 DS-GVO das Recht, zu erwirken, dass die personenbezogenen Daten direkt von einem Verantwortlichen an einen anderen Verantwortlichen übermittelt werden, soweit dies technisch machbar ist und sofern hiervon nicht die Rechte und Freiheiten anderer Personen beeinträchtigt werden.

Zur Geltendmachung des Rechts auf Datenübertragbarkeit kann sich die betroffene Person jederzeit an einen Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wenden.

</li>
<li>
<h4>g) Recht auf Widerspruch</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, aus Gründen, die sich aus ihrer besonderen Situation ergeben, jederzeit gegen die Verarbeitung sie betreffender personenbezogener Daten, die aufgrund von Art. 6 Abs. 1 Buchstaben e oder f DS-GVO erfolgt, Widerspruch einzulegen. Dies gilt auch für ein auf diese Bestimmungen gestütztes Profiling.

Die TU München, Lehrstuhl für Nachrichtentechnik verarbeitet die personenbezogenen Daten im Falle des Widerspruchs nicht mehr, es sei denn, wir können zwingende schutzwürdige Gründe für die Verarbeitung nachweisen, die den Interessen, Rechten und Freiheiten der betroffenen Person überwiegen, oder die Verarbeitung dient der Geltendmachung, Ausübung oder Verteidigung von Rechtsansprüchen.

Verarbeitet die TU München, Lehrstuhl für Nachrichtentechnik personenbezogene Daten, um Direktwerbung zu betreiben, so hat die betroffene Person das Recht, jederzeit Widerspruch gegen die Verarbeitung der personenbezogenen Daten zum Zwecke derartiger Werbung einzulegen. Dies gilt auch für das Profiling, soweit es mit solcher Direktwerbung in Verbindung steht. Widerspricht die betroffene Person gegenüber der TU München, Lehrstuhl für Nachrichtentechnik der Verarbeitung für Zwecke der Direktwerbung, so wird die TU München, Lehrstuhl für Nachrichtentechnik die personenbezogenen Daten nicht mehr für diese Zwecke verarbeiten.

Zudem hat die betroffene Person das Recht, aus Gründen, die sich aus ihrer besonderen Situation ergeben, gegen die sie betreffende Verarbeitung personenbezogener Daten, die bei der TU München, Lehrstuhl für Nachrichtentechnik zu wissenschaftlichen oder historischen Forschungszwecken oder zu statistischen Zwecken gemäß Art. 89 Abs. 1 DS-GVO erfolgen, Widerspruch einzulegen, es sei denn, eine solche Verarbeitung ist zur Erfüllung einer im öffentlichen Interesse liegenden Aufgabe erforderlich.

Zur Ausübung des Rechts auf Widerspruch kann sich die betroffene Person direkt jeden Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik oder einen anderen Mitarbeiter wenden. Der betroffenen Person steht es ferner frei, im Zusammenhang mit der Nutzung von Diensten der Informationsgesellschaft, ungeachtet der Richtlinie 2002/58/EG, ihr Widerspruchsrecht mittels automatisierter Verfahren auszuüben, bei denen technische Spezifikationen verwendet werden.
</li>
<li><h4>h) Automatisierte Entscheidungen im Einzelfall einschließlich Profiling</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, nicht einer ausschließlich auf einer automatisierten Verarbeitung — einschließlich Profiling — beruhenden Entscheidung unterworfen zu werden, die ihr gegenüber rechtliche Wirkung entfaltet oder sie in ähnlicher Weise erheblich beeinträchtigt, sofern die Entscheidung (1) nicht für den Abschluss oder die Erfüllung eines Vertrags zwischen der betroffenen Person und dem Verantwortlichen erforderlich ist, oder (2) aufgrund von Rechtsvorschriften der Union oder der Mitgliedstaaten, denen der Verantwortliche unterliegt, zulässig ist und diese Rechtsvorschriften angemessene Maßnahmen zur Wahrung der Rechte und Freiheiten sowie der berechtigten Interessen der betroffenen Person enthalten oder (3) mit ausdrücklicher Einwilligung der betroffenen Person erfolgt.

Ist die Entscheidung (1) für den Abschluss oder die Erfüllung eines Vertrags zwischen der betroffenen Person und dem Verantwortlichen erforderlich oder (2) erfolgt sie mit ausdrücklicher Einwilligung der betroffenen Person, trifft die TU München, Lehrstuhl für Nachrichtentechnik angemessene Maßnahmen, um die Rechte und Freiheiten sowie die berechtigten Interessen der betroffenen Person zu wahren, wozu mindestens das Recht auf Erwirkung des Eingreifens einer Person seitens des Verantwortlichen, auf Darlegung des eigenen Standpunkts und auf Anfechtung der Entscheidung gehört.

Möchte die betroffene Person Rechte mit Bezug auf automatisierte Entscheidungen geltend machen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.

</li>
<li><h4>i) Recht auf Widerruf einer datenschutzrechtlichen Einwilligung</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, eine Einwilligung zur Verarbeitung personenbezogener Daten jederzeit zu widerrufen.

Möchte die betroffene Person ihr Recht auf Widerruf einer Einwilligung geltend machen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.

</li>
</ul>
<h4>8. Rechtsgrundlage der Verarbeitung</h4>
Art. 6 I lit. a DS-GVO dient unserem Unternehmen als Rechtsgrundlage für Verarbeitungsvorgänge, bei denen wir eine Einwilligung für einen bestimmten Verarbeitungszweck einholen. Ist die Verarbeitung personenbezogener Daten zur Erfüllung eines Vertrags, dessen Vertragspartei die betroffene Person ist, erforderlich, wie dies beispielsweise bei Verarbeitungsvorgängen der Fall ist, die für eine Lieferung von Waren oder die Erbringung einer sonstigen Leistung oder Gegenleistung notwendig sind, so beruht die Verarbeitung auf Art. 6 I lit. b DS-GVO. Gleiches gilt für solche Verarbeitungsvorgänge die zur Durchführung vorvertraglicher Maßnahmen erforderlich sind, etwa in Fällen von Anfragen zur unseren Produkten oder Leistungen. Unterliegt unser Unternehmen einer rechtlichen Verpflichtung durch welche eine Verarbeitung von personenbezogenen Daten erforderlich wird, wie beispielsweise zur Erfüllung steuerlicher Pflichten, so basiert die Verarbeitung auf Art. 6 I lit. c DS-GVO. In seltenen Fällen könnte die Verarbeitung von personenbezogenen Daten erforderlich werden, um lebenswichtige Interessen der betroffenen Person oder einer anderen natürlichen Person zu schützen. Dies wäre beispielsweise der Fall, wenn ein Besucher in unserem Betrieb verletzt werden würde und daraufhin sein Name, sein Alter, seine Krankenkassendaten oder sonstige lebenswichtige Informationen an einen Arzt, ein Krankenhaus oder sonstige Dritte weitergegeben werden müssten. Dann würde die Verarbeitung auf Art. 6 I lit. d DS-GVO beruhen.
Letztlich könnten Verarbeitungsvorgänge auf Art. 6 I lit. f DS-GVO beruhen. Auf dieser Rechtsgrundlage basieren Verarbeitungsvorgänge, die von keiner der vorgenannten Rechtsgrundlagen erfasst werden, wenn die Verarbeitung zur Wahrung eines berechtigten Interesses unseres Unternehmens oder eines Dritten erforderlich ist, sofern die Interessen, Grundrechte und Grundfreiheiten des Betroffenen nicht überwiegen. Solche Verarbeitungsvorgänge sind uns insbesondere deshalb gestattet, weil sie durch den Europäischen Gesetzgeber besonders erwähnt wurden. Er vertrat insoweit die Auffassung, dass ein berechtigtes Interesse anzunehmen sein könnte, wenn die betroffene Person ein Kunde des Verantwortlichen ist (Erwägungsgrund 47 Satz 2 DS-GVO).


<h4>9. Berechtigte Interessen an der Verarbeitung, die von dem Verantwortlichen oder einem Dritten verfolgt werden</h4>
Basiert die Verarbeitung personenbezogener Daten auf Artikel 6 I lit. f DS-GVO ist unser berechtigtes Interesse die Durchführung unserer Geschäftstätigkeit zugunsten des Wohlergehens all unserer Mitarbeiter und unserer Anteilseigner.

<h4>10. Dauer, für die die personenbezogenen Daten gespeichert werden</h4>
Das Kriterium für die Dauer der Speicherung von personenbezogenen Daten ist die jeweilige gesetzliche Aufbewahrungsfrist. Nach Ablauf der Frist werden die entsprechenden Daten routinemäßig gelöscht, sofern sie nicht mehr zur Vertragserfüllung oder Vertragsanbahnung erforderlich sind.

<h4>11. Gesetzliche oder vertragliche Vorschriften zur Bereitstellung der personenbezogenen Daten; Erforderlichkeit für den Vertragsabschluss; Verpflichtung der betroffenen Person, die personenbezogenen Daten bereitzustellen; mögliche Folgen der Nichtbereitstellung</h4>
Wir klären Sie darüber auf, dass die Bereitstellung personenbezogener Daten zum Teil gesetzlich vorgeschrieben ist (z.B. Steuervorschriften) oder sich auch aus vertraglichen Regelungen (z.B. Angaben zum Vertragspartner) ergeben kann.
Mitunter kann es zu einem Vertragsschluss erforderlich sein, dass eine betroffene Person uns personenbezogene Daten zur Verfügung stellt, die in der Folge durch uns verarbeitet werden müssen. Die betroffene Person ist beispielsweise verpflichtet uns personenbezogene Daten bereitzustellen, wenn unser Unternehmen mit ihr einen Vertrag abschließt. Eine Nichtbereitstellung der personenbezogenen Daten hätte zur Folge, dass der Vertrag mit dem Betroffenen nicht geschlossen werden könnte.
Vor einer Bereitstellung personenbezogener Daten durch den Betroffenen muss sich der Betroffene an einen unserer Mitarbeiter wenden. Unser Mitarbeiter klärt den Betroffenen einzelfallbezogen darüber auf, ob die Bereitstellung der personenbezogenen Daten gesetzlich oder vertraglich vorgeschrieben oder für den Vertragsabschluss erforderlich ist, ob eine Verpflichtung besteht, die personenbezogenen Daten bereitzustellen, und welche Folgen die Nichtbereitstellung der personenbezogenen Daten hätte.


<h4>12. Bestehen einer automatisierten Entscheidungsfindung</h4>
Als verantwortungsbewusstes Unternehmen verzichten wir auf eine automatische Entscheidungsfindung oder ein Profiling.

Diese Datenschutzerklärung wurde durch den Datenschutzerklärungs-Generator der DGD Deutsche Gesellschaft für Datenschutz GmbH, die als [https://dg-datenschutz.de/datenschutz-dienstleistungen/externer-datenschutzbeauftragter/ Externer Datenschutzbeauftragter Berlin] tätig ist, in Kooperation mit den [https://www.wbs-law.de/ Datenschutz (DSGVO) Anwälten der Kanzlei WILDE BEUGER SOLMECKE | Rechtsanwälte] erstellt.

LNTwww:Datenschutz

2023-02-24T14:55:30Z

Bene: Text replacement - "sysadmin@lnt.ei.tum.de" to "sysadmin@ice.cit.tum.de"

<html>
<h4>Datenschutzerklärung</h4>

Eine Nutzung der Internetseiten der TU München, Lehrstuhl für Nachrichtentechnik ist grundsätzlich ohne jede Angabe personenbezogener Daten möglich. Sofern eine betroffene Person besondere Services unseres Unternehmens über unsere Internetseite in Anspruch nehmen möchte, könnte jedoch eine Verarbeitung personenbezogener Daten erforderlich werden. Ist die Verarbeitung personenbezogener Daten erforderlich und besteht für eine solche Verarbeitung keine gesetzliche Grundlage, holen wir generell eine Einwilligung der betroffenen Person ein.

Die Verarbeitung personenbezogener Daten, beispielsweise des Namens, der Anschrift, E-Mail-Adresse oder Telefonnummer einer betroffenen Person, erfolgt stets im Einklang mit der Datenschutz-Grundverordnung und in Übereinstimmung mit den für die TU München, Lehrstuhl für Nachrichtentechnik geltenden landesspezifischen Datenschutzbestimmungen. Mittels dieser Datenschutzerklärung möchte unser Unternehmen die Öffentlichkeit über Art, Umfang und Zweck der von uns erhobenen, genutzten und verarbeiteten personenbezogenen Daten informieren. Ferner werden betroffene Personen mittels dieser Datenschutzerklärung über die ihnen zustehenden Rechte aufgeklärt.

Die TU München, Lehrstuhl für Nachrichtentechnik hat als für die Verarbeitung Verantwortlicher zahlreiche technische und organisatorische Maßnahmen umgesetzt, um einen möglichst lückenlosen Schutz der über diese Internetseite verarbeiteten personenbezogenen Daten sicherzustellen. Dennoch können Internetbasierte Datenübertragungen grundsätzlich Sicherheitslücken aufweisen, sodass ein absoluter Schutz nicht gewährleistet werden kann. Aus diesem Grund steht es jeder betroffenen Person frei, personenbezogene Daten auch auf alternativen Wegen, beispielsweise telefonisch, an uns zu übermitteln.

<h4>1. Begriffsbestimmungen</h4>
Die Datenschutzerklärung der TU München, Lehrstuhl für Nachrichtentechnik beruht auf den Begrifflichkeiten, die durch den Europäischen Richtlinien- und Verordnungsgeber beim Erlass der Datenschutz-Grundverordnung (DS-GVO) verwendet wurden. Unsere Datenschutzerklärung soll für die Öffentlichkeit einfach lesbar und verständlich sein. Um dies zu gewährleisten, möchten wir vorab die verwendeten Begrifflichkeiten erläutern.

Wir verwenden in dieser Datenschutzerklärung unter anderem die folgenden Begriffe:

<ul style="list-style: none">
<li><h4>a) personenbezogene Daten</h4>
Personenbezogene Daten sind alle Informationen, die sich auf eine identifizierte oder identifizierbare natürliche Person (im Folgenden „betroffene Person“) beziehen. Als identifizierbar wird eine natürliche Person angesehen, die direkt oder indirekt, insbesondere mittels Zuordnung zu einer Kennung wie einem Namen, zu einer Kennnummer, zu Standortdaten, zu einer Online-Kennung oder zu einem oder mehreren besonderen Merkmalen, die Ausdruck der physischen, physiologischen, genetischen, psychischen, wirtschaftlichen, kulturellen oder sozialen Identität dieser natürlichen Person sind, identifiziert werden kann.
</li>
<li><h4>b) betroffene Person</h4>
Betroffene Person ist jede identifizierte oder identifizierbare natürliche Person, deren personenbezogene Daten von dem für die Verarbeitung Verantwortlichen verarbeitet werden.
</li>
<li><h4>c) Verarbeitung</h4>
Verarbeitung ist jeder mit oder ohne Hilfe automatisierter Verfahren ausgeführte Vorgang oder jede solche Vorgangsreihe im Zusammenhang mit personenbezogenen Daten wie das Erheben, das Erfassen, die Organisation, das Ordnen, die Speicherung, die Anpassung oder Veränderung, das Auslesen, das Abfragen, die Verwendung, die Offenlegung durch Übermittlung, Verbreitung oder eine andere Form der Bereitstellung, den Abgleich oder die Verknüpfung, die Einschränkung, das Löschen oder die Vernichtung.
</li>
<li><h4>d) Einschränkung der Verarbeitung</h4>
Einschränkung der Verarbeitung ist die Markierung gespeicherter personenbezogener Daten mit dem Ziel, ihre künftige Verarbeitung einzuschränken.
</li>
<li><h4>e) Profiling</h4>
Profiling ist jede Art der automatisierten Verarbeitung personenbezogener Daten, die darin besteht, dass diese personenbezogenen Daten verwendet werden, um bestimmte persönliche Aspekte, die sich auf eine natürliche Person beziehen, zu bewerten, insbesondere, um Aspekte bezüglich Arbeitsleistung, wirtschaftlicher Lage, Gesundheit, persönlicher Vorlieben, Interessen, Zuverlässigkeit, Verhalten, Aufenthaltsort oder Ortswechsel dieser natürlichen Person zu analysieren oder vorherzusagen.
</li>
<li><h4>f) Pseudonymisierung</h4>
Pseudonymisierung ist die Verarbeitung personenbezogener Daten in einer Weise, auf welche die personenbezogenen Daten ohne Hinzuziehung zusätzlicher Informationen nicht mehr einer spezifischen betroffenen Person zugeordnet werden können, sofern diese zusätzlichen Informationen gesondert aufbewahrt werden und technischen und organisatorischen Maßnahmen unterliegen, die gewährleisten, dass die personenbezogenen Daten nicht einer identifizierten oder identifizierbaren natürlichen Person zugewiesen werden.
</li>
<li><h4>g) Verantwortlicher oder für die Verarbeitung Verantwortlicher</h4>
Verantwortlicher oder für die Verarbeitung Verantwortlicher ist die natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle, die allein oder gemeinsam mit anderen über die Zwecke und Mittel der Verarbeitung von personenbezogenen Daten entscheidet. Sind die Zwecke und Mittel dieser Verarbeitung durch das Unionsrecht oder das Recht der Mitgliedstaaten vorgegeben, so kann der Verantwortliche beziehungsweise können die bestimmten Kriterien seiner Benennung nach dem Unionsrecht oder dem Recht der Mitgliedstaaten vorgesehen werden.
</li>
<li><h4>h) Auftragsverarbeiter</h4>
Auftragsverarbeiter ist eine natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle, die personenbezogene Daten im Auftrag des Verantwortlichen verarbeitet.
</li>
<li><h4>i) Empfänger</h4>
Empfänger ist eine natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle, der personenbezogene Daten offengelegt werden, unabhängig davon, ob es sich bei ihr um einen Dritten handelt oder nicht. Behörden, die im Rahmen eines bestimmten Untersuchungsauftrags nach dem Unionsrecht oder dem Recht der Mitgliedstaaten möglicherweise personenbezogene Daten erhalten, gelten jedoch nicht als Empfänger.
</li>
<li><h4>j) Dritter</h4>
Dritter ist eine natürliche oder juristische Person, Behörde, Einrichtung oder andere Stelle außer der betroffenen Person, dem Verantwortlichen, dem Auftragsverarbeiter und den Personen, die unter der unmittelbaren Verantwortung des Verantwortlichen oder des Auftragsverarbeiters befugt sind, die personenbezogenen Daten zu verarbeiten.
</li>
<li><h4>k) Einwilligung</h4>
Einwilligung ist jede von der betroffenen Person freiwillig für den bestimmten Fall in informierter Weise und unmissverständlich abgegebene Willensbekundung in Form einer Erklärung oder einer sonstigen eindeutigen bestätigenden Handlung, mit der die betroffene Person zu verstehen gibt, dass sie mit der Verarbeitung der sie betreffenden personenbezogenen Daten einverstanden ist.
</li>
</ul>

<h4>2. Name und Anschrift des für die Verarbeitung Verantwortlichen</h4>
Verantwortlicher im Sinne der Datenschutz-Grundverordnung, sonstiger in den Mitgliedstaaten der Europäischen Union geltenden Datenschutzgesetze und anderer Bestimmungen mit datenschutzrechtlichem Charakter ist die:

TU München, Lehrstuhl für Nachrichtentechnik
Arcisstr. 21
80333 München
Deutschland
Tel.: +49 89 289-23466
E-Mail: sysadmin@ice.cit.tum.de
Website: en.lntwww.de

<h4>3. Name und Anschrift des Datenschutzbeauftragten</h4>
Der Datenschutzbeauftragte des für die Verarbeitung Verantwortlichen ist:

TU München
Arcisstr. 21
80333 München
Deutschland
Tel.: +4989289-17052
E-Mail: sekretariat@datenschutz.tum.de
Website: https://www.datenschutz.tum.de/der-datenschutzbeauftragte/

Jede betroffene Person kann sich jederzeit bei allen Fragen und Anregungen zum Datenschutz direkt an unseren Datenschutzbeauftragten wenden.

<h4>4. Cookies</h4>
Die Internetseiten der TU München, Lehrstuhl für Nachrichtentechnik verwenden Cookies. Cookies sind Textdateien, welche über einen Internetbrowser auf einem Computersystem abgelegt und gespeichert werden.

Zahlreiche Internetseiten und Server verwenden Cookies. Viele Cookies enthalten eine sogenannte Cookie-ID. Eine Cookie-ID ist eine eindeutige Kennung des Cookies. Sie besteht aus einer Zeichenfolge, durch welche Internetseiten und Server dem konkreten Internetbrowser zugeordnet werden können, in dem das Cookie gespeichert wurde. Dies ermöglicht es den besuchten Internetseiten und Servern, den individuellen Browser der betroffenen Person von anderen Internetbrowsern, die andere Cookies enthalten, zu unterscheiden. Ein bestimmter Internetbrowser kann über die eindeutige Cookie-ID wiedererkannt und identifiziert werden.

Durch den Einsatz von Cookies kann die TU München, Lehrstuhl für Nachrichtentechnik den Nutzern dieser Internetseite nutzerfreundlichere Services bereitstellen, die ohne die Cookie-Setzung nicht möglich wären.

Mittels eines Cookies können die Informationen und Angebote auf unserer Internetseite im Sinne des Benutzers optimiert werden. Cookies ermöglichen uns, wie bereits erwähnt, die Benutzer unserer Internetseite wiederzuerkennen. Zweck dieser Wiedererkennung ist es, den Nutzern die Verwendung unserer Internetseite zu erleichtern. Der Benutzer einer Internetseite, die Cookies verwendet, muss beispielsweise nicht bei jedem Besuch der Internetseite erneut seine Zugangsdaten eingeben, weil dies von der Internetseite und dem auf dem Computersystem des Benutzers abgelegten Cookie übernommen wird. 

Die betroffene Person kann die Setzung von Cookies durch unsere Internetseite jederzeit mittels einer entsprechenden Einstellung des genutzten Internetbrowsers verhindern und damit der Setzung von Cookies dauerhaft widersprechen. Ferner können bereits gesetzte Cookies jederzeit über einen Internetbrowser oder andere Softwareprogramme gelöscht werden. Dies ist in allen gängigen Internetbrowsern möglich. Deaktiviert die betroffene Person die Setzung von Cookies in dem genutzten Internetbrowser, sind unter Umständen nicht alle Funktionen unserer Internetseite vollumfänglich nutzbar.

<h4>5. Erfassung von allgemeinen Daten und Informationen</h4>
Die Internetseite der TU München, Lehrstuhl für Nachrichtentechnik erfasst mit jedem Aufruf der Internetseite durch eine betroffene Person oder ein automatisiertes System eine Reihe von allgemeinen Daten und Informationen. Diese allgemeinen Daten und Informationen werden in den Logfiles des Servers gespeichert. Erfasst werden können die (1) verwendeten Browsertypen und Versionen, (2) das vom zugreifenden System verwendete Betriebssystem, (3) die Internetseite, von welcher ein zugreifendes System auf unsere Internetseite gelangt (sogenannte Referrer), (4) die Unterwebseiten, welche über ein zugreifendes System auf unserer Internetseite angesteuert werden, (5) das Datum und die Uhrzeit eines Zugriffs auf die Internetseite, (6) eine Internet-Protokoll-Adresse (IP-Adresse), (7) der Internet-Service-Provider des zugreifenden Systems und (8) sonstige ähnliche Daten und Informationen, die der Gefahrenabwehr im Falle von Angriffen auf unsere informationstechnologischen Systeme dienen.

Bei der Nutzung dieser allgemeinen Daten und Informationen zieht die TU München, Lehrstuhl für Nachrichtentechnik keine Rückschlüsse auf die betroffene Person. Diese Informationen werden vielmehr benötigt, um (1) die Inhalte unserer Internetseite korrekt auszuliefern, (2) die Inhalte unserer Internetseite sowie die Werbung für diese zu optimieren, (3) die dauerhafte Funktionsfähigkeit unserer informationstechnologischen Systeme und der Technik unserer Internetseite zu gewährleisten sowie (4) um Strafverfolgungsbehörden im Falle eines Cyberangriffes die zur Strafverfolgung notwendigen Informationen bereitzustellen. Diese anonym erhobenen Daten und Informationen werden durch die TU München, Lehrstuhl für Nachrichtentechnik daher einerseits statistisch und ferner mit dem Ziel ausgewertet, den Datenschutz und die Datensicherheit in unserem Unternehmen zu erhöhen, um letztlich ein optimales Schutzniveau für die von uns verarbeiteten personenbezogenen Daten sicherzustellen. Die anonymen Daten der Server-Logfiles werden getrennt von allen durch eine betroffene Person angegebenen personenbezogenen Daten gespeichert.

<h4>6. Routinemäßige Löschung und Sperrung von personenbezogenen Daten</h4>
Der für die Verarbeitung Verantwortliche verarbeitet und speichert personenbezogene Daten der betroffenen Person nur für den Zeitraum, der zur Erreichung des Speicherungszwecks erforderlich ist oder sofern dies durch den Europäischen Richtlinien- und Verordnungsgeber oder einen anderen Gesetzgeber in Gesetzen oder Vorschriften, welchen der für die Verarbeitung Verantwortliche unterliegt, vorgesehen wurde.

Entfällt der Speicherungszweck oder läuft eine vom Europäischen Richtlinien- und Verordnungsgeber oder einem anderen zuständigen Gesetzgeber vorgeschriebene Speicherfrist ab, werden die personenbezogenen Daten routinemäßig und entsprechend den gesetzlichen Vorschriften gesperrt oder gelöscht.

<h4>7. Rechte der betroffenen Person</h4>
<ul style="list-style: none;">
<li><h4>a) Recht auf Bestätigung</h4>
Jede betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber eingeräumte Recht, von dem für die Verarbeitung Verantwortlichen eine Bestätigung darüber zu verlangen, ob sie betreffende personenbezogene Daten verarbeitet werden. Möchte eine betroffene Person dieses Bestätigungsrecht in Anspruch nehmen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.
</li>
<li><h4>b) Recht auf Auskunft</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, jederzeit von dem für die Verarbeitung Verantwortlichen unentgeltliche Auskunft über die zu seiner Person gespeicherten personenbezogenen Daten und eine Kopie dieser Auskunft zu erhalten. Ferner hat der Europäische Richtlinien- und Verordnungsgeber der betroffenen Person Auskunft über folgende Informationen zugestanden:

<ul style="list-style: none;">
<li>die Verarbeitungszwecke</li>
<li>die Kategorien personenbezogener Daten, die verarbeitet werden</li>
<li>die Empfänger oder Kategorien von Empfängern, gegenüber denen die personenbezogenen Daten offengelegt worden sind oder noch offengelegt werden, insbesondere bei Empfängern in Drittländern oder bei internationalen Organisationen</li>
<li>falls möglich die geplante Dauer, für die die personenbezogenen Daten gespeichert werden, oder, falls dies nicht möglich ist, die Kriterien für die Festlegung dieser Dauer</li>
<li>das Bestehen eines Rechts auf Berichtigung oder Löschung der sie betreffenden personenbezogenen Daten oder auf Einschränkung der Verarbeitung durch den Verantwortlichen oder eines Widerspruchsrechts gegen diese Verarbeitung</li>
<li>das Bestehen eines Beschwerderechts bei einer Aufsichtsbehörde</li>
<li>wenn die personenbezogenen Daten nicht bei der betroffenen Person erhoben werden: Alle verfügbaren Informationen über die Herkunft der Daten</li>
<li>das Bestehen einer automatisierten Entscheidungsfindung einschließlich Profiling gemäß Artikel 22 Abs.1 und 4 DS-GVO und — zumindest in diesen Fällen — aussagekräftige Informationen über die involvierte Logik sowie die Tragweite und die angestrebten Auswirkungen einer derartigen Verarbeitung für die betroffene Person</li>

</ul>
Ferner steht der betroffenen Person ein Auskunftsrecht darüber zu, ob personenbezogene Daten an ein Drittland oder an eine internationale Organisation übermittelt wurden. Sofern dies der Fall ist, so steht der betroffenen Person im Übrigen das Recht zu, Auskunft über die geeigneten Garantien im Zusammenhang mit der Übermittlung zu erhalten.

Möchte eine betroffene Person dieses Auskunftsrecht in Anspruch nehmen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.
</li>
<li><h4>c) Recht auf Berichtigung</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, die unverzügliche Berichtigung sie betreffender unrichtiger personenbezogener Daten zu verlangen. Ferner steht der betroffenen Person das Recht zu, unter Berücksichtigung der Zwecke der Verarbeitung, die Vervollständigung unvollständiger personenbezogener Daten — auch mittels einer ergänzenden Erklärung — zu verlangen.

Möchte eine betroffene Person dieses Berichtigungsrecht in Anspruch nehmen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.</li>
<li>
<h4>d) Recht auf Löschung (Recht auf Vergessen werden)</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, von dem Verantwortlichen zu verlangen, dass die sie betreffenden personenbezogenen Daten unverzüglich gelöscht werden, sofern einer der folgenden Gründe zutrifft und soweit die Verarbeitung nicht erforderlich ist:

<ul style="list-style: none;">
<li>Die personenbezogenen Daten wurden für solche Zwecke erhoben oder auf sonstige Weise verarbeitet, für welche sie nicht mehr notwendig sind.</li>
<li>Die betroffene Person widerruft ihre Einwilligung, auf die sich die Verarbeitung gemäß Art. 6 Abs. 1 Buchstabe a DS-GVO oder Art. 9 Abs. 2 Buchstabe a DS-GVO stützte, und es fehlt an einer anderweitigen Rechtsgrundlage für die Verarbeitung.</li>
<li>Die betroffene Person legt gemäß Art. 21 Abs. 1 DS-GVO Widerspruch gegen die Verarbeitung ein, und es liegen keine vorrangigen berechtigten Gründe für die Verarbeitung vor, oder die betroffene Person legt gemäß Art. 21 Abs. 2 DS-GVO Widerspruch gegen die Verarbeitung ein.</li>
<li>Die personenbezogenen Daten wurden unrechtmäßig verarbeitet.</li>
<li>Die Löschung der personenbezogenen Daten ist zur Erfüllung einer rechtlichen Verpflichtung nach dem Unionsrecht oder dem Recht der Mitgliedstaaten erforderlich, dem der Verantwortliche unterliegt.</li>
<li>Die personenbezogenen Daten wurden in Bezug auf angebotene Dienste der Informationsgesellschaft gemäß Art. 8 Abs. 1 DS-GVO erhoben.</li>

</ul>
Sofern einer der oben genannten Gründe zutrifft und eine betroffene Person die Löschung von personenbezogenen Daten, die bei der TU München, Lehrstuhl für Nachrichtentechnik gespeichert sind, veranlassen möchte, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden. Der Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wird veranlassen, dass dem Löschverlangen unverzüglich nachgekommen wird.

Wurden die personenbezogenen Daten von der TU München, Lehrstuhl für Nachrichtentechnik öffentlich gemacht und ist unser Unternehmen als Verantwortlicher gemäß Art. 17 Abs. 1 DS-GVO zur Löschung der personenbezogenen Daten verpflichtet, so trifft die TU München, Lehrstuhl für Nachrichtentechnik unter Berücksichtigung der verfügbaren Technologie und der Implementierungskosten angemessene Maßnahmen, auch technischer Art, um andere für die Datenverarbeitung Verantwortliche, welche die veröffentlichten personenbezogenen Daten verarbeiten, darüber in Kenntnis zu setzen, dass die betroffene Person von diesen anderen für die Datenverarbeitung Verantwortlichen die Löschung sämtlicher Links zu diesen personenbezogenen Daten oder von Kopien oder Replikationen dieser personenbezogenen Daten verlangt hat, soweit die Verarbeitung nicht erforderlich ist. Der Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wird im Einzelfall das Notwendige veranlassen.
</li>
<li><h4>e) Recht auf Einschränkung der Verarbeitung</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, von dem Verantwortlichen die Einschränkung der Verarbeitung zu verlangen, wenn eine der folgenden Voraussetzungen gegeben ist:

<ul style="list-style: none;">
<li>Die Richtigkeit der personenbezogenen Daten wird von der betroffenen Person bestritten, und zwar für eine Dauer, die es dem Verantwortlichen ermöglicht, die Richtigkeit der personenbezogenen Daten zu überprüfen.</li>
<li>Die Verarbeitung ist unrechtmäßig, die betroffene Person lehnt die Löschung der personenbezogenen Daten ab und verlangt stattdessen die Einschränkung der Nutzung der personenbezogenen Daten.</li>
<li>Der Verantwortliche benötigt die personenbezogenen Daten für die Zwecke der Verarbeitung nicht länger, die betroffene Person benötigt sie jedoch zur Geltendmachung, Ausübung oder Verteidigung von Rechtsansprüchen.</li>
<li>Die betroffene Person hat Widerspruch gegen die Verarbeitung gem. Art. 21 Abs. 1 DS-GVO eingelegt und es steht noch nicht fest, ob die berechtigten Gründe des Verantwortlichen gegenüber denen der betroffenen Person überwiegen.</li>

</ul>
Sofern eine der oben genannten Voraussetzungen gegeben ist und eine betroffene Person die Einschränkung von personenbezogenen Daten, die bei der TU München, Lehrstuhl für Nachrichtentechnik gespeichert sind, verlangen möchte, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden. Der Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wird die Einschränkung der Verarbeitung veranlassen.
</li>
<li><h4>f) Recht auf Datenübertragbarkeit</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, die sie betreffenden personenbezogenen Daten, welche durch die betroffene Person einem Verantwortlichen bereitgestellt wurden, in einem strukturierten, gängigen und maschinenlesbaren Format zu erhalten. Sie hat außerdem das Recht, diese Daten einem anderen Verantwortlichen ohne Behinderung durch den Verantwortlichen, dem die personenbezogenen Daten bereitgestellt wurden, zu übermitteln, sofern die Verarbeitung auf der Einwilligung gemäß Art. 6 Abs. 1 Buchstabe a DS-GVO oder Art. 9 Abs. 2 Buchstabe a DS-GVO oder auf einem Vertrag gemäß Art. 6 Abs. 1 Buchstabe b DS-GVO beruht und die Verarbeitung mithilfe automatisierter Verfahren erfolgt, sofern die Verarbeitung nicht für die Wahrnehmung einer Aufgabe erforderlich ist, die im öffentlichen Interesse liegt oder in Ausübung öffentlicher Gewalt erfolgt, welche dem Verantwortlichen übertragen wurde.

Ferner hat die betroffene Person bei der Ausübung ihres Rechts auf Datenübertragbarkeit gemäß Art. 20 Abs. 1 DS-GVO das Recht, zu erwirken, dass die personenbezogenen Daten direkt von einem Verantwortlichen an einen anderen Verantwortlichen übermittelt werden, soweit dies technisch machbar ist und sofern hiervon nicht die Rechte und Freiheiten anderer Personen beeinträchtigt werden.

Zur Geltendmachung des Rechts auf Datenübertragbarkeit kann sich die betroffene Person jederzeit an einen Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik wenden.

</li>
<li>
<h4>g) Recht auf Widerspruch</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, aus Gründen, die sich aus ihrer besonderen Situation ergeben, jederzeit gegen die Verarbeitung sie betreffender personenbezogener Daten, die aufgrund von Art. 6 Abs. 1 Buchstaben e oder f DS-GVO erfolgt, Widerspruch einzulegen. Dies gilt auch für ein auf diese Bestimmungen gestütztes Profiling.

Die TU München, Lehrstuhl für Nachrichtentechnik verarbeitet die personenbezogenen Daten im Falle des Widerspruchs nicht mehr, es sei denn, wir können zwingende schutzwürdige Gründe für die Verarbeitung nachweisen, die den Interessen, Rechten und Freiheiten der betroffenen Person überwiegen, oder die Verarbeitung dient der Geltendmachung, Ausübung oder Verteidigung von Rechtsansprüchen.

Verarbeitet die TU München, Lehrstuhl für Nachrichtentechnik personenbezogene Daten, um Direktwerbung zu betreiben, so hat die betroffene Person das Recht, jederzeit Widerspruch gegen die Verarbeitung der personenbezogenen Daten zum Zwecke derartiger Werbung einzulegen. Dies gilt auch für das Profiling, soweit es mit solcher Direktwerbung in Verbindung steht. Widerspricht die betroffene Person gegenüber der TU München, Lehrstuhl für Nachrichtentechnik der Verarbeitung für Zwecke der Direktwerbung, so wird die TU München, Lehrstuhl für Nachrichtentechnik die personenbezogenen Daten nicht mehr für diese Zwecke verarbeiten.

Zudem hat die betroffene Person das Recht, aus Gründen, die sich aus ihrer besonderen Situation ergeben, gegen die sie betreffende Verarbeitung personenbezogener Daten, die bei der TU München, Lehrstuhl für Nachrichtentechnik zu wissenschaftlichen oder historischen Forschungszwecken oder zu statistischen Zwecken gemäß Art. 89 Abs. 1 DS-GVO erfolgen, Widerspruch einzulegen, es sei denn, eine solche Verarbeitung ist zur Erfüllung einer im öffentlichen Interesse liegenden Aufgabe erforderlich.

Zur Ausübung des Rechts auf Widerspruch kann sich die betroffene Person direkt jeden Mitarbeiter der TU München, Lehrstuhl für Nachrichtentechnik oder einen anderen Mitarbeiter wenden. Der betroffenen Person steht es ferner frei, im Zusammenhang mit der Nutzung von Diensten der Informationsgesellschaft, ungeachtet der Richtlinie 2002/58/EG, ihr Widerspruchsrecht mittels automatisierter Verfahren auszuüben, bei denen technische Spezifikationen verwendet werden.
</li>
<li><h4>h) Automatisierte Entscheidungen im Einzelfall einschließlich Profiling</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, nicht einer ausschließlich auf einer automatisierten Verarbeitung — einschließlich Profiling — beruhenden Entscheidung unterworfen zu werden, die ihr gegenüber rechtliche Wirkung entfaltet oder sie in ähnlicher Weise erheblich beeinträchtigt, sofern die Entscheidung (1) nicht für den Abschluss oder die Erfüllung eines Vertrags zwischen der betroffenen Person und dem Verantwortlichen erforderlich ist, oder (2) aufgrund von Rechtsvorschriften der Union oder der Mitgliedstaaten, denen der Verantwortliche unterliegt, zulässig ist und diese Rechtsvorschriften angemessene Maßnahmen zur Wahrung der Rechte und Freiheiten sowie der berechtigten Interessen der betroffenen Person enthalten oder (3) mit ausdrücklicher Einwilligung der betroffenen Person erfolgt.

Ist die Entscheidung (1) für den Abschluss oder die Erfüllung eines Vertrags zwischen der betroffenen Person und dem Verantwortlichen erforderlich oder (2) erfolgt sie mit ausdrücklicher Einwilligung der betroffenen Person, trifft die TU München, Lehrstuhl für Nachrichtentechnik angemessene Maßnahmen, um die Rechte und Freiheiten sowie die berechtigten Interessen der betroffenen Person zu wahren, wozu mindestens das Recht auf Erwirkung des Eingreifens einer Person seitens des Verantwortlichen, auf Darlegung des eigenen Standpunkts und auf Anfechtung der Entscheidung gehört.

Möchte die betroffene Person Rechte mit Bezug auf automatisierte Entscheidungen geltend machen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.

</li>
<li><h4>i) Recht auf Widerruf einer datenschutzrechtlichen Einwilligung</h4>
Jede von der Verarbeitung personenbezogener Daten betroffene Person hat das vom Europäischen Richtlinien- und Verordnungsgeber gewährte Recht, eine Einwilligung zur Verarbeitung personenbezogener Daten jederzeit zu widerrufen.

Möchte die betroffene Person ihr Recht auf Widerruf einer Einwilligung geltend machen, kann sie sich hierzu jederzeit an einen Mitarbeiter des für die Verarbeitung Verantwortlichen wenden.

</li>
</ul>
<h4>8. Rechtsgrundlage der Verarbeitung</h4>
Art. 6 I lit. a DS-GVO dient unserem Unternehmen als Rechtsgrundlage für Verarbeitungsvorgänge, bei denen wir eine Einwilligung für einen bestimmten Verarbeitungszweck einholen. Ist die Verarbeitung personenbezogener Daten zur Erfüllung eines Vertrags, dessen Vertragspartei die betroffene Person ist, erforderlich, wie dies beispielsweise bei Verarbeitungsvorgängen der Fall ist, die für eine Lieferung von Waren oder die Erbringung einer sonstigen Leistung oder Gegenleistung notwendig sind, so beruht die Verarbeitung auf Art. 6 I lit. b DS-GVO. Gleiches gilt für solche Verarbeitungsvorgänge die zur Durchführung vorvertraglicher Maßnahmen erforderlich sind, etwa in Fällen von Anfragen zur unseren Produkten oder Leistungen. Unterliegt unser Unternehmen einer rechtlichen Verpflichtung durch welche eine Verarbeitung von personenbezogenen Daten erforderlich wird, wie beispielsweise zur Erfüllung steuerlicher Pflichten, so basiert die Verarbeitung auf Art. 6 I lit. c DS-GVO. In seltenen Fällen könnte die Verarbeitung von personenbezogenen Daten erforderlich werden, um lebenswichtige Interessen der betroffenen Person oder einer anderen natürlichen Person zu schützen. Dies wäre beispielsweise der Fall, wenn ein Besucher in unserem Betrieb verletzt werden würde und daraufhin sein Name, sein Alter, seine Krankenkassendaten oder sonstige lebenswichtige Informationen an einen Arzt, ein Krankenhaus oder sonstige Dritte weitergegeben werden müssten. Dann würde die Verarbeitung auf Art. 6 I lit. d DS-GVO beruhen.
Letztlich könnten Verarbeitungsvorgänge auf Art. 6 I lit. f DS-GVO beruhen. Auf dieser Rechtsgrundlage basieren Verarbeitungsvorgänge, die von keiner der vorgenannten Rechtsgrundlagen erfasst werden, wenn die Verarbeitung zur Wahrung eines berechtigten Interesses unseres Unternehmens oder eines Dritten erforderlich ist, sofern die Interessen, Grundrechte und Grundfreiheiten des Betroffenen nicht überwiegen. Solche Verarbeitungsvorgänge sind uns insbesondere deshalb gestattet, weil sie durch den Europäischen Gesetzgeber besonders erwähnt wurden. Er vertrat insoweit die Auffassung, dass ein berechtigtes Interesse anzunehmen sein könnte, wenn die betroffene Person ein Kunde des Verantwortlichen ist (Erwägungsgrund 47 Satz 2 DS-GVO).


<h4>9. Berechtigte Interessen an der Verarbeitung, die von dem Verantwortlichen oder einem Dritten verfolgt werden</h4>
Basiert die Verarbeitung personenbezogener Daten auf Artikel 6 I lit. f DS-GVO ist unser berechtigtes Interesse die Durchführung unserer Geschäftstätigkeit zugunsten des Wohlergehens all unserer Mitarbeiter und unserer Anteilseigner.

<h4>10. Dauer, für die die personenbezogenen Daten gespeichert werden</h4>
Das Kriterium für die Dauer der Speicherung von personenbezogenen Daten ist die jeweilige gesetzliche Aufbewahrungsfrist. Nach Ablauf der Frist werden die entsprechenden Daten routinemäßig gelöscht, sofern sie nicht mehr zur Vertragserfüllung oder Vertragsanbahnung erforderlich sind.

<h4>11. Gesetzliche oder vertragliche Vorschriften zur Bereitstellung der personenbezogenen Daten; Erforderlichkeit für den Vertragsabschluss; Verpflichtung der betroffenen Person, die personenbezogenen Daten bereitzustellen; mögliche Folgen der Nichtbereitstellung</h4>
Wir klären Sie darüber auf, dass die Bereitstellung personenbezogener Daten zum Teil gesetzlich vorgeschrieben ist (z.B. Steuervorschriften) oder sich auch aus vertraglichen Regelungen (z.B. Angaben zum Vertragspartner) ergeben kann.
Mitunter kann es zu einem Vertragsschluss erforderlich sein, dass eine betroffene Person uns personenbezogene Daten zur Verfügung stellt, die in der Folge durch uns verarbeitet werden müssen. Die betroffene Person ist beispielsweise verpflichtet uns personenbezogene Daten bereitzustellen, wenn unser Unternehmen mit ihr einen Vertrag abschließt. Eine Nichtbereitstellung der personenbezogenen Daten hätte zur Folge, dass der Vertrag mit dem Betroffenen nicht geschlossen werden könnte.
Vor einer Bereitstellung personenbezogener Daten durch den Betroffenen muss sich der Betroffene an einen unserer Mitarbeiter wenden. Unser Mitarbeiter klärt den Betroffenen einzelfallbezogen darüber auf, ob die Bereitstellung der personenbezogenen Daten gesetzlich oder vertraglich vorgeschrieben oder für den Vertragsabschluss erforderlich ist, ob eine Verpflichtung besteht, die personenbezogenen Daten bereitzustellen, und welche Folgen die Nichtbereitstellung der personenbezogenen Daten hätte.


<h4>12. Bestehen einer automatisierten Entscheidungsfindung</h4>
Als verantwortungsbewusstes Unternehmen verzichten wir auf eine automatische Entscheidungsfindung oder ein Profiling.

Diese Datenschutzerklärung wurde durch den Datenschutzerklärungs-Generator der DGD Deutsche Gesellschaft für Datenschutz GmbH, die als <a href="https://dg-datenschutz.de/datenschutz-dienstleistungen/externer-datenschutzbeauftragter/" rel="nofollow">Externer Datenschutzbeauftragter Berlin</a> tätig ist, in Kooperation mit den <a href="https://www.wbs-law.de/" rel="nofollow">Datenschutz (DSGVO) Anwälten der Kanzlei WILDE BEUGER SOLMECKE | Rechtsanwälte</a> erstellt.

</html>

LNTwww:About LNTwww

2023-02-24T14:55:09Z

Bene: Text replacement - "LNTwww (at) ice.cit.tum.de" to "LNTwww@ice.cit.tum.de"

==Welcome to the English version of LNTwww==
 
»$\text{https://en.lntwww.de}$«  is an e-learning tutorial for Communications Engineering with nine didactic multimedia textbooks including exercises with solutions,  learning videos,  and interactive applets.  It is offered by the  »[https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]«  $\rm (LNT)$  of the  »[https://www.tum.de/en/ Technical University of Munich]«  $\rm (TUM)$. 

:⇒   '''It is freely accessible,  registration is not necessary and no system requirements are needed'''.

The German-language version   »$\text{https://www.lntwww.de}$«   ⇒   »$\rm L$erntutorial für $\rm N$achrichten$\rm T$echnik im $\rm w$orld $\rm w$ide $\rm w$eb«  was created between 2001 – 2021 by members of our Institute.  The toolbar entry  »Deutsch«  takes you to the German original. In spring 2020 we started the English translation,  and in spring 2023 we finished.

*The current version from 2023 is based on the software  [https://en.wikipedia.org/wiki/MediaWiki »MediaWiki«],  known by the encyclopaedia  "WIKIPEDIA";.   The following is a kind of "user guide" to our e–learning offer.  Corresponding links to this file  »About LNTwww«  can be found at the bottom of each page between  »Privacy policy« and  »Disclaimer«.

*You can find more information on our e-learning offer in the PDF document  [https://www.lntwww.de/downloads/Sonstiges/HER-Beitrag_LNTwww.pdf »LNTwww - Praxisbericht zum E-Learning aus den Ingenieurwissenschaften«],  a German-language contribution by Günter Söder in the special issue  »So gelingt E-Learning«,  Reader zum Higher Education Summit 2019,  Munich:  Pearson Studium.

*We consider the present version as final;  an extension is currently not planned.  But of course we will continue to improve detected errors or inaccuracies promptly.  So if you notice any inadequacies regarding content,  presentation or handling,  then please send a detailed message by mail to  »LNTwww@ice.cit.tum.de«.

*On the  [[LNTwww:Information|»Information«]]  page you will find notes about temporary restrictions  $($e.g. in case of unavailability due to service work$)$  and a list of bugs already detected by us,  but not yet fixed.   We wish  that in this list there are only few entries and only for a short time.

We would be pleased if we could arouse your interest in our e-learning offer.  We wish you a successful learning success.

$\text{Have fun and good luck!}$  

[https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29| $\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]]

Munich,  in spring 2023          

 

==Content==

===(A)   The didactic concept of LNTwww===

At the beginning of the work on  $\rm LNTwww$  in 2001,  we gave ourselves the following ten rules.  These still apply today:

'''(1)'''   The teaching area  "Information and Communication Technology"  $\text{(I&K)}$  including associated basic subjects  $($Signal Representation,  Fourier and Laplace Transform,  Stochastic Signal Theory, etc.$)$  is presented in a didactically and multimedia prepared form.

'''(2)'''   Nine subject areas were selected,  each of which is covered by a self-contained book in the scope of a one-semester course with three semester hours per week to five semester hours per week.

'''(3)'''   The target group of our online offer are students of  $\text{I&K}$  technology,  especially of communications engineering,  as well as practicing engineers  $($Keywords:  "professional training",  "lifelong learning"$)$.

'''(4)'''   In particular,  the interrelationships between different subfields of our extensive e-leatning offer should also be shown,  which is promoted by a nomenclature that is largely consistent in all books.

'''(5)'''   $\rm LNTwww$  offers two modes of learning:   Beginners should proceed sequentially  –   for advanced learners, use as a tutorial  $($work through tasks first,  jump to theory if deficits are identified$)$.

'''(6)'''   Theory is explained as in a traditional engineering textbook through texts,  graphics,  and mathematical derivations.  In addition,  each chapter includes at least one multimedia module.

'''(7)'''   $\rm LNTwww$  shall provide the user with multiple interaction options regarding the selection and presentation of theory chapters,  exercises,  learning videos as well as multimedia and calculation modules.

'''(8)'''   The methodology of hyperlinks typical of the  "world wide web"  is extensively used within
the  $\rm LNTwww$  and externally.  This is also intended to show connections between different teaching areas.

'''(9)'''   In order to prevent a user from getting lost in his learning environment and using  $\rm LNTwww$  only for  "surfing",  a purposeful path must be recognizable for him at all times despite certain freedoms.

'''(10)'''  For reasons of sustainability of learning success,  there are possibilities for printing the texts and graphics,  ignoring the fact that today's students generation often devalues this as a  "relapse into the analog age".
 

===(B)   Content and scope of LNTwww===

$\rm LNTwww$  is a virtual course totaling  $\text{36 sh/w}$  (semester hours per week) 
*with  $\text{23 sh/w}$  (quasi-)lectures

*and  $\text{13 sh/w}$  exercises. 

It is organized in book form.  Each book contains a one-semester course.  For example,  in the case of the third book,  it is indicated that this book corresponds to a face-to-face–course with three semester hours per week of LECTURE and two semester hours per week of EXERCISES.

# [[Signal_Representation|'''Signal Representation''']]   ⇒   [[LNTwww:General_notes_about_"Signal_Representation"|More Information]],
# [[Linear_and_Time_Invariant_Systems|'''Linear and Time Invariant Systems''']]   ⇒  [[LNTwww:General_notes_about_"Linear_and_Time_Invariant_Systems"|More Information]],
# [[Theory_of_Stochastic_Signals|'''Theory of Stochastic Signals''']]   ⇒  [[LNTwww:General_Notes_about_the_Book_"Stochastic_Signal_Theory"|More Information]],
# [[Information_Theory|'''Information Theory''']]   ⇒  [[LNTwww:General_notes_about_"Information_Theory"|More Information]],
# [[Modulation_Methods|'''Modulation Methods''']]   ⇒  [[LNTwww:General_notes_about_"Modulation_Methods"|More Information]],
# [[Digital_Signal_Transmission|'''Digital Signal Transmission''']]   ⇒  [[LNTwww:General_notes_about_"Digital_Signal_Transmission"|More Information]],
# [[Mobile_Communications|'''Mobile Communications''']]   ⇒  [[LNTwww:General_notes_about_"Mobile_Communications"|More Information]],
# [[Channel_Coding|'''Channel Coding''']]   ⇒  [[LNTwww:General_notes_about_"Channel_Coding"|More Information]],
#[[Examples_of_Communication_Systems|'''Examples of Communication Systems''']]   ⇒  [[LNTwww:General_notes_about_"Examples_of_Communication_Systems"|More Information]].

*The theory pages of all books result in the print version in approx.  $1500$  pages  $($DIN A4$)$  and contain on average one and a half graphics per page. 

*In addition, LNTwww provides via the link  "Biographies & Bibliography"  a subject-specific bibliography with approx.  $400$  entries,  plus links to the WIKIPEDIA biographies of important scientists.
 

===(C)   Design and structure of LNTwww===

One can reach the nine reference books and &bdquo;Biographies & Bibliography”  through the link  [[Book Overview|"Book Overview"]].  From this interface one can reach the individual books.  
*Each book is divided into several  $\text{main chapters}$, 
*each main chapter into several  $\rm chapters$,  and
*each chapter includes several  $\rm sections$.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
We consider the book  [[Signal Representation|"Signal Representation"]].  This contains five  "main chapters", including  "Basic Terms of Communications Engineering".
*By clicking on the first main chapter,  one can get to three  "chapters"  including the first chapter  [[Signal_Representation/Principles_of_Communication|"Principles of Communication"]].  Such a chapter corresponds to a saved MediaWiki–file.
*The exemplary chapter  "Principles of Communication"  contains ten  "sections".  The last two sections are almost the same in all chapters, namely  "Exercises for the chapter"  and  "References".}}
 

===(D)   Content overviews for LNTwww===

A brief overview of all books is available on the selection interface  [[Book Overview|"Book Overview"]].
*More information is provided by the  "first page"  of each book.
*The respective main chapter content can be found in the first sub–chapter on the first page of each.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The first page  ("title page")  of the book  [[Signal_Representation|"Signal Representation"]]  provides the following information:
* A brief summary;
* Scope of learning:  Lecture with two semester hours per week  $\rm (2\ sh/w)$  and additional  $\rm 1 sh/w$  exercise.  Five main chapters.  Nineteen chapters;
* Links to the five main chapters;
* Links to the assignments, learning videos, and interactive applets in the book  "Signal Representation";
* Recommended reading for the book;
* Other notes about the book  (Authors, Other contributors, Materials as a starting point of the book, List of sources).

The content of the first main chapter &bdquo;Time-variant transmission channels” can be found on the page 
[[Signal_Representation/Principles_of_Communication#OVERVIEW_OF_THE_FIRST_MAIN_CHAPTER|"OVERVIEW OF THE FIRST MAIN CHAPTER"
]]}}
 

===(E)   LNTwww exercises===

You can find the exercise overview for all books  (approx.  $640$  exercises, approx.  $3100$  subtasks)  on the home page via the link  [[Aufgaben:Aufgabensammlung|"Exercises"]].  Please note:
*Each exercise consists of several  "subtasks".  An exercise is only solved correctly if all subtasks are correct.
* For each exercise there is a detailed  "sample solution",  sometimes also with the indication of several ways to the goal.
* The exercise types used are:
# "Single Choice"   ⇒   only one of the  $n$  given answers is correct;      ⇒   alternative answers– mark:  ${\huge\circ}$
# "Multiple Choice"   ⇒   of the  $n$  given answers, between zero and  $n$  answers can be correct;      ⇒   alternative answers– mark:  $\square$
# "Arithmetic exercise"   ⇒   numerical value query,  possibly with sign;     small deviations  $($usually  $\pm 3\%)$  are allowed when checking real-valued results.
* We distinguish between  "exercises"  (e.g.  "Exercise 1.1") and  "additional exercises"  (e.g.  "Exercise 1.1Z").
* If you were able to solve all exercises of a chapter without any problems,  we believe that you are familiar with the chapter content.  If you have solved one exercise incorrectly,  you should also work on the following,  usually somewhat easier additional exercise.

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The  $93$  exercises/additional exercises of the book  "Theory of Stochastic Signals"  can be accessed via the link  [https://en.lntwww.de/Category:Theory_of_Stochastic_Signals:_Exercises "Theory of Stochastic Signals: Exercises"]. 

*From there you can proceed to the individual exercises,  for example  [[Aufgaben:Exercise_1.1:_A_Special_Dice_Game| Exercise 1.1: "A Special Dice Game"]].  This relatively simple exercise consists of a  "Multiple Choice"  and an  "Arithmetic Exercise".  You can see that the  "solution"  is described very detailed and even includes a short video.

*But there are also much more difficult exercises in  $\rm LNTwww$.  Although MediaWiki also calls arithmetic exercises  "quizzes",  answering them is usually much more difficult than with  "Jauch".   Because:  there are no predetermined answers in an arithmetic exercise,  and moreover,  integrals often have to be solved beforehand,  such as in  [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|Exercise 4.4: "Two-dimensional Gaussian PDF"]].
*We recommend:  Print the exercise first   ⇒   "$\text{Printable version}$"  and solve the exercise  "offline"  before checking  "online"      ⇒   Note:  In the  "printable version":  For links, the target addresses are always given in brackets.
}}
 

===(F)   LNTwww learning videos===

You can access approximately  $30$  learning videos via the link  "Videos"  on the start page.  The realization of a learning video required the following individual steps:  Writing the script and texts   –   Creating a set of slides with only slight differences between successive slides   –   Voicing texts, cutting and audio editing   –   Combining texts and images into a coherent video stream.
*Clicking on this link brings up a  "list"  of all learning videos,  grouped by textbook.  Some videos appear for multiple books.
*After selecting the desired learning video,  a wiki description page appears with a short content description and user interface.
*From here you can start the video in mp4 and ogv format.  The browser will search for the appropriate format.
*The videos can be played by many browsers  (Firefox, Chrome, Safari, ...)  as well as smartphones and tablets.
*The bottom link provides all available learning videos in alphabetical order.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
We'll take a look at   [[Analoge_und_digitale_Signale_(Lernvideo)|"Analog and digital signals"]]  as an example.  This provides a two-part video in mp4 and ogv format.
*Each video part can be started by single click and paused by another click.
*The playback speed of the videos can be changed:
** Firefox offers a submenu after right-clicking on the video.
** For Google Chrome you can install e.g. the plugin  "Video Speed Controller".
}}
 

===(G)   LNTwww applets===

You can access the provided interactive applets via the link  "Applets"  on the start page.
* If you click on it,  a  "list"  of all applets appears,  grouped by textbook.  We distinguish between the newer  $\text{HTML 5/JavaScript}$ applets  (in the respective lists above)  and the older  $\text{SWF}$ applets  (below).  '''Unfortunately, the latter do not work on smartphones and tablets'''.
*After selecting an  $\text{HTML 5/JS}$  applet,  a wiki description page appears with a description of the content,  an often longer theory section,  and then the execution of the experiment with sample solutions.  At the beginning and end of this page there are links to the actual HTML5 applet in German and English.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The didactic significance of the applets is to be demonstrated by means of   [[Applets:Eye_Pattern_and_Worst-Case_Error_Probability|"Eye Pattern and Worst-Case Error Probability"]].   The eye diagram is a proven tool of transmission technology to capture the influence of line dispersion on the quality characteristic  "error probability".  It is used to clarify more difficult issues,  in the example of the step-by-step construction of the eye diagram from the symbol sequence.

The program offers a lot of setting possibilities.  However,  not every setting brings the user a relevant learning success and even fewer lead to a so-called  "aha effect".  Therefore,  we guide the user specifically through the program on the basis of the experimental procedure.  He has to solve different tasks:  Predict and evaluate results, optimize parameters, etc.

Applets have a similar function as practical courses in mathematical-scientific courses:  Supplementing lecture/exercise with independent work by the student on the topic covered.  A "top 10%" student has of course the possibility to use the applet to set himself tasks that go beyond the execution of the experiment and thus penetrate very deeply into the presented subject matter. }}

In addition to these  $30$  or so HTML 5/JS applets, we also offer some of our  $50$  SWF  ('''S'''hock '''W'''ave '''F'''lash)  applets.  These were programmed for "Adobe Flash".  Since the Flashplayer Browser Plugin is no longer supported for security reasons, these applets have to be opened with the "Projector Version".

You do not have to install the projector version and it will not be integrated into your browser.  So there are no security concerns in that regard, as long as you trust our  $\rm LNTwww$.   On the corresponding wiki pages you can find the projector version of the flash player and of course the applet itself.
 

===(H)   The download area of LNTwww===

All texts for LNTww can be found as PDF under the link   [http://www.lntwww.de/downloads/ '''Zum Download-Verzeichnis''']

'''! Noch überarbeiten !'''
 

===(I)    History of LNTwww===
 
At the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  $\rm (LNT)$  of the  [https://www.tum.de Technical University of Munich]  $\rm (TUM)$  two  [[LNTwww:Über_LNTwww#Zu_unseren_fr.C3.BCheren_Arbeiten_bez.C3.BCglich_e.E2.80.93Learning|teaching software packages]]  $\text{(LNTsim, LNTwin)}$  were realized from 1984 to 1996, which were used in our practical courses.  Several other universities have also used these programs in teaching.

At the beginning of the first Internet euphoria, there were inquiries from students whether we could also provide such simulation and demonstration programs online.  After careful consideration  ("Is the expected big effort worth it?")  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]]  started planning "LNTww.v1"   (2001).  Co-responsible was  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]], who was already very active in the 1970s in "computer-assisted teaching" - that was the name of "e-learning" at that time.  The project was to be completed by 2011 at the latest.

The content was based on the teaching materials of Klaus Eichin and Günter Söder as well as  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  (Professorship "Conducted Transmission Technology"). Other lecture material, which was produced at the Institute of Communications Engineering under the last four chair holders, was also taken into account:
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|Hans Marko]]  (1962– 1993),
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]]  (1993– 2006),
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]  (2007–2009) and
::*Professor [https://www.ei.tum.de/en/lnt/people/professors/kramer/ Gerhard Kramer]  (since 2010).

Before we could start to implement our ideas, several dedicated and IT-savvy students had to develop the platform "LNTwww" as part of their final theses.  The authoring system was based on the http server "Apache", the database "MySQL" and the script language "Perl".  All entered entities  (texts and text fragments, equations, diagrams, hyperlinks, multimedia elements, etc.)  were stored in the database, which was huge for the time, along with various display features for color coding of definitions, examples, etc.

*We decided to use Shock Wave Flash (SWF) as the technical basis for the multimedia applications.  The decision was easy, because this tool was acknowledged to be the best at that time.

*The upcoming work in the following years was the adaptation of the manuscripts to online operation, the input into the database with the rather complicated LNTww syntax, the creation of the diagrams as well as the conception and realization of multimedia elements.

But only in 2016 - after fifteen years and five years after the planned completion - the desired final state of "LNTwww.v2" was reached.  At the same time, it became known that the base "SWF" of our multimedia applications would not be supported by relevant manufacturers in the future.

This fact and the criticism heard from some users about the meanwhile too staid design  (our authoring system was on the level of 2003)  were decisive for a new start with "LNTww.v3", based on MediaWiki (known by WIKIPEDIA).
*The conversion to "LNTww.v3" took more than four labor-intensive years.  For mathematical and scientific content, porting to another e-learning base  (like here from "LNTwww" to "MediaWiki")  is only possible manually due to many special characters, italics, superscripts and subscripts.
*The conversion of the learning videos  (from "swf" to "mp4" or "ogv")  could be largely automated.  In contrast, the conversion of the interactive applets  (from "swf" to "HTML5/JS")  required reprogramming, in which many of our students were involved, as in previous years.

After some control and correction iterations, our e-learning service  $\text{https://www.LNTwww.de}$  will now finally be released in March 2021, almost exactly twenty years after the first planning and ten years after the planned completion. 

In terms of content, this third version differs not only insignificantly from the second, but the multimedia elements in particular have been significantly improved.  We assume that "MediaWiki" will remain the quasi-standard for Internet applications for several years.  Then this effort would have been worthwhile.
 

===(J)   Acknowledgement===

Günter Söder, who is still responsible for  $\rm LNTwww$, would like to thank the many people involved in the creation of  $\rm LNTwww$, also on behalf of the Institute of Communications Engineering at the TU Munich and its director Gerhard Kramer,

*first of all to the two co-responsible persons  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]]  (until 2011, besides planning also co–author) and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29|Tasnád Kernetzky]] (since 2016, responsible for the system configuration and administration as well as the conversion to MediaWiki, HTML5/JS, MP4);

*to  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Martin_Winkler_.28Diplomarbeit_LB_2001.2C_danach_freie_Mitarbeit_bis_2003.29|Martin Winkler]]  and  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Yven_Winter_.28Diplomarbeit_LB_2004.2C_danach_freie_Mitarbeit_bis_2016.29|Yven Winter]],  who laid the technical foundations with their diploma theses in the early 2000s; the latter was still a volunteer system administrator until 2016;

* to Prof.  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]] (professorship "line-bound transmission technology" – co–author of several books and eager propagator of our learning opportunities in his lectures)  and  to his PhD students  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|Bernhard Göbel]],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29|Tasnád Kernetzky]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29|Benedikt Leible]];

* to former LNT colleagues Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|Ronald Böhnke]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|Joschi Brauchle]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|Thomas Hindelang]],  Prof. [[Biographies_and_Bibliographies/External_Contributors_to_LNTwww#Dr._Gianluigi_Liva|Gianluigi Liva]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|Tobias Lutz]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|Michael Mecking]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|Markus Stinner]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|Thomas Stockhammer]],  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|Johannes Zangl]]  and  Dr. [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Georg_Zeitler_.28at_LNT_from_2007-2012.29|Georg Zeitler]], who contributed as co–authors or experts or supervised student work;

* to all colleagues of the LNT, who actively supported us in many often tedious and nerve-racking tasks:  Doris Dorn (entered countless texts and equations in the complicated LNTwww syntax), Manfred Jürgens, Martin Kontny, Winfried Kretzinger, Robert Schetterer and Christin Wizemann;

*to many  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww|students involved in LNTwww]] – behind this link are almost fifty students who, between 2001 and 2021, have independently worked on subareas, designed learning videos and interactive applets elements or implemented the porting to the MediaWiki version within the framework of engineering practice, admission, diploma, bachelor and master theses or within the framework of a working student activity;

*to the  [https://www.https://www.ei.tum.de/en/welcome/ Department of Electrical and Computer Engineering]  and the  [https://www.tum.de/en/ Technical University of Munich]  for funding working students within the framework of the  [https://www.ei.tum.de/studium/studienzuschuesse/ "MoliTUM"]  and  [https://www.tum.de/en/studies/teaching/awards-and-competitions/ideas-competition "EXIni"]  funding programs, respectively, in the years since 2016.

LNTwww:Information

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===Here some notes on LNTwww===

*For detailed information about our e–Learning tutorial,  please see the page  »[[LNTwww:About_LNTwww|$\text{About LNTwww}$]]«  –  a kind of &bdquo;user guide”.

*On every LNTwww page,  there is a link to this file at the bottom  $($between &bdquo;Privacy” and &bdquo;Disclaimer”$)$.

*We consider this March 2023 version to be final;  no further revision or expansion is currently planned. 

*Therefore,  there will be no English translations of the German-language learning videos and SWF applets in the future.

*But, of course, we will continue to improve identified errors regarding content,  presentation or handling in a timely manner. 

*Should you notice any such inadequacies,  then please send a detailed message by mail to &bdquo;LNTwww@ice.cit.tum.de”.

*At the bottom of this page you will find  »Notes about temporary restrictions«  as well as a  »List of bugs we have already detected,  but not yet fixed«. 

*It is our wish that there are only very few entries in these two sections and only for a short time.

===Notes on temporary restrictions===
#   ...

===List of bugs we have already detected, but not yet fixed===

#   ...

LNTwww:General disclaimer

2023-02-24T14:53:50Z

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===Site notice===

'''Responsible for the project:''' 
:Prof. Dr. sc. techn. '''Gerhard Kramer''' 
:Institute for Communications Engineering (LNT)
:Technical University of Munich (TUM) 
:Theresienstrasse 90 
:D–80333 München 
:Tel: +49 89 289-23466 
:E-Mail:   [mailto:gerhard.kramer@tum.de gerhard.kramer@tum.de]

'''Authorized representative:'''
:The Technical University of Munich is a statutory body under public law. It is legally represented by the President, Prof. Dr. '''Thomas F. Hofmann'''.

'''Competent supervisory authority:'''
:Bavarian State Ministry of Sciences, Research and the Arts

'''Responsible for content''' 
:'''Francisco Javier Garcia Gomez''', M. Sc. 
:Prof. (i. R.) Dr.–Ing. habil. '''Günter Söder''' 
:Institute for Communications Engineering (address see above) 
:Tel: +49 89 289-23480 
:E-Mail:   [mailto:LNTwww@ice.cit.tum.de LNTwww@ice.cit.tum.de]

'''Responsible for MediaWiki presentation:''' 
:'''Tasnád Kernetzky''', M. Sc. 
:Associate Professorship of Line Transmission Technology (LÜT, same address as LNT) 
:Tel: +49 89 289-25002 
:E-Mail:   [mailto:LNTwww@ice.cit.tum.de LNTwww@ice.cit.tum.de]

'''Legal disclaimer:'''
:*In spite of taking great care to check, we do not accept any responsibility for the content of external links. The operators of these websites are solely responsible for their content.

:*Contributions in the discussion sections which are attributed by name reflect the opinion of the author. The authors themselves are solely responsible for the content of the contributions.

LNTwww:Impressum

2023-02-24T14:53:49Z

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===Impressum===
'''Verantwortlich für das Projekt:''' 
:Prof. Dr. sc. techn. '''Gerhard Kramer''' 
:Lehrstuhl für Nachrichtentechnik 
:Technische Universität München 
:Theresienstrasse 90 
:D–80333 München 
:Tel: +49 89 289-23466 
:E-Mail:   [mailto:gerhard.kramer@tum.de gerhard.kramer@tum.de]

'''Vertretungsberechtigt:'''
:Die Technische Universität München ist eine Körperschaft des Öffentlichen Rechts. Sie wird gesetzlich vertreten durch den Präsidenten, Prof. Dr. '''Thomas F. Hofmann'''.

'''Zuständige Aufsichtsbehörde:'''
:Bayerisches Staatsministerium für Wissenschaft, Forschung und Kunst

'''Verantwortlich für den Inhalt:''' 
:Prof. (i. R.) Dr.–Ing. habil. '''Günter Söder''' 
:Lehrstuhl für Nachrichtentechnik (Adresse siehe oben) 
:Tel: +49 89 289-23486 
:E-Mail:   [mailto:LNTwww@ice.cit.tum.de LNTwww@ice.cit.tum.de]

'''Verantwortlich für die MediaWiki–Umsetzung:''' 
:'''Tasnád Kernetzky''', M. Sc. 
:Professur „Leitungsgebundene Übertragungstechnik” (Adresse wie LNT) 
:Tel: +49 89 289-25002 
:E-Mail:   [mailto:LNTwww@ice.cit.tum.de LNTwww@ice.cit.tum.de]

'''Haftungshinweis:'''
:*Trotz sorgfältiger inhaltlicher Kontrolle übernehmen wir keine Haftung für die Inhalte externer Links. Für den Inhalt der verlinkten Seiten sind ausschließlich deren Betreiber verantwortlich.
:*Namentlich gekennzeichnete Beiträge in den Diskussionsbereichen geben die Meinung des Autors wieder. Für die Inhalte der Beiträge sind ausschließlich die Autoren zuständig.

===Site notice===

'''Responsible for the project:''' 
:Prof. Dr. sc. techn. '''Gerhard Kramer''' 
:Institute for Communications Engineering (LNT)
:Technical University of Munich (TUM) 
:Theresienstrasse 90 
:D–80333 München 
:Tel: +49 89 289-23466 
:E-Mail:   [mailto:gerhard.kramer@tum.de gerhard.kramer@tum.de]

'''Authorized representative:'''
:The Technical University of Munich is a statutory body under public law. It is legally represented by the President, Prof. Dr. '''Thomas F. Hofmann'''.

'''Competent supervisory authority:'''
:Bavarian State Ministry of Sciences, Research and the Arts

'''Responsible for content''' 
:Prof. (i. R.) Dr.–Ing. habil. '''Günter Söder''' 
:Institute for Communications Engineering (address see above) 
:Tel: +49 89 289-23486 
:E-Mail:   [mailto:LNTwww@ice.cit.tum.de LNTwww@ice.cit.tum.de]

'''Responsible for MediaWiki presentation:''' 
:'''Tasnád Kernetzky''', M. Sc. 
:Associate Professorship of Line Transmission Technology (LÜT, same address as LNT) 
:Tel: +49 89 289-25002 
:E-Mail:   [mailto:LNTwww@ice.cit.tum.de LNTwww@ice.cit.tum.de]

'''Legal disclaimer:'''
:*In spite of taking great care to check, we do not accept any responsibility for the content of external links. The operators of these websites are solely responsible for their content.

:*Contributions in the discussion sections which are attributed by name reflect the opinion of the author. The authors themselves are solely responsible for the content of the contributions.

LNTwww:Impressum

2023-02-24T14:53:34Z

Bene: Text replacement - "LNTwww@LNT.ei.tum.de" to "LNTwww@ice.cit.tum.de"

===Impressum===
'''Verantwortlich für das Projekt:''' 
:Prof. Dr. sc. techn. '''Gerhard Kramer''' 
:Lehrstuhl für Nachrichtentechnik 
:Technische Universität München 
:Theresienstrasse 90 
:D–80333 München 
:Tel: +49 89 289-23466 
:E-Mail:   [mailto:gerhard.kramer@tum.de gerhard.kramer@tum.de]

'''Vertretungsberechtigt:'''
:Die Technische Universität München ist eine Körperschaft des Öffentlichen Rechts. Sie wird gesetzlich vertreten durch den Präsidenten, Prof. Dr. '''Thomas F. Hofmann'''.

'''Zuständige Aufsichtsbehörde:'''
:Bayerisches Staatsministerium für Wissenschaft, Forschung und Kunst

'''Verantwortlich für den Inhalt:''' 
:Prof. (i. R.) Dr.–Ing. habil. '''Günter Söder''' 
:Lehrstuhl für Nachrichtentechnik (Adresse siehe oben) 
:Tel: +49 89 289-23486 
:E-Mail:   [mailto:lntwww@lnt.ei.tum.de LNTwww@ice.cit.tum.de]

'''Verantwortlich für die MediaWiki–Umsetzung:''' 
:'''Tasnád Kernetzky''', M. Sc. 
:Professur „Leitungsgebundene Übertragungstechnik” (Adresse wie LNT) 
:Tel: +49 89 289-25002 
:E-Mail:   [mailto:lntwww@lnt.ei.tum.de LNTwww@ice.cit.tum.de]

'''Haftungshinweis:'''
:*Trotz sorgfältiger inhaltlicher Kontrolle übernehmen wir keine Haftung für die Inhalte externer Links. Für den Inhalt der verlinkten Seiten sind ausschließlich deren Betreiber verantwortlich.
:*Namentlich gekennzeichnete Beiträge in den Diskussionsbereichen geben die Meinung des Autors wieder. Für die Inhalte der Beiträge sind ausschließlich die Autoren zuständig.

===Site notice===

'''Responsible for the project:''' 
:Prof. Dr. sc. techn. '''Gerhard Kramer''' 
:Institute for Communications Engineering (LNT)
:Technical University of Munich (TUM) 
:Theresienstrasse 90 
:D–80333 München 
:Tel: +49 89 289-23466 
:E-Mail:   [mailto:gerhard.kramer@tum.de gerhard.kramer@tum.de]

'''Authorized representative:'''
:The Technical University of Munich is a statutory body under public law. It is legally represented by the President, Prof. Dr. '''Thomas F. Hofmann'''.

'''Competent supervisory authority:'''
:Bavarian State Ministry of Sciences, Research and the Arts

'''Responsible for content''' 
:Prof. (i. R.) Dr.–Ing. habil. '''Günter Söder''' 
:Institute for Communications Engineering (address see above) 
:Tel: +49 89 289-23486 
:E-Mail:   [mailto:lntwww@lnt.ei.tum.de LNTwww@ice.cit.tum.de]

'''Responsible for MediaWiki presentation:''' 
:'''Tasnád Kernetzky''', M. Sc. 
:Associate Professorship of Line Transmission Technology (LÜT, same address as LNT) 
:Tel: +49 89 289-25002 
:E-Mail:   [mailto:lntwww@lnt.ei.tum.de LNTwww@ice.cit.tum.de]

'''Legal disclaimer:'''
:*In spite of taking great care to check, we do not accept any responsibility for the content of external links. The operators of these websites are solely responsible for their content.

:*Contributions in the discussion sections which are attributed by name reflect the opinion of the author. The authors themselves are solely responsible for the content of the contributions.

LNTwww:General disclaimer

2023-02-24T14:53:29Z

Bene: Text replacement - "LNTwww@LNT.ei.tum.de" to "LNTwww@ice.cit.tum.de"

===Site notice===

'''Responsible for the project:''' 
:Prof. Dr. sc. techn. '''Gerhard Kramer''' 
:Institute for Communications Engineering (LNT)
:Technical University of Munich (TUM) 
:Theresienstrasse 90 
:D–80333 München 
:Tel: +49 89 289-23466 
:E-Mail:   [mailto:gerhard.kramer@tum.de gerhard.kramer@tum.de]

'''Authorized representative:'''
:The Technical University of Munich is a statutory body under public law. It is legally represented by the President, Prof. Dr. '''Thomas F. Hofmann'''.

'''Competent supervisory authority:'''
:Bavarian State Ministry of Sciences, Research and the Arts

'''Responsible for content''' 
:'''Francisco Javier Garcia Gomez''', M. Sc. 
:Prof. (i. R.) Dr.–Ing. habil. '''Günter Söder''' 
:Institute for Communications Engineering (address see above) 
:Tel: +49 89 289-23480 
:E-Mail:   [mailto:lntwww@lnt.ei.tum.de LNTwww@ice.cit.tum.de]

'''Responsible for MediaWiki presentation:''' 
:'''Tasnád Kernetzky''', M. Sc. 
:Associate Professorship of Line Transmission Technology (LÜT, same address as LNT) 
:Tel: +49 89 289-25002 
:E-Mail:   [mailto:lntwww@lnt.ei.tum.de LNTwww@ice.cit.tum.de]

'''Legal disclaimer:'''
:*In spite of taking great care to check, we do not accept any responsibility for the content of external links. The operators of these websites are solely responsible for their content.

:*Contributions in the discussion sections which are attributed by name reflect the opinion of the author. The authors themselves are solely responsible for the content of the contributions.

Exercises:Exercise Overview

2022-09-06T14:04:57Z

Bene:

{{DISPLAYTITLE:List of Exercises}}

<html>
<div class="container-fluid">
<div class="row overview-rowitem">
<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Signal_Representation:_Exercises">Signal Representation</a> </h4>
58 exercises   ⇒   250 subtasks
</div>
[[Category:Signal_Representation:_Exercises]]

<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Lineare_zeitinvariante_Systeme">Lineare zeitinvariante Systeme</a> </h4>
54 Aufgaben   ⇒   245 Teilaufgaben   (German)
</div>
</div>

<div class="row overview-rowitem">
<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Stochastische_Signaltheorie">Stochastische Signaltheorie</a> </h4>
93 Aufgaben   ⇒   465 Teilaufgaben   (German)
</div>

<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Informationstheorie">Informationstheorie</a> </h4>
71 Aufgaben   ⇒   357 Teilaufgaben   (German)
</div>
</div>

<div class="row overview-rowitem">
<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Modulationsverfahren">Modulationsverfahren</a> </h4>
89 Aufgaben   ⇒   461 Teilaufgaben   (German)
</div>

<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Digitalsignalübertragung">Digitalsignalübertragung</a> </h4>
90 Aufgaben   ⇒   465 Teilaufgaben   (German)
</div>
</div>

<div class="row overview-rowitem">
<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Exercises_for_Mobile_Communications">Mobile Communications</a> </h4>
47 exercises   ⇒   231 subtasks
</div>

<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Kanalcodierung">Kanalcodierung</a> </h4>
98 Aufgaben   ⇒   482 Teilaufgaben   (German)
</div>
</div>

<div class="row overview-rowitem">
<div class="col-md-5 overview-colitem">
<h4> <a href=":Category:Aufgaben_zu_Beispiele_von_Nachrichtensystemen">Beispiele von Nachrichtensystemen</a> </h4>
38 Aufgaben   ⇒   191 Teilaufgaben   (German)
</div>
</div>
</div>
</html>

__NOTOC__
__NOEDITSECTION__

Biographies and Bibliographies/LNTwww members from LÜT

2022-08-16T08:39:27Z

Bene:

{{Header|
Untermenü=Beteiligte der Professur Leitungsgebundene Übertragungstechnik
|Vorherige Seite= An LNTwww beteiligte Mitarbeiter und Dozenten|
Nächste Seite= An LNTwww beteiligte Studierende
}}

== The Professorship "Line Transmission Technology" and the LNT==

The Department of  '''Line Transmission Technology'''  $\rm (LÜT)$  was established in 2004,  when the former LNT doctoral student  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  returned to the TU Munich and was appointed as its head.  But already since the 1960s, the Chair of Communications Engineering, headed by Professor  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|Hans Marko]],  had worked very intensively and also successfully in this field.

In 2014, this field became the "Line Transmission Technology".  More information can be found on the  [https://www.ei.tum.de/lnt/forschung/professur-fuer-leitungsgebundene-uebertragungstechnik// '''LÜT homepage'''].

Organizationally, three research groups of the  $\rm LNT$  are combined in one [http://www.lnt.ei.tum.de/en/home/ "teaching and research unit"], the  '''Institute for Communications Engineering'''  $\rm (ICE)$,  namely
* the research group  [http://www.lnt.ei.tum.de/en/research/chair-of-communications-engineering/ $\rm LNT$]  of  Professor  [http://www.lnt.ei.tum.de/en/people/professors/kramer/ Gerhard Kramer],
* the research group  [https://www.ei.tum.de/en/lnt/research/associate-professorship-of-line-transmission-technology/ $\rm LÜT$]  of Professor  [http://www.lnt.ei.tum.de/en/people/professors/hanik/ Norbert Hanik],  and
* the research group  [http://www.lnt.ei.tum.de/en/research/assistant-professorship-of-coding-for-communications-and-data-storage/ $\rm COD$]  of Professor  [http://www.lnt.ei.tum.de/en/people/professors/wachter-zeh/ Antonia Wachter-Zeh].

==Prof. Dr.-Ing. Norbert Hanik (at LNT from 1989-1995, at LÜT since 2004)==

[[File:n_hanik.jpg|165px|right|Norbert Hanik]]

Norbert Hanik was born in 1962 in the Bavarian town of Wemding in the Donau-Ries region and studied at the Faculty of Electrical Engineering and Information Technology at the Technical University of Munich from 1983 onwards, specializing in communications engineering.  In 1995, he received his doctorate from Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|Hans Marko]] at the LNT on "Nonlinear effects in optical signal transmission".  He then worked at the Technology Center of Deutsche Telekom AG in the field of optical transmission technology,  since 1999 as head of the research group "System Concepts of Photonic Networks ".  In 2002, he was a visiting professor at the COM Research Center of the Technical University of Denmark (TUD) in Copenhagen.

With effect from April 1, 2004, Norbert Hanik was appointed to the (current) professorship for "Line Transmission Technology" at the Faculty of Electrical Engineering and Information Technology at TUM.  He thus returned to his home chair after nine years in Berlin.  After the death of our chair holder Prof.  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962|Ralf Kötter]],  Norbert Hanik was appointed acting head of the LNT in Spring 2009.

His research focuses on modeling, simulation and optimization of components, subsystems and transmission links of optical transmission systems and optical networks.

[https://www.ce.cit.tum.de/en/lnt/people/professors/hanik/ $\text{Biography of Norbert Hanik on the LÜT homepage}$]

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Professor Hanik has been very supportive of the development of our learning tutorial and he has always been an extremely competent technical advisor.
*He was co-author on "Linear and Time-Invariant Systems" and on single chapters of "Digital Signal Transmission" and "Examples of Communication Systems".
*In particular, the initiators of $\rm LNTwww$ would like to thank Norbert  for his early and versatile use of our learning tutorial in his lectures.
}}

==Dr.-Ing. Bernhard Göbel (at LÜT from 2004-2010)==

[[File:bernhard.jpg|165px|right|Bernhard Göbel]]

Bernhard Göbel, born in Munich in 1978, finished his studies of electrical engineering and information technology at the Technical University of Munich in 2004 after semesters abroad in Southampton and Princeton with a diploma thesis on the investigation of genetic diseases using information theory.

From autumn 2004 until the end of 2010, Bernhard Göbel was an assistant to Prof.  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  in the department of "Line Transmission Technology".  After a research stay at Bell Labs in New Jersey, he received his PhD in 2010 on the topic of "Information-theoretical properties of fiber-optic communication channels".  In addition to supervising courses, his other responsibilities included managing the CITPER project,  which was initiated by the European Union.

After completing his doctorate, Dr. Göbel moved to Volkswagen AG in Wolfsburg,  where he began training as a patent attorney.  In 2014, he returned to Munich and is now working for BMW AG.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Bernhard was always consulted by us when the authors realized that some things could be done better with "MATLAB" than without.
*Furthermore, he was an expert advisor for several tutorial videos and interaction modules, for example "Attenuation of Copper Cables", "Time Response of Copper Cables" and "Viterbi Receivers".
*We would also like to thank Bernhard for making our learning tutorial known to many students of the TU Munich as an exercise assistant for "Line Transmission Technology".}}

==Tasnád Kernetzky, M.Sc. (at LÜT since 2014)==

[[File:Tasnad.png|165px|right|Tasnád Kernetzky]]

Tasnád Kernetzky was born in 1987 in Marosvásárhely  (today: Târgu Mureș, Romania).  He studied Electrical Engineering and Information Technology at the Technical University of Munich from 2009 and graduated in 2014 with a master thesis on the transmission characteristics of  "Powerline Communication" (PLC) systems.

Since December 2014, he has been working as a Ph.D. student with Prof. [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]] in the professorship "Line Transmission Technology systems" – initially in cooperation with SIEMENS AG continued on the topic  "PLC".  The focus of his current work is on the simulation of nonlinear optical transmission systems with multi-mode fibers and optical waveguides.

In teaching, Tasnád is responsible for the exercises for the lecture  "Fundamentals of Information Technology (LB)"  by Prof. Hanik.  Besides, he organizes the  "Advanced Seminar Digital Communication Systems".

Since the beginning of 2016, Tasnád has been involved in the system administrator of the chair computers.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  

For years, Tasnád has been intensively involved in the LNTwww team as a system/web administrator,  and is one of the project managers without whom nothing went:
*In 2016, he took over as successor to  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|Markus Stinner]]  assisted the student team in porting the  "old LNTwww"  to the present wiki form (version 3).
*He completed the pending move of the wiki to a new server in 2018, and also the associated update–work on the wiki.
*He converted the learning videos to modern formats (mp4, ogv).  These can now be played by many browsers, but also by smartphones.
*He was supervisor and contact person for all student work on porting the interactive applets to HTML5.
*He has done essential preliminary work to be able to generate the English  "$\rm en.LNTwww.de$"  from  "$\rm www.LNTwww.de$"  with reasonable effort.}}

==Benedikt Leible, M.Sc. (at LÜT since 2017)==

[[File:Leible.png|165px|right|Tasnád Kernetzky]]

Benedikt Leible, born in Kempten in 1988, studied electrical engineering and information technology at the Technical University of Ulm (Bachelor) and at the Technical University of Stuttgart (Master) from 2010. He graduated in 2016 with a master's thesis on "Parallelization of Channel Decoders for 5G Communication Systems". 

Since February 2017, he is working as a PhD student with Prof. Norbert Hanik in the professorship "Line Transmission Technology".  His current work focuses on the topic "Fiber optic communication using nonlinear Fourier transform".  Furthermore, he is responsible for the supervision of the lecture "Physical Layer Methods" and also conducts the corresponding tutorial.
 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He was supervisor of students who programmed interactive HTML5/JS applets for the LNTwww in their Bachelor Thesis/Engineering Practice.
*From 2021, Benedikt led the conversion to the English version  "[https://en.lntwww.de/Home $\text{https://en.lntwww.de}$]"  by the student translation team. }}

{{Display}}

Biographies and Bibliographies/Chair holders of the LNT since 1962

2022-08-16T08:38:21Z

Bene:

{{Header
|Untermenü=Lehrstuhlinhaber des LNT|
Vorherige Seite=Buchstaben N - Z|
Nächste Seite=An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten
}}

==Prof. Dr.-Ing. Dr.-Ing. E.h. Hans Marko (1962-1993)==
 
[[File:Marko.png|165px|right|Hans Marko]]

Hans Marko, born on February 24, 1925 in Kronstadt/Siebenbürgen, studied communications engineering at the TH Stuttgart and received his doctorate in 1953 under Ernst Feldtkeller.  He then worked at Standard Elektrik Lorenz AG, where he developed one of the first pulse code modulation systems in Germany.  At this time he was already lecturing at the universities of Stuttgart and Karlsruhe.  In 1961, he wrote his post-doctoral thesis on the utilization of telegraph channels for information transmission.

In 1962, at the age of only 37, Hans Marko succeeded Hans Piloty as head of the  "Lehrstuhl für Nachrichtentechnik"  (LNT)  at the  "Technische Hochschule München"  (today:  Technical University of Munich, TUM)  and worked successfully in teaching and research for 31 years until his retirement.  He supervised nine habilitations and 75 doctorates.

The scientific fields he and his institute worked on included

* the application of systems theory in technical, biological and cybernetic systems and its multidimensional extension for image processing and pattern recognition,
* the further development of Shannon's information theory to a bidirectional communication theory,
* theoretical investigations and practical realizations of high-rate digital transmission systems over cable and optical fiber.

Hans Marko is the author of several books and more than one hundred publications as well as numerous patents. He has received many high-ranking honors:

* He is a laureate of the  "Nachrichtentechnische Gesellschaft"  (NTG)  and a  "Fellow of the Institute of Electrical and Electronics Engineering"  (IEEE).
*In 1983, he was the first to be awarded the "Karl Küpfmüller Prize" of the Information Technology Society in the VDE.
*In 1985 he received an honorary doctorate from the TH Darmstadt and in 1994 the Cross of Merit of the Federal Republic of Germany.
*He is a founding member of the  "Academia Scientiarium et Artium Europaea"  in Salzburg.

After his retirement in 1993, Hans Marko has always remained connected to his former institute, both to his direct successor  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]]  as well as his successors  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Gerhard Kramer]].  Of particular note in this context was his participation in a workshop in May 2012, at which, at the age of 87, he discussed his findings on "bidirectional communication" obtained 40 years ago with the pioneer James Massey and other leading researchers in this field.

Professor Hans Marko passed away on Sept. 12, 2017, in Gräfelfing, Germany, at the age of 93.

{{BlaueBox|TEXT=
'''Hans Marko's contribution to the LNTwww'''  results from the fact that our authors  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]]  did their PhD with him. 
*Many of the statements in the books  "Signal Representation",  "Linear Time-Invariant Systems",  "Modulation Methods"  and  "Digital Signal Transmission"  are thus indirectly due to Professor Marko.}}

==Prof. Dr.-Ing. Dr.-Ing. E.h. Joachim Hagenauer (1993-2006)==
 
[[File:Hagenauer.jpg|165px|right|Joachim Hagenauer]]
 
'''So isses'''

[https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/ $\text{Hagenauer's Biography from the LNT website}$]

'''So soll es sein'''

[https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/ $\text{Hagenauer's Biography from the LNT website}$]
 
{{BlaueBox|TEXT=
'''The beginning of the LNTwww falls during Hagenauer's time as chair of the LNT'''. 
*All the teaching areas considered here were also the content of his courses and those of his PhD students. 
*Many of them were actively involved as co–authors or experts in the development of LNTwww.

'''We thank Professor Joachim Hagenauer for his constant support of our e-learning project'''.}}

==Prof. Dr. Ralf Kötter (2007-2009)==
 
[[File:P_ID1790_RalfKoetter1.jpg|165px|right|Ralf Kötter]]

Ralf Kötter, born on October 10, 1963 in Königstein/Taunus and deceased on February 2, 2009 in Munich, was a German professor in the field of "electrical engineering and information technology" whose numerous works in the field of network coding were of central importance for the further development of mobile communications despite his early death.

Ralf Kötter studied electrical and communications engineering at Darmstadt Technical University. After graduating in 1990, he subsequently worked at the University of Linköping in the Department of Electrical Engineering until 1996. There he received the degree of Ph.D. (''Teknisk Doktor'') in Electrical Engineering in 1996. In 1996/97, he was a visiting scientist at the IBM Almaden Research Laboratory in San José, California, and subsequently a professor at the Coordinated Science Laboratory and Department of Engineering at the University of Illinois at Urbana-Champaign. In October 2006, he accepted an appointment to the Chair of Communications Engineering in the Department of Electrical Engineering and Information Technology at the Technical University of Munich, succeeding [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].

Ralf Kötter worked in the field of algebraic coding theory and was one of the first scientists to use graph theory to develop error control codes. For his work on decoding Reed-Solomon codes, he was awarded the ''Best Paper Award of the IEEE Information Theory Society'' in 2004. In 2008, he received the ''Best Paper Award of the Signal Processing Society'' for his work on turbo equalization. In addition, he was awarded the 2008 Vodafone Foundation Innovation Award for Research for his "seminal work" on information and coding theory.

Ralf Kötter died at the age of only 45, leaving behind his wife Nuala (who also died of cancer just under 5 years after him) and his then 4-year-old son Finn.

The ''Department for Electrical and Computer Engineering'' at his former university in Illinois established the ''Ralf Koetter Memorial Fund in Electrical and Computer Engineering'' after his death to support students in the department. Ralf's parents Ruth and Hubert Koetter endowed the "Prof. Dr. Ralf Koetter Memorial Award" in 2010, which will be awarded annually until 2023.

'''Important Awards and Honors for Ralf Kötter''':

*IBM Invention Achievement Award (1997).
*NSF CAREER Award (2000)
*IBM Partnership Award (2001)
*Best Paper Award of the IEEE Information Theory Society (2004)
*University of Illinois College of Engineering XEROX Award for Faculty Research (2006)
*Best Paper Award of the Signal Processing Society (2008)
*Innovation Award of the Vodafone Foundation for Research (2008)

[https://www.ei.tum.de/en/lnt/people/former-employees/koetter/ $\text{Appreciation of Ralf Kötter from the LNT website}$]

{{BlaueBox|TEXT=
'''Ralf Kötter has been very supportive of the further development of the LNTwww'''  in his unfortunately very short time at LNT. 
*One can clearly recognise his  "scientific handwriting" especially in the books  "Information Theory"  and  "Channel Coding". 

'''We will always keep Ralf in good memory'''.}}

==Prof. Dr. sc. techn. Gerhard Kramer (seit 2010)==
 
[[File:Kramer4.jpg|165px|right|Gerhard Kramer]]

Gerhard Kramer, born in 1970 in Winnipeg, Canada, is Alexander von Humboldt Professor and has been full professor of the Department of Communications Engineering (LNT) at the Technical University of Munich (TUM) since 2010 and its Vice President since 2019. He received the B.Sc. in 1991 and the M.Sc. in 1992 in electrical engineering from the University of Manitoba, Winnipeg, Canada. In 1998, he was awarded the Dr. sc. techn. degree (Doctor of Technical Sciences) from the Swiss Federal Institute of Technology (ETH) Zurich.

From 1998 to 2000, Gerhard Kramer worked at Endora Tech AG, Basel, as a communications engineer. From 2000 to 2008 he was a Member of Technical Staff at the Math Center, Bell Laboratories, Alcatel-Lucent in Murray Hill/New Jersey. In 2009, he moved to the University of Southern California (USC) in Los Angeles/California as a professor.

Kramer's research focuses on information theory and communication theory with applications to both wireless and wired networks over copper and fibre respectively.

[https://www.ei.tum.de/en/lnt/people/professors/kramer/ $\text{Kramer's Biography from the LNT website}$]

{{BlaueBox|TEXT=
'''During Gerhard Kramer's tenure (since 2010) the  "German LNTwww"   was completed and he initiated the English version in 2020'''.
*All the teaching areas considered in our learning offer are also covered in Gerhard Kramer's lectures. 
*But not all of his lecture content is presented in the  "LNTww"   with the same depth and mathematical exactness. 

'''The LNTwww team would like to thank Gerhard Kramer for his great support of our e-learning project'''.}}

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Biographies and Bibliographies/Chair holders of the LNT since 1962

2022-08-16T08:37:43Z

Bene: Text replacement - "https://www.lnt.ei.tum.de" to "https://www.ce.cit.tum.de"

{{Header
|Untermenü=Lehrstuhlinhaber des LNT|
Vorherige Seite=Buchstaben N - Z|
Nächste Seite=An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten
}}

==Prof. Dr.-Ing. Dr.-Ing. E.h. Hans Marko (1962-1993)==
 
[[File:Marko.png|165px|right|Hans Marko]]

Hans Marko, born on February 24, 1925 in Kronstadt/Siebenbürgen, studied communications engineering at the TH Stuttgart and received his doctorate in 1953 under Ernst Feldtkeller.  He then worked at Standard Elektrik Lorenz AG, where he developed one of the first pulse code modulation systems in Germany.  At this time he was already lecturing at the universities of Stuttgart and Karlsruhe.  In 1961, he wrote his post-doctoral thesis on the utilization of telegraph channels for information transmission.

In 1962, at the age of only 37, Hans Marko succeeded Hans Piloty as head of the  "Lehrstuhl für Nachrichtentechnik"  (LNT)  at the  "Technische Hochschule München"  (today:  Technical University of Munich, TUM)  and worked successfully in teaching and research for 31 years until his retirement.  He supervised nine habilitations and 75 doctorates.

The scientific fields he and his institute worked on included

* the application of systems theory in technical, biological and cybernetic systems and its multidimensional extension for image processing and pattern recognition,
* the further development of Shannon's information theory to a bidirectional communication theory,
* theoretical investigations and practical realizations of high-rate digital transmission systems over cable and optical fiber.

Hans Marko is the author of several books and more than one hundred publications as well as numerous patents. He has received many high-ranking honors:

* He is a laureate of the  "Nachrichtentechnische Gesellschaft"  (NTG)  and a  "Fellow of the Institute of Electrical and Electronics Engineering"  (IEEE).
*In 1983, he was the first to be awarded the "Karl Küpfmüller Prize" of the Information Technology Society in the VDE.
*In 1985 he received an honorary doctorate from the TH Darmstadt and in 1994 the Cross of Merit of the Federal Republic of Germany.
*He is a founding member of the  "Academia Scientiarium et Artium Europaea"  in Salzburg.

After his retirement in 1993, Hans Marko has always remained connected to his former institute, both to his direct successor  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]]  as well as his successors  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Gerhard Kramer]].  Of particular note in this context was his participation in a workshop in May 2012, at which, at the age of 87, he discussed his findings on "bidirectional communication" obtained 40 years ago with the pioneer James Massey and other leading researchers in this field.

Professor Hans Marko passed away on Sept. 12, 2017, in Gräfelfing, Germany, at the age of 93.

{{BlaueBox|TEXT=
'''Hans Marko's contribution to the LNTwww'''  results from the fact that our authors  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]]  did their PhD with him. 
*Many of the statements in the books  "Signal Representation",  "Linear Time-Invariant Systems",  "Modulation Methods"  and  "Digital Signal Transmission"  are thus indirectly due to Professor Marko.}}

==Prof. Dr.-Ing. Dr.-Ing. E.h. Joachim Hagenauer (1993-2006)==
 
[[File:Hagenauer.jpg|165px|right|Joachim Hagenauer]]
 
'''So isses'''

[https://www.ce.cit.tum.de/en/people/professors/hagenauer/ $\text{Hagenauer's Biography from the LNT website}$]

'''So soll es sein'''

[https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/ $\text{Hagenauer's Biography from the LNT website}$]
 
{{BlaueBox|TEXT=
'''The beginning of the LNTwww falls during Hagenauer's time as chair of the LNT'''. 
*All the teaching areas considered here were also the content of his courses and those of his PhD students. 
*Many of them were actively involved as co–authors or experts in the development of LNTwww.

'''We thank Professor Joachim Hagenauer for his constant support of our e-learning project'''.}}

==Prof. Dr. Ralf Kötter (2007-2009)==
 
[[File:P_ID1790_RalfKoetter1.jpg|165px|right|Ralf Kötter]]

Ralf Kötter, born on October 10, 1963 in Königstein/Taunus and deceased on February 2, 2009 in Munich, was a German professor in the field of "electrical engineering and information technology" whose numerous works in the field of network coding were of central importance for the further development of mobile communications despite his early death.

Ralf Kötter studied electrical and communications engineering at Darmstadt Technical University. After graduating in 1990, he subsequently worked at the University of Linköping in the Department of Electrical Engineering until 1996. There he received the degree of Ph.D. (''Teknisk Doktor'') in Electrical Engineering in 1996. In 1996/97, he was a visiting scientist at the IBM Almaden Research Laboratory in San José, California, and subsequently a professor at the Coordinated Science Laboratory and Department of Engineering at the University of Illinois at Urbana-Champaign. In October 2006, he accepted an appointment to the Chair of Communications Engineering in the Department of Electrical Engineering and Information Technology at the Technical University of Munich, succeeding [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].

Ralf Kötter worked in the field of algebraic coding theory and was one of the first scientists to use graph theory to develop error control codes. For his work on decoding Reed-Solomon codes, he was awarded the ''Best Paper Award of the IEEE Information Theory Society'' in 2004. In 2008, he received the ''Best Paper Award of the Signal Processing Society'' for his work on turbo equalization. In addition, he was awarded the 2008 Vodafone Foundation Innovation Award for Research for his "seminal work" on information and coding theory.

Ralf Kötter died at the age of only 45, leaving behind his wife Nuala (who also died of cancer just under 5 years after him) and his then 4-year-old son Finn.

The ''Department for Electrical and Computer Engineering'' at his former university in Illinois established the ''Ralf Koetter Memorial Fund in Electrical and Computer Engineering'' after his death to support students in the department. Ralf's parents Ruth and Hubert Koetter endowed the "Prof. Dr. Ralf Koetter Memorial Award" in 2010, which will be awarded annually until 2023.

'''Important Awards and Honors for Ralf Kötter''':

*IBM Invention Achievement Award (1997).
*NSF CAREER Award (2000)
*IBM Partnership Award (2001)
*Best Paper Award of the IEEE Information Theory Society (2004)
*University of Illinois College of Engineering XEROX Award for Faculty Research (2006)
*Best Paper Award of the Signal Processing Society (2008)
*Innovation Award of the Vodafone Foundation for Research (2008)

[https://www.ei.tum.de/en/lnt/people/former-employees/koetter/ $\text{Appreciation of Ralf Kötter from the LNT website}$]

{{BlaueBox|TEXT=
'''Ralf Kötter has been very supportive of the further development of the LNTwww'''  in his unfortunately very short time at LNT. 
*One can clearly recognise his  "scientific handwriting" especially in the books  "Information Theory"  and  "Channel Coding". 

'''We will always keep Ralf in good memory'''.}}

==Prof. Dr. sc. techn. Gerhard Kramer (seit 2010)==
 
[[File:Kramer4.jpg|165px|right|Gerhard Kramer]]

Gerhard Kramer, born in 1970 in Winnipeg, Canada, is Alexander von Humboldt Professor and has been full professor of the Department of Communications Engineering (LNT) at the Technical University of Munich (TUM) since 2010 and its Vice President since 2019. He received the B.Sc. in 1991 and the M.Sc. in 1992 in electrical engineering from the University of Manitoba, Winnipeg, Canada. In 1998, he was awarded the Dr. sc. techn. degree (Doctor of Technical Sciences) from the Swiss Federal Institute of Technology (ETH) Zurich.

From 1998 to 2000, Gerhard Kramer worked at Endora Tech AG, Basel, as a communications engineer. From 2000 to 2008 he was a Member of Technical Staff at the Math Center, Bell Laboratories, Alcatel-Lucent in Murray Hill/New Jersey. In 2009, he moved to the University of Southern California (USC) in Los Angeles/California as a professor.

Kramer's research focuses on information theory and communication theory with applications to both wireless and wired networks over copper and fibre respectively.

[https://www.ei.tum.de/en/lnt/people/professors/kramer/ $\text{Kramer's Biography from the LNT website}$]

{{BlaueBox|TEXT=
'''During Gerhard Kramer's tenure (since 2010) the  "German LNTwww"   was completed and he initiated the English version in 2020'''.
*All the teaching areas considered in our learning offer are also covered in Gerhard Kramer's lectures. 
*But not all of his lecture content is presented in the  "LNTww"   with the same depth and mathematical exactness. 

'''The LNTwww team would like to thank Gerhard Kramer for his great support of our e-learning project'''.}}

{{Display}}

Biographies and Bibliographies/LNTwww members from LÜT

2022-08-16T08:37:35Z

Bene: Text replacement - "https://www.lnt.ei.tum.de" to "https://www.ce.cit.tum.de"

{{Header|
Untermenü=Beteiligte der Professur Leitungsgebundene Übertragungstechnik
|Vorherige Seite= An LNTwww beteiligte Mitarbeiter und Dozenten|
Nächste Seite= An LNTwww beteiligte Studierende
}}

== The Professorship "Line Transmission Technology" and the LNT==

The Department of  '''Line Transmission Technology'''  $\rm (LÜT)$  was established in 2004,  when the former LNT doctoral student  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  returned to the TU Munich and was appointed as its head.  But already since the 1960s, the Chair of Communications Engineering, headed by Professor  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|Hans Marko]],  had worked very intensively and also successfully in this field.

In 2014, this field became the "Line Transmission Technology".  More information can be found on the  [https://www.ei.tum.de/lnt/forschung/professur-fuer-leitungsgebundene-uebertragungstechnik// '''LÜT homepage'''].

Organizationally, three research groups of the  $\rm LNT$  are combined in one [http://www.lnt.ei.tum.de/en/home/ "teaching and research unit"], the  '''Institute for Communications Engineering'''  $\rm (ICE)$,  namely
* the research group  [http://www.lnt.ei.tum.de/en/research/chair-of-communications-engineering/ $\rm LNT$]  of  Professor  [http://www.lnt.ei.tum.de/en/people/professors/kramer/ Gerhard Kramer],
* the research group  [https://www.ei.tum.de/en/lnt/research/associate-professorship-of-line-transmission-technology/ $\rm LÜT$]  of Professor  [http://www.lnt.ei.tum.de/en/people/professors/hanik/ Norbert Hanik],  and
* the research group  [http://www.lnt.ei.tum.de/en/research/assistant-professorship-of-coding-for-communications-and-data-storage/ $\rm COD$]  of Professor  [http://www.lnt.ei.tum.de/en/people/professors/wachter-zeh/ Antonia Wachter-Zeh].

==Prof. Dr.-Ing. Norbert Hanik (at LNT from 1989-1995, at LÜT since 2004)==

[[File:n_hanik.jpg|165px|right|Norbert Hanik]]

Norbert Hanik was born in 1962 in the Bavarian town of Wemding in the Donau-Ries region and studied at the Faculty of Electrical Engineering and Information Technology at the Technical University of Munich from 1983 onwards, specializing in communications engineering.  In 1995, he received his doctorate from Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|Hans Marko]] at the LNT on "Nonlinear effects in optical signal transmission".  He then worked at the Technology Center of Deutsche Telekom AG in the field of optical transmission technology,  since 1999 as head of the research group "System Concepts of Photonic Networks ".  In 2002, he was a visiting professor at the COM Research Center of the Technical University of Denmark (TUD) in Copenhagen.

With effect from April 1, 2004, Norbert Hanik was appointed to the (current) professorship for "Line Transmission Technology" at the Faculty of Electrical Engineering and Information Technology at TUM.  He thus returned to his home chair after nine years in Berlin.  After the death of our chair holder Prof.  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962|Ralf Kötter]],  Norbert Hanik was appointed acting head of the LNT in Spring 2009.

His research focuses on modeling, simulation and optimization of components, subsystems and transmission links of optical transmission systems and optical networks.

[https://www.ce.cit.tum.de/en/people/professors/hanik/ $\text{Biography of Norbert Hanik on the LÜT homepage}$]

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Professor Hanik has been very supportive of the development of our learning tutorial and he has always been an extremely competent technical advisor.
*He was co-author on "Linear and Time-Invariant Systems" and on single chapters of "Digital Signal Transmission" and "Examples of Communication Systems".
*In particular, the initiators of $\rm LNTwww$ would like to thank Norbert  for his early and versatile use of our learning tutorial in his lectures.
}}

==Dr.-Ing. Bernhard Göbel (at LÜT from 2004-2010)==

[[File:bernhard.jpg|165px|right|Bernhard Göbel]]

Bernhard Göbel, born in Munich in 1978, finished his studies of electrical engineering and information technology at the Technical University of Munich in 2004 after semesters abroad in Southampton and Princeton with a diploma thesis on the investigation of genetic diseases using information theory.

From autumn 2004 until the end of 2010, Bernhard Göbel was an assistant to Prof.  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  in the department of "Line Transmission Technology".  After a research stay at Bell Labs in New Jersey, he received his PhD in 2010 on the topic of "Information-theoretical properties of fiber-optic communication channels".  In addition to supervising courses, his other responsibilities included managing the CITPER project,  which was initiated by the European Union.

After completing his doctorate, Dr. Göbel moved to Volkswagen AG in Wolfsburg,  where he began training as a patent attorney.  In 2014, he returned to Munich and is now working for BMW AG.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Bernhard was always consulted by us when the authors realized that some things could be done better with "MATLAB" than without.
*Furthermore, he was an expert advisor for several tutorial videos and interaction modules, for example "Attenuation of Copper Cables", "Time Response of Copper Cables" and "Viterbi Receivers".
*We would also like to thank Bernhard for making our learning tutorial known to many students of the TU Munich as an exercise assistant for "Line Transmission Technology".}}

==Tasnád Kernetzky, M.Sc. (at LÜT since 2014)==

[[File:Tasnad.png|165px|right|Tasnád Kernetzky]]

Tasnád Kernetzky was born in 1987 in Marosvásárhely  (today: Târgu Mureș, Romania).  He studied Electrical Engineering and Information Technology at the Technical University of Munich from 2009 and graduated in 2014 with a master thesis on the transmission characteristics of  "Powerline Communication" (PLC) systems.

Since December 2014, he has been working as a Ph.D. student with Prof. [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]] in the professorship "Line Transmission Technology systems" – initially in cooperation with SIEMENS AG continued on the topic  "PLC".  The focus of his current work is on the simulation of nonlinear optical transmission systems with multi-mode fibers and optical waveguides.

In teaching, Tasnád is responsible for the exercises for the lecture  "Fundamentals of Information Technology (LB)"  by Prof. Hanik.  Besides, he organizes the  "Advanced Seminar Digital Communication Systems".

Since the beginning of 2016, Tasnád has been involved in the system administrator of the chair computers.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  

For years, Tasnád has been intensively involved in the LNTwww team as a system/web administrator,  and is one of the project managers without whom nothing went:
*In 2016, he took over as successor to  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|Markus Stinner]]  assisted the student team in porting the  "old LNTwww"  to the present wiki form (version 3).
*He completed the pending move of the wiki to a new server in 2018, and also the associated update–work on the wiki.
*He converted the learning videos to modern formats (mp4, ogv).  These can now be played by many browsers, but also by smartphones.
*He was supervisor and contact person for all student work on porting the interactive applets to HTML5.
*He has done essential preliminary work to be able to generate the English  "$\rm en.LNTwww.de$"  from  "$\rm www.LNTwww.de$"  with reasonable effort.}}

==Benedikt Leible, M.Sc. (at LÜT since 2017)==

[[File:Leible.png|165px|right|Tasnád Kernetzky]]

Benedikt Leible, born in Kempten in 1988, studied electrical engineering and information technology at the Technical University of Ulm (Bachelor) and at the Technical University of Stuttgart (Master) from 2010. He graduated in 2016 with a master's thesis on "Parallelization of Channel Decoders for 5G Communication Systems". 

Since February 2017, he is working as a PhD student with Prof. Norbert Hanik in the professorship "Line Transmission Technology".  His current work focuses on the topic "Fiber optic communication using nonlinear Fourier transform".  Furthermore, he is responsible for the supervision of the lecture "Physical Layer Methods" and also conducts the corresponding tutorial.
 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He was supervisor of students who programmed interactive HTML5/JS applets for the LNTwww in their Bachelor Thesis/Engineering Practice.
*From 2021, Benedikt led the conversion to the English version  "[https://en.lntwww.de/Home $\text{https://en.lntwww.de}$]"  by the student translation team. }}

{{Display}}

Aufgaben:Exercise 1.4: Rayleigh PDF and Jakes PDS

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process}}

[[File:P_ID2119__Mob_A_1_4.png|right|frame| PDF and  $|z(t)|$  for Rayleigh fading with Doppler effect]]
We consider two different mobile radio channels with  [[Mobile_Communications/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Beispielhafte_Signalverl.C3.A4ufe_bei_Rayleigh.E2.80.93Fading|Rayleigh fading]]. In both cases the PDF of the magnitude  $a(t) = |z(t)| ≥ 0$  is
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)}
\hspace{0.05cm}.$$

The probability that this amount is not greater than a given value  $A$  is
:$${\rm Pr}(|z(t)| \le A) = 1 - {\rm e}^{ -{A^2}/(2\sigma^2)}
\hspace{0.05cm}.$$

The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power-spectral density  $\rm (PSD)$   ${\it \Phi}_z(f_{\rm D})$.

*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process|Jakes spectrum]].

*For a Doppler frequency  $f_{\rm D}$  with  $|f_{\rm D}| <f_{\rm D,\hspace{0.1cm}max}$  the Jakes spectrum is given by
:$${\it \Phi}_z(f_{\rm D}) = \frac{1}{\pi \hspace{-0.05cm}\cdot \hspace{-0.05cm}f_{\rm D, \hspace{0.1cm} max} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\sqrt{ 1 \hspace{-0.05cm}- \hspace{-0.05cm}(f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }
\hspace{0.05cm}.$$

*For Doppler frequencies outside this interval from  $-f_{\rm D,\hspace{0.1cm}max}$  to  $+f_{\rm D,\hspace{0.1cm}max}$,   we have ${\it \Phi}_z(f_{\rm D})=0$.

The corresponding descriptor in the time domain is the auto-correlation function  $\rm (ACF)$:
:$$\varphi_z ({\rm \delta}t) = 2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot {\rm \delta}t)\hspace{0.05cm}.$$

*Here,  ${\rm J_0}(.)$  is the Bessel function of the first kind and zeroth order.  We have  ${\rm J_0}(0) = 1$.
*The maximum Doppler frequency of the channel model  $\rm R$  is known to be   $f_{\rm D,\hspace{0.1cm}max} = 200 \ \rm Hz$.
* It is also known that the speeds  $v_{\rm R}$  and  $v_{\rm B}$  differ by the factor  $2$ .
*Whether  $v_{\rm R}$  is twice as large as  $v_{\rm B}$  or vice versa, you should decide based on the above graphs.

''Notes:''
* This task belongs to the topic of  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading|Statistical bindings within the Rayleigh process]].
* To check your results you can use the interactive applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions|PDF, CDF and Moments of Special Distributions]].

===Questionns===

<quiz display=simple>
{Determine the Rayleigh parameter  $\sigma$  for the channels  $\rm R$  and  $\rm B$.
|type="{}"}
$\sigma_{\rm R} \ = \ $ { 1 3% } $\ \ \rm $
$\sigma_{\rm B} \ = \ $ { 1 3% } $\ \ \rm $

{In each case, give the probability that  $20 \cdot {\rm lg} \ a ≤ -10 \ \ \ \rm dB$   ⇒   $a ≤ 0.316$.
|type="{}"}
Channel  ${\rm R}\text{:} \hspace{0.4cm} {\rm Pr}(a ≤ 0.316) \ = \ $ { 4.9 3% } $\ \rm \%$
Channel  ${\rm B}\text{:} \hspace{0.4cm} {\rm Pr}(a ≤ 0.316) \ = \ $ { 4.9 3% } $\ \rm \%$

{Which statements are correct regarding the driving speeds  $v$ ?
|type="[]"}
- $v_{\rm B}$  is twice as big as  $v_{\rm R}$.
+ $v_{\rm B}$  is half as big as  $v_{\rm R}$.
+ With  $v = 0$,   $|z(t)|$  would be constant.
- With  $v = 0$,   $|z(t)|$  would have a white spectrum.
- With  $v → ∞$,   $|z(t)|$  would be constant.
+ With  $v → ∞$,   $|z(t)|$  would be white.

{Which of the following statements are correct?
|type="[]"}
- The PDS value  ${\it \Phi_z}(f_{\rm D} = 0)$  is the same for both channels.
+ The ACF value  $\varphi_z(\Delta t = 0)$  is the same for both channels.
+ The area under  ${\it \Phi_z}(f_{\rm D})$  is the same for both channels.
- The area below  $\varphi_z(\Delta t)$  is the same for both channels.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  The maximum value of the PDF for both channels is  $0.6$  and occurs at  $a = 1$.
*The Rayleigh PDF and its derivative are
:$$f_a(a) \hspace{-0.1cm} = \hspace{-0.1cm} \frac{a}{\sigma^2} \cdot {\rm e}^{ -a^2/(2\sigma^2)} \hspace{0.05cm},$$
:$$\frac{{\rm d}f_a(a)}{{\rm d}a} \hspace{-0.1cm} = \hspace{-0.1cm} \frac{1}{\sigma^2} \cdot {\rm e}^{ -a^2/(2\sigma^2)}-
\frac{a^2}{\sigma^4} \cdot {\rm e}^{ -a^2/(2\sigma^2)} \hspace{0.05cm}.$$

*By setting the derivative to  $0$, you can show that the maximum of the PDF occurs at  $a = \sigma$.  Since the Rayleigh PDF applies to both channels, it follows that
:$$\sigma_{\rm R} = \sigma_{\rm B} \hspace{0.15cm} \underline{ = 1} \hspace{0.05cm}.$$

'''(2)'''  As they fading coefficients have the same PDF, the desired probability is also the same for both channels.
*Using the given equation, we have
:$${\rm Pr}(a \le 0.316) = {\rm Pr}(20 \cdot {\rm lg}\hspace{0.15cm} a \le -10\,\,{\rm dB}) = 1 - {\rm e}^{ -{0.316^2}/(2\sigma^2)}
= 1- 0.951 \hspace{0.15cm} \underline{ \approx 4.9 \%}
\hspace{0.05cm}.$$

'''(3)'''  The correct statements are 2, 3 and 6:
* The smaller speed  $v_{\rm B}$  can be recognized by the fact that the magnitude  $|z(t)|$  changes more slowly with the blue curve.
* When the vehicle is stationary, the PDS degenerates to  ${\it \Phi_z}(f_{\rm D}) = 2\sigma^2\cdot \delta(f_{\rm D})$,  and we have  $|z(t)| = A = \rm const.$, where the constant  $A$  is drawn from the Rayleigh distribution.
* At extremely high speed, the Jakes spectrum becomes flat and has an increasingly small magnitude over an increasingly wide range.  It then approaches the PDS of white noise.  However,  $v$  would have to be in the order of the speed of light.

'''(4)'''  Statements 2 and 3 are correct:
*The Rayleigh parameter  $\sigma = 1$  also determines the "power"  ${\rm E}[|z(t)|^2] = 2\sigma^2 = 2$  of the random process.
*This applies to both  $\rm R$  and  $\rm B$:
:$$\varphi_z ({\rm \delta}t = 0) = 2 \hspace{0.05cm}, \hspace{0.2cm} \int_{-\infty}^{+\infty}{\it \Phi}_z(f_{\rm D}) \hspace{0.15cm}{\rm d}f_{\rm D} = 2 \hspace{0.05cm}.$$
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^1.3 Rayleigh Fading with Memory^]]

Aufgaben:Exercise 5.2: Band Spreading and Narrowband Interferer

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation
}}

[[File:EN_Mod_A_5_2.png|right|frame|Considered model 
of band spreading]]
A spread spectrum system is considered according to the given diagram in the equivalent low-pass range:
*Let the digital signal  $q(t)$  possess the power-spectral density  ${\it \Phi}_q(f)$,  which is to be approximated as rectangular with bandwidth  $B = 1/T = 100\ \rm kHz$   (a rather unrealistic assumption):
:$${\it \Phi}_{q}(f) =
\left\{ \begin{array}{c} {\it \Phi}_{0} \\
0 \\ \end{array} \right.
\begin{array}{*{10}c} {\rm{for}}
\\ {\rm{otherwise}} \hspace{0.05cm}. \\ \end{array}\begin{array}{*{20}c}
|f| <B/2 \hspace{0.05cm}, \\
\\
\end{array}$$
*Thus,  in the low-pass range,  the bandwidth  (only the components at positive frequencies)  is equal to  $B/2$  and the bandwidth in the band-pass range is  $B$.
*The band spreading is done by multiplication with the PN sequence  $c(t)$  of the chip duration  $T_c = T/100$  ("PN" stands for "pseudo-noise").
*To simplify matters,  the following applies to the auto-correlation function:
:$$ {\it \varphi}_{c}(\tau) = \left\{ \begin{array}{c}1 - |\tau|/T_c \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ {\rm{otherwise}} \hspace{0.05cm}. \\ \end{array}\begin{array}{*{20}c} -T_c \le \tau \le T_c \hspace{0.05cm}, \\ \\ \end{array}$$
*At the receiver,  the same spreading sequence  $c(t)$  is again added phase-synchronously.
*The interference signal  $i(t)$  is to be neglected for the time being.
*In subtask  '''(4)'''   $i(t)$  denotes a narrowband interferer at carrier frequency  $f_{\rm T} = 30 \ \rm MHz = f_{\rm I}$  with power  $P_{\rm I}$.
*The influence of the  (always present)  AWGN noise  $n(t)$  is not considered in this exercise.

Note:
*This exercise belongs to the chapter  [[Modulation_Methods/PN–Modulation|Direct-Sequence Spread Spectrum Modulation]].

===Questions===

<quiz display=simple>
{What is the power-spectral density  ${\it \Phi}_c(f )$  of the spreading signal  $c(t)$?  What value results at the frequency  $f = 0$?
|type="{}"}
${\it \Phi}_c(f = 0) \ = \ $ { 0.1 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$

{Calculate the equivalent bandwidth  $B_c$  of the spread signal as the width of the equal-area  $\rm PDS$  rectangle.
|type="{}"}
$B_c \ = \ $ { 10 3% } $\ \rm MHz$

{Which statements are true for the bandwidths of the signals  $s(t)$   ⇒   $B_s$ and  $b(t)$   ⇒   $B_b$?  The (two-sided) bandwidth of  $q(t)$  is  $B$.
|type="[]"}
- $B_s$  is exactly equal to  $B_c$.
+ $B_s$  is approximately equal to  $B_c + B$.
- $B_b$  is exactly equal to  $B_s$.
- $B_b$  is equal to  $B_s + B_c = 2B_c + B$.
+ $B_b$  is exactly equal to  $B$.

{What is the effect of band spreading on a  "narrowband interferer"  at the carrier frequency?  Let  $f_{\rm I} = f_{\rm T}$.
|type="[]"}
+ The interfering influence is weakened by band spreading.
- The interfering power is only half as large.
- The interfering power is not changed by band spreading.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  The power-spectral density  $\rm (PDS)$  ${\it \Phi}_c(f)$  is the Fourier transform of the triangular ACF,  which can be represented with rectangles of width  $T_c$  as follows:
:$${\it \varphi}_{c}(\tau) = \frac{1}{T_c} \cdot {\rm rect} \big(\frac{\tau}{T_c} \big ) \star {\rm rect} \big(\frac{\tau}{T_c} \big ) \hspace{0.05cm}.$$
*From this follows  ${\it \Phi}_{c}(f) = {1}/{T_c} \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] = T_c \cdot {\rm sinc}^2 \left(f T_c \right ) \hspace{0.05cm}$  with maximum value
:$${\it \Phi}_{c}(f = 0) = T_c = \frac{T}{100}= \frac{1}{100 \cdot B} = \frac{1}{100 \cdot 10^5\,{\rm 1/s}} = 10^{-7}\,{\rm 1/Hz} \hspace{0.15cm}\underline {= 0.1 \cdot 10^{-6}\,{\rm 1/Hz}}\hspace{0.05cm}.$$

'''(2)'''  By definition,  with  $T_c = T/100 = 0.1\ \rm µ s$:
[[File:P_ID1869__Mod_A_5_2b.png|right|frame|Power density spectrum of the pseudo-noise spread signal]]
:$$B_c= \frac{1}{T_c} \cdot \hspace{-0.03cm} \int_{-\infty }^{+\infty} \hspace{-0.03cm} {\it \Phi}_{c}(f)\hspace{0.1cm} {\rm d}f = \hspace{-0.03cm} \int_{-\infty }^{+\infty} \hspace{-0.03cm} {\rm sinc}^2 \left(f T_c \right )\hspace{0.1cm} {\rm d}f $$
:$$\Rightarrow \hspace{0.3cm} B_c= \frac{1}{T_c}\hspace{0.15cm}\underline {= 10\,{\rm MHz}} \hspace{0.05cm}$$
The graph illustrates,
*that  $B_c$  is given by the first zero of the  $\rm sinc^2$ function in the equivalent low-pass range,
*but at the same time also gives the equivalent  (equal area)  bandwidth in the band-pass region.

'''(3)'''  Solutions 2 and 5  are correct:
*The PDS  ${\it \Phi}_s(f)$  results from the convolution of  ${\it \Phi}_q(f)$  and  ${\it \Phi}_c(f)$.  This actually gives  $B_s = B_c + B$  for the bandwidth of the transmitted signal.
*Since the spreading signal  $c(t) ∈ \{+1, –1\}$  multiplied by itself always gives the value  $1$,  naturally  $b(t) ≡ q(t)$  and consequently  $B_b = B$.
*Obviously, the bandwidth  $B_b$  of the band compressed signal is not equal to  $2B_c + B$,  although the convolution  ${\it \Phi}_s(f) ∗ {\it \Phi}_c(f)$  suggests this.
*This is due to the fact that the power density spectra must not be convolved, but the spectral functions  (amplitude spectra)  $S(f)$  and  $C(f)$  must be assumed, taking into account the phase relations.
*Only then can the PDS  $B(f)$  be determined from  ${\it \Phi}_b(f)$.  Clearly,  the following is also true:   $C(f) ∗ C(f) = δ(f)$.

'''(4)'''  Only the  first solution  is correct.  The solution shall be clarified by the diagram at the end of the page:
*In the upper diagram the PDS  ${\it \Phi}_i(f)$  of the narrowband interferer is approximated by two Dirac delta functions at  $±f_{\rm T}$  with weights  $P_{\rm I}/2$.   Also plotted is the bandwidth  $B = 0.1 \ \rm MHz$  (not quite true to scale).

*The receiver-side multiplication with  $c(t)$  – actually with the function of the band compression,  at least with respect to the useful part of  $r(t)$ –  causes a band spreading with respect to the interference signal  $i(t)$.  Without considering the useful signal,  $b(t) = n(t) = i(t) · c(t)$.  It follows:
:$${\it \Phi}_{n}(f) = {\it \Phi}_{i}(f) \star {\it \Phi}_{c}(f) = \frac{P_{\rm I}\cdot T_c}{2}\cdot {\rm sinc}^2 \left( (f - f_{\rm T}) \cdot T_c \right )+ \frac{P_{\rm I}\cdot T_c}{2}\cdot {\rm sinc}^2 \left( (f + f_{\rm T}) \cdot T_c \right ) \hspace{0.05cm}.$$
[[File:P_ID1870__Mod_A_5_2c.png|right|frame|Power density spectra before and after band spreading]]

*Note that  $n(t)$  is used here only as an abbreviation and does not denote AWGN noise.  
*In a narrow range around the carrier frequency  $f_{\rm T} = 30 \ \rm MHz$,  the PDS  ${\it \Phi}_n(f)$  is almost constant.  Thus,  the interference power after band spreading is:
:$$ P_{n} = P_{\rm I} \cdot T_c \cdot B = P_{\rm I}\cdot \frac{B}{B_c} = \frac{P_{\rm I}}{J}\hspace{0.05cm}. $$
*This means:   The interference power is reduced by the factor  $J = T/T_c$  by band spreading,  which is why  $J$  is often called  "spreading gain".
*However,  such a  "spreading gain"  is only given for a narrowband interferer.

{{ML-Fuß}}

[[Category:Modulation Methods: Exercises|^5.2 PN Modulation^]]

Aufgaben:Exercise 1.6: Autocorrelation Function and PDS with Rice Fading

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/Non-Frequency_Selective_Fading_With_Direct_Component}}

[[File:P_ID2132__Mob_A_1_6.png|right|frame|Rice PDF for different values of  $z_0^2$]]
One speaks of  "Rice fading"  if the complex factor  $z(t)$  describing the mobile radio channel contains besides the purely stochastic component  $x(t) +{\rm j} \cdot y(t)$  a deterministic part of the form  $x_0 + {\rm j} \cdot y_0$.

The equations of Rice fading can be summarized briefly as follows:
:$$r(t) = z(t) \cdot s(t) ,$$
:$$z(t) = x(t) + {\rm j} \cdot y(t) ,$$
:$$x(t) = u(t) + x_0 ,$$
:$$y(t) = v(t) + y_0 .$$

The following applies:
* The direct path is defined by the complex constant  $z_0 = x_0 + {\rm j} \cdot y_0$.  The magnitude of this time-invariant component is
:$$|z_0| = \sqrt{x_0^2 + y_0^2}\hspace{0.05cm}.$$
* $u(t)$  and  $v(t)$  are zero-mean Gaussian random processes, both with variance  $\sigma^2$  and uncorrelated with each other.  They model scattering, refraction and diffraction effects on a variety of indirect paths.
* The magnitude  $a(t) = |z(t)|$  has a Rice probability density function  $\rm (PDF)$, which gives this channel model its name.  For   $a ≥ 0$, the PDF is
:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm exp} [ -\frac{a^2 + |z_0|^2}{2\sigma^2}] \cdot {\rm I}_0 \left [ \frac{a \cdot |z_0|}{\sigma^2} \right ]\hspace{0.05cm}, \hspace{0.2cm}{\rm I }_0 (u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$

The graph shows the Rice PDF for  $|z_0|^2 = 0,\ 2, \ 4, \ 10$  and  $20$.  For all curves, we have   $\sigma = 1$   ⇒   $\sigma^2 = 1$.

In this task, however, we will not consider the PDF of the magnitude, but the auto-correlation function  $\rm (ACF)$  of the complex factor  $z(t)$,
:$$\varphi_z ({\rm \Delta}t) = {\rm E}\big [ z(t) \cdot z^{\star}(t + {\rm \Delta}t)\big ]
\hspace{0.05cm},$$

and the corresponding power-spectral density  $\rm (PSD)$
:$${\it \Phi}_z (f_{\rm D})
\hspace{0.3cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.3cm} \varphi_z ({\rm \Delta}t)
\hspace{0.05cm}.$$

''Notes:''
* This task belongs to chapter  [[Mobile_Communications/Non-frequency_selective_fading_with_direct_component|Non-frequency selective fading with direct component]].
* Reference is also made to the chapters  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|Auto-Correlation function (ACF)]]  and  [[Theory_of_Stochastic_Signals/Power_Density_Spectrum_(PSD)|Power Density Spectrum (PSD)]]  in the book "Stochastic Signal Theory".

===Questions===

<quiz display=simple>
{Which value of   $|z_0|^2$  corresponds to Rayleigh fading?
|type="{}"}
$|z_0|^2 \ = \ $ { 0. } $\ \rm $

{Let  $|z_0|^2 \ne 0$.  Which of the following functions depend only on  $|z_0|^2 = x_0^2$ + $y_0^2$,  but not on its components  $x_0^2$  and  $y_0^2$  alone?
|type="[]"}
- PDF  $f_x(x)$  of the real part,
- PDF  $f_y(y)$  of the imaginary part,
+ PDF  $f_a(a)$  of the magnitude,
- PDF  $f_{\rm \phi}(\phi)$  of the phase,
+ ACF  $\varphi_z(\Delta t)$  the complex quantity  $z(t)$,
+ PDS  ${\it \Phi}_z(f_{\rm D})$  the complex quantity  $z(t)$.

{Calculate the root mean square value  ${\rm E}\big[|z(t)|^2\big]$  for different values of  $|z_0|^2$.  Assume   $\sigma^2 = 1$.
|type="{}"}
$|z_0|^2 = 0\text{:} \hspace{0.52cm} {\rm E}\big[|z(t)|^2\big] \ = \ $ { 2 3% } $\ \ \rm $
$|z_0|^2 = 2\text{:} \hspace{0.52cm} {\rm E}\big[|z(t)|^2\big] \ = \ $ { 4 3% } $\ \ \rm $
$|z_0|^2 = 10\text{:} \hspace{0.3cm} {\rm E}\big[|z(t)|^2\big] \ = \ $ { 12 3% } $\ \ \rm $

{How differ the auto-correlation functions  $\rm (ACFs)$  of the black, the blue and the green channel?
|type="[]"}
+ The "blue" ACF is above the "black" ACF by about  $4$  units.
- The "blue" ACF is below the "black" ACF by about  $2$  units.
- The "green" ACF is wider than the "blue" by the factor  $2.5$ .

{How differ the power density spectra  $\rm (PDSs)$  among the black, blue, and green mobile radio channels?
|type="[]"}
+ The "black" PDS is purely continuous (no Dirac).
+ The "blue" and "green" PDSs contain one Dirac each.
+ The "green" Dirac has a greater weight than the "blue" one.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)''' Rayleigh fading  results from the Rice fading  with  $|z_0|^2 \ \underline {= \ 0}$.

'''(2)''' Options 3, 5 and 6 are correct:

It is obvious that
* $f_x(x)$  depends only on  $x_0$,
* $f_y(y)$  depends only on  $y_0$,
* $f_{\rm \phi}(\phi)$ depends only on the ratio $y_0/x_0$.

The given equation for the PDF  $f_a(a)$  shows that the magnitude  $a$  depends only on  $|z_0|$.

For the ACF, using  $z(t) = x(t) + {\rm j} \cdot y(t)$  we have
:$$\varphi_z ({\rm \Delta}t) = {\rm E}\left [ z(t) \cdot z^{\star}(t + {\rm \Delta}t)\right] = {\rm E}\left [ \left ( x(t) + {\rm j} \cdot y(t) \right )\cdot (x(t + {\rm \Delta}t) - {\rm j} \cdot (y(t+ {\rm \Delta}t)\right ]
\hspace{0.05cm}.$$

Because of the statistical independence between real and imaginary parts, the equation can be simplified as follows:
:$$\varphi_z ({\rm \Delta}t) = {\rm E}\left [ x(t) \cdot x(t + {\rm \Delta}t)\right ] +
{\rm E}\left [ y(t) \cdot y(t + {\rm \Delta}t)\right ] \hspace{0.05cm}.$$

With  $x(t) = u(t) + x_0$  and  $t' = t + \Delta t$, the first part results in  $x(t) = u(t) + x_0$:
:$${\rm E}\left [ x(t) \cdot x(t')\right ] = {\rm E}\left [ u(t) \cdot u(t')\right ] + x_0 \cdot {\rm E}\left [ u(t) \right ]
+ x_0 \cdot {\rm E}\left [ u(t') \right ] + x_0^2\hspace{0.05cm},$$
:$$\Rightarrow \hspace{0.3cm} {\rm E}\left [ x(t) \cdot x(t + {\rm \Delta}t)\right ] = {\rm E}\left [ u(t) \cdot u(t + {\rm \Delta}t)\right ] + x_0^2 = \varphi_u ({\rm \Delta}t) + x_0^2
\hspace{0.05cm}.$$

This takes into account that the Gaussian random variable  $u(t)$  has zero mean and has the variance  $\sigma^2$.

In the same way with  $y(t) = v(t) + y_0$  is obtained:
:$${\rm E}\left [ y(t) \cdot y(t + {\rm \Delta}t)\right ] = \ ... \ = \varphi_v ({\rm \Delta}t) + y_0^2 \hspace{0.3cm}
\Rightarrow \hspace{0.3cm} \varphi_z ({\rm \Delta}t) = \varphi_u ({\rm \Delta}t) + \varphi_v ({\rm \Delta}t) + x_0^2 + y_0^2
= 2 \cdot \varphi_u ({\rm \Delta}t) + |z_0|^2
\hspace{0.05cm}.$$

But if the ACF  $\varphi_z(\Delta t)$  only depends on  $|z_0^2|$, then this also applies to the Fourier transform   ⇒   power-spectral density  $\rm PDS$.

'''(3)''' The root mean square can be calculated from the PDF of the magnitude:
:$${\rm E}\left [ |z(t)|^2 \right ] = {\rm E}\left [ a^2 \right ] = \int_{0}^{\infty}a^2 \cdot f_a(a)\hspace{0.15cm}{\rm d}a
\hspace{0.05cm}.$$

At the same time, the root mean square value – i.e. the power – is also determined by the ACF:
:$${\rm E}\left [ |z(t)|^2 \right ] = \varphi_z ({\rm \delta}t = 0) = 2 \cdot \varphi_u ({\rm \delta}t = 0) + |z_0|^2 = 2 \cdot \sigma^2 + |z_0|^2
\hspace{0.05cm}.$$

With  $\sigma = 1$  you get the following numerical results:
:$$ \ |z_0|^2 = 0\text{:} \ \hspace{0.3cm}{\rm E}\left [ |z(t)|^2 \right ] = 2 + 0 \hspace{0.15cm} \underline{ = 2} \hspace{0.05cm},$$
:$$ \ |z_0|^2 = 2\text{:} \ \hspace{0.3cm}{\rm E}\left [ |z(t)|^2 \right ] = 2 + 2 \hspace{0.15cm} \underline{ = 4} \hspace{0.05cm},$$
:$$|z_0|^2 = 10\text{:} \ \hspace{0.3cm}{\rm E}\left [ |z(t)|^2 \right ] = 2 + 10 \hspace{0.15cm} \underline{ = 12} \hspace{0.05cm}.$$

'''(4)''' Correct is the  solution 1, as already derived in the solution  '''(2)'''.

The following statements would also be correct:
* The blue ACF is 4 over the black one.
* The green ACF is 6 over the blue one.

'''(5)''' All solution suggestions apply:
* The black PDS is a  [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process#ACF_und_PDS_with_Rayleigh.E2.80.93Fading|Jakes spectrum]]  and therefore continuous, i.e. all frequencies are present within an interval.
* In the auto-correlation function (ACF) of the blue or green channel, the constant  $|z_0|^2$  also occurs.
* In the power-spectral density (PSD), there are Dirac functions at the Doppler frequency  $f_{\rm D} = 0$  with the weight  $|z_0|^2$ because  of these constants in the ACF.
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^1.4 Fading with Direct Path Component^]]

Aufgaben:Exercise 4.8: Near-end and Far-end Crosstalk Disorders

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Lineare zeitinvariante Systeme/Kupfer–Doppelader
}}

[[File:EN_LZI_A_4_8.png|right|frame|Local and long-distance crosstalk]]
On the  $S_0$ bus at  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN|$\rm ISDN$]]  ("Integrated Services Digital Networks"),  data is transmitted separately according to transmission direction on a star quad.  The received signal of an ISDN device is therefore disturbed not only by connections on other wires but also by crosstalk from its own transmitted signal.

In this exercise,  two ISDN terminals with  $\text{50 m}$  distance are considered,  assuming:
* For the power-spectral density  $\rm (PSD)$  of the transmitter of each terminal,  let  ${\it\Phi}_{0} = 5 \cdot 10^{-9} \ \rm W/Hz$  be very simplified:
:$${\it\Phi}_{s}(f)= \left\{ \begin{array}{c} {\it\Phi}_{0} \\
0 \end{array} \right.
\begin{array}{c} {\rm{for}} \\ {\rm{for}}
\end{array}\begin{array}{*{20}c}
{ |f| \le f_0 = 100\:{\rm kHz} \hspace{0.05cm},} \\
{ |f| > f_0\hspace{0.05cm}.}
\end{array}$$
* The power transfer function on the  $S_0$ bus  $\text{(0.6 mm}$  copper two–wire line,  $\text{50 m)}$  is to be approximated in the considered range  $0 < |f| < 100 \ \rm kHz$  as follows  (very simplified):
:$$|H_{\rm K}(f)|^2 = 0.9 - 0.04 \cdot \frac{|f|}{\rm 1 \ MHz}\hspace{0.05cm}.$$
* The near–end crosstalk power transfer function is given as follows  $(\rm NEXT$ stands for  "near–end crosstalk"$)$:
:$$|H_{\rm NEXT}(f)|^2 = \left ( K_{\rm NEXT} \cdot |f|\right )^{3/2}\hspace{0.05cm},\hspace{0.2cm}K_{\rm
NEXT} = 6 \cdot 10^{-10}\,{\rm s}
\hspace{0.05cm}.$$

The diagram shows the system configuration under consideration. 
*Two twisted pairs connect subscribers  $1$  and  $2$   (one in each direction),
*while on two other twisted pairs  (not in the same star quad)  there is a connection between subscriber  $3$  and subscriber  $4$. 

''Notes:''
*The exercise belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Eigenschaften_von_Kupfer–Doppeladern|Properties of Balanced Copper Pairs]].
*It also refers to the chapter  [[Examples_of_Communication_Systems/ISDN-Basisanschluss|ISDN Basic Access]]  in the book "Examples of Communication Systems".

===Questions===

<quiz display=simple>
{Which of the following statements are true?
|type="[]"}
- Transmitter  $S_1$  leads to near-end crosstalk at receiver  $E_2$. 
+ Transmitter  $S_2$  leads to near-end crosstalk at receiver  $E_2$. 
- Transmitter  $S_3$  leads to near-end crosstalk at receiver  $E_2$. 
+ Near-end crosstalk is more unpleasant than far-end crosstalk.

{Calculate the transmission power using the simplified assumption given?   (German:  "Sendeleistung"   ⇒   subscript  "S").
|type="{}"}
$P_{\rm S} \ = \ $ { 1 3% } $\ \rm mW$

{What is the useful power arriving at the receiver?   (German:  "Empfangsleistung"   ⇒   subscript  "E").
|type="{}"}
$P_{\rm E} \ = \ $ { 0.88 3% } $\ \rm mW$

{Specify the power of the crosstalk interference. Note:  $ 1 \ \rm nW = 10^{-9} \ \rm W$.
|type="{}"}
$P_\text{NEXT} \ = \ $ { 0.186 3% } $\ \rm nW$

{What is the signal–to–crosstalk signal–to–noise ratio?
|type="{}"}
$\rm 10 \cdot \ lg\ {\it P}_E/{\it P}_\text{NEXT} \ = \ $ { 66.7 3% } $\ \rm dB$

</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  Crosstalk is caused by coupling of the transmitted signals to adjacent pairs of wires.
*In the case of near–end crosstalk  $\rm (NEXT)$,  the interfering transmitter and the interfered receiver are at the same end of the line.
* In the case of far–end crosstalk  $\rm (FEXT)$,  they are at different ends.
*However,  since the interfering signals are also very strongly attenuated on the copper wire pair,  NEXT is always by far the more dominant interfering effect compared with FEXT.

Solutions 2 and 4  are correct:
*Here,  the receiver  $E_2$  is particularly disturbed by its own transmitter  $S_2$,  i.e. by near–end crosstalk.
*The interference of  $E_2$  by  $S_3$  is far–end crosstalk,  while  $S_1$  provides the useful signal for  $E_2$.

'''(2)'''  The transmit power is equal to the integral over the power-spectral density:
:$$P_{\rm S} = {\it\Phi}_{0} \cdot 2 f_0 = 5 \cdot 10^{-9}\, {\rm W}/{\rm Hz} \cdot 2 \cdot 10^{5}\,{\rm Hz}\hspace{0.15cm}\underline{ = 1\,{\rm mW}}
\hspace{0.05cm}.$$

'''(3)'''  The following applies to the received power  (excluding the component due to near–end crosstalk):
:$$P_{\rm E} = \int_{-\infty}^{
+\infty} {\it\Phi}_{s}(f) \cdot |H_{\rm K}(f)|^2
\hspace{0.1cm}{\rm d}f = 2 {\it\Phi}_{0} \cdot \int_{0}^{
f_0} \left [ 0.9 - 0.04 \cdot \frac{f}{f_0} \right ]
\hspace{0.1cm}{\rm d}f $$
:$$\Rightarrow \hspace{0.3cm}P_{\rm E} =
2 {\it\Phi}_{0} \cdot \left [ 0.9 \cdot f_0 - \frac{0.04}{2} \cdot \frac{f_0^2}{f_0} \right ] = 2 {\it\Phi}_{0} \cdot 0.88 =
0.88 \cdot P_{\rm S}\hspace{0.15cm}\underline{ = 0.88 \,{\rm mW}}
\hspace{0.05cm}.$$

'''(4)'''  For the interfering power component of the crosstalk interference one obtains
:$$P_{\rm NEXT} = \int_{-\infty}^{
+\infty} {\it\Phi}_{s}(f) \cdot |H_{\rm NEXT}(f)|^2
\hspace{0.1cm}{\rm d}f = 2 {\it\Phi}_{0} \cdot {K_{\rm NEXT}\hspace{0.01cm}}^{3/2}
\cdot \int_{0}^{ f_0} f^{3/2}
\hspace{0.1cm}{\rm d}f $$
:$$\Rightarrow \hspace{0.3cm} P_{\rm NEXT}
= \frac{4}{5} \cdot {\it\Phi}_{0} \cdot {K_{\rm NEXT}\hspace{0.01cm}}^{3/2} \cdot f_0^{5/2}
= 0.8 \cdot 5 \cdot 10^{-9}\, \frac{\rm W}{\rm Hz}
\cdot \left ( 6 \cdot 10^{-10}\,{\rm s}\right )^{3/2} \cdot
\left ( 10^{5}\,{\rm Hz}\right )^{5/2}
= {0.186 \cdot 10^{-9}\,{\rm W}}
\hspace{0.05cm}
= \hspace{0.15cm}\underline{0.186 \,{\rm nW}}
\hspace{0.05cm}.$$

'''(5)'''    ${P_{\rm E}}/{P_{\rm NEXT}} \approx 4.73 \cdot 10^{6}$  is valid,  resulting in the logarithmic value of
:$$10 \cdot {\rm lg}\hspace{0.15cm} {P_{\rm E}}/{P_{\rm NEXT}} =
10 \cdot {\rm lg}\hspace{0.15cm}
(4.73 \cdot 10^{6})
\hspace{0.15cm}\underline{= 66.7\,\,{\rm dB}}
\hspace{0.05cm}.$$
{{ML-Fuß}}

[[Category:Linear and Time-Invariant Systems: Exercises|^4.3 Balanced Copper Twisted Pair^]]

Aufgaben:Exercise 5.8Z: Matched Filter for Rectangular PSD

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Matched_Filter
}}

[[File:P_ID647__Sto_Z_5_8.png|right|frame|Useful signal spectrum  $G(f)$  and PSD  ${\it \Phi}_n (f)$  of the interference]]
The interference power density effective on a system can be assumed to be constant in range:
:$$\it{\Phi} _n \left( f \right) = \left\{ \begin{array}{l} N_0 /2 \\ N_1 /2 \\ \end{array} \right.\quad \begin{array}{*{20}c} \rm{f\ddot{u}r} \\ \rm{f\ddot{u}r} \\\end{array}\quad \begin{array}{*{20}c} {\left| f \right| \le f_{\rm N} ,} \\ {\left| f \right| > f_{\rm N} .} \\\end{array}$$

*Here, let the interference power density  $N_1$  in the outer region  $|f| > f_{\rm N}$  always be much smaller than  $N_0$.
*For example, use the following values:
:$$N_0 = 2 \cdot 10^{ - 6} \;{\rm{V}}^{\rm{2}} /{\rm{Hz}},\quad N_1 = 2 \cdot 10^{ - 8} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$

Such an interference signal  $n(t)$  occurs, for example, when the dominant interference source contains only components below the frequency limit  $f_{\rm N}$.    Due to the unavoidable thermal noise, also for  $|f| > f_{\rm N}$  the interference power density is  ${\it \Phi}_n(f) \ne 0$.

Further, it holds:
*Let the spectrum  $G(f)$  of the useful signal also be rectangular according to the above diagram.
*Therefore, the corresponding useful pulse  $g(t)$  has the following curve with  $\Delta f = 2 \cdot f_{\rm G}$: 
:$$g(t) = G_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} \left( {{\rm{\pi }} \cdot \Delta f \cdot t} \right).$$

*Let the reception filter be optimally matched to the useful spectrum  $G(f)$  and the interference power-spectral density  ${\it \Phi}_n(f)$. 
*That is,  let  $H_{\rm E}(f) = H_{\rm MF}(f)$.  Let the detection time be simplified  $T_{\rm D} = 0$  (acausal system description).

''Notes:''
*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Matched_Filter|Matched Filter]].

*For numerical calculations always use the numerical values
:$$G_0 = 10^{ - 4} \;{\rm{V/Hz}}{\rm{, }}\quad \Delta f = 10\;{\rm{kHz}}.$$

===Questions===

<quiz display=simple>
{Which of the following statements are valid under the condition  $f_{\rm N} > f_{\rm G}$?
|type="[]"}
+ Applicable is the "matched filter" for "white noise".
- The MF output pulse is triangular.
+ The MF output pulse is  $\rm si$–shaped.
- The MF output pulse is  $\rm si^2$–shaped.

{What is the S/N ratio (in dB) for  $f_{\rm N} > f_{\rm G}$?
|type="{}"}
$10 \cdot \lg \; \rho_d \ = \ $ { 20 3% } $\ \rm dB$

{What SNR (in dB) results for  $f_{\rm N} = f_{\rm G}/2$?  Interpretation.
|type="{}"}
$10 \cdot \lg \; \rho_d \ = \ $ { 37.03 3% } $\ \rm dB$

</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  Solutions 1 and 3 are correct:
*For all frequencies  $|f| > f_{\rm G}$ at which the useful signal has spectral components  $(G_d(f) \ne 0)$, the interference power-spectral density is  ${\it}\Phi_n(f) = N_0/2$.
*Thus, the frequency response of the matched filter, assuming  $T_{\rm D} = 0$  is:
:$$H_{\rm MF} (f) = K_{\rm MF} \cdot G(f).$$
*In this case, the optimal frequency response  $H_{\rm MF}(f)$,  just like  $G(f)$,  is rectangular with width  $\Delta f$.
*Thus, for the useful component of the MF output signal holds:
:$$d_{\rm S}(t)\quad \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \quad G(f) \cdot H_{\rm MF} (f).$$
*The product of two rectangular functions of equal width again yields a rectangular function.
*It further follows that the output pulse of the matched filter is also  $\rm si$-shaped.

'''(2)'''  With white noise one obtains:
:$$\rho _d = \frac{1}{N_0 /2}\int_{ - \infty }^{ + \infty } {\left| {G(f)} \right|^2 \, {\rm{d}}f.}$$

*The integral yields the value  $G_0^2 \cdot \Delta f$.  It follows that:
:$$\rho _d = \frac{G_0 ^2 \cdot \Delta f }{N_0 /2} = \frac{ 10^{ - 8}\,(\rm V/Hz)^2 \;\cdot10^4 \;{\rm{Hz}} }{10^{ - 6}\,\rm V^2/Hz} = 10^2
\quad \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline { = 20\;{\rm{dB}}}.$$

[[File:P_ID648__Sto_Z_5_8_c.png|right|frame|Regarding subtask  '''(3)''']]
'''(3)'''  In general, the SNR for colored interference is:
:$$\rho _d = 2 \cdot \int_0^\infty \frac{\left| {G(f)} \right|^2 }{{\it \Phi}_n (f)} \, {\rm{d}}f.$$

*As can be seen from the accompanying qualitative diagram, the integrand is piecewise constant for the given frequency responses.
*Thus, with  $f_{\rm G} = 5 \; \rm kHz$  and  $f_{\rm N} = f_{\rm G}/2 = 2.5 \; \rm kHz$,  we obtain:
:$$\rho _d = 2 \cdot 2.5\;{\rm{kHz}}\left( { \frac{10^{ - 2}}{\rm{Hz}} + \frac{1}{{{\rm{Hz}}}} } \right) = 5.05 \cdot 10^3
\quad \Rightarrow \quad 10\cdot\lg \rho _d \hspace{0.15cm}\underline {= 37.03\;{\rm{dB}}}.$$

'''Interpretation''': 
*The matched filter frequency response  $H_{\rm MF}(f)$  has exactly the same shape as the integrand sketched above.
*If the constant  $K_{\rm MF}$  is chosen (arbitrarily) so that in the range  $f_{\rm N} \le |f| \le f_{\rm G}$  the MF frequency response has the value  $1$,  then for low frequencies  $(|f| < f_{\rm G})$:   $H_{\rm MF}(f) = 0.01$. This means:
:*The matched filter favors those frequencies that are only slightly affected by the interference  ${\it \Phi}_n(f)$. 
:*If instead we would use a filter  $H(f)$,  which gives equal weight to all frequencies of the wanted signal up to and including  $f_{\rm G}$  (purple curve in the sketch below), the following ratios would result:
::$$d_{\rm S}( {T_{\rm D} } ) = G_0 \cdot 2 \cdot f_{\rm G} = 1\;{\rm{V}}, \quad \sigma _d ^2 = 10^{ - 6} \frac{{{\rm{V}}^{\rm{2}} }}{{{\rm{Hz}}}} \cdot f_{\rm G} + 10^{ - 8} \frac{{{\rm{V}}^{\rm{2}} }}{{{\rm{Hz}}}} \cdot ( {f_{\rm G} - f_{\rm N} } ) = 2.5 \cdot 1.01 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}}$$
:$$ \Rightarrow \hspace{0.3cm} \rho _d = \frac {d_{\rm S}( {T_{\rm D} } )^2}{\sigma _d ^2} = \frac{1 \;{\rm{V}}^{\rm{2}}}{2.525 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}}} = 396 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \, \rho _d = 25.98 \, {\rm dB}.$$
:*The signal–to–noise ratio is thus about  $11\ \rm dB$  worse than when using the matched filter for colored interference.
{{ML-Fuß}}

[[Category:Theory of Stochastic Signals: Exercises|^5.4 Matched Filter^]]

Aufgaben:Exercise 2.10Z: Noise with DSB-AM and SSB-AM

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Modulationsverfahren/Einseitenbandmodulation
}}

[[File:EN_Mod_Z_2_9_c.png|right|frame|Shared block diagram for DSB-AM and SSB-AM]]
Now the influence of noise on the sink-to-noise ratio  $10 · \lg ρ_v$  for both   DSB–AM  and  SSB–AM transmission will be compared.   The illustration shows the underlying block diagram.

The differences between the two system variants are highlighted in red on the image, namely the modulator  (DSB or SSB)  as well as the dimensionless constant
:$$ K = \left\{ \begin{array}{c} 2/\alpha_{\rm K} \\ 4/\alpha_{\rm K} \\ \end{array} \right.\quad \begin{array}{*{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\rm DSB} \hspace{0.05cm}, \\ {\rm SSB} \hspace{0.05cm} \\ \end{array}$$
of the receiver-side carrier signal  $z_{\rm E}(t) = K · \cos(ω_{\rm T} · t)$, which is assumed to be frequency and phase synchronous with the carrier signal  $z(t)$  at the transmitter.

The system parameters captured by the shared performance parameter are labelled in green:
:$$\xi = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}}$$

Further note:
* The cosine signal  $q(t)$  with frequency  $B_{\rm NF}$  stands for a source signal with bandwidth  $B_{\rm NF}$ composed of multiple frequencies.
* DSB–AM with carrier is parameterized by a modulation depth of  $m = A_{\rm N}/A_{\rm T}$ , while SSB-AM is determined by the sideband-to-carrier ratio  $μ = A_{\rm N}/(2 · A_{\rm T})$ .
* The frequency-independent channel transmission factor  $α_{\rm K}$  is balanced by the constant  $K$ , so that in the noise-free case  $(N_0 = 0)$ , the sink signal  $v(t)$  matches the source signal  $q(t)$ .
* The sink SNR can thus be given as follows $(T_0$ indicates the period of the source signal$)$:
:$$ \rho_{v } = \frac{P_{q}}{P_{\varepsilon }}\hspace{0.5cm}{\rm mit}\hspace{0.5cm} P_{q} = \frac{1}{T_{\rm 0}}\cdot\int_{0}^{ T_{\rm 0}} {q^2(t)}\hspace{0.1cm}{\rm d}t, \hspace{0.5cm}P_{\varepsilon} = \int_{-B_{\rm NF}}^{ +B_{\rm NF}} \hspace{-0.1cm}{\it \Phi_{\varepsilon}}(f)\hspace{0.1cm}{\rm d}f\hspace{0.05cm}.$$

''Hints:''
*This exercise belongs to the chapter  [[Modulation_Methods/Single-Sideband_Modulation|Single-sideband Modulation]].
*Particular reference is made to the page  [[Modulation_Methods/Single-Sideband_Modulation#Sideband-to-carrier_ratio|Sideband-to-carrier ratio]].
*The results for DSB–AM can be found on the page [[Modulation_Methods/Synchronous_Demodulation#Sink_SNR_and_the_performance_parameter|Sink SNR and the performance parameter]].

===Questions===

<quiz display=simple>
{Which kind of demodulation is considered here?
|type="()"}
+ Synchronous demodulation.
- Envelope demodulation.

{Which relationship holds between the quantities  $ρ_v$  and  $ξ$  for  DSB–AM without a carrier  as $(m → ∞)$?
|type="[]"}
-  $ρ_v = 2 · ξ$.
+  $ρ_v = ξ$.
-  $ρ_v = ξ/2$.

{Which relationship holds between  $ρ_v$  and  $ξ$  for  SSB–AM without a carrier  as $(μ → ∞)$?
|type="[]"}
-  $ρ_v = 2 · ξ$.
+  $ρ_v = ξ$.
-  $ρ_v = ξ/2$.

{Let  $ξ = 10^4$.  Calculate the sink-to-noise ratio of   DSB–AM without a carrier  for modulation depths  $m = 0.5$  and  $m = 1$.
|type="{}"}
$m = 0.5\text{:} \ \ 10 · \lg \ ρ_v \ = \ $ { 30.46 3% } $\ \rm dB$
$m = 1.0\text{:} \ \ 10 · \lg \ ρ_v \ = \ $ { 35.23 3% } $\ \rm dB$

{Further let  $ξ = 10^4$.  Calculate the sink-to-noise ratio of   SSB–AM  for the parameters  $μ = 0.354$  and  $μ = 0.707$.
|type="{}"}
$μ = 0.354\text{:} \ \ \ 10 · \lg \ ρ_v \ = \ $ { 30.46 3% } $\ \rm dB$
$μ = 0.707\text{:} \ \ \ 10 · \lg \ ρ_v \ = \ $ { 35.23 3% } $\ \rm dB$

</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  We are dealing with a   synchronous demodulator.  Answer 1 is correct.

'''(2)'''  Answer 2 is correct:
*For DSB–AM without a carrier,   $P_{\rm S} = P_q/2$.  This is simultaneously the power of the useful component of the sink signal  $v(t)$.
*The power-spectral density   ${\it Φ}_ε(f)$  of the noise component of   $v(t)$  results from the convolution :
[[File:P_ID1048__Mod_Z_2_9_b.png|right|frame|Noise power density in DSB-AM]]
:$${\it \Phi}_\varepsilon(f) = {\it \Phi}_{z{\rm E} }(f) \star {\it \Phi}_n (f) = \frac{1}{\alpha_{\rm K}^2} \cdot \big[\delta(f - f_{\rm T}) + \delta(f + f_{\rm T}) \big]\star {\it \Phi}_n (f) \hspace{0.05cm}.$$
*The expression  $\big[$ ... $\big]$  describes the power-spectral density of a cosine signal with the signal amplitude   $K = 2$.
*The correction of channel damping is considered with   $1/α_K^2$ .
*Thus, taking   ${\it \Phi}_n(f) = N_0/2$  into account, we get:
:$${\it \Phi}_\varepsilon(f) = \frac{N_0}{\alpha_{\rm K}^2}
\hspace{0.3cm} \Rightarrow \hspace{0.3cm} P_\varepsilon = \int_{-B_{\rm NF}}^{+B_{\rm NF}} {{\it \Phi}_\varepsilon(f) }\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0 \cdot B_{\rm NF}}{\alpha_{\rm K}^2}\hspace{0.05cm}.$$
*From this, it follows for the the signal-to-noise power ratio (SNR):
:$$\rho_{v } = \frac{P_{q}}{P_{\varepsilon }} = \frac{2 \cdot P_{\rm S}}{2 \cdot N_0 \cdot B_{\rm NF}/\alpha_{\rm K}^2} = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}}\hspace{0.15cm}\underline { = \xi} \hspace{0.05cm}.$$

[[File:P_ID1049__Mod_Z_2_9_c.png|right|frame|Noise power density in  USB-AM]]
'''(3)'''  Answer 2 is correct:
*In contrast to DSB,   $P_S = P_q/4$  holds for SSB, as well as
:$${\it \Phi}_\varepsilon(f) = {\it \Phi}_{z{\rm E} }(f) \star {\it \Phi}_n (f) = \frac{4}{\alpha_{\rm K}^2} \cdot \big[\delta(f - f_{\rm T}) + \delta(f + f_{\rm T}) \big]\star {\it \Phi}_n (f) \hspace{0.05cm}.$$
*Taking   $B_{\rm HF} = B_{\rm NF}$  into account (see adjacent diagram for USB modulation), we now get:
:$${\it \Phi}_\varepsilon(f) = \frac{2 \cdot N_0}{\alpha_{\rm K}^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} P_\varepsilon = \frac{4 \cdot N_0 \cdot B_{\rm NF}}{\alpha_{\rm K}^2}\hspace{0.05cm}.$$

*This means:  when the carrier is not transmitted, single-sideband modulation demonstrates the same noise behaviour as DSB-AM.

'''(4)'''  Assuming a cosine carrier with amplitude   $A_{\rm T}$  and a similarly cosine message signal   $q(t)$ , for DSB with carrier, we get:
:$$ s(t) = \big (q(t) + A_{\rm T}\big ) \cdot \cos( \omega_{\rm T} \cdot t)
= A_{\rm T} \cdot \cos( \omega_{\rm T} \cdot t) + \frac{A_{\rm N}}{2}\cdot \cos\big(( \omega_{\rm T}+ \omega_{\rm N}) \cdot t \big)+ \frac{A_{\rm N}}{2}\cdot \cos\big(( \omega_{\rm T}- \omega_{\rm N}) \cdot t\big)\hspace{0.05cm}.$$
*The transmit power is thus given by
:$$ P_{\rm S}= \frac{A_{\rm T}^2}{2} + 2 \cdot \frac{(A_{\rm N}/2)^2}{2} = \frac{A_{\rm T}^2}{2} + \frac{A_{\rm N}^2}{4} \hspace{0.05cm}.$$
*Taking   $P_q = A_{\rm N}^2/2$  and  $m = A_{\rm N}/A_{\rm T}$  into account, this can also be written as:
:$$P_{\rm S}= \frac{A_{\rm N}^2}{4} \cdot \left[ 1 + \frac{2 \cdot A_{\rm T}^2}{A_{\rm N}^2}\right] = \frac{P_q}{2} \cdot \left[ 1 + {2 }/{m^2}\right]\hspace{0.05cm}.$$
*With a noise power   $P_ε$  according to subtask  '''(2)'''  we thus obtain:
:$$\rho_{v } = \frac{P_{q}}{P_{\varepsilon }} = \frac{2 \cdot P_{\rm S}\cdot (1 + 2/m^2)}{2 \cdot N_0 \cdot B_{\rm NF}/\alpha_{\rm K}^2} = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}} \cdot \frac{1}{1 +{2 }/{m^2}} \hspace{0.05cm}.$$
*And in logarithmic representation:
:$$ 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } = 10 \cdot {\rm lg} \hspace{0.15cm}\xi - 10 \cdot {\rm lg} \hspace{0.15cm}\left[{1 +{2 }/{m^2}}\right] \hspace{0.05cm}.$$
:$$\Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \ (m = 0.5) = 40 \,{\rm dB} - 10 \cdot {\rm lg} (9) \hspace{0.15cm}\underline {= 30.46\, {\rm dB}}$$
:$$\Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \ (m = 1.0) = 40 \,{\rm dB} - 10 \cdot {\rm lg} (3) \hspace{0.15cm}\underline {= 35.23\, {\rm dB} \hspace{0.05cm}}.$$

'''(5)'''  In SSB–AM there is only one sideband.
*Therefore, considering the sideband-to-carrier ratio   $μ = A_{\rm N}/(2A_{\rm T})$ gives:
:$$ P_{\rm S}= \frac{A_{\rm T}^2}{2} + \frac{(A_{\rm N}/2)^2}{2} = {A_{\rm N}^2}/{8} \cdot \big[ 1 + {4 \cdot A_{\rm T}^2}/{A_{\rm N}^2}\big] = {P_q}/{4} \cdot \big[ 1 + {1 }/{\mu^2}\big] \hspace{0.05cm}.$$
*Thus, with the noise power from subtask  '''(3)''' we obtain:
:$$\rho_{v } = \frac{P_{q}}{P_{\varepsilon }} = \frac{4 \cdot P_{\rm S}\cdot (1 + 1/\mu^2)}{4 \cdot N_0 \cdot B_{\rm NF}/\alpha_{\rm K}^2} = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}} \cdot \frac{1}{1 +{1 }/{\mu^2}}\hspace{0.3cm}
\Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } = 10 \cdot {\rm lg} \hspace{0.15cm}\xi - 10 \cdot {\rm lg} \hspace{0.15cm}\big[{1 +{1 }/{\mu^2}}\big] \hspace{0.05cm}.$$
*So we get the same result with SSB-AM as in DSB-AM with a modulation depth of  $m = \sqrt{2} · μ$.
*From this, it further follows:
:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}({\rm SSB,} \hspace{0.1cm}\mu = {0.5}/{\sqrt{2}}) = 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}({\rm DSB,} \hspace{0.1cm}m=0.5) \hspace{0.15cm}\underline {=30.46\,{\rm dB}},$$
:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}({\rm SSB,} \hspace{0.1cm}\mu = {1.0}/{\sqrt{2}}) = 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}({\rm DSB,} \hspace{0.1cm}m=1.0) \hspace{0.15cm}\underline {=35.23\,{\rm dB}}\hspace{0.05cm}.$$
{{ML-Fuß}}

[[Category:Modulation Methods: Exercises|^2.4 Single Sideband Amplitude Modulation^]]

Aufgaben:Exercise 2.7: Coherence Bandwidth

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/The_GWSSUS_Channel_Model}}

[[File:P_ID2172__Mob_A_2_7.png|right|frame|Delay power-spectral density and frequency correlation function]]
For the delay power-spectral density, we assume an exponential behavior.  With  ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$  we have
:$${{\it \phi}_{\rm V}(\tau)}/{{\it \phi}_{\rm 0}} = {\rm e}^{ -\tau / \tau_0 } \hspace{0.05cm}.$$

The constant  $\tau_0$  can be determined from the tangent in the point  $\tau = 0$  according to the upper graph.  Note that  ${\it \Phi}_{\rm V}(\tau)$  has unit  $[1/\rm s]$ .  Furthermore,
* The probability density function  $\rm (PDF)$  $f_{\rm V}(\tau)$  has the same form as  ${\it \Phi}_{\rm V}(\tau)$, but is normalized to area  $1$ .
* The  average excess delay or mean excess delay  $m_{\rm V}$  is equal to the linear expectation  $E\big [\tau \big]$  and can be determined from the PDF $f_{\rm V}(\tau)$ .
* The  multipath spread or delay spread  $\sigma_{\rm V}$  gives the standard deviation of the random variable  $\tau$ .  In the theory part we also use the term  $T_{\rm V}$ for this.
* The displayed frequency correlation function  $\varphi_{\rm F}(\Delta f)$  can be calculated as the Fourier transform of the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau)$ :
:$$\varphi_{\rm F}(\Delta f)
\hspace{0.2cm} {\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau)\hspace{0.05cm}.$$
* The coherence bandwidth  $B_{\rm K}$  is the value of $\Delta f$ at which the frequency correlation function  $\varphi_{\rm F}(\Delta f)$  has dropped to half in absolute value.

''Notes:''
* This exercise belongs to the chapter  [[Mobile_Communications/The_GWSSUS_Channel_Model|The GWSSUS Channel Model]].
* This exercise requires knowledge of  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments|Expected values and moments]]  from the book "Theory of Stochastic Signals".
* In addition, the following Fourier transform is given:
:$$x(t) = \left\{ \begin{array}{c} {\rm e}^{- \lambda \hspace{0.05cm}\cdot \hspace{0.05cm} t}\\
0 \end{array} \right.\quad
\begin{array}{*{1}c} \hspace{-0.35cm} {\rm for} \hspace{0.15cm} t \ge 0
\\ \hspace{-0.35cm} {\rm for} \hspace{0.15cm} t < 0 \\ \end{array}
\hspace{0.4cm} {\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.4cm} X(f) = \frac{1}{\lambda + {\rm j} \cdot 2\pi f}\hspace{0.05cm}.$$

===Questionnaire===
<quiz display=simple>
{What is the probability density function  $f_{\rm V}(\tau)$  of the delay time?
|type="()"}
- $f_{\rm V}(\tau) = {\rm e}^{-\tau/\tau_0}$.
+ $f_{\rm V}(\tau) = 1/\tau_0 \cdot {\rm e}^{-\tau/\tau_0}$,
- $f_{\rm V}(\tau) = {\it \Phi}_0 \cdot {\rm e}^{-\tau/\tau_0}$.

{Determine the average delay time for  $\tau_0 = 1 \ \ \rm µ s$.
|type="{}"}
$m_{\rm V} \ = \ ${ 1 3% } $\ \rm µ s$

{Which value results for the multipath spread with  $\tau_0 = 1 \ \ \rm µ s$?
|type="{}"}
$\sigma_{\rm V} \ = \ ${ 1 3% } $\ \rm µ s$

{What is the frequency–correlation function  $\varphi_{\rm F}(\Delta f)$?
|type="()"}
+ $\varphi_{\rm F}(\Delta f) = \big[1/\tau_0 + {\rm j} \ 2 \pi \cdot \delta f \big]^{-1}$,
- $\varphi_{\rm F}(\Delta f) = {\rm e}^ {-(\tau_0 \hspace{0.05cm}\cdot \hspace{0.05cm}\Delta f)^2}$.

{Determine the coherence bandwidth  $B_{\rm K}$.
|type="{}"}
$B_{\rm K} \ = \ ${ 276 3% } $\ \ \rm kHz$
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  Let  ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$.  The integral of the delay power-spectral density gives
:$$\int_{0}^{+\infty} {\it \Phi}_{\rm V}(\tau) \hspace{0.15cm}{\rm d} \tau =
{\it \Phi}_{\rm 0} \cdot \int_{0}^{+\infty} {\rm e}^{-\tau / \tau_0} \hspace{0.15cm}{\rm d} \tau =
{\it \Phi}_{\rm 0} \cdot \tau_0
\hspace{0.05cm}. $$

*Then the probability density function is
:$$f_{\rm V}(\tau) = \frac{{\it \Phi}_{\rm V}(\tau) }{{\it \Phi}_{\rm 0} \cdot \tau_0}= \frac{1}{\tau_0} \cdot {\rm e}^{-\tau / \tau_0}
\hspace{0.05cm}.$$

*Solution 2 is correct.

'''(2)'''  The  $k$-th moment of an  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|one-sided exponential random variable]]  is  $m_k = k! \cdot \tau_0^k$.
*With  $k = 1$, this results in the linear mean value  $m_1 = m_{\rm V}$:
:$$m_{\rm V} = \tau_0 \hspace{0.1cm} \underline {= 1\,{\rm µ s}}
\hspace{0.05cm}. $$

'''(3)'''  According to the  [[Theory_of_Stochastic_Signals/Erwartungswerte_und_Momente#Some common central moments| Steiner's Theorem]], the variance of any random variable is  $\sigma^2 = m_2 \, -m_1^2$.
*This yields  $m_2 = 2 \cdot \tau_0^2$, and therefore
:$$\sigma_{\rm V}^2 = m_2 - m_1^2 = 2 \cdot \tau_0^2 - (\tau_0)^2 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}
\sigma_{\rm V} = \tau_0 \hspace{0.1cm} \underline {= 1\,{\rm µ s}}
\hspace{0.05cm}. $$

'''(4)'''  ${\it \Phi}_{\rm V}(\tau)$  is identical to  $x(t)$  in the given Fourier transform pair if  $t$  is replaced by  $\tau$  and  $\lambda$  by  $1/\tau_0$.
*Thus,  $\varphi_{\rm F}(\Delta f)$  is equal to  $X(f)$  with the substitution  $f → \Delta f$:
:$$\varphi_{\rm F}(\Delta f) = \frac{1}{1/\tau_0 + {\rm j} \cdot 2\pi \Delta f}
= \frac{\tau_0}{1 + {\rm j} \cdot 2\pi \cdot \tau_0 \cdot \Delta f}\hspace{0.05cm}.$$

*The first expression is correct.

'''(5)'''  The coherence bandwidth is implicit in the following equation:
:$$|\varphi_{\rm F}(\Delta f = B_{\rm K})| \stackrel {!}{=} \frac{1}{2} \cdot |\varphi_{\rm F}(\Delta f = 0)| = \frac{\tau_0}{2}\hspace{0.3cm}
\Rightarrow \hspace{0.3cm}|\varphi_{\rm F}(\Delta f = B_{\rm K})|^2
= \frac{\tau_0^2}{1 + (2\pi \cdot \tau_0 \cdot B_{\rm K})^2} \stackrel {!}{=} \frac{\tau_0^2}{4}$$
:$$\Rightarrow \hspace{0.3cm}(2\pi \cdot \tau_0 \cdot B_{\rm K})^2 = 3 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}
B_{\rm K}= \frac{\sqrt{3}}{2\pi \cdot \tau_0} \approx \frac{0.276}{ \tau_0}\hspace{0.05cm}. $$

*With  $\tau_0 = 1 \ \ \rm µ s$, the coherence bandwidth is  $B_{\rm K} \ \underline {= 276 \ \ \rm kHz}$.
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^2.3 The GWSSUS Channel Model^]]

Aufgaben:Exercise 1.5: Reconstruction of the Jakes Spectrum

2022-02-17T11:41:31Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process}}

[[File:P_ID2124__Mob_A_1_5.png|right|frame|Considered Jakes spectrum]]
In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.

This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }
\hspace{0.05cm}.$$

What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]  of the PDS.

With the  [[Applets:Bessel_Functions_of_the_First_Kind|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get
:$$\varphi_z ({\rm \Delta}t) = 2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot {\rm \Delta}t)\hspace{0.05cm}.$$

To take into account the Doppler effect and thus a relative movement between transmitter and receiver in a system simulation, two digital filters are inserted in the  [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|Rayleigh channel model]], each with the frequency response  $H_{\rm DF}(f_{\rm D})$.

The dimensioning of these filters is part of this task.
*We restrict ourselves here to the branch for generating the real part  $x(t)$.  The ratios derived here are also valid for the imaginary part  $y(t)$.
*At the input of the left digital filter of the  [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Modeling_of_non-frequency_selective_fading|Rayleigh channel model]] , there is white Gaussian noise  $n(t)$  with variance  $\sigma^2 = 0.5$.
*The real component is then obtained from the following convolution
:$$x(t) = n(t) \star h_{\rm DF}(t) \hspace{0.05cm}.$$

''Notes:''
* This task belongs to the topic of  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process|Statistical bindings within the Rayleigh process]].
* The digital filter is treated in detail in chapter  [[Theory_of_Stochastic_Signals/Digitale_Filter|Digital Filter]]  of the book "Stochastic Signal Theory".

===Questions===

<quiz display=simple>
{What is the value of the Jakes spectrum of the real part at the Doppler frequency $f_{\rm D} = 0$?
|type="{}"}
${\it \Phi}_x(f_{\rm D} = 0)\ = \ $ { 1.59 } $\ \cdot 10^{\rm –3} \ {\rm Hz}^{-1}$

{Which dimensioning is correct, where  $K$  is an appropriately chosen constant?
|type="[]"}
- It holds  $H_{\rm DF}(f_{\rm D}) = K \cdot {\it \Phi}_x(f_{\rm D})$.
+ It applies  $|H_{\rm DF}(f_{\rm D})|^2 = K \cdot {\it \Phi}_x(f_{\rm D})$

{From which condition can the constant  $K$  be determined?
|type="[]"}
- $K$  can be selected as desired.
- The integral over  $|H_{\rm DF}(f_{\rm D})|$  must equal  $1$ .
+ The integral over  $|H_{\rm DF}(f_{\rm D})|^2$  must be  $1$ .

{Is  $H_{\rm DF}(f)$  unambiguously defined by the two conditions according to  '''(2)'''  and  '''(3)'''?
|type="()"}
- Yes.
+ No.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  The Jakes spectrum of the real part is half the resulting spectrum  ${\it \Phi}_z(f)$:
:$${\it \Phi}_x(f_{\rm D} = 0) = {\it \Phi}_y(f_{\rm D} = 0) = \frac{{\it \Phi}_z(f_{\rm D} = 0)}{2}= \frac{\sigma^2}{\pi \cdot f_{\rm D, \hspace{0.05cm} max}} =
\frac{0.5}{\pi \cdot 100\,\,{\rm Hz}} \hspace{0.15cm} \underline{ = 1.59 \cdot 10^{-3}\,\,{\rm Hz^{-1}}}
\hspace{0.05cm}.$$

'''(2)'''  Solution 2 is correct:
*The input signal  $n(t)$  has a white (constant) PDS  ${\it \Phi}_n(f_{\rm D})$.
*The PDS at the output is then
:$${\it \Phi}_x(f_{\rm D}) = {\it \Phi}_n(f_{\rm D}) \cdot | H_{\rm DF}(f_{\rm D}|^2
\hspace{0.05cm}.$$

'''(3)'''  Solution 3 is correct.
*Only if this condition is fulfilled, the signal  $x(t)$  has the same variance  $\sigma^2$  as the noise signal  $n(t)$.

'''(4)'''  No:
*The two conditions after subtasks  '''(2)'''  and  '''(3)'''  only refer to the magnitude of the digital filter.
*There is no constraint for the phase of the digital filter.
*This phase can be chosen arbitrarily.  Usually it is chosen in such a way that a minimum phase network results.
*In this case, the impulse response  $h_{\rm DF}(t)$  then has the lowest possible duration.

The graph shows the result of the approximation.  The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.  You can see:
[[File:EN_Mob_A_1_5d.png|right|frame|Approximation of the Jakes spectrum and the auto-correlation function]]

* The Jakes PDS (left graph) can only be reproduced very inaccurately due to the vertical drop at  $± f_{\rm D, \ max}$.
* For the time domain, this means that the ACF decreases much faster than the theory suggests.
*For small values of  $\Delta t$, however, the approximation is very good (right graph).
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^1.3 Rayleigh Fading with Memory^]]

Aufgaben:Exercise 2.7Z: Coherence Bandwidth of the LTI Two-Path Channel

2022-02-17T11:41:20Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/The_GWSSUS_Channel_Model}}

[[File:EN_Mob_A_2_7Z.png|right|frame|Two two-path channels]]
For the GWSSUS model, two parameters are given, which both statistically capture the resulting delay  $\tau$.  More information on the topic "Multipath Propagation" can be found in the section  [[Mobile_Communications/The_GWSSUS_Channel_Model#Parameters_of_the_GWSSUS_model| Parameters of the GWSSUS model]]  of the theory part.
* The  delay spread  $T_{\rm V}$  is by definition equal to the standard deviation of the random variable  $\tau$. This can be determined from the probability density function  $f_{\rm V}(\tau)$.  The PDF  $f_{\rm V}(\tau)$  has the same shape as the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau)$.
* The  coherence bandwidth  $B_{\rm K}$  describes the same situation in the frequency domain. It is defined as the value of $\Delta f$ at which the magnitude of the frequency correlation function  $\varphi_{\rm F}(\Delta f)$  first drops to half its maximum value:
:$$|\varphi_{\rm F}(\Delta f = B_{\rm K})| \stackrel {!}{=} {1}/{2} \cdot |\varphi_{\rm F}(\Delta f = 0)| \hspace{0.05cm}.$$

The relationship between  ${\it \Phi}_{\rm V}(\tau)$  and  $\varphi_{\rm F}(\Delta f)$  is given by the Fourier transform:
:$$\varphi_{\rm F}(\Delta f)
\hspace{0.2cm} {\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau)\hspace{0.05cm}.$$

*Both definitions are only partially suitable for a time-invariant channel.
*For a time-invariant two-path channel (i.e. one with constant path weights according to the above graph), the following approximation for the coherence bandwidth is often used:
:$$B_{\rm K}\hspace{0.01cm}' = \frac{1}{\tau_{\rm max} - \tau_{\rm min}} \hspace{0.05cm}.$$

In this task we want to clarify
* why there are different definitions for the coherence bandwidth in the literature,
* which connection exists between  $B_{\rm K}$  and  $B_{\rm K}\hspace{0.01cm}'$,  and
* which definitions make sense for which boundary conditions.

''Notes:''
*This exercise belongs to the chapter  [[Mobile_Communications/The_GWSSUS_Channel_Model|The GWSSUS Channel Model]].
*This task also refers to some theory pages in chapter  [[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications| Multi-Path Reception in Mobile Communications]].

===Questionnaire===
<quiz display=simple>
{What is the approximate coherence bandwidth  $B_{\rm K}\hspace{0.01cm}'$  for channels  $\rm A$  and  $\rm B$?
|type="{}"}
Channel  ${\rm A} \text{:} \hspace{0.4cm} B_{\rm K}\hspace{0.01cm}' \ = \ ${ 1000 3% } $\ \ \rm kHz$
Channel  ${\rm B} \text{:} \hspace{0.4cm} B_{\rm K}\hspace{0.01cm}' \ = \ ${ 1000 3% } $\ \ \rm kHz$

{Let  $G$  be the weight of the second path.  What is the PDF  $f_{\rm V}(\tau)$?
|type="()"}
- $f_{\rm V}(\tau) = \delta(\tau) + G \cdot \delta(\tau \, –\tau_0)$,
- $f_{\rm V}(\tau) = \delta(\tau) + G^2 \cdot \delta(\tau \, –\tau_0)$,
+ $f_{\rm V}(\tau) = 1/(1 + G^2) \cdot \delta(\tau) + G^2/(1 + G^2) \cdot \delta(\tau \, –\tau_0)$.

{Calculate the delay spread  $ T_{\rm V}$.
|type="{}"}
Channel  ${\rm A} \text{:} \hspace{0.4cm} T_{\rm V} \ = \ ${ 0.5 3% } $\ \rm µ s$
Channel  ${\rm B} \text{:} \hspace{0.4cm} T_{\rm V} \ = \ ${ 0.4 3% } $\ \rm µ s$

{What is the coherence bandwidth  $B_{\rm K}$  of channel  ${\rm A}$ ?
|type="()"}
+   $B_{\rm K} = 333 \ \rm kHz$.
-   $B_{\rm K} = 500 \ \rm kHz$.
-   $B_{\rm K} = 1 \ \rm MHz$.
- $B_{\rm K}$  cannot be calculated according to this definition.

{What is the coherence bandwidth  $B_{\rm K}$  of channel  ${\rm B}$ ?
|type="()"}
-   $B_{\rm K} = 333 \ \rm kHz$.
-   $B_{\rm K} = 500 \ \rm kHz$.
-   $B_{\rm K} = 1 \ \ \rm MHz$.
+ $B_{\rm K}$  cannot be calculated according to this definition.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  For both channels, the delay difference is  $\Delta \tau = \tau_{\rm max} \, - \tau_{\rm min} = 1 \ \ \rm µ s$.
* That's why both channels have the same value:
:$$B_{\rm K}\hspace{0.01cm}' \ \ \underline {= 1000 \ \rm kHz}.$$

'''(2)'''  The graphs refer to the impulse response  $h(\tau)$.
*To obtain the delay–power-spectral density, the weights must be squared:
:$${\it \Phi}_{\rm V}(\tau) = 1^2 \cdot \delta(\tau) + G^2 \cdot \delta(\tau - \tau_0) \hspace{0.05cm}.$$

*The integral of  ${\it \Phi}_{\rm V}(\tau)$  is therefore  $1 + G^2$.
*The probability density function  $\rm (PDF)$, however, must have "area $1$" $($i.e., the sum of the two Dirac weights must be $1)$.  From this follows:
:$$f_{\rm V}(\tau) = \frac{1}{1 + G^2} \cdot \delta(\tau) + \frac{G^2}{1 + G^2} \cdot \delta(\tau - \tau_0) \hspace{0.05cm}.$$

*Only solution 3 is correct.
*The first option does not describe the PDF  $f_{\rm V}(\tau)$, but the impulse response  $h(\tau)$.
*The second equation specifies the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau)$.

'''(3)'''  For channel  $\rm A$  the two impulse weights are equal.  This means that the mean value  $m_{\rm V}$  and the standard deviation  $\sigma_{\rm V} = T_{\rm V}$  can be computed simply:
:$$m_{\rm V} = \frac{\tau_0}{2} \hspace{0.15cm} {= 0.5\,{\rm µ s}}\hspace{0.05cm},
\hspace{0.2cm}T_{\rm V} = \sigma_{\rm V} =\frac{\tau_0}{2} \hspace{0.15cm}\underline {= 0.5\,{\rm µ s}}
\hspace{0.05cm}.$$

For channel  $\rm B$  the Dirac weights are  $1/(1+0.5^2) = 0.8$  $($for  $\tau = 0)$  and  $0.2$  $($for  $\tau = 1 \ \rm µ s)$.
* According to the  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments|basic laws]]  of statistics, the non-central first and second order moments are:
:$$m_{\rm 1} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 0.8 \cdot 0 + 0.2 \cdot 1\,{\rm µ s} = 0.2\,{\rm µ s} \hspace{0.05cm},\hspace{0.5cm}
m_{\rm 2} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 0.8 \cdot 0^2 + 0.2 \cdot (1\,{\rm µ s})^2 = 0.2\,({\rm µ s})^2 \hspace{0.05cm}.$$

*To get the result you are looking for you can use the  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Some_common_central_moments| Steiner's Theorem]]:
:$$\sigma_{\rm V}^2 = m_{\rm 2} - m_{\rm 1}^2 = 0.2\,({\rm µ s})^2 - (0.2\,{\rm µ s})^2 = 0.16\,({\rm µ s})^2
\hspace{0.3cm}\Rightarrow \hspace{0.3cm}T_{\rm V} = \sigma_{\rm V} \hspace{0.15cm}\underline {= 0.4\,{\rm µ s}}\hspace{0.05cm}.$$

'''(4)'''  The frequency correlation function is the Fourier transform of  ${\it \Phi}_{\rm V}(\tau) = \delta(\tau) + \delta(\tau \, – \tau_0)$:
:$$\varphi_{\rm F}(\Delta f) = 1 + {\rm exp}(-{\rm j} \cdot 2\pi \cdot \Delta f \cdot \tau_0) = 1 + {\rm cos}(2\pi \cdot \Delta f \cdot \tau_0) -{\rm j} \cdot {\rm sin}(2\pi \cdot \Delta f \cdot \tau_0) $$
[[File:P_ID2186__Mob_Z_2_7d.png|right|frame|Frequency correlation function and coherence bandwidth]]
:$$\Rightarrow \hspace{0.3cm} |\varphi_{\rm F}(\Delta f)| = \sqrt{2 + 2 \cdot {\rm cos}(2\pi \cdot \Delta f \cdot \tau_0) }\hspace{0.05cm}.$$

*The maximum at  $\Delta f = 0$  is equal to  $2$.
*Therefore the equation to determine $B_{\rm K}$ is
:$$|\varphi_{\rm F}(B_{\rm K})| = 1 \hspace{0.3cm}
\Rightarrow \hspace{0.3cm}|\varphi_{\rm F}(B_{\rm K})|^2 = 1 $$
:$$ \Rightarrow \hspace{0.3cm}2 + 2 \cdot {\rm cos}(2\pi \cdot B_{\rm K} \cdot \tau_0) = 1 \hspace{0.3cm}
\Rightarrow \hspace{0.3cm}{\rm cos}(2\pi \cdot B_{\rm K} \cdot \tau_0) = -0.5 \hspace{0.3cm} $$
:$$\Rightarrow \hspace{0.3cm}2\pi \cdot B_{\rm K} \cdot \tau_0 = \frac{2\pi}{3}\hspace{0.3cm} \Rightarrow \hspace{0.3cm}B_{\rm K} = \frac{1}{3\tau_0} = 333\,{\rm kHz}\hspace{0.05cm}.$$

Solution 1 is correct.  The graph (blue curve) illustrates the result.

'''(5)'''  For channel  ${\rm B}$  the corresponding equations are
:$${\it \Phi}_{\rm V}(\tau) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1^2 \cdot \delta(\tau) + (-0.5)^2 \cdot \delta(\tau - \tau_0) \hspace{0.05cm},\hspace{0.05cm}$$
:$$\varphi_{\rm F}(\Delta f) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1 + 0.25 \cdot {\rm cos}(2\pi \cdot \Delta f \cdot \tau_0) -{\rm j} \cdot 0.25 \cdot {\rm sin}(2\pi \cdot \Delta f \cdot \tau_0)\hspace{0.05cm},$$
:$$|\varphi_{\rm F}(\Delta f)| \hspace{-0.1cm} \ = \ \sqrt{\frac{17}{16} + \frac{1}{2} \cdot {\rm cos}(2\pi \cdot \Delta f \cdot \tau_0) }\hspace{0.3cm}$$
:$$\Rightarrow \hspace{0.3cm}{\rm Max}\hspace{0.1cm}|\varphi_{\rm F}(\Delta f)| = 1.25\hspace{0.05cm},\hspace{0.2cm}{\rm Min}\hspace{0.1cm}|\varphi_{\rm F}(\Delta f)| = 0.75\hspace{0.05cm}.$$

You can see from this result that the  $50\%$–coherence bandwidth cannot be specified here   ⇒   solution 4 is correct.

This result is the reason why there are different definitions for the coherence bandwidth in the literature, for example
* the  $90\%$–coherence bandwidth  $($in the example  $B_{\rm K, \hspace{0.03cm} 90\%} =184 \ \ \rm kHz$$)$,
* the very simple approximation  $B_{\rm K}\hspace{0.01cm}'$  given above  $($in the example  $B_{\rm K}\hspace{0.01cm}' =1 \ \ \rm MHz)$.

You can see from these numerical values that all the information on this topic is very vague and that the individual "coherence bandwidths" can be very different.

{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^2.3 The GWSSUS Channel Model^]]

Aufgaben:Exercise 2.9: Coherence Time

2022-02-17T11:41:20Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/The_GWSSUS_Channel_Model}}

[[File:P_ID2180__Mob_A_2_9.png|right|frame|Doppler power-spectral density and correlation function]]
In the frequency domain, the influence of Rayleigh fading is described by the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading|Jakes spectrum]].  If the Rayleigh parameter is  $\sigma = \sqrt{0.5}$  the Jakes spectrum is
:$${\it \Phi}_{\rm D}(f_{\rm D}) = \frac{1}{ \pi \cdot f_{\rm D,\hspace{0.05cm}max} \cdot \sqrt{1 - \left (\frac{f_{\rm D}}{f_{\rm D,\hspace{0.05cm}max}} \right )^2} } $$
in the Doppler frequency range  $(|f_{\rm D}| ≤ f_{\rm D, \ max})$,  and zero otherwise.  This function is sketched for  $f_{\rm D, \ max} = 50 \ \rm Hz$  (blue curve) and  $f_{\rm D, \ max} = 100 \ \rm Hz$  (red curve).

The correlation function  $\varphi_{\rm Z}(\Delta t)$  is the inverse Fourier transform of the Doppler power-spectral density  ${\it \Phi}_{\rm D}(f)$:
:$$\varphi_{\rm Z}(\Delta t ) = {\rm J}_0(2 \pi \cdot f_{\rm D,\hspace{0.05cm}max} \cdot \Delta t ) \hspace{0.05cm}.$$

${\rm J}_0$  denotes the zeroth-order Bessel function of the first kind.  The correlation function  $\varphi_{\rm Z}(\Delta t)$  which is also symmetrical, is drawn below, but for space reasons only the right half.

A characteristic value can be derived from each of these two description functions:
* The  Doppler spread  $B_{\rm D}$  refers to the Doppler PDS  ${\it \Phi}_{\rm D}(f_{\rm D})$  and is equal to the standard deviation  $\sigma_{\rm D}$  of the Doppler frequency  $f_{\rm D}$. 
::Note that the Jakes spectrum is zero-mean, so that the variance  $\sigma_{\rm D}^2$  according to Steiner's theorem is equal to the second moment  ${\rm E}\big[f_{\rm D}^2\big]$.  The calculation is analogous to the determination of the delay spread  $T_{\rm V}$  from the delay PDS  ${\it \Phi}_{\rm V}(\tau)$   ⇒   [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth| Exercise 2.7]].

* The  coherence time $T_{\rm D}$  refers to the time correlation function  $\varphi_{\rm Z}(\Delta t)$ .
:: $T_{\rm D}$  is the value of  $\Delta t$  at which the magnitude  $|\varphi_{\rm Z}(\Delta t)|$  first drops to half of the maximum  $($at  $\Delta t = 0)$ . One recognizes the analogy with the determination of the coherence bandwidth  $B_{\rm K}$  from the frequency correlation function  $\varphi_{\rm F}(\Delta f)$   ⇒   [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth| Exercise 2.7]].

''Notes:''
*This exercise belongs to the chapter [[Mobile_Communications/The_GWSSUS_Channel_Model|The GWSSUS Channel Model]].
* Reference is also made to chapter  [[Mobile_Communications/General_Description_of_Time_Variant_Systems| General Description of Time–Variant Systems]].
* The following indefinite integral is given:
:$$\int \frac{u^2}{\sqrt{1-u^2}} \hspace{0.15cm}{\rm d} u = -\frac{u}{2} \cdot \sqrt{1-u^2} + \frac{1}{2} \cdot {\rm arcsin}\,(u)
\hspace{0.05cm}.$$
* We also provide the following few values of the zeroth-order Bessel function of the first kind $({\rm J}_0)$:
:$${\rm J}_0(\pi/2) = 0.472\hspace{0.05cm},\hspace{0.4cm}{\rm J}_0(1.52) = 0.500\hspace{0.05cm},\hspace{0.4cm}{\rm J}_0(\pi) = -0.305\hspace{0.05cm},\hspace{0.4cm} {\rm J}_0(2\pi) = 0.221
\hspace{0.05cm}.$$

===Questionnaire===
<quiz display=simple>
{Which statements apply to the probability density function (PDF) of the Doppler frequency in this example?
|type="[]"}
+ The Doppler PDF is always identical in shape to the Doppler PDS.
+ The Doppler PDF is identical with the Doppler PDS in this example.
- Doppler PDF and Doppler PDS differ fundamentally.

{Determine the Doppler spread $B_{\rm D}$.
|type="{}"}
$f_{\rm D, \ max} = 50 \ {\rm Hz} \text{:} \ \hspace{0.6cm} B_{\rm D} \ = \ ${ 35.35 3% } $\ \ \rm Hz$
$f_{\rm D, \ max} = 100 \ {\rm Hz} \text{:} \ \hspace{0.4cm} B_{\rm D} \ = \ ${ 70.7 3% } $\ \ \rm Hz$

{What is the time correlation value for  $\Delta t = 5 \ \ \rm ms$?
|type="{}"}
$f_{\rm D, \ max} = 50 \ {\rm Hz} \text{:} \ \hspace{0.6cm} \varphi_{\rm Z}(\Delta t = 5 \ \rm ms) \ = \ ${ 0.472 3% }
$f_{\rm D, \ max} = 100 \ {\rm Hz} \text{:} \ \hspace{0.4cm} \varphi_{\rm Z}(\Delta t = 5 \ \rm ms) \ = \ ${ -0.31415--0.29585 }

{What are the coherence times  $T_{\rm D}$  for both parameter sets?
|type="{}"}
$f_{\rm D, \ max} = 50 \ {\rm Hz} \text{:} \ \hspace{0.6cm} T_{\rm D} \ = \ ${ 4.84 3% } $\ \ \rm ms$
$f_{\rm D, \ max} = 100 \ {\rm Hz} \text{:} \ \hspace{0.4cm} T_{\rm D} \ = \ ${ 2.42 3% } $\ \ \rm ms$

{What is the relationship between the Doppler spread  $B_{\rm D}$  and the coherence time  $T_{\rm D}$, when the Doppler PDS is the Jakes spectrum?
|type="()"}
- $B_{\rm D} \cdot T_{\rm D} \approx $1,
- $B_{\rm D} \cdot T_{\rm D} \approx 0.5$,
+ $B_{\rm D} \cdot T_{\rm D} \approx $0.171.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  Solutions 1 and 2 are correct:
*Doppler PDF and Doppler PDS are generally only identical in shape.
*But since in the example considered the integral over  ${\it \Phi}_{\rm D}(f_{\rm D})$  is equal to  $1$   ⇒  this can be easily seen from the correlation value  $\varphi_{\rm Z}(\Delta t = 0) = 1$, the Doppler PDF and the Doppler PDS are identical in this example.
*If the Rayleigh parameter  $\sigma$  had been chosen differently, this would not apply.

'''(2)'''  From the axial symmetry of  ${\it \Phi}_{\rm D}(f_{\rm D})$  you can see that the mean Doppler shift is  $m_{\rm D} = {\rm E}\big [f_{\rm D}\big] = 0$.
*The variance of the random variable  $f_{\rm D}$  can thus be calculated directly as a mean square value:
:$$\sigma_{\rm D}^2 = \int_{-\infty}^{+\infty} f_{\rm D}^2 \cdot {\it \Phi}_{\rm D}(f_{\rm D}) \hspace{0.15cm}{\rm d} f_{\rm D} = \int_{-f_{\rm D,\hspace{0.05cm}max}}^{+f_{\rm D,\hspace{0.05cm}max}} \frac{f_{\rm D}^2}{ \pi \cdot f_{\rm D,\hspace{0.05cm}max} \cdot \sqrt{1 - \left ({f_{\rm D}}/{f_{\rm D,\hspace{0.05cm}max}} \right )^2} } \hspace{0.15cm}{\rm d} f_{\rm D}
\hspace{0.05cm}.$$

*Using symmetry and the substitution  $u = f_{\rm D}/f_{\rm D, \ max}$, we obtain
:$$\sigma_{\rm D}^2 = \frac{2}{\pi} \cdot f_{\rm D,\hspace{0.05cm}max}^2 \cdot \int_{0}^{1} \frac{u^2}{\sqrt{1-u^2}} \hspace{0.15cm}{\rm d} u
\hspace{0.05cm}. $$

*With the integral provided in the task description, you get further:
:$$\sigma_{\rm D}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{2}{\pi} \cdot f_{\rm D,\hspace{0.05cm}max}^2 \cdot \left [ -\frac{u}{2} \cdot \sqrt{1-u^2} + \frac{1}{2} \cdot {\rm arcsin}\,(u) \right ]_0^1 = \frac{2}{\pi} \cdot f_{\rm D,\hspace{0.05cm}max}^2 \cdot \frac{2}{2}\cdot \frac{\pi}{2} = \frac{f_{\rm D,\hspace{0.05cm}max}^2}{2}
\hspace{0.05cm}.$$

*The Doppler spread is equal to the standard deviation of  $f_{\rm D}$, i.e., the square root of its variance:
:$$B_{\rm D} = \sigma_{\rm D} = \frac{f_{\rm D,\hspace{0.05cm}max}}{\sqrt{2}}= \left\{ \begin{array}{c} \underline{35.35\,{\rm Hz}}\\
\underline{70.7\,{\rm Hz}} \end{array} \right.\quad
\begin{array}{*{1}c} {\rm f\ddot{u}r} \hspace{0.15cm}f_{\rm D,\hspace{0.05cm}max} = 50\,{\rm Hz}
\\ {\rm f\ddot{u}r} \hspace{0.15cm}f_{\rm D,\hspace{0.05cm}max} = 100\,{\rm Hz} \\ \end{array}
\hspace{0.05cm}. $$

'''(3)'''  With the Bessel values given, one obtains
* for  $f_{\rm D, \ max} = 50 \ \rm Hz$:
:$$\varphi_{\rm Z}(\Delta t = 5\,{\rm ms}) = {\rm J}_0(2 \pi \cdot 50\,{\rm Hz} \cdot 5\,{\rm ms} ) = {\rm J}_0(\pi/2) \hspace{0.1cm} \underline {= 0.472} \hspace{0.05cm},$$
* for  $f_{\rm D, \ max} = 100 \ \rm Hz$:
:$$\varphi_{\rm Z}(\Delta t = 5\,{\rm ms}) = {\rm J}_0(\pi) \hspace{0.1cm} \underline {= -0.305} \hspace{0.05cm}.$$

'''(4)'''  The coherence time  $T_{\rm D}$  is derived from the correlation function  $\varphi_{\rm Z}(\Delta t)$.  $T_{\rm D}$  is the specific value  of $\Delta t$  where  $|\varphi_{\rm Z}(\Delta t)|$  has decayed to half of its maximum value.  It must hold:
:$$\varphi_{\rm Z}(\Delta t = T_{\rm D}) = {\rm J}_0(2 \pi \cdot f_{\rm D,\hspace{0.05cm}max} \cdot T_{\rm D}) \stackrel {!}{=} 0.5 \hspace{0.3cm}
\Rightarrow \hspace{0.3cm} 2 \pi \cdot f_{\rm D,\hspace{0.05cm}max} \cdot T_{\rm D} = 1.52
\hspace{0.3cm} \Rightarrow \hspace{0.3cm} T_{\rm D} = \frac{1.52}{2 \pi f_{\rm D,\hspace{0.05cm}max}} = \frac{0.242}{ f_{\rm D,\hspace{0.05cm}max}}$$
:$$\Rightarrow \hspace{0.3cm} f_{\rm D,\hspace{0.05cm}max} = 50\,{\rm Hz}\text{:} \ \hspace{-0.1cm}\hspace{0.2cm} T_{\rm D} \hspace{0.1cm} \underline {\approx 4.84\,{\rm ms}} \hspace{0.05cm},\hspace{0.8cm} f_{\rm D,\hspace{0.05cm}max} = 100\,{\rm Hz}\text{:} \ \hspace{-0.1cm}\hspace{0.2cm} T_{\rm D} \hspace{0.1cm} \underline {\approx 2.42\,{\rm ms}} \hspace{0.05cm}. $$

'''(5)'''  In the subtasks  '''(2)'''  and  '''(4)'''  we obtained:
:$$B_{\rm D} = \frac{ f_{\rm D,\hspace{0.05cm}max}}{\sqrt{2}}\hspace{0.05cm}, \hspace{0.2cm} T_{\rm D} = \frac{1.52}{2 \pi f_{\rm D,\hspace{0.05cm}max}}\hspace{0.3cm}
\Rightarrow \hspace{0.3cm} B_{\rm D} \cdot T_{\rm D} = \frac{1.52}{\sqrt{2} \cdot 2 \pi } \hspace{0.1cm}\underline {\approx 0.171}\hspace{0.05cm}.$$

Therefore, the last option is the correct one.
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^2.3 The GWSSUS Channel Model^]]

Aufgaben:Exercise 2.6Z: Signal-to-Noise Ratio

2022-02-17T11:41:20Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Modulationsverfahren/Synchrondemodulation
}}

[[File:P_ID1017__Mod_Z_2_6.png|right|frame|Spectra and power density spectra]]
In the following exercise,  we assume:
*a cosine source signal:
:$$ q(t) = 4 \,{\rm V} \cdot \cos(2 \pi \cdot 5\,{\rm kHz} \cdot t )\hspace{0.05cm},$$
* DSB–AM by multiplication with
:$$z(t) = 1 \cdot \cos(2 \pi \cdot 20\,{\rm kHz} \cdot t )\hspace{0.05cm},$$
* a frequency-independent attenuation on the channel corresponding to  $α_{\rm K} = 10^{–4}$,
* additive white input noise with power density  $N_0 = 4 · 10^{–19} \ \rm W/Hz$,
* phase-synchronous and frequency-synchronous demodulation by multiplication the same  $z(t)$  as at the transmitter,
* a rectangular low-pass at the synchronous demodulator with cutoff frequency  $f_{\rm E} = 5 \ \rm kHz$.

In the graph,  these specifications are shown in the spectral domain.  It should be explicitly mentioned that the power-spectral density  ${\it Φ}_z(f)$  of the cosine oscillation  $z(t)$  is composed of two Dirac lines at  $±f_{\rm T}$,  as in the amplitude spectrum   $Z(f)$,  but with weight  $A^2/4$  instead of  $A/2$.  The amplitude should always be set to  $A=1$  in this exercise.

The sink signal  $v(t)$  is composed of the useful component  $α · q(t)$  and the noise component  $ε(t)$ .  Thus, the general rule for the signal-to-noise power ratio to be determined is:
:$$ \rho_{v } = \frac{\alpha^2 \cdot P_q}{P_\varepsilon}\hspace{0.05cm}.$$
This important quality criterion is often abbreviated to  $\rm SNR$  ("signal–to–noise power ratio").

Hints:
*This exercise belongs to the chapter  [[Modulation_Methods/Synchronous_Demodulation|Synchronous Demodulation]].
*Particular reference is made to the pages  [[Modulation_Methods/Synchronous_Demodulation#Calculating_noise_power|Calculating noise power]]  and  [[Modulation_Methods/Synchronous_Demodulation#Relationship_between_the_powers_from_source_signal_and_transmitted signal|Relationship between the powers from source and_transmitted signal]].
*Please note that the variables  $α$  and  $α_{\rm K}$  need not be the same.
*All powers refer to a resistance of  $R = 50 \ \rm Ω$  with the exception of subtask   '''(1)'''.
*For DSB-AM without carrier,  $P_q$  also represents the transmit power  $P_{\rm S}$.

===Questions===

<quiz display=simple>

{Calculate the transmit power with respect to the unit resistance  $R = 1 \ \rm Ω$.
|type="{}"}
$P_q \ = \ $ { 8 3% } $\ \rm V^2$

{What is the power  $P_q$  in watts for the resistor  $R = 50 \ \rm Ω$?
|type="{}"}
$P_q \ = \ $ { 0.16 3% } $\ \rm W$

{Which attenuation factor  $α$  results for the whole system?
|type="{}"}
$α \ = \ $ { 0.5 3% } $\ \cdot 10^{-4}$

{Calculate the power density of the noise component  $ε(t)$  at the output.  What is the value when  $f = 0$?  Let  $H_{\rm E}(f = 0) = 1$.
|type="{}"}
${\it Φ}_ε(f = 0) \ = \ $ { 4 3% } $\ \cdot 10^{-19} \ \rm W/Hz$

{What is the noise power of the sink signal?
|type="{}"}
$P_ε \ = \ $ { 4 3% } $\ \cdot 10^{-15} \ \rm W$

{What is the signal-to-noise power ratio (SNR) at the sink? What is the resulting dB value?
|type="{}"}
$ρ_v \ = \ $ { 100000 3% }
$10 · \lg ρ_v \ = \ $ { 50 3% } $\ \rm dB$
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  The power-spectral density of a cosine signal with amplitude   $A$  consists of two Dirac lines,  each with weight   $A^2/4$.
*The power is obtained from the integral over the PDS and is thus equal to the sum of the two Dirac weights. 
*Thus,  when   $A = 4 \ \rm V$,  we obtain the power of the source signal:
:$$ P_q = \frac{A^2}{2} \hspace{0.15cm}\underline {= 8\,{\rm V^2}} \hspace{0.05cm}.$$
*For the modulation method  "DSB-AM without a carrier",  this is also the transmit power $P_{\rm S}$ in reference to the unit resistance  $1\ \rm Ω$.

'''(2)'''  According to the elementary laws of Electrical Engineering:
:$$P_q = \frac{8\,{\rm V^2}}{50\,{\Omega}} \hspace{0.15cm}\underline {= 0.16\,{\rm W}} \hspace{0.05cm}.$$

'''(3)'''  In the theory section,  it is shown that  $v(t) = q(t)$  holds under ideal conditions.  However,  the following should be taken into account:
*From the graph,  it can be seen that  $Z_{\rm E}(f) = Z(f)$  holds.  Thus,  the receiver-side carrier signal   $z_{\rm E}(t)$  has like   $z(t)$  the amplitude   $1$.
*Ideally,  however,  the receiver-side carrier signal   $z_{\rm E}(t)$  should have amplitude   $2$.
*Therefore,  $υ(t) = q(t)/2$  applies here.
*If we further consider the channel attenuation   $α_{\rm K} = 10^{–4}$,  we obtain the final result:  $α\hspace{0.15cm}\underline { = 0.5 · 10^{–4}}.$

'''(4)'''  The power-spectral density of the product   $n(t) · z(t)$  is obtained by convolving the two power density spectra of   $n(t)$  and  $z(t)$:
:$$ {\it \Phi}_\varepsilon \hspace{0.01cm} '(f) = {\it \Phi}_n (f) \star {\it \Phi}_{z }(f)= \frac{N_0}{2} \star \left[\delta(f - f_{\rm T}) + \delta(f + f_{\rm T}) \right]= N_0 \hspace{0.05cm}.$$
*For the power-spectral density of the signal   $ε(t)$  after the low-pass filter,  we obtain a rectangular shape with the same value at   $f = 0$:
:$${\it \Phi}_\varepsilon (f) = {\it \Phi}_\varepsilon \hspace{0.01cm} '(f) \cdot |H_{\rm E}(f)|^2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} {\it \Phi}_\varepsilon (f=0)= N_0\hspace{0.15cm}\underline {= 4 \cdot 10^{-19}\,{\rm W/Hz}} \hspace{0.05cm}.$$

'''(5)'''  The noise power is the integral over the noise power density:
:$$ P_{\varepsilon} = \int_{-f_{\rm E}}^{ + f_{\rm E}} {{\it \Phi}_\varepsilon (f)}\hspace{0.1cm}{\rm d}f = N_0 \cdot 2 f_{\rm E} = 4 \cdot 10^{-19}\,\frac{ \rm W}{\rm Hz} \cdot 10^{4}\,{\rm Hz} \hspace{0.15cm}\underline {= 4 \cdot 10^{-15}\,{\rm W}}\hspace{0.05cm}.$$

'''(6)'''  From the results of subtasks  '''(2)''',  '''(3)'''  and  '''(5)'''  it follows that:
:$$\rho_{v } = \frac{\alpha^2 \cdot P_q}{P_\varepsilon} = \frac{(0.5 \cdot 10^{-4})^2 \cdot 0.16\,{\rm W}}{4 \cdot 10^{-15}\,{\rm W}} \hspace{0.15cm}\underline {= 100000} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}10 \cdot {\rm lg }\hspace{0.1cm}\rho_{v } \hspace{0.15cm}\underline {= 50\,{\rm dB}}\hspace{0.05cm}.$$

{{ML-Fuß}}

[[Category:Modulation Methods: Exercises|^2.2 Synchronous Demodulation^]]

Aufgaben:Exercise 2.6: Dimensions in GWSSUS

2022-02-17T11:41:20Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/The_GWSSUS_Channel_Model}}

[[File:P_ID2167__Mob_A_2_6.png|right|frame|Overview of the GWSSUS functions]]
The mobile radio channel can be described in very general terms by four system functions, whereby the relationship between each pair of functions is described by
* the Fourier transform or
* the inverse Fourier transform.

We denote all of the functions with  $\eta_{i_1i_2}$. The indices $i_1$ and $i_2$ are defined as follows:
* $\boldsymbol{\rm V}$  $($because of German  $\rm V\hspace{-0.05cm}$erzögerung$)$  stands for delay time  $\tau$  $($index  $i_1)$,
* $\boldsymbol{\rm F}$  stands for frequency  $f$  $($index  $i_1)$,
* $\boldsymbol{\rm Z}$  $($because of German  $\rm Z\hspace{-0.05cm}$eit$)$  stands for the time  $t$  $($index  $i_2)$,
* $\boldsymbol{\rm D}$  stands for the Doppler frequency $f_{\rm D}$  $($index  $i_2)$.

The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:
* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral|Fourier transform]].
* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]  (opposite direction).

For example:
:$$\eta_{\rm VZ}(\tau, t)
\hspace{0.2cm} \stackrel{\tau, \hspace{0.02cm}f}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} \eta_{\rm FZ}(f,t)\hspace{0.05cm},
\hspace{0.4cm}\eta_{\rm FZ}(f,t)
\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$

*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power-spectral density  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra are labeled in blue. The GWSSUS model is always assumed.

Let us consider here the system function  $\eta_{\rm VZ}(\tau, t)$, i.e. the time–variant impulse response  $h(\tau, t)$.  We define:
:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm},$$
:$$\Delta \tau = \tau_2 - \tau_1 \hspace{0.05cm}, \hspace{0.2cm} \Delta t = t_2 - t_1
\hspace{0.3cm} \Rightarrow \hspace{0.3cm}
\varphi_{\rm VZ}(\Delta \tau, \Delta t) \hspace{0.05cm}, $$
:$$\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.$$
:$${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)\hspace{0.05cm}. $$

''Note:''   This exercise belongs to the chapter  [[Mobile_Communications/The_GWSSUS_Channel_Model| The GWSSUS Channel Model]].

===Questionnaire===
<quiz display=simple>
{Which of the following specified dimensions of the system functions are correct?
|type="[]"}
+ $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$.
+ $\eta_{\rm FZ}(f, t)$  is without unit.
+ $\eta_{\rm VD}(\tau, f_{\rm D})$  is without unit.
+ $\eta_{\rm FD}(f, f_{\rm D})$  has the unit  $[1/\rm Hz]$.

{Which of the following statements are correct?
|type="[]"}
- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s]$.
+ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$  has the unit  $[1/\rm s]$.
+ ${\it \phi}_{\rm V}(\tau)$  has the unit  $[1/\rm s]$.

{Which of the following statements are correct?
|type="[]"}
+ $\varphi_{\rm FZ}(\Delta f, \Delta t),  \varphi_{\rm F}(\Delta f)$  and  $\varphi_{\rm Z}(\Delta t)$  have no unit.
- ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  has the unit  $[1/\rm s]$.
+ ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  and  ${\it \Phi}_{\rm D}(f_{\rm D})$  have the unit  $[1/\rm Hz]$ each.
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  All statements are correct:
*$\eta_{\rm VZ}(\tau, t)$  is the time-variant impulse response, for which the term  $h(\tau, t)$  is also common.  Like every impulse response,  $h(\tau, t)$  has the unit  $[1/\rm s]$.
*By Fourier transform of the function  $\eta_{\rm VZ}(\tau, t)$  with respect to the delay  $\tau$  one obtains
:$$\eta_{\rm FZ}(f, t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau
\hspace{0.05cm}. $$

*Due to the integration over  $\tau$  $($unit:  $\rm s)$, the time-variant transfer function  $\eta_{\rm FZ}(f, t)$  is dimensionless.  In some literature,  $H(f, t)$  is also used instead of  $\eta_{\rm FZ}(f, t)$.

*The delay–Doppler representation  $\eta_{\rm VD}(\tau, f_{\rm D})$  is dimensionless, too.  This function results from the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  by Fourier transform with respect to  $t$:
:$$\eta_{\rm VD}(\tau, f_{\rm D}) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} t}\hspace{0.15cm}{\rm d}t
\hspace{0.05cm}.$$

*The function  $\eta_{\rm FD}(t, f_{\rm D})$  is obtained from the dimensionless functions  $\eta_{\rm VD}(\tau, f_{\rm D})$  and  $\eta_{\rm FZ}(f, t)$  respectively by a Fourier transform, which results in the unit  $[\rm s] = [1/\rm Hz]$.

'''(2)'''  Solutions 2 and 3 are correct:
*The auto-correlation  $\rm (ACF)$  function is by definition the following expected value:
:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$

*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.

*The Dirac function  ${\rm \delta}(\Delta \tau)$  has the unit  $[1/\rm s]$, since the integral over all  $\tau$  $($with unit  $[\rm s])$ must be  $1$.  Therefore, both the delay–time cross power-spectral density  ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$  and the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$  have unit  $[1/\rm s]$.

'''(3)'''  Statements 1 and 3 are correct:
*The function  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  has unit  $[1/\rm s]$.  Its Fourier transform with respect to  $\tau$  is  $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to  $t$  is  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.  Both  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  are therefore without unit.

*The frequency–Doppler cross power-spectral density has the unit  $[\rm s] = [1/\rm Hz]$, because
:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^2.3 The GWSSUS Channel Model^]]

Aufgaben:Exercise 2.8: COST Delay Models

2022-02-17T11:41:20Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Mobile_Communications/The_GWSSUS_Channel_Model}}

[[File:Mob_A_2_8_version2.png|right|frame|COST delay models]]
On the right, four delay power density spectra are plotted logarithmically as a function of the delay time  $\tau$ 
:$$10 \cdot {\rm lg}\hspace{0.15cm} ({{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0}) \hspace{0.05cm}.$$

Here the abbreviation  $\phi_0 = \phi_{\rm V}(\tau = 0)$  is used.  These are the so-called COST delay models.

The upper sketch contains the two profiles  ${\rm RA}$  ("Rural Area") and  ${\rm TU}$  ("Typical Urban").  Both of these are exponential:
:$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$

The value of the parameter  $\tau_0$  (time constant of the auto-correlation function) should be determined from the graphic in subtask  '''(1)'''.  Note the specified values of   $\tau_{-30}$ for  ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:
:$${\rm RA}\text{:}\hspace{0.15cm}\tau_{-30} = 0.75\,{\rm µ s} \hspace{0.05cm},\hspace{0.2cm}
{\rm TU}\text{:}\hspace{0.15cm}\tau_{-30} = 6.9\,{\rm µ s} \hspace{0.05cm}. $$

The lower graph applies to less favourable conditions in
* urban areas ("Bad Urban",  ${\rm BU}$):
:$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}
= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\
0.5 \cdot {\rm e}^{ (5\,{\rm \mu s}-\tau) / \tau_0} \end{array} \right.\quad
\begin{array}{*{1}c} \hspace{-0.55cm} {\rm if}\hspace{0.15cm}0 < \tau < 5\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm},
\\ \hspace{-0.15cm} {\,\, \,\, \rm if}\hspace{0.15cm}5\,{\rm µ s} < \tau < 10\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}, \end{array}$$

* in rural areas ("Hilly Terrain",  ${\rm HT}$):
:$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}
= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\
{0.04 \cdot \rm e}^{ (15\,{\rm \mu s}-\tau) / \tau_0} \end{array} \right.\quad
\begin{array}{*{1}c} \hspace{-0.55cm} {\rm if}\hspace{0.15cm}0 < \tau < 2\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 0.286\,{\rm µ s} \hspace{0.05cm},
\\ \hspace{-0.35cm} {\rm if}\hspace{0.15cm}15\,{\rm µ s} < \tau < 20\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}. \end{array}$$

For the models  ${\rm RA}$,  ${\rm TU}$  and  ${\rm BU}$  the following parameters are to be determined:
* The  delay spread  $T_{\rm V}$  is the standard deviation of the delay  $\tau$. If the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau)$  has an exponential course as with the profiles  ${\rm RA}$  and  ${\rm TU}$, then  $T_{\rm V} = \tau_0$, see  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].

* The coherence bandwidth  $B_{\rm K}$  is the value of  $\Delta f$ at which the magnitude of the frequency correlation function  $\varphi_{\rm F}(\Delta f)$  has dropped to half its value for the first time. With exponential  ${\it \Phi}_{\rm V}(\tau)$  as with  ${\rm RA}$  and  ${\rm TU}$  the product is  $T_{\rm V} \cdot B_{\rm K} \approx 0.276$, see  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].

''Notes:''
*This exercise belongs to the chapter [[Mobile_Communications/The_GWSSUS_Channel_Model|The GWSSUS Channel Model]].
* The following integrals are given:
:$$\frac{1}{\tau_0} \cdot \int_{0}^{\infty}\hspace{-0.15cm} {\rm e}^{ -\tau / \tau_0} \hspace{0.15cm}{\rm d} \tau = 1
\hspace{0.05cm},\hspace{0.6cm}
\frac{1}{\tau_0} \cdot \int_{0}^{\infty}\hspace{-0.15cm} {\tau} \cdot{\rm e}^{ -\tau / \tau_0}\hspace{0.15cm}{\rm d} \tau = \tau_0
\hspace{0.05cm},\hspace{0.6cm}
\frac{1}{\tau_0} \cdot \int_{0}^{\infty} \hspace{-0.15cm}{\tau^2} \cdot{\rm e}^{ -\tau / \tau_0}\hspace{0.15cm}{\rm d} \tau = 2\tau_0^2\hspace{0.05cm}.$$

===Questionnaire===
<quiz display=simple>
{Specify the parameter  $\tau_0$  of the delay power-spectral density for the profiles  ${\rm RA}$  and  ${\rm TU}$ .
|type="{}"}
${\rm RA} \text{:} \ \hspace{0.4cm} \tau_0 \ = \ ${ 0.109 3% } $\ \rm µ s$
${\rm TU} \text{:} \ \hspace{0.4cm} \tau_0 \ = \ ${ 1 3% } $\ \rm µ s$

{How large is the delay spread  $T_{\rm V}$  of these channels?
|type="{}"}
${\rm RA} \text{:} \ \hspace{0.4cm} T_{\rm V} \ = \ ${ 0.109 3% } $\ \rm µ s$
${\rm TU} \text{:} \ \hspace{0.4cm} T_{\rm V} \ = \ ${ 1 3% } $\ \rm µ s$

{What is the coherence bandwidth  $B_{\rm K}$  of these channels?
|type="{}"}
${\rm RA} \text{:} \ \hspace{0.4cm} B_{\rm K} \ = \ ${ 2500 3% } $\ \ \rm kHz$
${\rm TU} \text{:} \ \hspace{0.4cm} B_{\rm K} \ = \ ${ 276 3% } $\ \ \rm kHz$

{For which channel does frequency selectivity play a greater role?
|type="()"}
- Rural Area  $({\rm RA})$.
+ Typical urban  $({\rm TU})$.

{How large is the (normalized) power density for  "Bad Urban"  $({\rm BU})$   with   $\tau = 5.001 \ \rm µ s$  and with   $\tau = 4.999 \ \rm µ s$?
|type="{}"}
${\it \Phi}_{\rm V}(\tau = 5.001 \ \rm µ s) \ = \ ${ 0.5 3% } $\ \cdot {\it \Phi}_0$
${\it \Phi}_{\rm V}(\tau = 4.999 \ \rm µ s) \ = \ ${ 0.00674 3% } $\ \cdot {\it \Phi}_0$

{We consider  ${\rm BU}$ again. Let $P_1$ be the power of the signal between $0$  and  $5 \ \rm µ s$, and let $P_2$ be the remaining signal power. What percentage of the total signal power comes from the interval
$0< t < 5 \ \rm µ s$?
|type="{}"}
$P_1/(P_1 + P_2) \ = \ ${ 66.7 3% } $\ \rm \%$

{Calculate the delay spread  $T_{\rm V}$  of the profile  ${\rm BU}$. ''Note'':  The average delay is  $m_{\rm V} = E[\tau] = 2.667 \ \rm µ s$.
|type="{}"}
$T_{\rm V} \ = \ ${ 2.56 3% } $\ \rm µ s$
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  The following property can be seen from the graph:
:$$10 \cdot {\rm lg}\hspace{0.1cm} (\frac{{\it \Phi}_{\rm V}(\tau_{\rm -30})}{{\it \Phi}_0}) =
10 \cdot {\rm lg}\hspace{0.1cm}\left [{\rm exp}[ -\frac{\tau_{\rm -30}}{ \tau_{\rm 0}}]\right ] \stackrel {!}{=} -30\,{\rm dB}$$
:$$\Rightarrow \hspace{0.3cm} {\rm lg}\hspace{0.1cm}\left [{\rm exp}[ -\frac{\tau_{\rm -30}}{ \tau_{\rm 0}}]\right ] = -3
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} {\rm ln}\hspace{0.1cm}\left [{\rm exp}[ -\frac{\tau_{\rm -30}}{ \tau_{\rm 0}}]\right ] = -3 \cdot
{\rm ln}\hspace{0.1cm}(10)\hspace{0.3cm}
\Rightarrow \hspace{0.3cm} \tau_{\rm 0} = \frac{\tau_{\rm -30}}{ 3 \cdot {\rm ln}\hspace{0.1cm}(10)}\approx \frac{\tau_{\rm -30}}{ 6.9}
\hspace{0.05cm}.$$

Here  $\tau_{-30}$  denotes the delay that leads to the logarithmic ordinate value  $-30 \ \rm dB$.  Thus one obtains
* for "Rural Area"  $\rm (RA)$  with  $\tau_{–30} = 0.75 \ \rm µ s$:
:$$\tau_{\rm 0} = \frac{0.75\,{\rm µ s}}{ 6.9} \hspace{0.1cm}\underline {\approx 0.109\,{\rm µ s}}
\hspace{0.05cm},$$
* for urban and suburban areas  ⇒   "Typical Urban" $\rm (TU)$  with  $\tau_{–30} = 6.9 \ \rm µ s$:
:$$\tau_{\rm 0} = \frac{6.9\,{\rm µ s}}{ 6.9} \hspace{0.1cm}\underline {\approx 1\,{\rm µ s}}
\hspace{0.05cm},$$

'''(2)'''  In  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is  $T_{\rm V} =\tau_0$  when the delay power-spectral density decreases exponentially according to  ${\rm e}^{-\tau/\tau_0}$.  Thus the following applies:
* for "Rural Area":  $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm µ s}$,
* for "Typical Urban":  $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm µ s}$.

'''(3)'''  In Exercise 2.7 it was also shown that for the coherence bandwidth  $B_{\rm K} \approx 0.276/\tau_0$  applies.  It follows:
*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ ("Rural Area"),
*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ ("Typical Urban")

'''(4)''' The second solution is correct:
*Frequency selectivity of the mobile radio channel is present if the signal bandwidth  $B_{\rm S}$  is larger than the coherence bandwidth  $B_{\rm K}$  (or at least of the same order of magnitude).
*The smaller  $B_{\rm K}$  is, the more often this happens.

'''(5)'''  According to the given equation, we have  ${\it \Phi}_{\rm V}(\tau = 5.001 \ \rm µ s)/{\it \Phi}_0 \hspace{0.15cm}\underline{\approx0.5}$.
*On the other hand, for slightly smaller  $\tau$  $($for example $\tau = 4.999 \ \rm µ s)$  we have approximately
:$${{\it \Phi}_{\rm V}(\tau = 4.999\,{\rm \mu s})}/{{\it \Phi}_{\rm 0}} = {\rm e}^{ -{4.999\,{\rm µ s}}/{ 1\,{\rm \mu s}}}
\approx {\rm e}^{-5} \hspace{0.1cm}\underline {= 0.00674 }\hspace{0.05cm}.$$

'''(6)'''  The power  $P_1$  of all signal components with delays between  $0$  and  $5 \ µ\rm   s$  is:
:$$P_1 = {\it \Phi}_{\rm 0} \cdot \int_{0}^{5\,{\rm \mu s}} {\rm exp}[ -{\tau}/{ \tau_0}] \hspace{0.15cm}{\rm d} \tau \hspace{0.15cm} \approx \hspace{0.15cm}
{\it \Phi}_{\rm 0} \cdot \int_{0}^{\infty} {\rm e}^{ -{\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau
= {\it \Phi}_{\rm 0} \cdot \tau_0 \hspace{0.05cm}.$$

*The power outside  $[0\;µ \mathrm{s}, 5\;µ \mathrm{s}]$  is
:$$P_2 = \frac{{\it \Phi}_{\rm 0}}{2} \cdot \int_{5\,{\rm µ s}}^{\infty} {\rm exp}[ \frac{5\,{\rm µ s} -\tau}{ \tau_0}] \hspace{0.15cm}{\rm d} \tau \hspace{0.15cm} \approx \hspace{0.15cm}
\frac{{\it \Phi}_{\rm 0}}{2} \cdot \int_{0}^{\infty} {\rm exp}[ -{\tau}/{ \tau_0}] \hspace{0.15cm}{\rm d} \tau
= \frac{{\it \Phi}_{\rm 0} \cdot \tau_0}{2} \hspace{0.05cm}. $$

[[File:P_ID2184__Mob_A_2_8f.png|right|frame|Delay power density of the COST profiles  ${\rm BU}$  and  ${\rm HT}$]]

*Correspondingly, the percentage of power between  $0$  and  $5 \ µ\rm   s$  is
:$$\frac{P_1}{P_1+ P_2} = \frac{2}{3} \hspace{0.15cm}\underline {\approx 66.7\%}\hspace{0.05cm}.$$

The figure shows  ${\it \Phi}_{\rm V}(\tau)$  in linear scale.  The areas  $P_1$  and  $P_2$  are labeled.
*The left graph is for  ${\rm BU}$, the right graph is for  ${\rm HT}$.
*For the latter, the power percentage of all later echoes  $($later than  $15 \ \rm µ s)$  is only about  $12\%$.
 
'''(7)'''  The area of the entire power-spectral density gives  $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
[[File:EN_Mob_A_2_8.png|right|frame|Delay PDF of profile  ${\rm BU}$ ]]

*Normalizing  ${\it \Phi}_{\rm V}(\tau)$  to this value yields the probability density function  $f_{\rm V}(\tau)$, as shown in the graph on the right (left diagram).

*With  $\tau_0 = 1 \ \ \rm µ s$  and  $\tau_5 = 5 \ \ \rm µ s$, the mean is:
:$$m_{\rm V}= \int_{0}^{\infty} f_{\rm V}(\tau) \hspace{0.15cm}{\rm d} \tau$$
:$$\Rightarrow \hspace{0.3cm}m_{\rm V}= \frac{2}{3\tau_0} \cdot \int_{0}^{\tau_5} \tau \cdot {\rm e}^{ - {\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau \ + $$
:$$ \hspace{1.7cm}+\ \frac{1}{3\tau_0} \cdot \int_{\tau_5}^{\infty} \tau \cdot {\rm e}^{ (\tau_5 -\tau)/\tau_0}\hspace{0.15cm}{\rm d} \tau \hspace{0.05cm}. $$

*The first integral is equal to  $2\tau_0/3$  according to the provided expression.

*With the substitution  $\tau' = \tau \, -\tau_5$  you finally obtain using the integral solutions given above:
:$$m_{\rm V} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{2\tau_0}{3} + \frac{1}{3\tau_0} \cdot \int_{0}^{\infty} (\tau_5 + \tau') \cdot{\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau ' = \frac{2\tau_0}{3} +
\frac{\tau_5}{3\tau_0} \cdot \int_{0}^{\infty} \cdot{\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau ' +
\frac{1}{3\tau_0} \cdot \int_{0}^{\infty} \tau' \cdot \cdot{\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau ' $$
:$$\Rightarrow \hspace{0.3cm}m_{\rm V}= \frac{2\tau_0}{3} + \frac{\tau_5}{3}+ \frac{\tau_0}{3} = \tau_0 + \frac{\tau_5}{3}
\hspace{0.15cm}\underline {\approx 2.667\,{\rm µ s}}
\hspace{0.05cm}. $$

*The variance  $\sigma_{\rm V}^2$  is equal to the second moment (mean of the square) of the zero-mean random variable  $\theta = \tau \, –m_{\rm V}$, whose PDF is shown in the right graph
*From this  $T_{\rm V} = \sigma_{\rm V}$  can be specified.

A second possibility is to first calculate the mean square value of the random variable  $\tau$  and from this the variance  $\sigma_{\rm V}^2$  using Steiner's theorem.
*With the substitutions and approximations already described above, one obtains
:$$m_{\rm V2} \hspace{-0.1cm} \ \approx \ \hspace{-0.1cm} \frac{2}{3\tau_0} \cdot \int_{0}^{\infty} \tau^2 \cdot {\rm e}^{ - {\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau + \frac{1}{3\tau_0} \cdot \int_{0}^{\infty} (\tau_5 + \tau')^2 \cdot {\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau ' $$
:$$\Rightarrow \hspace{0.3cm}m_{\rm V2} = \frac{2}{3} \cdot \int_{0}^{\infty} \frac{\tau^2}{\tau_0} \cdot {\rm e}^{ - {\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau + \frac{\tau_5^2}{3} \cdot \int_{0}^{\infty} \frac{1}{\tau_0} \cdot {\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau ' +\frac{2\tau_5}{3} \cdot \int_{0}^{\infty} \frac{\tau '}{\tau_0} \cdot {\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau ' + \frac{1}{3} \cdot \int_{0}^{\infty} \frac{{\tau '}^2}{\tau_0} \cdot {\rm e}^{ - {\tau}'/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau '
\hspace{0.05cm}. $$

*With the integrals given above, we have
:$$m_{\rm V2} \approx \frac{2}{3} \cdot 2 \tau_0^2 + \frac{\tau_5^2}{3} \cdot 1 + \frac{2\tau_5}{3} \cdot \tau_0 +
\frac{1}{3} \cdot 2 \tau_0^2 = 2 \tau_0^2 + \frac{\tau_5^2}{3} + \frac{2 \cdot \tau_0 \cdot \tau_5}{3} $$
:$$\Rightarrow \hspace{0.3cm} \sigma_{\rm V}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} m_{\rm V2} - m_{\rm V}^2 = 2 \tau_0^2 + \frac{\tau_5^2}{3} + \frac{2 \cdot \tau_0 \cdot \tau_5}{3}
- (\tau_0 + \frac{\tau_5}{3})^2 =\tau_0^2 + \frac{2\tau_5^2}{9} = (1\,{\rm µ s})^2 + \frac{2\cdot (5\,{\rm µ s})^2}{9} = 6.55\,({\rm µ s})^2$$
:$$\Rightarrow \hspace{0.3cm} T_{\rm V} = \sigma_{\rm V} \hspace{0.15cm}\underline {\approx 2.56\,{\rm µ s}}\hspace{0.05cm}.$$

The above graph shows the parameters  $T_{\rm V}$ and $\sigma_{\rm V}$.
{{ML-Fuß}}

[[Category:Mobile Communications: Exercises|^2.3 The GWSSUS Channel Model^]]

Aufgaben:Exercise 5.8: Matched Filter for Colored Interference

2022-02-17T11:41:07Z

Bene: Text replacement - "power spectral density" to "power-spectral density"

{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Matched_Filter
}}

[[File:P_ID646__Sto_A_5_8.png|right|frame|Spectrum  $G(f)$  of the useful signal and PSD  ${\it \Phi}_n (f)$  of the interference ]]
At the input of a filter there is a Gaussian pulse
:$$g(t) = g_0 \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {t/\Delta t} \right)^2 }$$

with amplitude  $g_0 = 2 \hspace{0.08cm}\rm V$  and equivalent pulse duration  $\Delta t = 1 \hspace{0.08cm}\rm ms$. 
*The corresponding spectral function  $G(f)$  is sketched above.
*The energy of this Gaussian pulse is given as follows:
:$$E_g = \int_{ - \infty }^{ + \infty } {g^2(t) \ {\rm{d}}t = \frac{g_0 ^2 \cdot \Delta t}{\sqrt 2 }} = 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}.$$

An interference  $n(t)$  is superimposed on the impulse  $g(t)$,  which largely covers the impulse.  

Two alternatives are considered for this purpose:
*Let the bilateral interference power density be constant (only for the first subtask):
:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$
*Let the interference signal  $n(t)$  be colored with the following interference power density:
:$${\it \Phi}_n (f) = \frac{N_0 /2}{{1 + \left( {f/f_0 } \right)^2 }},\quad f_0 = 500\;{\rm{Hz}}.$$

This second PSD response can be modeled, for example, from white noise by a shape filter with frequency response
:$$H_{\rm N}(f) = \frac{1}{{1 + {\rm{j}}\cdot f/f_0 }}\quad\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\, \quad h_{\rm N}(t) = 2{\rm{\pi }}f_0 \cdot {\rm{e}}^{ - 2{\rm{\pi }}f_0 t} $$
(first order low pass). Further, let it hold:

*Let the filter be optimally matched to the transmission pulse shape  $g(t)$  and the interference power-spectral density  ${\it \Phi}_n (f)$,  respectively:   $H(f) = H_{\rm MF}(f)$.
*The filter constant  $K_{\rm MF}$  is to be chosen such that  $H(f= 0) =1$. 
*Let the detection time be  $T_{\rm D}= 0$  for simplicity (acausal system description).

''Notes:''
*The chapter belongs to the chapter  [[Theory_of_Stochastic_Signals/Matched_Filter|Matched Filter]].

*Given is also the following definite integral:
:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$

===Questions===

<quiz display=simple>
{What is the S/N ratio (in dB) in the white noise case?
|type="{}"}
$10 \cdot \lg \hspace{0.1cm} \rho_{d,\hspace{0.08cm} \rm WR} \ = \ $ { 37.53 3% } $\ \rm dB$

{Calculate the MF frequency response in the presence of colored interference.  What is the magnitude of the value of  $H_\text{MF}(f)$  at frequency  $f = 1 \hspace{0.08cm} \rm kHz$? 
|type="{}"}
$|H_\text{MF}(f = 1 \hspace{0.08cm} \rm kHz)| \ = \ $ { 0.216 3% }

{What is the S/N ratio at the receiver in the case of the given colored interference?  Give reasons for the better result.
|type="{}"}
$10 \cdot \lg \hspace{0.1cm} \rho_d \ = \ $ { 38.73 3% } $\ \rm dB$

</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''  For white noise, according to the general equations in the theory section:
:$$\rho_{d,\ \rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad
\Rightarrow \quad 10\lg \cdot \rho _{d,\ \rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$

'''(2)'''  For the frequency response in the case of colored interferences, under the condition  $T_{\rm D}= 0$:
:$$H_\text{MF} (f) = K_\text{MF}\cdot \frac{G^{\star} (f)}{\left| {H_{\rm N} (f)} \right|^2 }\hspace{0.2cm}{\rm mit}\hspace{0.15cm}
G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } =1+\left( f/f_0 \right)^2. $$

*From the condition  $H_\text{MF}(f = 0) = 1$  it follows  $K_\text{MF} = 1/(g_0 \cdot \Delta t)$.  Thus one obtains:
:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$

*For white (frequency-independent) noise, the matched filter would be given by the first term alone, which causes the matching to the useful pulse  $g(t)$. 
*For colored interference  ⇒  interference power-spectral density  ${\it \Phi}_n(f)$,  higher frequencies are raised by the correction term  $1+\left( f/f_0 \right)^2$  because in this range the interference is lower.
*For  $f = 1/\Delta t = 2f_0 = 1\hspace{0.08cm} \rm kHz$  we obtain:
:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$

'''(3)'''  In general, the S/N ratio at the output of the matched filter is:
:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$

*The first summand is equal to the S/N ratio in white noise.  For the second summand, we obtain:
:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$

*After substituting  $x = 2 \cdot \pi^{0.5}\cdot f \cdot \Delta t$,  this integral becomes:
:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$

*This particular integral was given in the front; it has the value  $1$.  The first factor again describes the S/N ratio in white noise.
*This gives the following equations:
:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}
\rho _d = \rho _{d,\rm WR} + \Delta \rho _d = \rho _{d, \rm WR} \left( {1 + \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}} \right).$$
:$$\Rightarrow \hspace{0.3cm} \Delta t \cdot f_0 = 0.5 : \hspace{0.3cm}
\rho _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$

Conclusion:
*There is a  $1.2 \; \rm dB$  better result than with white noise, because here  ${\it \Phi}_n(f)$  is smaller than  $N_0/2$ in the whole frequency range except for the frequency  $f = 0$  $($here the equal sign applies$)$. 
*This fact is also used by the matched filter.
{{ML-Fuß}}

[[Category:Theory of Stochastic Signals: Exercises|^5.4 Matched Filter^]]

Mobile Communications/LTE-Advanced - a Further Development of LTE

2022-02-17T11:39:28Z

Bene: Text replacement - "List of Sources" to "References"

{{LastPage}}
{{Header
|Untermenü=LTE – Long Term Evolution
|Vorherige Seite=Physical Layer for LTE
|Nächste Seite=
}}

== How fast is LTE really? ==
 
Consumers are accustomed to being able to use (at least to a large extent) the speed offered by established cable-based services such as  [[Examples_of_Communication_Systems/General_Description_of_DSL|$\rm DSL$]]  ("Digital Subscriber Line").
*But what is the situation with LTE? 

*What data rates can the individual LTE user actually reach?

It is much more difficult for the providers of mobile radio systems to provide concrete data rate information, since many influences that are difficult to predict have to be taken into account for a radio connection. 

As already described in chapter  [[Mobile_Communications/Technical Innovations of LTE# Multiple Antenna Systems|Technical Innovations of LTE]]  according to the planning for 2011, data rates of up to 326 Mbit/s are possible in LTE downlink and approx. 86 Mbit/s in uplink.  These figures are only maximum achievable values.  In reality, however, the speed is determined by a variety of factors.  In the following we refer to the downlink, see  [Gut10]<ref name='Gut10'>Gutt, E.:  LTE - a new dimension of mobile broadband use.  [http://www.ltemobile.de/uploads/media/LTE_Einfuehrung_V1.pdf PDF document on the Internet], 2010.</ref>:
*Since LTE is a so-called  "Shared Medium" , all users of a cell have to share the entire data rate.  Note that voice transmission or normal use of the Internet generates less traffic than, for example,  "File Sharing"  or similar. 

*The faster a user moves, the lower the available data rate will be.  An elementary component of the LTE specification is that for mobility up to 15 km/h the highest data rates are guaranteed and up to 300 km/h at least still "good functionality". 

*The highest data rate is achieved in close proximity to the base station.  The further away a user is from the base station, the lower the data rate assigned to him, which can be explained by switching from "64–QAM" or "16–QAM" to "4–QAM" (QPSK), among other things.

*Shielding by walls and buildings or sources of interference of any kind limit the achievable data rate enormously.  Optimal would be a "Line of Sight" connection between receiver and base station, a scenario that is rather unusual. 

The reality in summer 2011 was as follows:  LTE is already available in some countries  (at least for testing purposes).  In addition to LTE pioneer Sweden, these include the USA and Germany.  In various tests, download speeds of between 5 and 12 Mbit/s were achieved, and in very good conditions up to 40 Mbit/s.  Details can be found in the Internet article  [Gol11]<ref name='Gol11'>Goldman, D.:  AT&T launching 'new' new 4G network.  http://money.cnn.com/2011/05/25/technology/att_4g_lte/index.htm PDF document on the Internet], 2011</ref>. 

Moreover, the 2011 LTE network did not seem ready to replace the established wired Internet connections due to excessive delay times and the resulting occasional connection interruptions.  However, the development in this area progressed with giant strides, so that this information from summer 2011 was not relevant for very long. 

== Some system improvements through LTE-Advanced==
 
While the first LTE systems corresponding to Release 8 of December 2008 slowly came onto the market in summer 2011, the successor was already on the doorstep.  The Release 10 of the  "3GPP"  completed in June 2011 is  "Long Term Evolution–Advanced", or in short  $\rm LTE-A$.  It is the first technology to meet the requirements of the ITU ("International Telecommunication Union") for a 4G standard.  A summary of these requirements, also called "IMT–Advanced", can be found in great detail in an [http://www.itu.int/dms_pub/itu-r/opb/rep/R-REP-M.2134-2008-PDF-E.pdf ITU article  (PDF).] 

Without claiming to be exhaustive, some of the features of LTE–Advanced are mentioned here:
*The data rate should be up to 1 Gbit/s with little movement of the user and up to 100 Mbit/s with fast movement.  In order to achieve these high data rates, some new technical specifications have been made, which will be briefly discussed here. 

*LTE–Advanced supports bandwidths up to 100 MHz maximum, while the LTE specification  (after Release 8)  provides only 20 MHz.  The FDD spectra no longer have to be divided symmetrically between uplink and downlink.  For example, a higher channel bandwidth can be used for the downlink than for the uplink, which corresponds to the normal use of the mobile Internet with a smartphone. 

*In the uplink of LTE–Advanced   [[Mobile_Communications/The_Application_of_OFDMA_and_SC-FDMA_in_LTE#Functionality_of_SC.E2.80.93FDMA|SC–FDMA]]  is also used.  Since the 3GPP consortium was not satisfied with the SC–FDMA transmission in LTE, some essential improvements in the process were developed. 

*Another interesting novelty is the introduction of so-called  "Relay Nodes".  Such a Relay Node  $\rm (RN)$  is placed at the edge of a cell to provide better transmission quality at the boundaries of a cell and thus increase the range of the cell. 

[[File:P ID2295 LTE T 4 5 S2 v1.png|right|frame|Functionality of Relay Nodes]]
A relay node looks like a normal base station for a LTE terminal device  ("eNodeB").  However, it only has to supply a relatively small area of operation and therefore does not need to be connected to the backbone in a complicated way.  In most cases a relay node is connected to the next base station via directional radio.

In this way, high data rates and good transmission quality without interruptions are guaranteed without great effort.  By increasing the physical proximity to the base stations, the reception quality in buildings is also improved.

Another feature added to LTE–A is known as  "Coordinated Multiple Point Transmission and Reception"  $\rm (CoMP)$.  This is an attempt to reduce the disturbing influence of intercell interference.  With intelligent scheduling across several base stations, it is even possible to make intercell interference usable.  The information for a terminal device is available at two adjacent base stations and can be transmitted simultaneously.  Details on CoMP–technology can be found, for example, in the internet article  [Wan13]<ref name='Wan13'>Wannstrom, J.:  LTE–Advanced.  http://www.3gpp.org/technologies/keywords-acronyms/97-lte-advanced PDF–Document on the Internet, 2011]</ref>  from 3gpp. 

{{BlaueBox|TEXT=
$\text{Intermediate status of 2011:}$ 
*Thanks to the above measures in combination with many other improvements, primarily the introduction of  "4×4"  MIMO for the uplink and  "8×8"  MIMO in the downlink, it is possible to significantly increase the spectral efficiency  (i.e. the transferable flow of information in one Hertz bandwidth within one second)  of LTE–A compared to LTE, namely in the downlink from 15 bit/s/Hz  to  $\text{30 bit/s/Hz}$  and in the uplink from 3. 75 bit/s/Hz to  $\text{15 bit/s/Hz}$. 

*Of course, backwards compatibility with the previous LTE standard and previous mobile phone systems must also be guaranteed.  Also with a UMTS cell phone one should be able to dial into a LTE network, even if one cannot use the LTE specific features. 

*At the beginning of June 2011 the first tests of LTE–Advanced were conducted.  Sweden, which has already set up the first commercial LTE network, once again took the lead.  Ericsson demonstrated for the first time a test system with practical, commercially available terminals and began commercial use of LTE–Advanced in 2013.
*In a YouTube video, an LTE test can be seen in a moving minibus, in which data rates of over 900 Mbit/s in the downlink and 300 Mbit/s in the uplink were achieved.}} 

== Standards in competition with LTE or LTE-Advanced ==
 
In addition to the LTE specified by the 3GPP consortium, there are other standards that are intended to serve the purpose of fast mobile data transmission.  The two most important ones are briefly discussed here: 

$\rm cdma2000$  (or "IS–2000")  and its further development $\rm UMB$  ("Ultra Mobile Broadband"): 

This is a third-generation mobile communications standard that was specified and further developed by  [http://www.3gpp2.org/ 3GPP2]  ("Third Generation Partnership Project 2").  Further information on cdma2000 can be found in the section  [[Mobile_Communications/Characteristics_of_UMTS#The_IMT-2000_standard|IMT–2000 standard]]. 

Far less is known about the further development of this standard than about LTE.  It is worth mentioning that for cdma2000 and UMB there is a sub–standard specified exclusively for data transmission.  The Cologne telecommunications provider  "NetCologne" has been offering mobile Internet in the 450 MHz range on this basis since 2011.  Furthermore, cdma2000 is insignificant in Germany. 

Note:   The "3GPP2" was founded almost at the same time as the almost identically named  [http://www.3gpp.org/ 3GPP]  in December 1998, obviously due to ideological differences. 

$\rm WiMAX$  (Worldwide Interoperability for Microwave Access): 

This term refers to a wireless transmission technology based on the IEEE standard 802.16.  It belongs to the family of the 802 standards like WLAN (802.11) and Ethernet (802.3).  There are two different sub–specifications to WiMAX, namely
*one for operating a static connection that does not allow handover, and 

*one for the mobile operation, which is to compete with UMTS and LTE. 

The potential of the static WiMAX connections lies mainly in the long range with nevertheless comparatively high data rate.  For this reason, static WiMAX was initially traded as DSL alternative for thinly populated areas.  For example, with a Line of Sight (LoS) connection between transmitter and receiver over 15 kilometers, about 4.5 Mbit/s are possible.  In urban areas without line of sight, WiMAX still has a range of about 600 meters, a much better value than the 100 meters typically offered by WLAN. 

At the moment  (2011)  it also worked on a further development called  "WiMAX2".  According to the initiators,  WiMAX2 in the mobile version is a 4G standard which, just like LTE–Advanced, can achieve data rates of up to 1 Gbit/s.  WiMAX2 was implemented in practice by the end of 2011.

In Germany, WiMAX does not play a major role (in 2011), since both the German government in its broadband offensive and all major mobile phone operators have declared  "Long Term Evolution"  (LTE or LTE–A)  to be the future of mobile data transmission. 

== Milestones in the development of LTE and LTE-Advanced ==
 
Finally, a brief overview of some milestones in the development towards LTE from the perspective of 2011:
*'''2004'''    The Japanese telecommunications company  [https://www.nttdocomo.co.jp/english/index.html NTT DoCoMo]  proposes LTE as the new international mobile communications standard. 

*'''09/2006'''    Nokia Siemens Networks (NSN) presents together with  [http://www.nomor.de/ Nomor Research]  for the first time an emulator of an LTE network.  For demonstration purposes, a HD–video is transmitted and two users play an interactive online game. 

*'''02/2007'''    At the  "3GSM World Congress", the world's largest mobile phone trade fair, the Swedish company  [https://www.ericsson.com/ Ericsson]  demonstrates an LTE system with 144 Mbit/s. 

*'''04/2008'''    [https://de.wikipedia.org/wiki/NTT_DOCOMO DoCoMo]  demonstrates an LTE data rate of 250 Mbit/s.  Almost simultaneously   [https://en.wikipedia.org/wiki/Nortel Nortel Networks Corp.]  (Canada) achieves a data rate in vehicle speed of 100 km/h of at least 50 Mbit/s. 

*'''10/2008'''    Test of the first working LTE modem by Ericsson in Stockholm.  This date is the starting point for the commercial use of LTE. 

*'''12/2008'''    Completion of Release 8 of 3GPP, synonymous with LTE.  The company  [http://www.lg.com/de LG Electronics]  develops the first LTE chip for cell phones. 

*'''03/2009'''    At the CeBIT in Hanover, Germany,  [https://www.t-mobile.de/ T–Mobile]  shows Video conferencing and online games from a moving car. 

*'''12/2009'''    The world's first commercial LTE network starts in downtown Stockholm, only 14 months after the start of the test phase. 

*'''04/2010'''    3GPP begins with the specification of Release 10, synonymous with LTE–A. 

*'''05/2010'''    The LTE frequency auction in Germany ends.  At 4.4 billion euros, the proceeds are significantly lower than experts had expected and politicians had hoped for. 

*'''08/2010'''    T-Mobile is building Germany's first commercially usable LTE base station in Kyritz.  For a functioning operation, suitable terminals are still missing. 

*'''12/2010'''    In Germany, the first major pilot tests are running on the networks of Telekom,  [https://www.o2online.de/ O2]  and  [http://www.vodafone.de/ Vodafone].  In the meantime, corresponding LTE routers are available.

*'''02/2011'''    In South Korea the first successful tests with the successor LTE–Advanced are being conducted. 

*'''03/2011'''    The 3GPP Release 10 is completed. 

*'''06/2011'''    Launch of the first German LTE network in Cologne.  By middle 2012, Deutsche Telekom will ensure that LTE network is rolled out across a wide area in 100 additional cities. 

==Exercise to chapter==
 
[[Aufgaben:Exercise 4.5: LTE vs LTE-Advanced]]

==References==

<references/>

{{Display}}

Linear and Time Invariant Systems/Nonlinear Distortions

2022-02-17T11:39:28Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Signal Distortion and Equalization
|Vorherige Seite=Classification_of_the_Distortions
|Nächste Seite=Linear_Distortions
}}

==Properties of nonlinear systems==
 
The system description by means of the frequency response  $H(f)$  and/or the impulse response  $h(t)$  is only possible for an  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|LTI System]] .   However, if the system contains nonlinear components, as it is assumed for this chapter, no frequency response and no impulse response can be stated.  The model must be designed in a more general way.

[[File:EN_LZI_T_2_2_S1.png|frame| Description of a nonlinear systems|class=fit]]

Also in this nonlinear system,  we denote the signals at the input and the output by  $x(t)$  and  $y(t)$  respectively,  and the corresponding spectral functions by  $X(f)$  and  $Y(f)$.

An observer will note the following here:
*The transmission characteristics are now also  '''dependent on the amplitude of the input signal'''.  If  $x(t)$  results in the output signal  $y(t)$,  it can now no longer be concluded that the input signal  $K · x(t)$  will always result in the signal  $K · y(t)$.
*This also implies that the  '''superposition principle is no longer applicable'''.  Consequently, the result  $x_1(t) + x_2(t)$   ⇒   $y_1(t) + y_2(t)$  cannot be reasoned from the two correspondences   $x_1(t) ⇒ y_1(t)$  and  $x_2(t) ⇒ y_2(t)$.
*Due to nonlinearities  '''new frequencies occur'''.  If  $x(t)$  is a harmonic oscillation with the frequency  $f_0$, the output signal  $y(t)$  also contains components at multiples of  $f_0$.  In Communications Engineering, these are referred to as  '''harmonics'''.
*In practice, an information signal usually contains many frequency components. The harmonics of the low-frequency signal components now fall into the range of higher-frequency useful components.  This results in '''nonreversible signal falsifications'''.

Before mentioning  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Constellations_which_result_in_nonlinear_distortions|"constellations which result in nonlinear distortions"]]  at the end of the section,  the problem of nonlinear distortions is captured mathematically.
*We assume here that  '''the system has no memory'''   so that the output value  $y = y(t_0)$  depends only on the instantaneous input value  $x = x(t_0)$, 
*but not on the signal curve  $x(t)$  for  $t < t_0$.

==Description of nonlinear systems==
 
{{BlaueBox|TEXT=
$\text{Definition:}$ 
A system is said to be  '''nonlinear''' if the following relationship exists between the signal value  $x = x(t)$  at the input and the output  $y = y(t)$ :
$$y = g(x) \ne {\rm const.} \cdot x.$$
The curve shape  $y = g(x)$  is called the  '''nonlinear characteristic curve'''  of the system.}}

[[File:P_ID888__LZI_T_2_2_S2_neu.png |right|frame| Nonlinear characteristic curve|class=fit]]

*In the diagram,  as an example,  the green curve is the nonlinear characteristic curve  $y = g(x)$  which is shaped according to the first quarter of a sine function.
*In this diagram the special case of a linear system with the characteristic curve  $y = x$  can be seen dashed in red.

Since every characteristic curve can be developed into a Taylor series around the operating point the output signal can also be represented as follows:
$$y(t) = \sum_{i=0}^{\infty}\hspace{0.1cm} c_i \cdot x^{i}(t) = c_0 + c_1 \cdot x(t) + c_2 \cdot x^{2}(t) + c_3 \cdot x^{3}(t) + \hspace{0.05cm}\text{...}$$
If  $x(t)$  has a unit,  for exmple "Volt”,  then the coefficients of the Taylor series also have appropriate and different units:
*$c_0$  with  $\rm V$,
*$c_1$  without units,
*$c_2$  with  $\rm 1/V$, etc.

In the above diagram, the operating point is identical to the zero point and  $c_0 = 0$  holds.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The properties of nonlinear systems listed on the  [[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen#Eigenschaften_nichtlinearer_Systeme|first page of this section]]  are illustrated here using the characteristic curve  $y = g(x) = \sin(x)$  shown in the centre of the diagram.
*Here, the direct (DC) signal  $x(t) = 0.5$  results in the constant output signal  $y(t) = 0.479$ .
* For  $x(t) = 1$  the input signal results in the output signal  $y(t) = 0.841 ≠ 2 · 0.479$.
*Thus, doubling   $x(t)$  does not cause the doubling of  $y(t)$    ⇒   the superposition principle is violated.

[[File:P_ID889__LZI_T_2_2_S2b_neu.png|center|frame|Effects of a nonlinear characteristic curve|class=fit]]

The outer diagrams show – each in blue – cosine-shaped input signals  $x(t)$  with different amplitudes  $A$  and the corresponding distorted output signals  $y(t)$ in red.  It can be seen that the nonlinear distortions increase with increasing amplitude,  which are quantified by the distortion factor  $K$  defined on the next page.

*The diagram on the upper right-hand corner for  $A = 1.5$  clearly shows that  $y(t)$  is no longer cosine-shaped;  the half-waves run rounder than the ones of the cosine function.
*But also for  $A = 0.5$  and  $A = 1.0$  the signals  $y(t)$  deviate - although less strongly - from the cosine form due to the harmonics.  That is, new frequency components at multiples of the cosine frequency  $f_0$  arise.

*In the picture on the bottom right-hand corner the characteristic curve is operated unilaterally due to an additional direct component.  Now an unbalance in the signal  $y(t)$  can be seen, too.  The lower half-wave is more peaked than the upper one.  The distortion factor here is  $K \approx 22\%$.}}

==The distortion factor==
 
To quantitatively capture the nonlinear distortions we assume that the input signal  $x(t)$  is cosine-shaped with the amplitude  $A_x$.  The output signal contains harmonics due to the nonlinear distortions and the following is generally true:       $y(t) = A_0 + A_1 \cdot \cos(\omega_0 t) + A_2 \cdot \cos(2\omega_0 t) +
A_3 \cdot \cos(3\omega_0 t) + \hspace{0.05cm}\text{...}$

[[File:EN_LZI_T_2_2_S3.png|center|frame|Definition of the distortion factor|class=fit]]

{{BlaueBox|TEXT=
$\text{Definition:}$ 
With these amplitude values  $A_i$  the equation for the  '''distortion factor''' is:
:$$K = \frac {\sqrt{A_2^2+ A_3^2+ A_4^2+ \hspace{0.05cm}\text{...} } }{A_1} = \sqrt{K_2^2+
K_3^2+K_4^2+ \hspace{0.05cm}\text{...} }.$$
In the second equation
*$K_2 = A_2/A_1$  denotes the distortion factor of second order,
*$K_3 = A_3/A_1$  denotes the distortion factor of third order, etc.}}

It is explicitly pointed out that the amplitude  $A_x$  of the input signal is not taken into account when computing the distortion factor.  Also a resulting direct (DC) component  $A_0$  remains unconsidered.

In the last section  ([[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion#Description_of_nonlinear_systems|$\text{Example 1}$]])  the distortion factors were specified with values between about  $1\%$  and  $20\%$ .
*These values are already significantly above the distortion factors of low-cost audio equipment,  for which  $K < 0.1\%$ applies.
*In HiFi equipment, particular emphasis is placed on linearity and a very low distortion factor is also reflected in the price.

A comparison with the page  [[Linear_and_Time_Invariant_Systems/Klassifizierung_der_Verzerrungen#Ber.C3.BCcksichtigung_von_D.C3.A4mpfung_und_Laufzeit|Consideration of attenuation and runtime]]  reveals that for the special case of a cosine-shaped input signal the signal–to–distortion–power ratio defined is equal to the reciprocal of the distortion factor squared:
:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} = \left(\frac{ A_{1}}{A_x} \right)^2 \cdot
\frac{ {1}/{2} \cdot A_{x}^2}{{1}/{2} \cdot (A_{2}^2 + A_{3}^2 + A_{4}^2 + \hspace{0.05cm}...) } = \frac{1}{K^2}\hspace{0.05cm}.$$

[[File:P_ID1090__LZI_T_2_2_S3b_neu.png |frame|Influence of a nonlinearity on a cosine signal | right|class=fit]]
{{GraueBox|TEXT=
$\text{Example 2:}$ 
We now consider an averaged cosine signal:
:$$x(t) = {1}/{2} + {1}/{2}\cdot \cos (\omega_0 \cdot t).$$

$x(t)$  takes values between  $0$  and  $1$,  and is drawn as the blue curve.  The signal power is
:$$P_x = 1/4 + 1/8 = 0.375.$$

If we apply this signal to a nonlinearity with the characteristic curve
:$$y=g(x) = \sin(x) \approx x - {x^3}/{6} \hspace{0.05cm},$$
then the output signal is:
:$$y(t) = A_0 + A_1 \cdot \cos (\omega_0 \cdot t)+ A_2 \cdot \cos (2\omega_0 \cdot t)+ A_3 \cdot \cos (3\omega_0 \cdot t)\hspace{0.05cm},$$
:$$\Rightarrow \hspace{0.3cm} A_0 = {86}/{192},\hspace{0.3cm}A_1 = {81}/{192},\hspace{0.3cm}A_2 = - {6}/{192},\hspace{0.3cm}A_3 = -
{1}/{192}\hspace{0.05cm}.$$

The trigonometric transformations for  $\cos^2(α)$  and  $\cos^3(α)$  were used to calculate the Fourier coefficients.  The distortion factor is thus given by
:$$K = \frac {\sqrt{A_2^{\hspace{0.05cm}2} + A_3^{\hspace{0.05cm}2} } }{A_1}\approx 7.5\%\hspace{0.05cm}.$$

It can be further seen that the signal  $y(t)$  sketched in red is almost equal to the signal  $α · x(t)$  sketched in green with  $α = \sin(1) ≈ 5/6$ .

*Defining the error signal as  $ε_1(t) = y(t) - α · x(t)$,  with its power
:$$P_{\varepsilon 1} = \frac {(80-86)^2}{192^2} + \frac {6^2 + (-1)^2}{2 \cdot 192^2}\approx 1.48 \cdot 10^{-3}$$
:the following is obtained for the signal–to–noise–power ratio:
:$$\rho_{{\rm V} 1} = \frac {\alpha^2 \cdot P_x}{P_{\varepsilon 1} } = \frac {(5/6)^2 \cdot 0.375}{1.48 \cdot 10^{-3} }\approx
176 = {1}/{K^2}\hspace{0.05cm}.$$

*In contrast, the SNR is significantly lower if we do not consider the attenuation factor,  that is,  if we assume the error signal  $ε_2 = y(t) - x(t)$ :
:$$P_{\varepsilon 2} = \frac {(86-96)^2}{192^2} + \frac {(81-96)^2 + 6^2 + (-1)^2}{2 \cdot 192^2}\approx 6.3 \cdot 10^{-3} \hspace{0.3cm}
\Rightarrow
\hspace{0.3cm}\rho_{{\rm V} 2} = \frac { P_x}{P_{\varepsilon 2}}= \frac {0.375}{6.3 \cdot 10^{-3}}
\approx 60 \hspace{0.05cm}.$$}}

==Clirr measurement==
 
A major disadvantage of the definition of the distortion factor is the thereby specification to cosine-shaped test signals, i.e. to conditions remote from reality.
[[File:EN_LZI_T_2_2_S4.png|right|frame|Principle of clirr measurement|class=fit]]

*In the so-called clirr measurement the signal  $x(t)$  to be transmitted is modelled by white noise with the noise power density  ${\it \Phi}_x(f)$.
*In addition, a narrow band-stop filter  $\rm (BS)$  with centre frequency  $f_{\rm M}$  and  (very small)  bandwidth  $B_{\rm BS}$  is introduced into the system.

In a linear system, the output spectrum  ${\it \Phi}_y(f)$  would not be wider than  $B_x$  and also in the region around  $f_{\rm M}$  there would be no components. 

These result solely from frequency conversion products  ("intermodulation components")  of different spectral components, i.e. from nonlinear distortions.

By varying the centre frequency  $f_{\rm M}$  and integrating over all these small interfering components the distortion power can thus be determined. More details on this method can be found, for example, in  [Kam04]<ref>Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.
 

==Constellations which result in nonlinear distortions==
 
As an example of the occurrence of nonlinear distortions in analog message transmission systems some constellations which result in such distortions shall be mentioned here.  In terms of content, this anticipates the book  [[Modulation_Methods]].

[[File:EN_LZI_T_2_2_S5.png |center|frame|General block diagram of a communication system|class=fit]]

Nonlinear distortions of the sink signal  $v(t)$  with respect to the source signal  $q(t)$  occur when
*nonlinear distortions already occur on the channel – i.e. with respect to the transmission signal  $s(t)$  and the received signal  $r(t)$,
*an envelope demodulator is used for  [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#ZSB-Amplitudenmodulation_mit_Tr.C3.A4ger|Double-Sideband Amplitude Modulation]]  (DSB–AM)  with modulation factor  $m > 1$,
*for DSB–AM and envelope demodulation there is a linearly distorting channel,  even with a modulation factor  $m < 1$,
*the combination of  [[Modulation_Methods/Single-Sideband_Modulation|Single-Sideband Modulation]]  and  [[Modulation_Methods/Hüllkurvendemodulation|Envelope Demodulation]]  is used (regardless of the sideband–to–carrier ratio),
*an  [[Modulation_Methods/Phasenmodulation_(PM)#Gemeinsamkeiten_zwischen_Phasen.E2.80.93_und_Frequenzmodulation|Angle Modulation]]  (generic term for frequency and phase modulation)  is applied and the available bandwidth is finite.

==Exercises for the chapter==
 
[[Aufgaben:Exercise_2.3:_Sinusoidal_Characteristic|Exercise 2.3: Sinusoidal Characteristic]]

[[Aufgaben:Exercise_2.3Z:_Asymmetrical_Characteristic_Operation|Exercise 2.3Z: Asymmetrical Characteristic Operation]]

[[Aufgaben:Exercise_2.4:_Distortion_Factor_and_Distortion_Power|Exercise 2.4: Distortion Factor and Distortion Power]]

[[Exercise_2.4Z:_Characteristics_Measurement|Exercise 2.4Z: Characteristics Measurement]]

==References==
<references/>

{{Display}}

Mobile Communications/Technical Innovations of LTE

2022-02-17T11:39:28Z

Bene: Text replacement - "List of Sources" to "References"

{{Header
|Untermenü=LTE – Long Term Evolution
|Vorherige Seite=General Information on the LTE Mobile Communications Standard
|Nächste Seite=The Application of OFDMA and SC-FDMA in LTE
}}

== For speech transmission with LTE ==
 
Unlike previous mobile phone standards, LTE only supports  '''packet-oriented transmission'''.  However, for speech transmission (sometimes the term "voice transmission" is used for this), a connection-oriented transmission with fixed reservation of resources would be better, since a "fragmented transmission", as is the case with the packet-oriented method, is relatively complicated. 

The problem of integrating speech transmission methods was one of the major challenges in the development of LTE, as speech transmission remains the largest source of revenue for network operators.  There were a number of approaches, as it can be seen in the internet article   [Gut10]<ref name='Gut10'>Gutt, E.:  LTE - a new dimension of mobile broadband use.  [http://www.ltemobile.de/uploads/media/LTE_Einfuehrung_V1.pdf PDF Internet document], 2010.</ref>

'''(1)'''   A very simple and obvious method is  "Circuit Switched Fallback"  $\rm (CSFB)$.  Here a wireline transmission is used for the speech transmission.  The principle is:
*The terminal device logs on to the LTE network and in parallel also to a GSM or UMTS network.  When an incoming call is received, the terminal device receives a message from the  "Mobile Management Entity"  $\text{(MME}$,  control node in the LTE network for user authentication$)$, whereupon a wireline transmission via the GSM or the UMTS network is established.
*A disadvantage of this solution  (actually it is a "problem concealment")  is the greatly delayed connection establishment.  In addition,  CSFB prevents the complete conversion of the network to LTE.

'''(2)'''   Another possibility for the integration of speech/voice in a packet-oriented transmission system is offered by  "Voice over LTE via GAN"  $\rm (VoLGA)$, which is based on the from  [[Mobile_Communications/General Information on the LTE Mobile Communications Standard#3GPP_. E2.80.93_Third_Generation_Partnership_Project| 3GPP]]  developed by  [https://en.wikipedia.org/wiki/Generic_Access_Network Generic Access Network].  In brief, the principle can be described as follows:
* GAN enables line-based services via a packet-oriented network (IP network), for example $\rm WLAN$  ("Wireless Local Area Network").  With compatible end devices one can register oneself in the GSM network over a WLAN connection and use line-based services.  VoLGA uses this functionality by replacing WLAN with LTE.
* The fast implementation of VoLGA is advantageous, as no lengthy new development or changes to the core network are necessary.  However, a so-called  "VoLGA Access Network Controller"  $\rm (VANC)$  must be added to the network as hardware.  This takes care of the communication between the end device and the  "Mobile Management Entity"  or the core network. 

Even though VoLGA does not need to use a GSM or UMTS network for voice connections like CSFB, it was considered by the majority of the mobile community as an (unsatisfactory) bridge technology due to its user-friendliness.  T–Mobile has long been a proponent of the VoLGA technology, but they also stopped further development in February 2011. 

In the following we describe a better solution proposal.  Keywords are  "IP Multimedia Subsystem"  $\rm (IMS)$  and  "Voice over LTE"  $\rm (VoLTE)$.  The operators in Germany switched to this technology relatively late:   Vodafone and O2 Telefonica at the beginning of 2015, Telekom at the beginning of 2016.

This is also the reason why the switch to LTE in Germany  (and in Europe in general)  was slower than in the US.  Many customers did not want to pay the higher prices for LTE as long as there was no well functioning solution for integrating voice transmission. 

== VoLTE - Voice over LTE ==
 
From today's point of view (2016), the most promising approach to integrating voice services into the LTE network, some of which are already established, is  "Voice over LTE"  $\rm (VoLTE)$.  This standard, officially adopted by the  [http://www.gsma.com/aboutus/ $\rm GSMA$],  the worldwide industry association of more than 800 mobile network operators and over 200 manufacturers of cell phones and network infrastructure, is exclusively IP packet-oriented and is based on the  "IP Multimedia Subsystem"  $\rm (IMS)$, which was already defined in the UMTS Release 9 in 2010. 

The technical facts about IMS are:
*The IMS basic protocol is the one from  "Voice over IP"  known  [https://de.wikipedia.org/wiki/Session_Initiation_Protocol "Session Initiation Protocol"]  $\rm (SIP)$.  This is a network protocol that can be used to establish and control connections between two users.
* This protocol enables the development of a completely  (for data and voice)  IP based network and is therefore future-proof. 

The reason why the introduction of  VoLTE  has been delayed by four years compared to LTE establishment in data traffic is due to the difficult interaction of "4G" with the older predecessor standards  GSM  ("2G") and  UMTS  ("3G").  Here is an example:
*If a mobile phone user leaves his LTE cell and switches to an area without 4G coverage, an immediate switch to the next best standard (3G) must be made.

*Speech is transmitted here technically completely differently, no longer by many small data packets   ⇒   "packet-switched" but sequentially in the logical and physical channels reserved especially for the user   ⇒   "circuit-switched". 

*This implementation must be so fast and smooth that the end customer does not notice anything.  And this implementation must work for all mobile phone standards and technologies.

According to all the experts, VoLTE will have a positive impact on mobile telephony in the same way that LTE has driven the mobile Internet forward since 2011.  Key benefits for users are:
*A higher voice quality, as VoLTE uses  [[Examples_of_Communication_Systems/Nachrichtentechnische_Aspekte_von_UMTS#Verbesserungen_bez.C3.BCglich_Sprachcodierung| AMR Wideband Codecs]]  with 12.65 or 23.85 kbit/s.  Furthermore, the VoLTE data packets are prioritized for lowest possible latencies. 

*An enormously accelerated connection setup within one or two seconds, whereas with  "Circuit Switched Fallback" (CSFB) it takes an unpleasantly long time to establish a connection. 

*A low battery consumption, significantly lower than "2G" and "3G", associated with a longer battery life.  Also in comparison to the usual VoIP services the power consumption is up to 40% lower. 

From the provider's point of view, the following advantages result:
*A better spectral efficiency:   Twice as many calls are possible in the same frequency band than with "3G".  In other words:   More capacity is available for data services for the same number of calls. 

*An easy implementation of  [https://en.ryte.com/wiki/Rich_Media Rich Media Services]  $\rm (RCS)$, e.g. for video telephony or future applications that can be used to attract new customers. 

*A better acceptance of the higher provisioning costs by LTE customers if you don't need to outsource to a "low-value" network like "2G" or "3G" for telephony.

== Bandwidth flexibility ==
 
LTE can be adapted to frequency bands of different widths with relatively little effort by using  [[Modulation_Methods/Allgemeine_Beschreibung_von_OFDM#Das_Prinzip_von_OFDM_.E2.80.93_Systembetrachtung_im_Zeitbereich_.281.29|$\rm OFDM$]]  ("Orthogonal Frequency Division Multiplex").  This fact is an important feature for various reasons, see  [Mey10]<ref name='Mey10'>Meyer, M.:  Siebenmeilenfunk.  c't 2010, issue 25, 2010.</ref>, especially for network operators:
*The frequency bands for LTE may vary in size depending on the legal requirements in different countries.  The outcome of the state-specific auctions of LTE frequencies  (separated into FDD and TDD)  has also influenced the width of the spectrum. 

*Often LTE is operated in the "frequency neighborhood" of established radio transmission systems, which are expected to be switched off soon.  If the demand increases, LTE can be gradually expanded to the frequency range that is becoming available. 

*For example, the migration of television channels after digitalization:   A part of the LTE network will be located in the VHF frequency range around 800 MHz, which has now been freed up, see  [[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard#LTE_frequency_band_splitting|Frequency Band Splitting Graphic]]. 

*Actually the bandwidths could be selected with a degree of fineness of up to 15 kHz  (corresponding to an OFDMA subcarrier).  However, since this would unnecessarily produce overhead, a duration of  '''one millisecond'''  and a bandwidth of  '''180 kHz'''  has been specified as the smallest addressable LTE resource.  Such a block corresponds to twelve subcarriers (180 kHz divided by 15 kHz). 

In order to keep the complexity and effort of hardware standardization as low as possible, a whole range of permissible bandwidths between 1.4 MHz and 20 MHz has been agreed upon. The following list – taken from  [Ges08]<ref name='Ges08'>Gessner, C.:  UMTS Long Term Evolution (LTE): Technology Introduction.  Rohde&Schwarz, 2008.</ref>  – specifies the standardized bandwidths, the number of available blocks and the "overhead":
*6 available blocks in the bandwidth 1.4 MHz   ⇒   relative overhead about 22.8%, 

*15 available blocks in the bandwidth 3 MHz   ⇒   relative overhead about 10%, 

*25 available blocks in the bandwidth 5 MHz   ⇒   relative overhead about 10%, 

*50 available blocks in the bandwidth 10 MHz   ⇒   relative overhead about 10%, 

*75 available blocks in the bandwidth 15 MHz   ⇒   relative overhead about 10%, 

*100 available blocks in the bandwidth 20 MHz   ⇒   relative overhead about 10%. 

Since otherwise some LTE specific functions would not work, at least six blocks must be provided.
*The relative overhead is comparatively high at small channel bandwidth (1.4 MHz):   (1.4 – 6 · 0.18)/1.4 ≈ 22.8%.
*From a bandwidth of 3 MHz the relative overhead is constant 10%.
*It also applies that all end devices must also support the maximum bandwidth of 20 MHz   [Ges08]<ref name='Ges08'></ref>

== FDD, TDD and half-duplex method==
 
[[File:EN_Mob_T_4_2_S3a.png|right|frame|transmission scheme for FDD (top) or TDD (bottom)|class=fit]]
Another important innovation of LTE is the half–duplex procedure, which is a mixture of the two from UMTS already known [[Examples_of_Communication_Systems/General_Description_of_UMTS#Full Duplex Procedure|duplex procedures]]:

*$\text{Frequency Division Duplex}$  $\rm (FDD)$, and 
*$\text{Time Division Duplex}$  $\rm (TDD)$ . 

Such duplexing is necessary to ensure that uplink and downlink are clearly separated from each other and that transmission runs smoothly.  The diagram illustrates the difference between FDD based and TDD based transmission. 

Using the FDD and TDD methods, LTE can be operated in paired and unpaired frequency ranges.
 
The two methods are present opposing requirements:
*$\rm FDD$  requires a paired spectrum, i.e. one frequency band for transmission from the base station to the terminal ("downlink") and one for transmission in the opposite direction ("uplink").  Downlink and uplink can be used at the same time. 

*$\rm TDD$  was designed for unpaired spectra.  Now only one band is needed for uplink and downlink.  However, transmitter and receiver must now alternate during transmission.  The main problem of TDD is the required synchronicity of the networks. 

In the graphic above the differences between FDD and TDD can be seen. In TDD a  "Guard Period"  has to be inserted when changing from downlink to uplink (or vice versa) to avoid an overlapping of the signals. 

Although FDD is likely to be used more in practice  (and FDD frequencies were much more expensive for the providers), there are several reasons for TDD:
*Frequencies are a rare and expensive commodity, as the 2010 auction has shown.  But TDD needs only half of the frequency bandwidth. 

*The TDD technique allows different modes, which determine how much time should be used for downlink or uplink and can be adjusted to individual requirements. 

For the actual innovation, the  $\text{half–duplex}$  method, you need a paired spectrum as with FDD  (see the folloing graphic):
[[File:P ID2276 Mob T 4 2 S4b v1.png|right|frame|Transmission scheme for half-duplex|class=fit]]
*Base station transmitter and receiver still alternate like TDD.  Each terminal device can either transmit or receive at a given time.
*Through a second connection to another end device with swapped downlink/uplink grid, the entire available bandwidth can still be fully used. 

*The main advantage of the half–duplex process is that the use of the TDD concept reduces the demands on the end devices and thus allows them to be produced at a lower cost. 

The fact that this aspect was of great importance in the standardization can also be seen in the use of  "OFDMA"  in the downlink and of  "SC–FDMA"  in the uplink:
*This results in a longer battery life of the end devices and allows the use of cheaper components.
*More about this can be found in chapter  [[Mobile_Communications/The_Application_of_OFDMA_and_SC-FDMA_in_LTE | The Application of OFDMA and SC-FDMA in LTE]].

== Multiple Antenna Systems==
 
If a radio system uses several transmitting and receiving antennas, one speaks of  $\text{Multiple Input Multiple Output}$  $\rm (MIMO)$.  This is not an LTE specific development.  WLAN, for example, also uses this technology.

{{GraueBox|TEXT=
$\text{Example 1:}$  The principle of multiple antenna systems is illustrated in the following graphic using the example of 2×2–MIMO (two transmitting and two receiving antennas). 

[[File:EN_Mob_T_4_2_S3b.png|right|frame|The difference between SISO and MIMO|class=fit]]
*The new thing about LTE is not the actual use of  "Multiple Input Multiple Output", but the particularly intensive one, namely  2×2 MIMO  in the uplink and maximum  4×4 MIMO  in the downlink.

*In the successor  [[Mobile_Communications/LTE-Advanced - a Further Development of LTE|"LTE Advanced"]]  the use of MIMO is even more pronounced, namely "4×4" in the uplink and "8×8" in the opposite direction.}}

A MIMO system has advantages compared to  "Single Input Single Output"  (SISO, only one transmitting and one receiving antenna).  A distinction is made between several gains depending on the channel:
*Power gain  according to the number of receiving antennas:   If the radio signals arriving via several antennas are combined in a suitable way   ⇒   [https://en.wikipedia.org/wiki/Maximal-ratio_combining Maximal-ratio Combining], the reception power is increased and the radio connection is improved.  By doubling the antennas, a power gain of maximum 3 dB is achieved. 

*Diversity gain through  [https://en.wikipedia.org/wiki/Antenna_diversity "Spatial Diversity"]:  If several spatially separated receiving antennas are used in an environment with strong multipath propagation, the fading at the individual antennas is mostly independent from each other and the probability that all antennas are affected by fading at the same time is very low. 

*Data rate gain:   This increases the efficiency of MIMO, especially in an environment with increased multipath propagation, especially when transmitter and receiver do not have a direct line of sight and the transmission is done via reflections.  Tripling the number of antennas for the transmitter and receiver results in approximately twice the data rate. 

However, it is not possible for all advantages to occur simultaneously.  Depending on the nature of the channel, it can also happen that one does not even have the choice of which advantage one wants to use. 

In addition to the MIMO systems there are also the following intermediate stages:
*MISO systems  (only one receiving antenna, therefore no power gain is possible), and 

*SIMO systems  (only one transmitting antenna, therefore only small diversity gain). 

{{GraueBox|TEXT=
$\text{Example 2:}$  The term  "MIMO"  summarizes multi-antenna techniques with different properties, each of which can be useful in certain situations.  The following description is based on the four diagrams shown here. 

[[File:EN_Mob_T_4_2_S5b.png|right|frame|Four multi-antenna procedures with different properties|class=fit]]

*If the mostly independent channels of a MIMO system are assigned to a single user (top left diagram), one speaks of  "Single–User MIMO".  With 2×2 MIMO, the data rate is doubled compared to SISO operation and with four transmit and receiving antennas each, the data rate can be doubled again under good channel conditions. 

::LTE allows maximum 4×4 MIMO but only in the downlink.  Due to the complexity of multi-antenna systems, only laptops with LTE modems can be used as receivers (end devices) for 4×4 MIMO.  For a mobile phone, the use is generally limited to 2×2 MIMO.

*Contrary to Single–User MIMO, the goal with the  '''Multi–User MIMO'''  is not the maximum data rate for a receiver, but the maximization of the number of end devices that can use the network simultaneously  (top right diagram).
::This involves transmitting different data streams to different users.  This is particularly useful in places with high demand, such as airports or soccer stadiums. 

*Multi-antenna operation is not only used to maximize the number of users or data rate, but also in the event of poor transmission conditions, multiple antennas can combine their power to transmit data to a single user to improve the quality of reception.  One then speaks of  '''Beamforming'''   (diagram below left), which also increases the range of a transmitting station. 

*The fourth possibility is  '''Antenna diversity'''   (diagram below right).  This increases the redundancy (regarding the system design) and makes the transmission more robust against interferences.
::A simple example:   There are four channels that all transmit the same data.  If one channel fails, there are still three channels for information transport.}}

== System Architecture==
 
The LTE architecture enables a transmission system based entirely on the IP protocol.  In order to achieve this goal, the system architecture specified for UMTS not only had to be changed in detail, but in some cases completely redesigned.  In the process, other IP based technologies such as  "mobile WiMAX"  or  "WLAN"  were also integrated in order to be able to switch to these networks without any problems. 

[[File:EN_Mob_T_4_2_S6.png|right|frame|System Architecture for UMTS  $\rm (UTRAN)$  and LTE  $\rm (EUTRAN)$|class=fit]]

In UMTS networks (left graphic), the  "Radio Network Controller"  $\rm (RNC)$  is inserted between a base station ("NodeB") and the core network, which is mainly responsible for switching between different cells and which can lead to latency times of up to 100 milliseconds. 

*The redesign of the base stations  ("eNodeB"  instead of  "NodeB")  and the interface  "X2"  are the decisive further developments from UMTS towards LTE. 
*The graphic on the right illustrates in particular the reduction in complexity compared to UMTS that goes hand in hand with the new technology (left graphic).

The  $\text{LTE system architecture}$  can be divided into two major areas, too:
*the LTE core network  "Evolved Packet Core"  $\rm (EPC)$, 
*the air interface  "Evolved UMTS Terrestrial Radio Access Network"  $\rm (EUTRAN)$, a further development of  [[Examples_of_Communication_Systems/UMTS_Network_Architecture#Access_level_architecture|"UMTS Terrestrial Radio Access Network"]]  $\rm (UTRAN)$. 

EUTRAN transmits the data between the terminal and the LTE base station  ("eNodeB")  via the so-called  "S1"  interface with two connections, one for the transmission of user data and a second for the transmission of signalling data.  You can see from the above graphic:
*The base stations are connected not only to the EPC but also to the neighboring base stations.  These connections  ("X2"  interfaces)  have the effect that as few packets as possible are lost when the terminal device moves from the vicinity of one base station towards another. 

*For this purpose, the base station whose service area the user is just leaving can pass on any cached data directly and quickly to the "new" base station.  This ensures (largely) continuous transmission. 

*The functionality of the  "RNC"  is partly transferred to the base station and partly to the  "Mobility Management Entity"  $\rm (MME)$  in the core network.  This reduction of the interfaces significantly shortens the signal throughput time in the network and the handover to 20 milliseconds. 

*The LTE system architecture is also designed so that future  "Inter–NodeB procedures"  (such as  "Soft Handover"  or  "Cooperative Interference Cancellation")  can be easily integrated. 

== LTE core network:  Backbone and Backhaul ==
 
The LTE core network  "Evolved Packet Core"  $\rm (EPC)$  of a network operator  (in the technical language  "Backbone")  consists of various network components.  The EPC is connected to the base stations via the  "Backhaul".  This means the connection of an upstream, usually hierarchically subordinated network node to a central network node. 

Currently, the  "Backhaul"  consists mainly of directional radio and so-called  "E1" lines.  These are copper lines and allow a throughput of about 2 Mbit/s.  For GSM and UMTS networks these connections were still sufficient, however, for the large-scale conceived  [[Examples_of_Communication_Systems/Further Developments_of_UMTS#High.E2.80.93Speed_Downlink_Packet_Access| $\rm HSDPA$]]  such data rates are no longer adequate.  For LTE such a  "Backhaul"  is completely unusable:
*The slow cable network would slow down the fast wireless connections;  overall, there would be no increase in speed. 

*Due to the low capacities of the lines with "E1" standard, an expansion with further lines of the same construction would not be economical.

In the course of the introduction of LTE, the backhaul had to be redesigned.  It was important to keep an eye on future security, since the next generation  "LTE Advanced"  was already in place before the introduction.  If one believes the experts' propaganda  "Moore's Law" for mobile phone bandwidths, the most important factor for future security is the expensive new installation of better cables. 

Due to the purely packet-oriented transmission technology, the Ethernet standard, which is also IP based, is suitable for the LTE backhaul, which is realized with the help of optical fibers.  In 2009, the company Fujitsu presented in the study  [Fuj09]<ref name='Fuj09'>Fujitsu Network Communications Inc.:  4G Impacts to Mobile Backhaul.  [http://www.fujitsu.com/downloads/TEL/fnc/whitepapers/4Gimpacts.pdf PDF Internet document].</ref>,  also the thesis that the current infrastructure will continue to play an important role for LTE backhaul for the next ten to fifteen years. 

There are two approaches for the generation change to an Ethernet based  backhaul :
*the parallel operation of the lines with  "E1" and Ethernet standard, 

*the immediate migration to an Ethernet based  backhaul. 

The former would have the advantage that the network operators could continue to run voice traffic over the old lines and would only have to handle bandwidth-intensive data traffic over the more powerful lines.

The second option raises some technical problems:
*The services previously transported through the slow "E1" standard lines would have to be switched immediately to a packet-based procedure.

*Ethernet does not offer  (unlike the current standard)  any  end–to–end  synchronization, which can lead to severe delays or even service interruptions when changing radio cells, thus a huge loss of service quality.
* However, in the concept  [https://en.wikipedia.org/wiki/Synchronous_Ethernet "Synchronous Ethernet"]  $\rm (SyncE)$, the Cisco company has already made suggestions as to how synchronization could be realized. 

For conurbations, a direct conversion of the backhaul would certainly be worthwhile, as relatively few new cables would have to be laid for a comparatively high number of new users.

In rural areas, however, major excavation work would quickly result in high costs.  However, this is exactly the area which must be covered first, according to the  [[Mobile_Communications/General Information on the LTE Mobile Communications Standard#LTE frequency band splitting|agreement reached]]  between the federal government and the (German) mobile phone operators.  Here, the mostly existing microwave radio link would have to be (and probably will be) extended to high data rates. 

==Exercises for Chapter==
 
[[Aufgaben:Exercise 4.2: FDD, TDD and Half-Duplex]]

[[Aufgaben:Exercise 4.2Z: MIMO Applications in LTE]]

==References==

<references/>

{{Display}}

Information Theory/Further Source Coding Methods

2022-02-17T11:39:17Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Source coding - Data compression
|Vorherige Seite=Entropy Coding According to Huffman
|Nächste Seite=Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables
}}

==The Shannon-Fano algorithm==
 
The Huffman coding from 1952 is a special case of  "entropy coding".  It attempts to represent the source symbol  $q_μ$  by a code symbol  $c_μ$  of length  $L_μ$,  aiming for the following construction rule:

:$$L_{\mu} \approx -{\rm log}_2\hspace{0.15cm}(p_{\mu})
\hspace{0.05cm}.$$

Since  $-{\rm log}_2\hspace{0.15cm}(p_{\mu})$  is in contrast to  $L_μ$  not always an integer, this does not succeed in any case.

Already three years before David A. Huffman,  [https://en.wikipedia.org/wiki/Claude_Shannon Claude E. Shannon]  and  [https://en.wikipedia.org/wiki/Robert_Fano Robert Fano] gave a similar algorithm, namely:
:#   Order the source symbols according to decreasing symbol probabilities (identical to Huffman).
:#   Divide the sorted characters into two groups of equal probability.
:#   The binary symbol  '''1'''  is assigned to the first group,  '''0'''  to the second (or vice versa).
:#   If there is more than one character in a group, the algorithm is to be applied recursively to this group.

{{GraueBox|TEXT=
$\text{Example 1:}$  As in the  [[Information_Theory/Entropiecodierung_nach_Huffman#The_Huffman_algorithm|introductory example for the Huffman algorithm]]  in the last chapter, we assume  $M = 6$  symbols and the following probabilities:

:$$p_{\rm A} = 0.30 \hspace{0.05cm},\hspace{0.2cm}p_{\rm B} = 0.24 \hspace{0.05cm},\hspace{0.2cm}p_{\rm C} = 0.20 \hspace{0.05cm},\hspace{0.2cm}
p_{\rm D} = 0.12 \hspace{0.05cm},\hspace{0.2cm}p_{\rm E} = 0.10 \hspace{0.05cm},\hspace{0.2cm}p_{\rm F} = 0.04
\hspace{0.05cm}.$$

Then the Shannon-Fano algorithm is:
:#   $\rm AB$   →   '''1'''x  (probability 0.54),   $\rm CDEF$   →   '''0'''x  (probability 0.46),
:#   $\underline{\rm A}$   →   '''11'''  (probability 0.30),   $\underline{\rm B}$   →   '''10'''  (probability 0.24),
:#   $\underline{\rm C}$   →   '''01'''  (probability 0.20),   $\rm DEF$ → '''00'''x,  (probability 0.26),
:#   $\underline{\rm D}$   →   '''001'''  (probability 0.12),   $\rm EF$   →   '''000'''x  (probability 0.14),
:#   $\underline{\rm E}$   →   '''0001'''  (probability 0.10),   $\underline{\rm F}$   →   '''0000'''  (probability 0.04).

''Notes'':
*An „x” again indicates that bits must still be added in subsequent coding steps.
*This results in a different assignment than with  [[Information_Theory/Entropiecodierung_nach_Huffman#The_Huffman_algorithm|Huffman coding]], but exactly the same average codeword length:

:$$L_{\rm M} = (0.30\hspace{-0.05cm}+\hspace{-0.05cm} 0.24\hspace{-0.05cm}+ \hspace{-0.05cm}0.20) \hspace{-0.05cm}\cdot\hspace{-0.05cm} 2 + 0.12\hspace{-0.05cm} \cdot \hspace{-0.05cm} 3 + (0.10\hspace{-0.05cm}+\hspace{-0.05cm}0.04) \hspace{-0.05cm}\cdot \hspace{-0.05cm}4 = 2.4\,{\rm bit/source\hspace{0.15cm} symbol}\hspace{0.05cm}.$$}}

With the probabilities corresponding to  $\text{Example 1}$,  the Shannon-Fano algorithm leads to the same avarage codeword length as Huffman coding.  Similarly, for most other probability profiles, Huffman and Shannon-Fano are equivalent from an information-theoretic point of view.

However, there are definitely cases where the two methods differ in terms of (mean) codeword length, as the following example shows.

{{GraueBox|TEXT=
$\text{Example 2:}$  We consider  $M = 5$  symbols with the following probabilities:
:$$p_{\rm A} = 0.38 \hspace{0.05cm}, \hspace{0.2cm}p_{\rm B}= 0.18 \hspace{0.05cm}, \hspace{0.2cm}p_{\rm C}= 0.16 \hspace{0.05cm},\hspace{0.2cm}
p_{\rm D}= 0.15 \hspace{0.05cm}, \hspace{0.2cm}p_{\rm E}= 0.13 \hspace{0.3cm}
\Rightarrow\hspace{0.3cm} H = 2.19\,{\rm bit/source\hspace{0.15cm} symbol}
\hspace{0.05cm}. $$

[[File:EN_Inf_T_2_4_S1.png|right|frame|Tree structures according to Shannon-Fano and Huffman]]

The diagram shows the respective code trees for Shannon-Fano (left) and Huffman (right).  The results can be summarised as follows:
* The Shannon-Fano algorithm leads to the code
:    $\rm A$   →   '''11''',   $\rm B$   →   '''10''',   $\rm C$   →   '''01''',   $\rm D$   →   '''001''',   $\rm E$   →   '''000''' 
:and thus to the mean code word length

:$$L_{\rm M} = (0.38 + 0.18 + 0.16) \cdot 2 + (0.15 + 0.13) \cdot 3 $$
:$$\Rightarrow\hspace{0.3cm} L_{\rm M} = 2.28\,\,{\rm bit/source\hspace{0.15cm} symbol}\hspace{0.05cm}.$$

*Using "Huffman", we get 
:    $\rm A$   →   '''1''',   $\rm B$   →   '''001''',   $\rm C$   →   '''010''',   $\rm D$   →   '''001''',   $\rm E$   →   '''000''' 
:and a slightly smaller codeword length:

:$$L_{\rm M} = 0.38 \cdot 1 + (1-0.38) \cdot 3 $$
:$$\Rightarrow\hspace{0.3cm} L_{\rm M} = 2.24\,\,{\rm bit/source\hspace{0.15cm} symbol}\hspace{0.05cm}. $$

*There is no set of probabilities for which  "Shannon–Fano"  provides a better result than  "Huffman",  which always provides the best possible entropy encoder.
*The graph also shows that the algorithms proceed in different directions in the tree diagram, namely once from the root to the individual symbols  (Shannon–Fano), and secondly from the individual symbols to the root  (Huffman).}}

The (German language) interactive applet  [[Applets:Huffman_Shannon_Fano|Huffman- und Shannon-Fano-Codierung  ⇒   $\text{SWF}$ version]]  illustrates the procedure for two variants of entropy coding.

==Arithmetic coding ==
 
Another form of entropy coding is arithmetic coding.  Here, too, the symbol probabilities  $p_μ$  must be known.  For the index applies  $μ = 1$, ... ,  $M$.

Here is a brief outline of the procedure:
*In contrast to Huffman and Shannon-Fano coding, a symbol sequence of length  $N$  is coded together in arithmetic coding.  We write abbreviated  $Q = 〈\hspace{0.05cm} q_1, q_2$, ... , $q_N \hspace{0.05cm} 〉$.
*Each symbol sequence  $Q_i$  is assigned a real number interval  $I_i$  which is identified by the beginning  $B_i$  and the interval width  ${\it Δ}_i$ .
*The  "code"  for the sequence  $Q_i$  is the binary representation of a real number value from this interval:   $r_i ∈ I_i = \big [B_i, B_i + {\it Δ}_i\big)$.  This notation says that  $B_i$  belongs to the interval  $I_i$    (square bracket), but  $B_i + {\it Δ}_i$  just does not  (round bracket).
*It is always  $0 ≤ r_i < 1$.  It makes sense to select  $r_i$  from the interval  $I_i$  in such a way that the value can be represented with as few bits as possible.  However, there is always a minimum number of bits, which depends on the interval width  ${\it Δ}_i$.

The algorithm for determining the interval parameters  $B_i$  and  ${\it Δ}_i$  is explained later in  $\text{Example 4}$ , as is a decoding option.
*First, there is a short example for the selection of the real number   $r_i$  with regard to the minimum number of bits.
*More detailed information on this can be found in the description of  [[Aufgaben:Aufgabe_2.11Z:_Nochmals_Arithmetische_Codierung|Exercise 2.11Z]].

{{GraueBox|TEXT=
$\text{Example 3:}$  For the two parameter sets of the arithmetic coding algorithm listed below, yields the following real results  $r_i$  and the following codes belong to the associated interval  $I_i$ :
* $B_i = 0.25, {\it Δ}_i = 0.10 \ ⇒ \ I_i = \big[0.25, 0.35\big)\text{:}$

:$$r_i = 0 \cdot 2^{-1} + 1 \cdot 2^{-2} = 0.25 \hspace{0.3cm}\Rightarrow\hspace{0.3cm}
{\rm Code} \hspace{0.15cm} \boldsymbol{\rm 01} \in I_i
\hspace{0.05cm},$$

* $B_i = 0.65, {\it Δ}_i = 0.10 \ ⇒ \ I_i = \big[0.65, 0.75\big);$  note:   $0.75$  does not belong to the interval:

:$$r_i = 1 \cdot 2^{-1} + 0 \cdot 2^{-2} + 1 \cdot 2^{-3} + 1 \cdot 2^{-4} = 0.6875 \hspace{0.3cm}\Rightarrow\hspace{0.3cm}
{\rm Code} \hspace{0.15cm} \boldsymbol{\rm 1011} \in I_i\hspace{0.05cm}. $$

However, to organise the sequential flow, one chooses the number of bits constant to  $N_{\rm Bit} = \big\lceil {\rm log}_2 \hspace{0.15cm} ({1}/{\it \Delta_i})\big\rceil+1\hspace{0.05cm}. $
*With the interval width  ${\it Δ}_i = 0.10$  results  $N_{\rm Bit} = 5$.
*So the actual arithmetic codes would be   '''01000'''   and   '''10110''' respectively.}}

{{GraueBox|TEXT=
$\text{Example 4:}$  Now let the symbol set size be  $M = 3$  and let the symbols be denoted by  $\rm X$,  $\rm Y$  and  $\rm Z$:
*The character sequence  $\rm XXYXZ$   ⇒   length of the source symbol sequence:   $N = 5$.
*Assume the probabilities  $p_{\rm X} = 0.6$,  $p_{\rm Y} = 0.2$  und  $p_{\rm Z} = 0.2$.

[[File:EN_Inf_T_2_4_S2.png|right|frame|About the arithmetic coding algorithm]]

The diagram on the right shows the algorithm for determining the interval boundaries.
*First, the entire probability range  $($between  $0$  and  $1)$  is divided according to the symbol probabilities  $p_{\rm X}$,  $p_{\rm Y}$  and  $p_{\rm Z}$  into three areas with the boundaries  $B_0$,  $C_0$,  $D_0$  and  $E_0$.
*The first symbol present for coding is  $\rm X$.  Therefore, in the next step, the probability range from  $B_1 = B_0 = 0$  to  $E_1 = C_0 = 0.6$  is again divided in the ratio  $0.6$  :  $0.2$  :  $0.2$.
*After the second symbol  $\rm X$ , the range limits are  $B_2 = 0$,  $C_2 = 0.216$,  $D_2 = 0.288$  and  $E_2 = 0.36$.  Since the symbol  $\rm Y$  is now pending, the range is subdivided between  $0.216$ ... $0.288$.
*After the fifth symbol  $\rm Z$  , the interval  $I_i$  for the considered symbol sequence  $Q_i = \rm XXYXZ$  is fixed.  A real number  $r_i$  must now be found for which the following applies::   $0.25056 ≤ r_i < 0.2592$.
*The only real number in the interval  $I_i = \big[0.25056, 0.2592\big)$, that can be represented with seven bits is 
:$$r_i = 1 · 2^{–2} + 1 · 2^{–7} = 0.2578125.$$
*Thus the coder output is fixed:   '''0100001'''.}}

Seven bits are therefore needed for these  $N = 5$  symbols, exactly as many as with Huffman coding with the assignment $\rm X$   →   '''1''', $\rm Y$   →   '''00''',   $\rm Z$   →   '''01'''.
*However, arithmetic coding is superior to Huffman coding when the actual number of bits used in Huffman deviates even more from the optimal distribution, for example, when a character occurs extremely frequently.
*Often, however, only the middle of the interval – in the example  $0.25488$ – is represented in binary:   '''0.01000010011''' .... The number of bits is obtained as follows:

:$${\it Δ}_5 = 0.2592 - 0.25056 = 0.00864 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}N_{\rm Bit} = \left\lceil {\rm log}_2 \hspace{0.15cm} \frac{1}{0.00864} \right\rceil + 1\hspace{0.15cm} =
\left\lceil {\rm log}_2 \hspace{0.15cm} 115.7 \right\rceil + 1 = 8
\hspace{0.05cm}.$$

*Thus the arithmetic code for this example with  $N = 5$  input characters is:   '''01000010'''.

*The decoding process can also be explained using the above graphic. The incoming bit sequence  '''0100001'''  is converted to  $r = 0.2578125$ .
*This lies in the first and second step respectively in the first area   ⇒   symbol $\rm X$, in the third step in the second area   ⇒   symbol $\rm Y$,  and so on.

Further information on arithmetic coding can be found in  [https://en.wikipedia.org/wiki/Arithmetic_coding WIKIPEDIA]  and in  [BCK02]<ref> Bodden, E.; Clasen, M.; Kneis, J.:  Algebraische Kodierung.  Algebraic Coding. Proseminar, Chair of Computer Science IV, RWTH Aachen University, 2002.</ref>.

==Run–Length coding ==
 
We consider a binary source  $(M = 2)$  with the symbol set  $\{$ $\rm A$,  $\rm B$ $\}$,  where one symbol occurs much more frequently than the other.  For example, let  $p_{\rm A} \gg p_{\rm B}$.

*Entropy coding only makes sense here when applied to  $k$–tuples.
*A second possibility is  '''Run-length Coding'''  $\rm (RLC)$, which considers the rarer character  $\rm B$  as a separator and returns the lengths  $L_i$  of the individual sub-strings  $\rm AA\text{...}A$  as a result.

[[File:P_ID2470__Inf_T_2_4_S4_neu.png|right|frame|To illustrate the run-length coding]]
{{GraueBox|TEXT=
$\text{Example 5:}$  The graphic shows an example sequence
*with the probabilities  $p_{\rm A} = 0.9$  and  $p_{\rm B} = 0.1$   ⇒   source entropy $H = 0.469$ bit/source symbol.

The example sequence of length  $N = 100$  contains the symbol  $\rm B$  exactly ten times and the symbol  $\rm A$ ninety times, i.e. the relative frequencies here correspond exactly to the probabilities.
 
You can see from this example:
*The run-length coding of this sequence results in the sequence  $ \langle \hspace{0.05cm}6, \ 14, \ 26, \ 11, \ 4, \ 10, \ 3,\ 9,\ 1,\ 16 \hspace{0.05cm} \rangle $.
*If one represents the lengths  $L_1$, ... , $L_{10}$  with five bits each, one thus requires  $5 · 10 = 50$  bits.
*The RLC data compression is thus not much worse than the theoretical limit that results according to the source entropy to  $H · N ≈ 47$  bits.
*The direct application of entropy coding would not result in any data compression here; rather, one continues to need  $100$  bits.
*Even with the formation of triples,  $54$  bits would still be needed with Huffman, i.e. more than with run-length coding.}}

However, the example also shows two problems of run-length coding:
*The lengths  $L_i$  of the substrings are not limited.  Special measures must be taken here if a length  $L_i$  is greater than  $2^5 = 32$  $($valid fo  $N_{\rm Bit} = 5)$, for example the variant  '''Run–Length Limited Coding'''  $\rm (RLLC)$.  See also  [Meck09]<ref>Mecking, M.: Information Theory.  Lecture manuscript, Chair of Communications Engineering, Technical University of Munich, 2009.</ref>  and  [[Aufgaben:Aufgabe_2.13:_Burrows-Wheeler-Rücktransformation|Exercise 2.13]].
*If the sequence does not end with  $\rm B$  – which is rather the normal case with small probability  $p_{\rm B}$  one must also provide special treatment for the end of the file.

==Burrows–Wheeler transformation==
 
To conclude this source coding chapter, we briefly discuss the algorithm published in 1994 by  [https://en.wikipedia.org/wiki/Michael_Burrows Michael Burrows]  and  [https://en.wikipedia.org/wiki/David_Wheeler_(computer_scientist) David J. Wheeler]  [BW94]<ref>Burrows, M.; Wheeler, D.J.:  A Block-sorting Lossless Data Compression Algorithm.  Technical Report. Digital Equipment Corp. Communications, Palo Alto, 1994.</ref>,
[[File:EN_Inf_T_2_4_S3_vers2.png|frame|Example of the BWT (forward transformation)]]

*which, although it has no compression potential on its own,
*but it greatly improves the compression capability of other methods.

 The Burrows–Wheeler Transformation accomplishes a blockwise sorting of data, which is illustrated in the diagram using the example of the text  $\text{ANNAS_ANANAS}$  (meaning:  Anna's pineapple)  of length  $N = 12$:

*First, an  $N×N$ matrix is generated from the string of length  $N$  with each row resulting from the preceding row by cyclic left shift.
*Then the BWT matrix is sorted lexicographically.  The result of the transformation is the last column  ⇒   $\text{L–Spalte}$.  In the example, this results in  $\text{_NSNNAANAAAS}$.
*Furthermore, the primary index  $I$  must also be passed on. This value indicates the row of the sorted BWT matrix that contains the original text  (marked red in the graphic).
*Of course, no matrix operations are necessary to determine the  $\text{L–column}$  and the primary index.  Rather, the BWT result can be found very quickly with pointer technology.
 
{{BlaueBox|TEXT=
$\text{Furthermore, it should be noted about the BWT procedure:}$ 

*Without an additional measure   ⇒   a downstream „real compression” – the BWT does not lead to any data compression.  
*Rather, there is even a slight increase in the amount of data, since in addition to the  $N$  characters, the primary index  $I$  now also be transmitted.
*For longer texts,  however,  this effect is negligible.  Assuming 8 bit–ASCII–characters  (one byte each)  and the block length  $N = 256$  the number of bytes per block only increases from  $256$  to  $257$, i.e. by only  $0.4\%$.}}

We refer to the detailed descriptions of BWT in  [Abel04]<ref>Abel, J.:  Grundlagen des Burrows-Wheeler-Kompressionsalgorithmus.  In: Informatik Forschung & Entwicklung, no. 2, vol. 18, S. 80-87, Jan. 2004 </ref>.

[[File:EN_Inf_T_2_4_S3b.png|frame|Example for BWT (reverse transformation)]]

Finally, we will show how the original text can be reconstructed from the  $\text{L}$–column  (from  "Last")  of the BWT matrix.
* For this, one still needs the primary index $I$, as well as the first column of the BWT matrix.
*This  $\text{F}$–column  (from  "First")  does not have to be transferred, but results from the  $\text{L}$–column very simply through lexicographic sorting.

The graphic shows the reconstruction procedure for the example under consideration:
*One starts in the line with the primary index  $I$.  The first character to be output is the  $\rm A$  marked in red in the  $\text{F}$–column.  This step is marked in the graphic with a yellow (1).
*This  $\rm A$  is the third  $\rm A$ character in the  $\text{F}$–column.  Now look for the third  $\rm A$  in the  $\text{L}$–column,  find it in the line marked with  '''(2)'''  and output the corresponding  '''N'''  of the  $\text{F}$–column.
*The last  '''N'''  of the  $\text{L}$–column is found in line  '''(3)'''.  The character of the F column is output in the same line, i.e. an  '''N''' again.

After  $N = 12$  decoding steps, the reconstruction is completed.
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*This example has shown that the  Burrows–Wheeler transformation is nothing more than a sorting algorithm for texts.  What is special about it is that the sorting is uniquely reversible.

*This property and additionally its inner structure are the basis for compressing the BWT result by means of known and efficient methods such as  [[Information_Theory/Entropiecodierung_nach_Huffman|Huffman]]  (a form of entropy coding) and  [[Information_Theory/Further_Source_Coding_Methods#Run.E2.80.93Length_coding|run–length coding ]].}}

==Application scenario for the Burrows-Wheeler transformation==
 
As an example for embedding the  [[Information_Theory/Weitere_Quellencodierverfahren#Burrows.E2.80.93Wheeler_Transformation|Burrows–Wheeler Transformation]]  (BWT) into a chain of source coding methods, we choose a structure proposed in  [Abel03]<ref>Abel, J.:  Verlustlose Datenkompression auf Grundlage der Burrows-Wheeler-Transformation.  In: PIK - Praxis der Informationsverarbeitung und Kommunikation, no. 3, vol. 26, pp. 140-144, Sept. 2003.</ref>.  We use the same text example  $\text{ANNAS_ANANAS}$  as on the last page.  The corresponding strings after each block are also given in the graphic.

[[File:EN_Inf_T_2_4_S5_v2.png|center|frame|Scheme for Burrows-Wheeler data compression]]

*The  $\rm BWT$  result is:     $\text{_NSNNAANAAAS}$.  BWT has not changed anything about the text length  $N = 12$  but there are now four characters that are identical to their predecessors  (highlighted in red in the graphic).  In the original text, this was only the case once.
*In the next block  $\rm MTF$  ("Move–To–Front") , each input character from the set  $\{$ $\rm A$,  $\rm N$,  $\rm S$,  '''_'''$\}$  becomes an index  $I ∈ \{0, 1, 2, 3\}$.  However,  this is not a simple mapping, but an algorithm given in  [[Exercise_2.13Z:_Combination_of_BWT_and_MTF|Exercise 1.13Z]].
*For our example, the MTF output sequence is  $323303011002$, also with length  $N = 12$.  The four zeros in the MTF sequence  (also in red font in the diagram)  indicate that at each of these positions the BWT character is the same as its predecessor.
*In large ASCII files, the frequency of the  $0$  may well be more than  $50\%$,  while the other  $255$  indices occur only rarely.  Run-length coding  $\rm (RCL)$  is an excellent way to compress such a text structure.
*The block  $\rm RCL0$  in the above coding chain denotes a special  [[Information_Theory/Further_Source_Coding_Methods#Run.E2.80.93Length_coding|run-length coding]]  for zeros.  The grey shading of the zeros indicates that a long zero sequence has been masked by a specific bit sequence   (shorter than the zero sequence).
*The entropy encoder  $\rm (EC$, for example "Huffman"$)$  provides further compression.  BWT  and  MTF  only have the task in the coding chain of increasing the efficiency of  "RLC0"  and  "EC"  through character preprocessing.  The output file is again binary.

==Exercises for the chapter ==
 
[[Aufgaben:Exercise_2.10:_Shannon-Fano_Coding|Exercise 2.10: Shannon-Fano Coding]]

[[Aufgaben:Exercise_2.11:_Arithmetic_Coding|Exercise 2.11: Arithmetic Coding]]

[[Aufgaben:Exercise_2.11Z:_Arithmetic_Coding_once_again|Exercise 2.11Z: Arithmetic Coding once again]]

[[Exercise_2.12:_Run–Length_Coding_and_Run–Length_Limited_Coding|Exercise 2.12: Run–Length Coding and Run–Length Limited Coding]]

[[Aufgaben:Exercise_2.13:_Inverse_Burrows-Wheeler_Transformation|Exercise 2.13: Inverse Burrows-Wheeler Transformation]]

[[Exercise_2.13Z:_Combination_of_BWT_and_MTF|Exercise 2.13Z: Combination of BWT and MTF]]

==References==
<references />

{{Display}}

Mobile Communications/Physical Layer for LTE

2022-02-17T11:39:17Z

Bene: Text replacement - "List of Sources" to "References"

{{Header
|Untermenü=LTE – Long Term Evolution
|Vorherige Seite=The Application of OFDMA and SC-FDMA in LTE
|Nächste Seite=LTE-Advanced - a Further Development of LTE
}}

== General Description==
 
The  "Physical Layer"  is the lowest layer in the OSI layer model of the  "International Organization for Standardization"  $\rm (ISO)$, which is also called  "Bit Transmission Layer".  It describes the physical transmission of bit sequences in LTE and the operation of the various channels according to the 3GPP specification.  All specifications are valid for Frequency Division Duplex  $\rm (FDD)$  as well as for Time Division Duplex  $\rm (TDD)$.

[[File:EN_LTE_T_4_4_S1b_v1.png|right|frame|Protocol architecture for LTE|class=fit]]
The diagram shows the layers of the LTE protocol architecture.  The communication between the individual layers takes place via three different types of channels:
*Logical channels, 
*Transport channels, 
*Physical channels. 

This chapter deals with the communication between transmitter and receiver in the lowest  (red highlighted)  "Physical Layer".  Basically it should be noted:
*Exactly like the Internet, LTE uses exclusively packet-based transmission, i.e. without specifically assigning resources to a single user. 

*The design of the LTE physical layer is therefore characterized by the principle of dynamically allocated network resources. 

*The physical layer plays a key role in the efficient allocation and utilization of available system resources.

According to this graphic the physical layer communicates with
*the block  "Medium Access Control"  $\rm (MAC)$  and exchanges information about the users and the regulation (control) of the network via the transport channels, 

*the block  "Radio Resource Control"  $\rm (RRC)$,  where control commands and measurements are continuously exchanged to adapt the transmission to the channel quality. 

[[File:EN_LTE_T_4_4_S1.png|left|frame|Communication between the individual layers in the LTE downlink|class=fit]]
 
The complexity of the LTE transmission is to be indicated by the left diagram, which has been directly adopted by the  "European Telecommunications Standards Institute"  $\rm (ETSI)$.  It shows the communication between the individual layers (channels) and applies exclusively to the downlink. 

*On the following pages we will take a closer look at the physical layer and the physical channels.  We distinguish between uplink and downlink, but we will limit ourselves to the essentials.

*In reality, the individual channels take over a number of other functions, but their description would go beyond the scope of this tutorial.

*If you are interested, you can find a detailed description in  [HT09]<ref name= 'HT09'>Holma, H.; Toskala, A.:  LTE for UMTS - OFDMA and SC-FDMA Based Radio Access.  Wiley & Sons, 2009.</ref>.
 
== Physical channels in uplink==
 
LTE uses the multiple access method  [[Mobile_Communications/The_Application_of_OFDMA_and_SC-FDMA_in_LTE#Functionality_of_SC.E2.80.93FDMA|$\rm SC–FDMA$]]  in the uplink transmission from the terminal device to the base station.  Accordingly, the following physical channels exist in the 3GPP specification:
*Physical Uplink Shared Channel  $\rm (PUSCH)$, 
*Physical Random Access Channel  $\rm (PRACH)$, 
*Physical Uplink Control Channel  $\rm (PUCCH)$. 

The user data are transmitted in the physical channel  $\rm (PUSCH)$.  The transmission speed depends on how much bandwidth is available to the user at that moment.  The transmission is based on dynamically allocated resources in time and frequency range with a resolution of one millisecond or 180 kHz.  This allocation is performed by the [[Mobile_Communications/Physical_Layer_for_LTE# Scheduling for LTE| Scheduler]]  in the base station  ("eNodeB").  A terminal device cannot transmit any data without instructions from the base station. 

The exception is the use of the physical channel  $\rm PRACH$, the only channel in the LTE uplink with non–synchronized transmission.  The function of this channel is the request for permission to send data via one of the other two physical channels.  By sending a  "Cyclic Prefix"  and a signature on the PRACH, the terminal and base station are synchronized and are thus ready for further transmissions. 

The third uplink channel  $\rm PUCCH$  is used exclusively for the transmission of control signals. By this one understands
*positive and negative acknowledgements of receipt  $\rm (ACK/NACK)$, 
*requests for repeated transmission  $($in case of  $\rm NACK)$, and 
*the exchange of channel quality information between the terminal and the base station. 

If, in addition to the control data, user data are sent from the terminal to the base station at the same time, such control signals can also be transmitted via the PUSCH.  If no user data is to be transmitted, PUCCH is used instead. 

A simultaneous use of PUSCH and PUCCH is not possible due to restrictions of the transmission method  "SC–FDMA".  If only one  "Shared Channel"  had been selected for all control information, one would have had to choose between
*intermittent problems with user data transmission, or 
*permanently too few resources for the control information. 

The information about the channel quality is obtained by means of so-called "reference symbols".  As indicators for the channel quality this information is then sent to
*  "Channel Quality Indicator"  $\rm (CQI)$, and 

*  "Rank Indicator"  $\rm (RI)$. 

A detailed explanation of the quality guarantee can be found, for example, in  [HR09]<ref name='HR09'>Homayounfar, K.; Rohani, B.:  CQI Measurement and Reporting in LTE: A New Framework. 
IEICE Technical Report, Vol. 108, No. 445, 2009.</ref>  and  [HT09]<ref name='HT09'></ref>. 

{{GraueBox|TEXT=
$\text{Example 1:}$  The reference symbols or channel quality information are distributed in the PUSCH according to the following graphic.

[[File:EN_LTE_T_4_4_S2.png|right|frame|Distribution of reference symbols and user data in PUSCH|class=fit]]

The graphic describes the arrangement of the useful information and the signaling data in a "virtual subcarrier".
*"Virtual" because SC–FDMA does not have subcarriers like OFDMA.

*The reference symbols are necessary to estimate the channel quality. 

*This information is also transferred as  "Channel Quality Indicator"  $\rm (CQI)$  or as  "Rank Indicator"  $\rm (RI)$  via the PUSCH.}} 

== Physical channels in downlink==
 
In contrast to the uplink, LTE uses the multiple access method  [[Mobile_Communications/The_Application_of_OFDMA_and_SC-FDMA_in_LTE#Differences_between_OFDMA_and_SC.E2.80.93FDMA|$\rm OFDMA$]]  in the downlink, i.e. during transmission from the base station to the terminal.  Accordingly, the 3GPP consortium specified the following physical channels for this purpose:
*Physical Downlink Shared Channel  $\rm (PDSCH)$, 

* Physical Downlink Control Channel  $\rm (PDCCH)$, 

* Physical Control Format Indicator Channel  $\rm (PCFICH)$, 

* Physical Hybrid ARQ Indicator Channel  $\rm (PHICH)$, 

* Physical Broadcast Channel  $\rm (PBCH)$, 

* Physical Multicast Channel  $\rm (PMCH)$. 

The user data are transmitted via the  $\rm PDSCH$.  The resource allocation is done both in the time domain  (with a resolution of one millisecond)  and in the frequency domain  (resolution:  180 kHz).  Due to the use of OFDMA as transmission method, the individual speed of each user depends on the number of assigned resource blocks (à 180 kHz).  A "eNodeB" allocates the resources related to the channel quality of each individual user. 

The  $\rm PDCCH$  contains all information regarding the allocation of resource blocks or bandwidth for both the uplink and the downlink.  A terminal device thereby receives information about how many resources are available. 

[[File:EN_LTE_T_4_4_S3.png|right|frame|Division between PDCCH and PDSCH in the LTE downlink]]
The diagram on the right shows an example of the division between the channels PDCCH and PDSCH:
*The PDCCH can occupy up to four symbols per subframe  (in the graphic:  two). 
*This leaves twelve time slots for the user data  (i.e. for the channel PDSCH).

Via channel  $\rm PCFICH$  the terminal device is informed how many symbols are to be assigned to the control information of the PDCCH.  The purpose of this dynamic division between control and user data is as follows:
*For example, many users can be supported in this way, each with a low data rate.  This scenario requires more tuning, which means that in this case the PDCCH would have to contain three or four symbols. 

*On the other hand, the overhead caused by PDCCH can be reduced by assigning a high data rate to only a few concurrent users.

[[File:EN_LTE_T_4_4_S3b.png|left|frame|Distribution of reference symbols in the LTE downlink|class=fit]]
 
In addition to the  $\rm PDCCH$,  reference symbols are also required in the downlink to estimate the channel quality and calculate the  "Channel Quality Indicator"  (CQI). . These reference symbols are distributed over the subcarriers (different frequencies) or symbols (different times) as shown in the left graphic.
 
Regarding the other physical channels of the LTE downlink is to be noted:
*The only purpose of the downlink channel  $\rm PHICH$  ("Physical Hybrid ARQ Indicator Channel")  is to signal whether a packet sent in the uplink has arrived. 

*On the broadcast channel  $\rm PBCH$  ("Physical Broadcast Channel")  the base stations send system information with operating parameters as well as synchronization signals, which are required for registration in the network, to all mobile terminals in the radio cell approximately every 40 milliseconds. 

*The multicast channel  $\rm PMCH$  ("Physical Multicast Channel")  has a similar purpose, information for so-called multicast transmissions is sent to several receivers simultaneously through this channel. This could be, for example, mobile television via LTE, which is planned for a future release, or something similar. 

== Processes on the physical layer==
 
By  "Processes in the physical layer"  one understands different methods and procedures, which are used in the bit transmission layer. Among them fall among other things:
:*Timing Advance, 
:*Paging, 
:*Random Access, 
:*Channel Feedback Reporting, 
:*Power Control, 
:*Hybrid Adaptive Repeat and Request. 

A complete list with the corresponding description can be found in  [HT09]<ref name='HT09'></ref>.  Only the last two procedures will be discussed in more detail here. 

== Power control with LTE==
 
By  "Power Control"  one understands generally the control of the transmission power with the goal,
*to improve the transmission quality, 
*to increase the network capacity, and 
*to reduce the power consumption. 

With regard to the last point, the standardization of LTE had to take this into account:
*On the one hand, the power consumption in the end devices was to be minimized in order to guarantee longer battery runtimes for them. 
*On the other hand, it should be avoided that the base stations have to provide too much power. 

With LTE,  power control  is only applied in the uplink, whereas it is more of an "slow" power control.  This means that the procedure specified in LTE does not have to react as quickly as for example in UMTS ("W–CDMA" ).  The reason is that by using the orthogonal carrier system "SC–FMDA" the so-called  [[Examples_of_Communication_Systems/Nachrichtentechnische_Aspekte_von_UMTS#Near.E2.80.93Far.E2.80.93Effekt| Near–Far problem]] does not exist.

*To be precise, for LTE the power control does not control the absolute power, but the spectral power density, i.e. the power per bandwidth. 

*Instead of trying to smooth power peaks by temporarily reducing the transmission power, power peaks can also be used to increase the data rate for a short time. 

All in all,  LTE power control is intended to find the optimum balance between the lowest possible power and at the same time interference that is still acceptable for the transmission quality (QoS).  This is specifically achieved by estimating the loss during transmission and by calculating a correction factor according to the current site characteristics.  The statements made here are largely taken from  [DFJ08]<ref name ='DFJ08'>Dahlman, E., Furuskär A., Jading Y., Lindström M., Parkvall, S.:  Key Features of the LTE Radio Interface.  Ericsson Review No. 2, 2008.</ref>.

== Hybrid Adaptive Repeat and Request ==
 
Every communication system needs a scheme for retransmission of lost data due to transmission errors to ensure sufficient transmission quality.  In LTE  "Hybrid Adaptive Repeat and Request"  $\rm (HARQ)$  was specified for this purpose.  This procedure is also used in  [[Examples_of_Communication_Systems/Weiterentwicklungen_von_UMTS#HARQ.E2.80.93Verfahren_und_Node_B_Scheduling| UMTS]]  in a similar form. 

The procedure based on the  "stop–and–wait"  technique is as follows:
*After a terminal device has received a packet from the base station, it is decoded and feedback is sent via the  [[Mobile_Communications/Physical_Layer_for_LTE#Physical channels in uplink| $\rm PUCCH$]]. 
*In case of a failed transmission  ("NACK")  the packet is resent.  Only if the transmission was successful  (Feedback:  "ACK"), the next packet is sent. 

In order to ensure continuous data transfer despite the  "stop–and–wait"  procedure,  LTE requires several simultaneous HARQ processes.  In LTE, eight parallel processes are used both in the uplink and in the downlink. 

{{GraueBox|TEXT=
$\text{Example 2:}$  The graphic illustrates how it works with eight simultaneous HARQ processes:

[[File:EN_LTE_T_4_4_S4a.png|right|frame|HARQ in LTE with eight simultaneous processes|class=fit]]
*In this example, the first process fails in the first attempt to transfer packet  '''1'''.
*The receiver tells this "Fail" to the transmitter by a "NACK".  In contrast, the second parallel process is successful with its first packet:   "Pass". 
*In the next step  (i.e. after the other seven HARQ processes have sent)  the first HARQ retransmits its last sent packet due to the acknowledgement "NACK". 

*The second process sends a new packet due to the acknowledgement "ACK" now. 

The other processes, which were ignored in this example, proceed in the same way.}} 

== Modulation for LTE ==
 
LTE uses the modulation method  [[Modulation_Methods/Quadratur%E2%80%93Amplitudenmodulation#Allgemeine_Beschreibung_und_Signalraumzuordnung|Quadrature Amplitude Modulation]]  $\rm (QAM)$.  Different variants are available in the uplink as well as in the downlink, namely

[[File:P ID2290 LTE T 4 4 S5a v2.png|right|frame|Possible QAM signal space constellations in LTE|class=fit]]

*4–QAM  (identical to QPSK)   ⇒   two bits per symbol, 
*16–QAM   ⇒   four bits per symbol, 
*64–QAM   ⇒   six bits per symbol. 

Note:   QAM is not an LTE specific development, but is also used in many already established wired transmission methods, such as those of  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL|$\rm DSL$]]  ("Digital Subscriber Line").

[[File:P ID2291 LTE T 4 4 S5b v1.png|left|frame|Modulation method depends on distance from base station]]
 
The signal space constellations of these variants are shown in the left graphic.  Depending on the environment and distance to the base station, the  [[Mobile_Communications/Physical_Layer_for_LTE#Scheduling_for_LTE|"Scheduler"]]  selects the appropriate QAM method:

*64–QAM allows the best data rates, but is also the most susceptible to interference and is therefore only used near the base stations.  The weaker the connection, the simpler the modulation method must be, which also reduces the spectral efficiency (in bit/s per Hertz). 
*Very robust is 4–QAM with only two bits per symbol  (one each for real and imaginary part).  This can be used for much larger distances than for example 16–QAM. 
*Due to the exact same signal space constellation the 4–QAM is often called  "Quaternary Phase Shift Keying"  $\rm (QPSK)$.  The four signal space points are arranged in a square pattern (QAM principle).  But they also lie on a circle (characteristic of "PSK").
 
[[File:EN_LTE_T_4_4_S5c_neu.png|right|frame|Throughput depending on SNR|class=fit]]

The graphic on the right from  [MG08]<ref name='MG08'>Myung, H.; Goodman, D.:  Single Carrier FDMA - A New Air Interface for Long Term Evolution.  West Sussex: John Wiley & Sons, 2008.</ref> gives the following facts:
*With 4–QAM or QPSK (two bit/symbol) a throughput of almost one Mbit/s is achieved in the LTE uplink with the assumptions made in  [MG08].

*Only above a certain "Signal–to–Noise Ratio"  $\rm (SNR)$  a higher level QAM is used, for example 16–QAM  (4  bit/symbol)  or 64–QAM  (8 bit/symbol). 

*If the SNR is sufficiently large, increasing the number of stages will lead to better results regarding the data throughput.

It should be noted that the low-rate QPSK (4–QAM) is always used in the control channels, since,
#on the one hand, this information do not require high data rates due to their small size, and,
#on the other hand, should be received (almost) error-free due to their importance. 

An exception is the channel  [[Mobile_Communications/Physical Layer for LTE#Physical channels in uplink| $\rm PUSCH$]]  in the uplink, which transmits both user and control data.  For this reason, the same modulation type is used here for both signals.
 

== Scheduling for LTE ==
 
[[File:EN_LTE_T_4_4_S6.png|right|frame|Functionality of the scheduler in the LTE uplink|class=fit]]
All LTE base stations contain a scheduler that can be switched between
*a total transfer rate as high as possible 
*with sufficiently good Quality of Service  $\rm (QoS)$. 

A possible QoS criterion is for example the  "packet delay duration".  So the scheduler tries to optimize the overall situation by using algorithms. 

Scheduling is necessary to ensure a fair distribution of resources.  A concrete example is that a user who currently has a poor channel and therefore low efficiency must still be allocated sufficient resources, otherwise the desired (and guaranteed) transmission quality cannot be maintained. 

The scheduler controls the selection of the modulation method and the subcarrier–mapping.  The functionality of the scheduler is illustrated by the graphic for the uplink.  Similar statements apply to the downlink. 

{{BlueBox|TEXT=
$\text{Conclusion:}$  Based on  [SABM06]<ref name ='SABM06'>Schmidt, M.; Ahn, N.; Braun, V.; Mayer, H.P.:  Performance of QoS and Channel-aware Packet Scheduling for LTE Downlink.  Alcatel-Lucent, 2006. </ref>,  [WGM07]<ref name ='WGM07'>Wang, X.; Giannakis, G.B.; Marques, A.G.:  A Unified Approach to QoS - Guaranteed Scheduling or Channel-Adaptive Wireless Networks.  Proceedings of the IEEE, Vol. 95, No. 12, Dec. 2007.</ref>  and  [MG08]<ref name ='MG08'></ref>  should be noted in summary:
*Scheduler algorithms are often very complicated due to the many optimization criteria, parameters and possible scenarios.  Therefore, the design is usually based on an optimal system in which each base station knows the channel transmission functions sufficiently well at all times and transmission delays are unproblematic. 

*From these boundary conditions, different approaches are created with the help of mathematical analyzes  [WGM07]<ref name ='WGM07'></ref>, whose effectiveness can only be verified by practical tests.  A detailed description of such tests can be found in  [MG08]<ref name ='MG08'></ref>. 

*In principle, the overall transmission rate can be increased by channel-dependent scheduling (exploiting frequency selectivity), but this involves a large overhead, since test signals must be sent over the entire bandwidth.  The information has to be distributed to all end devices if the full optimization potential is to be exploited. 

*In various tests, the clear and significant advantages  (doubling of throughput)  of channel based scheduling were shown, but also the expected losses with faster moving users.  More about this in the recommended document  [SABM06]<ref name ='SABM06'></ref>.}} 

Due to many advantages, scheduling is an integral part of the LTE Release 8, specified by 3GPP. 

==Exercises to chapter==
 
[[Aufgaben:Exercise 4.4: Modulation in LTE]]

[[Aufgaben:Exercise 4.4Z: Physical Channels in LTE]]

==References==

<references/>

{{Display}}

Mobile Communications/Non-Frequency-Selective Fading With Direct Component

2022-02-17T11:39:17Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Time-Variant Transmission Channels
|Vorherige Seite=Statistical Bonds Within the Rayleigh Process
|Nächste Seite=General description of time variant systems
}}
== Channel model and Rice PDF ==
 
The  [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Modeling_of_non-frequency_selective_fading| Rayleigh distribution]]  describes the mobile communication channel under the assumption that there is no direct path and thus the multiplicative factor  $z(t) = x(t) + {\rm j} \cdot y(t)$  is solely composed of diffusely scattered components.

If a direct component  $($Line of Sight,  $\rm LoS)$  is present, it is necessary to add direct components   $x_0$  and/or  $y_0$  to the zero mean Gaussian processes   $x(t)$  and  $y(t)$:
[[File:EN_Mob_T_1_4_S1.png|right|frame|Rice fading channel model|class=fit]]

::<math>x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},</math>

::<math>z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.</math>
The graph shows this  '''Rice fading channel model'''.  As a special case, the Rayleigh model results when   $x_0 = y_0= 0$.
 
The Rice fading model can be summarized as follows, see also  [Hin08]<ref name = 'Hin08'>Hindelang, T.:  Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2008.</ref>:
*The real part  $x(t)$  is gaussian distributed with mean value  $x_0$  and variance  $\sigma ^2$.
*The imaginary part  $y(t)$  is also gaussian distributed  $($mean  $y_0$,  equal variance  $\sigma ^2)$  and independent of  $x(t)$. 

*For  $z_0 \ne 0$  the value  $|z(t)|$  has a [[Theory_of_Stochastic_Signals/Further_distributions#Rice_PDF| Rice PDF]], from which the term  "Rice fading"  is derived.
*To simplify the notation we set  $|z(t)| = a(t)$.   For  $a < 0$  it's PDF is  $f_a(a) \equiv 0$,  for  $a \ge 0$ the following equation applies, where  $\rm I_0(\cdot)$  denotes the  "modified Bessel–function" of zero order:

::<math>f_a(a) = \frac{a}{\sigma^2} \cdot {\rm exp} \big [ -\frac{a^2 + |z_0|^2}{2\sigma^2}\big ] \cdot {\rm I}_0 \left [ \frac{a \cdot |z_0|}{\sigma^2} \right ]\hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.</math>

*The greater the direct path power  $(|z_0|^2)$  compared to the power of the stray components  $(2\sigma^2)$  the better suited for digital signal transmission is the mobile communication channel.

*If   $|z_0| \gg \sigma$  $($factor  $3$  or more$)$, the Rice PDF can be approximated accurately by a Gaussian distribution with mean  $|z_0|$  and variance  $\sigma^2$. 

*In contrast to  Rayleigh fading   ⇒   $z_0 \equiv 0$, the phase at  Rice fading  is not equally distributed, but there is a preferred direction  $\phi_0 = \arctan(y_0/x_0)$.  Often one sets  $y_0 = 0$   ⇒   $\phi_0 = 0$. 

== Example of signal behaviour with Rice fading==
 
[[File:P ID2129 Mob T 1 4 S2 v1.png|right|frame|Comparison of Rayleigh fading (blue) and Rice fading (red)|class=fit]]
The diagram shows typical signal characteristics and density functions of two mobile communication channels:
*Rayleigh fading  (blue curves)  with 
:$${\rm E}\big [|z(t))|^2\big ] = 2 \cdot \sigma^2 = 1,$$

*Rice fading  (red curves)  with same  $\sigma$  and
:$$x_0 = 0.707,\ \ y_0 = -0.707.$$

For the generation of the signals according to the above model, the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|maximum Doppler frequency]]  $f_\text{D, max} = 100 \ \rm Hz$  was used as reference.

The auto-correlation function  $\rm (ACF)$  and power-spectral density  $\rm (PSD)$  of Rayleigh and Rice differ only slightly, other than adjusted parameter values.  The following applies:

::<math>\varphi_z ({\rm \Delta}t)\Bigg |_{\hspace{0.1cm}{\rm Rice}} \hspace{-0.5cm} = \varphi_z ({\rm \Delta}t)\Bigg |_{\hspace{0.1cm}{\rm Rayleigh}} \hspace{-0.8cm} + |z_0|^2 \hspace{0.05cm},</math>
::<math> {\it \Phi}_z(f_{\rm D})\Bigg |_{\hspace{0.1cm}{\rm Rice}} \hspace{-0.5cm} = {\it \Phi}_z(f_{\rm D})\Bigg |_{\hspace{0.1cm}{\rm Rayleigh}} \hspace{-0.8cm} + |z_0|^2 \cdot \delta (f_{\rm D}) \hspace{0.05cm}.</math>

It is taken into account that the spectral representation of a DC component leads to a Dirac function. 
 
It should be noted about this graph:
*The real parts  $x(t)$  of Rayleigh (blue) and Rice (red) only differ by the constant  $x_0 = 0.707$.   The statistical properties are otherwise the same:   Gaussian PDF $f_x(x)$  with standard deviation  $\sigma = 0.707$, either zero-mean (Rayleigh) or with mean  $x_0$  (Rice). 

*In the imaginary part  $y(t)$  of the Rice distribution one can additionally recognize the direct component  $y_0 = -0.707$.  The (here not shown) PDF $f_y(y)$  is thus a Gaussian curve with the standard deviation  $\sigma = 0. 707$  around the mean value  $ y_0 = -0.707$, thus axisymmetrical to the shown PDF $f_x(x)$. 

*The (logarithmic) representation of   ⇒   $a(t) =|z(t)|$ shows that the red curve is usually above the blue one.  This can also be read from the PDF $f_a(a)$ .
*For the Rice channel, the error probability is lower than for Rayleigh when AWGN is taken into account, since the receiver gets some usable energy via the Rice direct path.

*The PDF $f_\phi(\phi)$  shows the preferred angle  $\phi \approx -45^\circ$  of the given Rice channel   The complex factor  $z(t)$  is located mainly in the fourth quadrant because of  $x_0 > 0$  and  $y_0 < 0$ , whereas in the Rayleigh channel all quadrants are equally probable. 

==Exercises to the chapter==
 
[[Aufgaben:Exercise 1.6: Autocorrelation Function and PSD with Rice Fading]]

[[Aufgaben:Exercise 1.6Z: Comparison of Rayleigh and Rice]]

[[Aufgaben:Exercise 1.7: PDF of Rice Fading]]

==References==

{{Display}}

Information Theory/Discrete Memoryless Sources

2022-02-17T11:39:17Z

Bene: Text replacement - "List of sources" to "References"

{{FirstPage}}
{{Header
|Untermenü=Entropy of Discrete Sources
|Vorherige Seite=
|Nächste Seite=Discrete Sources with Memory
}}

== # OVERVIEW OF THE FIRST MAIN CHAPTER # ==
 
This first chapter describes the calculation and the meaning of entropy.  According to the Shannonian information definition, entropy is a measure of the mean uncertainty about the outcome of a statistical event or the uncertainty in the measurement of a stochastic quantity.  Somewhat casually expressed, the entropy of a random quantity quantifies its "randomness".

In detail are discussed:

:*The  »information content«  of a symbol and the  »entropy«  of a discrete memoryless source,
:*the  »binary entropy function«  and its application to non-binary sources,
:*the entropy calculation for  »sources with memory«  and suitable approximations,
:*the special features of  »Markov sources«  regarding the entropy calculation,
:*the procedure for sources with a large number of symbols, for example  »natural texts«,
:*the  »entropy estimates«  according to Shannon and Küpfmüller.

== Model and requirements ==
 
We consider a discrete message source  $\rm Q$, which gives a sequence  $ \langle q_ν \rangle$  of symbols.
*For the variable  $ν = 1$, ... , $N$, where  $N$  should be "sufficiently large".
*Each individual source symbol  $q_ν$  comes from a symbol set  $\{q_μ \}$  where  $μ = 1$, ... , $M$, where  $M$  denotes the symbol set size:

:$$q_{\nu} \in \left \{ q_{\mu} \right \}, \hspace{0.25cm}{\rm with}\hspace{0.25cm} \nu = 1, \hspace{0.05cm} \text{ ...}\hspace{0.05cm} , N\hspace{0.25cm}{\rm and}\hspace{0.25cm}\mu = 1,\hspace{0.05cm} \text{ ...}\hspace{0.05cm} , M \hspace{0.05cm}.$$

The figure shows a quaternary message source  $(M = 4)$  with the alphabet  $\rm \{A, \ B, \ C, \ D\}$  and an exemplary sequence of length  $N = 100$.

[[File:EN_Inf_T_1_1_S1a.png|frame|Quaternary source]]

The following requirements apply:
*The quaternary source is fully described by  $M = 4$  symbol probabilities  $p_μ$.  In general it applies:
:$$\sum_{\mu = 1}^M \hspace{0.1cm}p_{\mu} = 1 \hspace{0.05cm}.$$
*The message source is memoryless, i.e., the individual sequence elements are  [[Theory_of_Stochastic_Signals/Statistical Dependence and Independence#General_definition_of_statistical_dependence|statistically independent of each other]]:
:$${\rm Pr} \left (q_{\nu} = q_{\mu} \right ) = {\rm Pr} \left (q_{\nu} = q_{\mu} \hspace{0.03cm} | \hspace{0.03cm} q_{\nu -1}, q_{\nu -2}, \hspace{0.05cm} \text{ ...}\hspace{0.05cm}\right ) \hspace{0.05cm}.$$
*Since the alphabet consists of symbols  (and not of random variables) , the specification of  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments|expected values]]  (linear mean, quadratic mean, standard deviation, etc.)  is not possible here, but also not necessary from an information-theoretical point of view.

These properties will now be illustrated with an example.

[[File:Inf_T_1_1_S1b_vers2.png|right|frame|Relative frequencies as a function of  $N$]]
{{GraueBox|TEXT=
$\text{Example 1:}$ 
For the symbol probabilities of a quaternary source applies:
:$$p_{\rm A} = 0.4 \hspace{0.05cm},\hspace{0.2cm}p_{\rm B} = 0.3 \hspace{0.05cm},\hspace{0.2cm}p_{\rm C} = 0.2 \hspace{0.05cm},\hspace{0.2cm}
p_{\rm D} = 0.1\hspace{0.05cm}.$$
For an infinitely long sequence  $(N \to \infty)$
*the  [[Theory_of_Stochastic_Signals/From_Random_Experiment_to_Random_Variable#Bernoulli's_Law_of_Large_Numbers|relative frequencies]]  $h_{\rm A}$,  $h_{\rm B}$,  $h_{\rm C}$,  $h_{\rm D}$   ⇒   a-posteriori parameters
*were identical to the  [[Theory_of_Stochastic_Signals/Some_Basic_Definitions#Event_and_Event_set|probabilities]]  $p_{\rm A}$,  $p_{\rm B}$,  $p_{\rm C}$,  $p_{\rm D}$   ⇒   a-priori parameters.

With smaller  $N$  deviations may occur, as the adjacent table (result of a simulation) shows.

*In the graphic above an exemplary sequence is shown with  $N = 100$  symbols.
*Due to the set elements  $\rm A$,  $\rm B$,  $\rm C$  and  $\rm D$  no mean values can be given.

However, if you replace the symbols with numerical values, for example  $\rm A \Rightarrow 1$,   $\rm B \Rightarrow 2$,   $\rm C \Rightarrow 3$,   $\rm D \Rightarrow 4$, then you will get after     »time averaging«   ⇒   crossing line     or     »ensemble averaging«   ⇒   expected value formation
*for the  [[Theory_of_Stochastic_Signals/Moments of a Discrete Random Variable#Linear_Average_-_Direct_Component|linear average]]:
:$$m_1 = \overline { q_{\nu} } = {\rm E} \big [ q_{\mu} \big ] = 0.4 \cdot 1 + 0.3 \cdot 2 + 0.2 \cdot 3 + 0.1 \cdot 4
= 2 \hspace{0.05cm},$$
*for the  [[Theory_of_Stochastic_Signals/Moments of a Discrete Random Variable#Square_mean_.E2.80.93_Variance_.E2.80.93_Scattering |square mean]]:
:$$m_2 = \overline { q_{\nu}^{\hspace{0.05cm}2} } = {\rm E} \big [ q_{\mu}^{\hspace{0.05cm}2} \big ] = 0.4 \cdot 1^2 + 0.3 \cdot 2^2 + 0.2 \cdot 3^2 + 0.1 \cdot 4^2
= 5 \hspace{0.05cm},$$
*for the  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Some_often_used_Central_Moments|standard deviation]]  according to the  »Theorem of Steiner«:
:$$\sigma = \sqrt {m_2 - m_1^2} = \sqrt {5 - 2^2} = 1 \hspace{0.05cm}.$$}}

==Maximum entropy of a discrete source==
 
[https://de.wikipedia.org/wiki/Claude_Shannon Claude Elwood Shannon]  defined in 1948 in the standard work of information theory  [Sha48]<ref name='Sha48'>Shannon, C.E.: A Mathematical Theory of Communication. In: Bell Syst. Techn. J. 27 (1948), pp. 379-423 and pp. 623-656.</ref>  the concept of information as  "decrease of uncertainty about the occurrence of a statistical event".

Let us make a mental experiment with  $M$  possible results, which are all equally probable:   $p_1 = p_2 = \hspace{0.05cm} \text{ ...}\hspace{0.05cm} = p_M = 1/M \hspace{0.05cm}.$

Under this assumption applies:
*Is  $M = 1$, then each individual attempt will yield the same result and therefore there is no uncertainty about the output.
*On the other hand, an observer learns about an experiment with  $M = 2$, for example the  "coin toss"  with the set of events  $\big \{\rm \boldsymbol{\rm Z}(ahl), \rm \boldsymbol{\rm W}(app) \big \}$  and the probabilities  $p_{\rm Z} = p_{\rm W} = 0. 5$, a gain in information.  The uncertainty regarding  $\rm Z$  resp.  $\rm W$  is resolved.
*In the experiment  »dice«  $(M = 6)$  and even more in  »roulette«  $(M = 37)$  the gained information is even more significant for the observer than in the  »coin toss«  when he learns which number was thrown or which ball fell.
*Finally it should be considered that the experiment  »triple coin toss«  with  $M = 8$  possible results  $\rm ZZZ$,  $\rm ZZW$,  $\rm ZWZ$,  $\rm ZWW$,  $\rm WZZ$,  $\rm WZW$,  $\rm WWZ$,  $\rm WWW$  provides three times the information as the single coin toss  $(M = 2)$.

The following definition fulfills all the requirements listed here for a quantitative information measure for equally probable events, indicated only by the symbol set size  $M$.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''maximum average information content'''   of a message source depends only on the symbol set size  $M$  and results in

:$$H_0 = {\rm log}\hspace{0.1cm}M = {\rm log}_2\hspace{0.1cm}M \hspace{0.15cm} {\rm (in \ “bit")}
= {\rm ln}\hspace{0.1cm}M \hspace{0.15cm}\text {(in “nat")}
= {\rm lg}\hspace{0.1cm}M \hspace{0.15cm}\text {(in “Hartley")}\hspace{0.05cm}.$$

*Since  $H_0$  indicates the maximum value of the  [[Information_Theory/Sources with Memory#Information_Content_and_Entropy|entropy]]  $H$,  $H_\text{max}=H_0$  is also used in our tutorial as short notation. }}

Please note our nomenclature:
*The logarithm will be called  »log«  in the following, independent of the base.
*The relations mentioned above are fulfilled due to the following properties:

:$${\rm log}\hspace{0.1cm}1 = 0 \hspace{0.05cm},\hspace{0.2cm}
{\rm log}\hspace{0.1cm}37 > {\rm log}\hspace{0.1cm}6 > {\rm log}\hspace{0.1cm}2\hspace{0.05cm},\hspace{0.2cm}
{\rm log}\hspace{0.1cm}M^k = k \cdot {\rm log}\hspace{0.1cm}M \hspace{0.05cm}.$$

* Usually we use the logarithm to the base  $2$   ⇒   »logarithm dualis«    $\rm (ld)$,  where the pseudo unit  "bit"  $($more precisely:  "bit/symbol"$)$  is then added:

:$${\rm ld}\hspace{0.1cm}M = {\rm log_2}\hspace{0.1cm}M = \frac{{\rm lg}\hspace{0.1cm}M}{{\rm lg}\hspace{0.1cm}2}
= \frac{{\rm ln}\hspace{0.1cm}M}{{\rm ln}\hspace{0.1cm}2}
\hspace{0.05cm}.$$

*In addition, you can find in the literature some additional definitions, which are based on the natural logarithm  $\rm (ln)$  or the logarithm of the tens  $\rm (lg)$.

==Information content and entropy ==
 
We now waive the previous requirement that all  $M$  possible results of an experiment are equally probable.  In order to keep the spelling as compact as possible, we define for this page only:

:$$p_1 > p_2 > \hspace{0.05cm} \text{ ...}\hspace{0.05cm} > p_\mu > \hspace{0.05cm} \text{ ...}\hspace{0.05cm} > p_{M-1} > p_M\hspace{0.05cm},\hspace{0.4cm}\sum_{\mu = 1}^M p_{\mu} = 1 \hspace{0.05cm}.$$

We now consider the '''information content'''  of the individual symbols, where we denote the  "logarithm dualis"  with  $\log_2$:

:$$I_\mu = {\rm log_2}\hspace{0.1cm}\frac{1}{p_\mu}= -\hspace{0.05cm}{\rm log_2}\hspace{0.1cm}{p_\mu}
\hspace{0.5cm}{\rm (unit\hspace{-0.15cm}: \hspace{0.15cm}bit\hspace{0.15cm}or\hspace{0.15cm}bit/Symbol)}
\hspace{0.05cm}.$$

You can see:
*Because of  $p_μ ≤ 1$  the information content is never negative.  In the borderline case  $p_μ \to 1$  goes  $I_μ \to 0$.
*However, for  $I_μ = 0$   ⇒   $p_μ = 1$   ⇒   $M = 1$  the information content is also  $H_0 = 0$.
*For decreasing probabilities  $p_μ$  the information content increases continuously:

:$$I_1 < I_2 < \hspace{0.05cm} \text{ ...}\hspace{0.05cm} < I_\mu <\hspace{0.05cm} \text{ ...}\hspace{0.05cm} < I_{M-1} < I_M \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$  '''The more improbable an event is, the greater is its information content'''.  This fact is also found in daily life:
*"6 right ones" in the lottery are more likely to be noticed than "3 right ones" or no win at all.
*A tsunami in Asia also dominates the news in Germany for weeks as opposed to the almost standard Deutsche Bahn delays.
*A series of defeats of Bayern Munich leads to huge headlines in contrast to a winning series.  With 1860 Munich exactly the opposite is the case.}}

However, the information content of a single symbol (or event) is not very interesting.  On the other hand one of the central quantities of information theory is obtained,
*by ensemble averaging over all possible symbols  $q_μ$  bzw. 
*by time averaging over all elements of the sequence  $\langle q_ν \rangle$.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''entropy'''  $H$  of a discrete source indicates the  '''mean information content of all symbols''':

:$$H = \overline{I_\nu} = {\rm E}\hspace{0.01cm}[I_\mu] = \sum_{\mu = 1}^M p_{\mu} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{p_\mu}=
-\sum_{\mu = 1}^M p_{\mu} \cdot{\rm log_2}\hspace{0.1cm}{p_\mu} \hspace{0.5cm}\text{(unit: bit, more precisely: bit/symbol)}
\hspace{0.05cm}.$$

The overline marks again a time averaging and  $\rm E[\text{...}]$  an ensemble averaging.}}

Entropy is among other things a measure for
*the mean uncertainty about the outcome of a statistical event,
*the  "randomness"  of this event,  and
*the average information content of a random variable.

==Binary entropy function ==
 
At first we will restrict ourselves to the special case  $M = 2$  and consider a binary source, which returns the two symbols  $\rm A$  and  $\rm B$.  The symbol probabilities are   $p_{\rm A} = p$  and   $p_{\rm B} = 1 - p$.

For the entropy of this binary source applies:

:$$H_{\rm bin} (p) = p \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm}} + (1-p) \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{1-p} \hspace{0.5cm}{\rm (unit\hspace{-0.15cm}: \hspace{0.15cm}bit\hspace{0.15cm}or\hspace{0.15cm}bit/symbol)}
\hspace{0.05cm}.$$

This function is called  $H_\text{bin}(p)$  the  '''binary entropy function'''.  The entropy of a source with a larger symbol set size  $M$  can often be expressed using  $H_\text{bin}(p)$ .

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The figure shows the binary entropy function for the values  $0 ≤ p ≤ 1$  of the symbol probability of  $\rm A$  $($or also of  $\rm B)$.  You can see:

[[File:EN_Inf_T_1_1_S4.png|frame|Binary entropy function as a function of  $p$|right]]
*The maximum value  $H_\text{max} = 1\; \rm bit$  results for  $p = 0.5$, thus for equally probable binary symbols.  Then   $\rm A$  and  $\rm B$  contribute the same amount to the entropy.
* $H_\text{bin}(p)$  is symmetrical around  $p = 0.5$.  A source with  $p_{\rm A} = 0.1$  and  $p_{\rm B} = 0. 9$  has the same entropy  $H = 0.469 \; \rm bit$  as a source with  $p_{\rm A} = 0.9$  and  $p_{\rm B} = 0.1$.
*The difference  $ΔH = H_\text{max} - H$ gives  the  »redundancy«  of the source and  $r = ΔH/H_\text{max}$  the  »relative redundancy«.   In the example,  $ΔH = 0.531\; \rm bit$  and  $r = 53.1 \rm \%$.
*For  $p = 0$  this results in  $H = 0$, since the symbol sequence  $\rm B \ B \ B \text{...}$  can be predicted with certainty   ⇒   symbol set size only  $M = 1$.  The same applies to  $p = 1$   ⇒   symbol sequence  $\rm A \ A \ A \text{...}$.
*$H_\text{bin}(p)$  is always a  "concave function",  since the second derivative after the parameter  $p$  is negative for all values of  $p$ :
:$$\frac{ {\rm d}^2H_{\rm bin} (p)}{ {\rm d}\,p^2} = \frac{- 1}{ {\rm ln}(2) \cdot p \cdot (1-p)}< 0
\hspace{0.05cm}.$$}}

==Non-binary sources==
 
In the  [[Information_Theory/Sources with Memory#Model_and_Prerequisites|first section]]  of this chapter we considered a quaternary message source  $(M = 4)$  with the symbol probabilities  $p_{\rm A} = 0. 4$,   $p_{\rm B} = 0.3$,   $p_{\rm C} = 0.2$  and  $ p_{\rm D} = 0.1$.  This source has the following entropy:

:$$H_{\rm quat} = 0.4 \cdot {\rm log}_2\hspace{0.1cm}\frac{1}{0.4} + 0.3 \cdot {\rm log}_2\hspace{0.1cm}\frac{1}{0. 3} + 0.2 \cdot {\rm log}_2\hspace{0.1cm}\frac{1}{0.2}+ 0.1 \cdot {\rm log}_2\hspace{0.1cm}\frac{1}{0.1}.$$

For numerical calculation, the detour via the decimal logarithm  $\lg \ x = {\rm log}_{10} \ x$  is often necessary, since the  "logarithm dualis"  $ {\rm log}_2 \ x$  is mostly not found on pocket calculators.

:$$H_{\rm quat}=\frac{1}{{\rm lg}\hspace{0.1cm}2} \cdot \left [ 0.4 \cdot {\rm lg}\hspace{0.1cm}\frac{1}{0.4} + 0.3 \cdot {\rm lg}\hspace{0.1cm}\frac{1}{0. 3} + 0.2 \cdot {\rm lg}\hspace{0.1cm}\frac{1}{0.2} + 0.1 \cdot {\rm lg}\hspace{0.1cm}\frac{1}{0.1} \right ] = 1.845\,{\rm bit}
\hspace{0.05cm}.$$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
Now there are certain symmetries between the symbol probabilities:
[[File:EN_Inf_T_1_1_S5.png|frame|Entropy of binary source and quaternary source]]

:$$p_{\rm A} = p_{\rm D} = p \hspace{0.05cm},\hspace{0.4cm}p_{\rm B} = p_{\rm C} = 0.5 - p \hspace{0.05cm},\hspace{0.3cm}{\rm with} \hspace{0.15cm}0 \le p \le 0.5 \hspace{0.05cm}.$$

In this case, the binary entropy function can be used to calculate the entropy:

:$$H_{\rm quat} = 2 \cdot p \cdot {\rm log}_2\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm} } + 2 \cdot (0.5-p) \cdot {\rm log}_2\hspace{0.1cm}\frac{1}{0.5-p}$$
$$\Rightarrow \hspace{0.3cm} H_{\rm quat} = 1 + H_{\rm bin}(2p) \hspace{0.05cm}.$$

The graphic shows as a function of  $p$
*the entropy of the quaternary source (blue)
*in comparison to the entropy course of the binary source (red).

For the quaternary source only the abscissa  $0 ≤ p ≤ 0.5$  is allowed.
 
You can see from the blue curve for the quaternary source:
*The maximum entropy  $H_\text{max} = 2 \; \rm bit/symbol$  results for  $p = 0.25$   ⇒   equally probable symbols:   $p_{\rm A} = p_{\rm B} = p_{\rm C} = p_{\rm A} = 0.25$.
*With  $p = 0$  the quaternary source degenerates to a binary source with  $p_{\rm B} = p_{\rm C} = 0. 5$,   $p_{\rm A} = p_{\rm D} = 0$   ⇒   $H = 1 \; \rm bit/symbol$.  Similar applies to $p = 0.5$.
*The source with  $p_{\rm A} = p_{\rm D} = 0.1$  and  $p_{\rm B} = p_{\rm C} = 0.4$  has the following characteristics (each with the pseudo unit "bit/symbol"):

:     '''(1)'''   entropy:   $H = 1 + H_{\rm bin} (2p) =1 + H_{\rm bin} (0.2) = 1.722,$

:     '''(2)'''   Redundancy:   ${\rm \Delta }H = {\rm log_2}\hspace{0.1cm} M - H =2- 1.722= 0.278,$

:     '''(3)'''   relative redundancy:   $r ={\rm \delta }H/({\rm log_2}\hspace{0.1cm} M) = 0.139\hspace{0.05cm}.$

*The redundancy of the quaternary source with  $p = 0.1$  is  $ΔH = 0.278 \; \rm bit/symbol$   ⇒   exactly the same as the redundancy of the binary source with  $p = 0.2$.}}

== Exercises for the chapter==
 
[[Aufgaben:Exercise_1.1:_Entropy_of_the_Weather|Exercise 1.1: Entropy of the Weather]]

[[Aufgaben:Exercise_1.1Z:_Binary_Entropy_Function|Exercise 1.1Z: Binary Entropy Function]]

[[Aufgaben:Exercise_1.2:_Entropy_of_Ternary_Sources|Exercise 1.2: Entropy of Ternary Sources]]

==References==
<references />

{{Display}}

Linear and Time Invariant Systems/Some Results from Line Transmission Theory

2022-02-17T11:39:17Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Properties of Electrical Cables
|Vorherige Seite=Laplace–Rücktransformation
|Nächste Seite=Properties of Coaxial Cables
}}

== # OVERVIEW  OF  THE  FOURTH  MAIN  CHAPTER # ==
 
A special case of causal and time-invariant systems are electrical cables.  Here,  due to the Hilbert transformation,  a complex-valued frequency response  $H(f)$  and strongly unbalanced impulse responses  $h(t)$  must always be assumed.

The fourth chapter presents a summary of  »conducted transmission channels«,  specifically

*important  »results and descriptive quantities of Line Transmission Theory«,  in particular primary line parameters,  complex propagation function  (per unit length),  attenuation function (per unit length),  phase function (per unit length),  wave impedance, and the operational attenuation to take account of mismatches and reflections,
*the  »frequency responses and impulse responses of coaxial cables«,  in which,  due to good shielding,  all other noise is negligible compared to thermal noise  (Gaussian distributed and white),  and
*the  »description of symmetrical copper cables«,  which are the main transmission medium in the  »access network of telecommunication systems«.  Since many pairs run in parallel in a cable,  crosstalk occurs due to capacitive and inductive couplings.

==Equivalent circuit diagram of a short transmission line section==
 
To derive the line equations,  first a very short line section of length  ${\rm d}x$  is considered,  so that the values for voltage and current
*at the beginning of the line  $(U$  resp.   $I$  at  $x)$  and
*the end of the line  $(U + {\rm d}U$  and  $I + {\rm d}I$  at  $x + {\rm d}x)$ 

differ only slightly.  The diagram below shows the underlying model.

{{BlaueBox|TEXT=
$\text{Or in other words:}$ 

Let the line length  ${\rm d}x$  be very small compared to the wavelength  $\lambda$  of the electromagnetic wave propagating along the line,  which results because
*there is a magnetic field connected to the current,
*the voltage between the conductors causes an electric field. }}

[[File:P_ID1792__LZI_T_4_1_S1_neu.png |right|frame| Equivalent circuit diagram of a short line section]]
All infinitesimal  "components"  in the equivalent circuit sketched on the right are location-independent for homogeneous lines:
*The inductance of the considered line section is  $L\hspace{0.05cm}' \cdot {\rm d}x$, where  $L'$  is called  '''serial inductance'''  per unit length.
*Similarly,  the  '''parallel capacitance'''  per unit length  $(C\hspace{0.05cm}')$  is an infinitesimally small quantity which,  similar to  $L'$,  depends only slightly on frequency.
*The  '''parallel conductance'''  per unit length  $(G\hspace{0.05cm}')$  takes into account the losses of the dielectric between the wires.  It increases approximately proportionally with frequency.
*By far the largest (negative) influence on signal transmission is the  '''serial resistance'''  per unit length  $(R\hspace{0.05cm}')$,  which increases almost proportionally with the root of the frequency for high frequencies due to the so-called  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference#Frequency_response_of_a_coaxial_cable|$\text{Skin effect}$]].

From the mesh and node equations of the line section,  $ω = 2πf$  results in the two difference equations:
:$$ U = I \cdot (R\hspace{0.05cm}' + {\rm j} \cdot \omega L\hspace{0.05cm}') \cdot {\rm d}x + (U + {\rm d}U)\hspace{0.05cm},$$
:$$ I = (U + {\rm d}U) \cdot (G\hspace{0.05cm}' + {\rm j} \cdot \omega C\hspace{0.05cm}') \cdot {\rm d}x + (I + {\rm d}I)\hspace{0.05cm}. $$

For very short line sections  $($infinitesimally small  ${\rm d}x)$  and neglecting the small second order quantities  $($for example  ${\rm d}U \cdot {\rm d}x)$ , one can now form two differential quotients whose joint consideration leads to a second order linear differential equation:
:$$\frac{ {\rm d}U}{ {\rm d}x} = - (R\hspace{0.05cm}' + {\rm j} \cdot \omega L\hspace{0.05cm}') \cdot I,\hspace{0.5cm} \frac{ {\rm d}I}{ {\rm d}x} = - (G\hspace{0.05cm}' + {\rm j} \cdot \omega C\hspace{0.05cm}')
\cdot U\hspace{0.3cm}
\Rightarrow \hspace{0.3cm}\frac{{\rm d}^2U}{{\rm d}x^2} = (R\hspace{0.05cm}' + {\rm j} \cdot \omega L\hspace{0.05cm}') \cdot (G\hspace{0.05cm}' + {\rm j} \cdot \omega C\hspace{0.05cm}')
\cdot U\hspace{0.05cm}.$$
The solution of this differential equation is:
:$$U(x) = U_{\rightarrow}(x=0) \cdot {\rm e}^{-\hspace{0.02cm}\gamma \hspace{0.03cm} \cdot \hspace{0.05cm}x} + U_{\leftarrow}(x=0) \cdot {\rm e}^{\gamma \hspace{0.03cm} \cdot \hspace{0.05cm}x} \hspace{0.05cm}.$$

The voltage curve depends not only on the location  $x$  but also on the frequency  $f$  which is not explicitly noted in the equation given here.

Formula-wise, this frequency dependence is captured by the  '''complex propagation function'''  per unit length:
:$$\gamma(f) = \sqrt{(R\hspace{0.05cm}' + {\rm j} \cdot 2\pi f \cdot L\hspace{0.05cm}') \cdot (G\hspace{0.05cm}' + {\rm j} \cdot 2\pi f \cdot C\hspace{0.05cm}')} = \alpha (f) + {\rm j} \cdot \beta (f)\hspace{0.05cm}.$$

The last two equations together describe the voltage curve along the line,  which results from the superposition of a wave  $U_→(x)$  running in the positive  $x$ direction and the wave  $U_←(x)$  in the opposite direction.

*The real part  $α(f)$  of the complex propagation function  $γ(f)$  attenuates the propagating wave and is therefore called  '''attenuation function'''  per unit length.  This always even function   ⇒   $α(-f) = α(f)$  results from the above  $γ(f)$ equation as follows:
:$$\alpha(f) = \sqrt{{1}/{2}\cdot \left (R\hspace{0.05cm}' \cdot G\hspace{0.05cm}' - \omega^2 \cdot L\hspace{0.05cm}' \cdot C\hspace{0.05cm}'\right)+ {1}/{2} \cdot \sqrt{(R\hspace{0.05cm}'\hspace{0.05cm}^2 + \omega^2 \cdot L\hspace{0.05cm}'\hspace{0.05cm}^2) \cdot (G\hspace{0.05cm}'\hspace{0.05cm}^2 + \omega^2 \cdot C\hspace{0.05cm}'\hspace{0.05cm}^2)}} \bigg |_{\hspace{0.05cm}\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}.$$
*The odd imaginary part   ⇒   $β(- f) = - β(f)$  is called  '''phase function'''  per unit length and describes the phase rotation of the wave along the line:
:$$\beta(f) = \sqrt{ {1}/{2}\cdot \left (-R\hspace{0.05cm}' \cdot G\hspace{0.05cm}' + \omega^2 \cdot L\hspace{0.05cm}' \cdot C\hspace{0.05cm}'\right)+ {1}/{2} \cdot \sqrt{(R\hspace{0.05cm}'\hspace{0.05cm}^2 + \omega^2 \cdot L\hspace{0.05cm}'\hspace{0.05cm}^2) \cdot (G\hspace{0.05cm}'\hspace{0.05cm}^2 + \omega^2 \cdot C\hspace{0.05cm}'\hspace{0.05cm}^2)}} \bigg |_{\hspace{0.05cm}\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}.$$

==Wave impedance and reflections==
 
We consider a homogeneous line of length  $l$  with a harmonic oscillation  $U_0(f)$  of frequency $f$  applied to its input.
*The transmitter has the internal resistance  $Z_1$,  the receiver has the input impedance  $Z_2$  (this is also the terminating resistor of the line).
*We assume for simplicity that  $Z_1$  and  $Z_2$  are real resistors.

[[File:P_ID1793__LZI_T_4_1_S2a_neu.png |right|frame| Line of length  $l$  with wiring]]

The current and voltage of the outgoing and return waves are linked via the wave impedance  $Z_{\rm W}(f)$ :
:$$I_{\rightarrow}(x, f) = \frac{U_{\rightarrow}(x, f)}{Z_{\rm W}(f)}\hspace{0.05cm}, $$
:$$ I_{\leftarrow}(x, f) = \frac{U_{\leftarrow}(x, f)}{Z_{\rm W}(f)}\hspace{0.05cm}.$$
The following applies to the wave impedance:
:$$Z_{\rm W}(f) = \sqrt{\frac {R\hspace{0.05cm}' + {\rm j} \cdot \omega L\hspace{0.05cm}'}{G\hspace{0.05cm}' + {\rm j} \cdot \omega C\hspace{0.05cm}'}} \hspace{0.1cm}\bigg |_{\hspace{0.05cm}\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}.$$

*The wave traveling in the positive  $x$ direction is generated by the AC voltage source at the beginning of the line  $($so at  $x = 0)$ .
*The backward wave is only generated by the reflection of the forward wave at the end of the line  $(x = l)$:
:$$U_{\leftarrow}(x = l) = {U_{\rightarrow}(x = l)}\cdot \frac{Z_2 -Z_{\rm W}(f)}{Z_2 + Z_{\rm W}(f)}\hspace{0.05cm}.$$
*At this point, the terminating resistor  $Z_2$  forces a fixed relationship between the voltage and the current:
:$$U_2(f) = Z_2 · I_2(f).$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
It can be seen from the above equation that there is  '''no returning wave only'''  for  $Z_2 = Z_{\rm W}(f)$ .
*Such resistance matching is always sought in Communications Engineering.
*This matching is not possible over a larger frequency range with fixed termination  $Z_2$  due to the frequency dependence of the wave impedance  $Z_{\rm W}(f)$ . }}

In the following,  these equations are explained with an example.

[[File:P_ID2844__LZI_T_4_1_S2c_neu.png |right|frame| Model to describe the wave reflection]]
{{GraueBox|TEXT=
$\text{Example 1:}$ 
We consider the case shown on the right:
*The terminating resistor  $Z_2$  of the line  (at the same time the input impedance of the following receiver)  differs from the wave impedance  $Z_{\rm W}(f)$.
*We disregard the mismatch at the beginning of the line.

The diagram below from [Han17]<ref name='Han17'>Hanik, N.:  Leitungsgebundene Übertragungstechnik.  Lecture notes. Chair of Communications Engineering, TU München, 2017.</ref> is intended to make clear how the resulting wave  $U(x)$  - shown as a solid curve -  differs from the outgoing wave  $U_→(x)$.

[[File:P_ID2840__LZI_T_4_1_S2b_V2.png |left|frame| Incoming, returning and resulting wave]]
 
To clarify:
*Marked in red is the outgoing wave  $U_→(x)$  which starting from the transmitter   ⇒   $U_→(x = 0)$ , attenuates along the line.  $U_→(x = l)$  denotes the wave at the end of the line.
*Due to the mismatch   ⇒   reflection results in the returning wave  $U_←(x)$  from the end of the line to the transmitter, marked in green.  For this the following applies at the end of the line  $(x = l)$:
::$$U_{\leftarrow}(x = l) = {U_{\rightarrow}(x = l)}\cdot \frac{Z_2 -Z_{\rm W}(f)}{Z_2 + Z_{\rm W}(f)}\hspace{0.05cm}.$$
*The resulting (blue) wave  $U(x)$  results from the in-phase addition of these two parts which are not visible by themselves.

*As  $x$  increases,  $U(x)$  becomes smaller, as does  $U_→(x)$  due to line attenuation.  Also the returning wave  $U_←(x)$  is attenuated with increasing length  (here from right to left). }}

==Lossless and low-loss lines==
 
{{BlaueBox|TEXT=
$\text{Definition:}$  For very short coaxial lines,  such as those used for connections of high-frequency measuring instruments in the laboratory,  from  $R\hspace{0.05cm}' \approx 0$  and  $G\hspace{0.05cm}' \approx 0$  can be assumed.  One then speaks of a  "'''lossless line'''".  For such a line, the above equations simplify:
*'''Attenuation function per unit length''':       $($please note the difference between  "a"  and  "alpha"$)$.
:$$\alpha(f) = \frac{\text{attenuation function }\ a(f)} {\text{length }\ l } = 0\hspace{0.05cm}.$$
*'''Phase function per unit length''':
:$$\beta(f) = \frac{\text{phase function }\ b(f)} {\text{length }\ l } = 2\pi \cdot f \cdot \sqrt{L\hspace{0.05cm}' \cdot C\hspace{0.05cm}' }\hspace{0.05cm}, $$
*'''Wave impedance''':
:$$Z_{\rm W}(f) = \sqrt{ {L\hspace{0.05cm}'}/{ C\hspace{0.05cm}'} }\hspace{0.05cm}.$$}}

If  $L\hspace{0.05cm}'$  and  $C\hspace{0.08cm}'$  are constant in the considered frequency range,  the (real) wave impedance  $Z_{\rm W}(f)=Z_{\rm W}$  is also frequency independent and the phase function per unit length   ⇒   $β(f)$  is proportional to the frequency.
*This means that a lossless line is always free of distortion.
*The output signal can only have a transit time  ("delay")  compared to the input signal.
*Common wave impedances are  $Z_{\rm W} = 50 \ \rm Ω$,  $Z_{\rm W} = 75 \ \rm Ω$ and  $Z_{\rm W} = 150 \ \rm Ω$.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
One speaks of a  '''low-loss line'''  when the line is somewhat longer,  but cannot yet be called  "long". }}

The given formula for the attenuation function per unit length   ⇒   $\alpha(f)$  is now to be evaluated for the case of constant primary line parameters,  which does not quite correspond to reality.
*Above a  '''characteristic frequency'''  $f_∗$ that depends on  $R\hspace{0.05cm}', \ L\hspace{0.05cm}', \ G\hspace{0.08cm}'$  and  $C\hspace{0.08cm}'$,  the serial resistance per unit length  $(R\hspace{0.05cm}')$  can be assumed to be very small compared to  $ωL\hspace{0.05cm}'$  and the parallel conductance per unit length  $(G\hspace{0.05cm}')$  can be assumed to be very small compared to  $ωC\hspace{0.08cm}'$.
*This gives the approximate formula often referred to as  '''weak attenuation''',  valid for  $(f \gg f_∗)$:

:$$\alpha_{_{ {\rm I} } }(f) = {1}/{2} \cdot \left [R\hspace{0.05cm}' \cdot \sqrt{{C\hspace{0.05cm}'}/{ L\hspace{0.05cm}'} } + G\hspace{0.08cm}' \cdot \sqrt{{L\hspace{0.05cm}'}/{ C\hspace{0.08cm}'} }\right ] \hspace{0.05cm}.$$

[[File:P_ID1795__LZI_T_4_1_S3_kleiner_neu.png |frame| Attenuation function  $α(f)$  ( per unit length) and upper bounds | right]]
*On the other hand,  for small frequencies  $(f < f_∗)$  the serial resistance per unit length  $(R\hspace{0.05cm}')$  is much larger than  $ωL\hspace{0.05cm}'$  and the parallel conductance per unit length  $(G\hspace{0.05cm}')$  is much larger than  $ ωC\hspace{0.08cm}'$. 

*A second upper bound is obtained,  often referred to in literature as  '''strong attenuation''',  valid for  $(f \ll f_∗)$:
:$$\alpha_{_{ {\rm II} } }(f) = \sqrt{ 1/2 \cdot \omega \cdot {R\hspace{0.05cm}' \cdot C\hspace{0.08cm}'} }\hspace{0.1cm} \bigg |_{\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}\hspace{0.05cm}.$$

The diagram shows the attenuation function per unit length  $α(f)$  at constant primary line parameters according to the exact  (but more complicated)  formula and the two bounds  $α_{\rm I}(f)$  and  $α_{\rm II}(f)$.

One recognizes from this representation:
*Both  $α_{\rm I}(f)$  and  $α_{\rm II}(f)$  are upper bounds for  $α(f)$.
*The characteristic frequency  $f_∗$  is the intersection of  $α_{\rm I}(f)$  and  $α_{\rm II}(f)$.
*For  $f \gg f_∗$  holds  $α(f) ≈ α_{\rm I}(f)$,  whereas for  $f \ll f_∗$   $α(f) ≈ α_{\rm II}(f)$.
 
==Influence of reflections - operational attenuation==
 
The choice of the terminating resistor  $Z_2(f) = Z_{\rm W}(f)$  prevents the generation of a reflected wave at the end of the line.  In practice,  however,  an exact matching of these resistors is usually possible only in a very limited frequency range,  for example
[[File:P_ID1796__LZI_T_4_1_S4_neu.png |right|frame|Line of length  $l$  with ohmic terminations]]
*due to the complicated frequency dependence of the wave impedance,
*with cables of different designs along a connection,
*when taking into account manufacturing tolerances.

Therefore,  in real systems,  the internal resistance  $R_1$  of the source and the terminating resistance  $R_2$  are usually chosen to be real and constant.  For example, in  [[Examples_of_Communication_Systems/General_Description_of_ISDN|$\rm ISDN$]]  ("Integrated Services Digital Network")  $R_1 = R_2 = 150 \ \rm Ω$  has been set.

This circuit simplification has the following effects:

'''(1)'''   The input impedance  (German:  "Eingangswiderstand"   ⇒   subscipt "E")     $Z_{\rm E}(f)$  of the line as seen from the source depends
*on the complex propagation function per unit length   ⇒   $γ(f)$,
*the line length  $l$,
*the wave impedance  $Z_{\rm W}(f)$,  and
*the terminating resistance  $R_2$:
:$$Z_{\rm E}(f) = Z_{\rm W}(f)\cdot \frac {R_2 + Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}
{Z_{\rm W}(f)+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)} \hspace{0.05cm}, \hspace{0.5cm}
{\rm with}\hspace{0.5cm}{\rm tanh}(x) = \frac {{\rm sinh}(x)}{{\rm cosh}(x)} = \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm e}^{-x}}\hspace{0.05cm}, \hspace{0.3cm}x \in {\cal C} \hspace{0.05cm}.$$
'''(2)'''   This circuit-related simplification   ⇒   $Z_{\rm 2}(f) = R_2$  results in reflections at the end of the line. These reduce the power available at the receiver and thus increase the line attenuation.

'''(3)'''   For the evaluation of such a mismatched system, the  '''operational attenuation'''  ("attenuation in operation",  German:  "Betriebsdämpfung"   ⇒   subscipt "B")  has been defined as follows:
:$${ a}_{\rm B}(f) \hspace{0.15cm}{\rm in} \hspace{0.15cm}{\rm Neper}\hspace{0.15cm}{\rm (Np)} = {\rm ln}\hspace{0.1cm}\frac {|U_0(f)|}{2 \cdot |U_2(f)|} \cdot \sqrt{{R_2}/{R_1}}
= \alpha (f ) \cdot l + {\rm ln}\hspace{0.1cm}|q_1(f)| + {\rm ln}\hspace{0.1cm}|q_2(f)| + {\rm ln}\hspace{0.1cm}|1 - r_1(f) \cdot r_2(f) \cdot {\rm e}^{-\gamma(f) \hspace{0.05cm} \cdot \hspace{0.05cm}l}| \hspace{0.05cm}.$$

{{GraueBox|TEXT=
$\text{Example 2:}$ 
This equation will now be discussed using the above block diagram  $($valid for ISDN$)$.  We consider the general case of a mismatched system:
:$$R_2 ≠ Z_{\rm W}(f).$$
*The  "operational attenuation"  defined above relates the actual active power transmitted from the sender to the receiver to the best possible case  $($negligible line length,  full matching$)$.
*For resistance matching,  the operational attenuation is equal to the  '''wave attenuation'''.  In this case,  only the first term of the above equation is effective:
:$$a_{\rm B}(f) = \alpha (f ) \cdot l \hspace{0.05cm}.$$
*The second and third terms take into account the power losses due to reflection at the transitions from the transmitter to the line and from the line to the receiver. For these two  '''reflection losses'''  holds:
:$$q_1(f)= \frac {R_1 + Z_{\rm W}(f)}{2 \cdot \sqrt{R_1 \cdot Z_{\rm W}(f)} } \hspace{0.05cm}, \hspace{0.3cm}q_2(f)= \frac {R_2 + Z_{\rm W}(f)}{2 \cdot \sqrt{R_2 \cdot Z_{\rm W}(f)} } \hspace{0.05cm}.$$
*The  '''interaction attenuation'''  (fourth term) describes the effect of a multiple reflected wave,  which  –  depending on the line length  –  is constructively or destructively superimposed on the useful signal at the receiver.  The following applies to these reflection factors:
:$$r_1(f)= \frac {R_1 - Z_{\rm W}(f)}{R_1 + Z_{\rm W}(f)} \hspace{0.05cm}, \hspace{0.3cm}r_2(f)= \frac {R_2 - Z_{\rm W}(f)}{R_2 + Z_{\rm W}(f)} \hspace{0.05cm}.$$

The various components of the operational attenuation  $a_{\rm B}(f)$  are calculated in  [[Aufgaben:Exercise_4.3:_Operational_Attenuation|Exercise 4.3]]  for a practical example.}}

{{BlaueBox|TEXT=
$\text{Please note:}$  In the following chapters for
*Coaxial Cables, and
*Balanced Copper Pairs

only the wave attenuation  $α(f)\cdot l$  is further considered and thus the effects of error matching are neglected.}}

==Exercises for the chapter==
 
[[Aufgaben:Exercise_4.1:_Attenuation_Function |Exercise 4.1: Attenuation Function]]

[[Aufgaben:Exercise_4.1Z:_Transmission_Behavior_of_Short_Cables |Exercise 4.1Z: Transmission Behavior of Short Cables]]

[[Aufgaben:Aufgabe_4.2:_Fehlangepasste_Leitung| Exercise 4.2: Mismatched Line]]

[[Aufgaben:Exercise_4.3:_Operational_Attenuation|Exercise 4.3: Operational Attenuation]]

==References==
<references/>

{{Display}}

Information Theory/Application to Digital Signal Transmission

2022-02-17T11:39:17Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Mutual Information Between Two Discrete Random Variables
|Vorherige Seite=Verschiedene Entropien zweidimensionaler Zufallsgrößen
|Nächste Seite=Differentielle Entropie
}}

==Information-theoretical model of digital signal transmission ==
 
The entropies defined so far in general terms are now applied to digital signal transmission, whereby we assume a discrete memoryless channel  $\rm (DMC)$  according to the graphic:

[[File:P_ID2779__Inf_T_3_3_S1a_neu.png|right|frame|Digital signal transmission model under consideration]]

*The set of source symbols is characterised by the discrete random variable  $X$ , where  $|X|$  indicates the source symbol set size:

:$$X = \{\hspace{0.05cm}x_1\hspace{0.05cm}, \hspace{0.15cm} x_2\hspace{0.05cm},\hspace{0.15cm} \text{...}\hspace{0.1cm} ,\hspace{0.15cm} x_{\mu}\hspace{0.05cm}, \hspace{0.15cm}\text{...}\hspace{0.1cm} , \hspace{0.15cm} x_{|X|}\hspace{0.05cm}\}\hspace{0.05cm}.$$

*Correspondingly,  $Y$  characterises the set of sink symbols with the symbol set size  $|Y|$:

:$$Y = \{\hspace{0.05cm}y_1\hspace{0.05cm}, \hspace{0.15cm} y_2\hspace{0.05cm},\hspace{0.15cm} \text{...}\hspace{0.1cm} ,\hspace{0.15cm} y_{\kappa}\hspace{0.05cm}, \hspace{0.15cm}\text{...}\hspace{0.1cm} , \hspace{0.15cm} Y_{|Y|}\hspace{0.05cm}\}\hspace{0.05cm}.$$

*Usually  $|Y| = |X|$is valid.  Also possible is  $|Y| > |X|$, for example with the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC|Binary Erasure Channel]]  (BEC):
:$$X = \{0,\ 1\},\hspace{0.5cm} Y = \{0,\ 1,\ \ \text{E}\}\ ⇒ \ |X| = 2, \ |Y| = 3.$$

*The sink symbol  $\rm E$  indicates an  "erasure".  The event  $Y=\text{E}$  indicates that a decision for  $0$  or for  $1$  would be too uncertain.

*The symbol probabilities of the source and sink are accounted for in the graph by the probability mass functions  $P_X(X)$  and  $P_Y(Y)$:
:$$P_X(x_{\mu}) = {\rm Pr}( X = x_{\mu})\hspace{0.05cm}, \hspace{0.3cm}
P_Y(y_{\kappa}) = {\rm Pr}( Y = y_{\kappa})\hspace{0.05cm}.$$

*Let it hold:  The prtobability mass functions  $P_X(X)$  and  $P_Y(Y)$  contain no zeros   ⇒   $\text{supp}(P_X) = P_X$  and  $\text{supp}(P_Y) = P_Y$.  This prerequisite facilitates the description without loss of generality.

*All transition probabilities of the discrete memoryless channel   $\rm (DMC)$  are captured by the  conditional probability function  $P_{Y|X}(Y|X)$. . With  $x_μ ∈ X$  and  $y_κ ∈ Y$,  the following definition applies to this:

:$$P_{Y\hspace{0.01cm}|\hspace{0.01cm}X}(y_{\kappa}\hspace{0.01cm} |\hspace{0.01cm} x_{\mu}) = {\rm Pr}(Y\hspace{-0.1cm} = y_{\kappa}\hspace{0.03cm} | \hspace{0.03cm}X \hspace{-0.1cm}= x_{\mu})\hspace{0.05cm}.$$

The green block in the graph marks  $P_{Y|X}(⋅)$  with  $|X|$  inputs and  $|Y|$  outputs.  Blue connections mark transition probabilities  $\text{Pr}(y_i | x_μ)$  starting from  $x_μ$  with  $1 ≤ i ≤ |Y|$,  while all red connections end at  $y_κ$:    $\text{Pr}(y_κ | x_i)$  with  $1 ≤ i ≤ |X|$.

Before we give the entropies for the individual probability functions, viz.
:$$P_X(X) ⇒ H(X),\hspace{0.5cm} P_Y(Y) ⇒ H(Y), \hspace{0.5cm} P_{XY}(X) ⇒ H(XY), \hspace{0.5cm} P_{Y|X}(Y|X) ⇒ H(Y|X),\hspace{0.5cm} P_{X|Y}(X|Y) ⇒ H(X|Y),$$
the above statements are to be illustrated by an example.

[[File:P_ID2780__Inf_T_3_3_S1b_neu.png|right|frame|Channel model  "Binary Erasure Channel"  $\rm (BEC)$]]
{{GraueBox|TEXT=
$\text{Example 1}$:  In the book  "Channel Coding"  we also deal with the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC|Binary Erasure Channel]]  $\rm (BEC)$, which is sketched in a somewhat modified form on the right.   The following prerequisites apply:
*Let the input alphabet be binary  ⇒   $X = \{0,\ 1 \}$   ⇒   $\vert X\vert = 2$  while three values are possible at the output   ⇒   $Y = \{0,\ 1,\ \text{E} \}$   ⇒   $\vert Y\vert = 3$.
*The symbol  $\text{E}$  indicates the case that the receiver cannot decide for one of the binary symbols  $0$  or  $1$  due to too much channel interference.  "E"  stands for erasure.
*With the  $\rm BEC$  according to the above sketch, both a transmitted  $0$  and a  $1$  are erased with probability  $λ$  while the probability of a correct transmission is  $1 – λ$  in each case.
*In contrast, transmission errors are excluded by the BEC model   ⇒   the conditional probabilities  $\text{Pr}(Y = 1 \vert X = 0)$  and  $\text{Pr}(Y = 0 \vert X = 1)$  are both zero.

At the transmitter, the  "zeros"  and  "ones"  would not necessarily be equally probable.  Rather, we use the probability mass functions
:$$\begin{align*}P_X(X) & = \big ( {\rm Pr}( X = 0)\hspace{0.05cm},\hspace{0.2cm} {\rm Pr}( X = 1) \big )\hspace{0.05cm},\\
P_Y(Y) & = \big ( {\rm Pr}( Y = 0)\hspace{0.05cm},\hspace{0.2cm} {\rm Pr}( Y = 1)\hspace{0.05cm},\hspace{0.2cm} {\rm Pr}( Y = {\rm E}) \big )\hspace{0.05cm}.\end{align*}$$

From the above model we then get:

:$$\begin{align*}P_Y(0) & = {\rm Pr}( Y \hspace{-0.1cm} = 0) = P_X(0) \cdot ( 1 - \lambda)\hspace{0.05cm}, \\
P_Y(1) & = {\rm Pr}( Y \hspace{-0.1cm} = 1) = P_X(1) \cdot ( 1 - \lambda)\hspace{0.05cm}, \\
P_Y({\rm E}) & = {\rm Pr}( Y \hspace{-0.1cm} = {\rm E}) = P_X(0) \cdot \lambda \hspace{0.1cm}+\hspace{0.1cm} P_X(1) \cdot \lambda \hspace{0.05cm}.\end{align*}$$

If we now take  $P_X(X)$  and  $P_Y(Y)$  to be vectors, the result can be represented as follows:

:$$P_{\hspace{0.05cm}Y}(Y) = P_X(X) \cdot P_{\hspace{0.05cm}Y\hspace{-0.01cm}\vert \hspace{-0.01cm}X}(Y\hspace{-0.01cm} \vert \hspace{-0.01cm} X) \hspace{0.05cm},$$

where the transition probabilities  $\text{Pr}(y_κ\vert x_μ)$  are accounted for by the following matrix:

:$$P_{\hspace{0.05cm}Y\hspace{-0.01cm} \vert \hspace{-0.01cm}X}(Y\hspace{-0.01cm} \vert \hspace{-0.01cm} X) =
\begin{pmatrix}
1 - \lambda & 0 & \lambda\\
0 & 1 - \lambda & \lambda
\end{pmatrix}\hspace{0.05cm}.$$

Note:
*We have chosen this representation only to simplify the description.
*$P_X(X)$  and  $P_Y(Y)$  are not vectors in the true sense and  $P_{Y \vert X}(Y\vert X)$  is not a matrix either.}}

==Directional diagram for digital signal transmission ==
 
All entropies defined in the  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen|last chapter]]  also apply to digital signal transmission.  However, it is expedient to choose the right-hand diagram instead of the diagram used so far, corresponding to the left-hand diagram, in which the direction from source  $X$  to sink  $Y$  is recognisable.

[[File:EN_Inf_T_3_3_S2_vers2.png|center|frame|Two information-theoretical models for digital signal transmission]]

Let us now interpret the right graph starting from the general  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Information-theoretical_model_of_digital_signal_transmission|$\rm DMC$  model]]:
*The  '''source entropy'''  $H(X)$  denotes the average information content of the source symbol sequence.   With the symbol set size  $|X|$  applies:

:$$H(X) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_X(X)}\right ] \hspace{0.1cm}
= -{\rm E} \big [ {\rm log}_2 \hspace{0.1cm}{P_X(X)}\big ] \hspace{0.2cm}
=\hspace{0.2cm} \sum_{\mu = 1}^{|X|}
P_X(x_{\mu}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_X(x_{\mu})} \hspace{0.05cm}.$$

*The  '''equivocation'''  $H(X|Y)$  indicates the average information content that an observer who knows exactly about the sink  $Y$  gains by observing the source  $X$ :

:$$H(X|Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}X\hspace{-0.01cm}|\hspace{-0.01cm}Y}(X\hspace{-0.01cm} |\hspace{0.03cm} Y)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}X\hspace{-0.01cm}|\hspace{0.03cm}Y}
(\hspace{0.05cm}x_{\mu}\hspace{0.03cm} |\hspace{0.05cm} y_{\kappa})}
\hspace{0.05cm}.$$

*The equivocation is the portion of the source entropy  $H(X)$  that is lost due to channel interference  (for digital channel: transmission errors).  The  '''mutual information'''  $I(X; Y)$  remains, which reaches the sink:

:$$I(X;Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{P_{XY}(X, Y)}{P_X(X) \cdot P_Y(Y)}\right ] \hspace{0.2cm} = H(X) - H(X|Y) \hspace{0.05cm}.$$

*The  '''irrelevance'''  $H(Y|X)$  indicates the average information content that an observer who knows exactly about the source  $X$  gains by observing the sink  $Y$:

:$$H(Y|X) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{-0.01cm}X}(Y\hspace{-0.01cm} |\hspace{0.03cm} X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{0.03cm}X}
(\hspace{0.05cm}y_{\kappa}\hspace{0.03cm} |\hspace{0.05cm} x_{\mu})}
\hspace{0.05cm}.$$

*The  '''sink entropy'''  $H(Y)$, the mean information content of the sink.  $H(Y)$  is the sum of the useful mutual information  $I(X; Y)$  and the useless irrelevance  $H(Y|X)$, which comes exclusively from channel errors:

:$$H(Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_Y(Y)}\right ] \hspace{0.1cm}
= -{\rm E} \big [ {\rm log}_2 \hspace{0.1cm}{P_Y(Y)}\big ] \hspace{0.2cm} =I(X;Y) + H(Y|X)
\hspace{0.05cm}.$$

==Calculation of the mutual information for the binary channel==
 
These definitions will now be illustrated by an example.   We deliberately avoid simplifying the calculations by exploiting symmetries.

[[File:P_ID2782__Inf_T_3_3_S3a.png|right|frame|General model of the binary channel]]
{{GraueBox|TEXT=
$\text{Example 2}$:  We consider the general binary channel without memory according to the sketch.  Let the falsification probabilities be:

:$$\begin{align*}\varepsilon_0 & = {\rm Pr}(Y\hspace{-0.1cm} = 1\hspace{0.05cm}\vert X \hspace{-0.1cm}= 0) = 0.01\hspace{0.05cm},\\
\varepsilon_1 & = {\rm Pr}(Y\hspace{-0.1cm} = 0\hspace{0.05cm} \vert X \hspace{-0.1cm}= 1) = 0.2\hspace{0.05cm}\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.05cm}Y\hspace{-0.01cm} \vert \hspace{-0.01cm}X}(Y\hspace{-0.01cm} \vert \hspace{-0.01cm} X) =
\begin{pmatrix}
1 - \varepsilon_0 & \varepsilon_0\\
\varepsilon_1 & 1 - \varepsilon_1
\end{pmatrix} =
\begin{pmatrix}
0.99 & 0.01\\
0.2 & 0.8
\end{pmatrix} \hspace{0.05cm}.$$

Furthermore, we assume source symbols that are not equally probable:

:$$P_X(X) = \big ( p_0,\ p_1 \big )=
\big ( 0.1,\ 0.9 \big )
\hspace{0.05cm}.$$

With the  [[Information_Theory/Gedächtnislose_Nachrichtenquellen#Binary_entropy_function|binary entropy function]]  $H_{\rm bin}(p)$,  we thus obtain for the source entropy:

:$$H(X) = H_{\rm bin} (0.1) = 0.4690 \hspace{0.12cm}{\rm bit}
\hspace{0.05cm}.$$

For the probability mass function of the sink as well as for the sink entropy we thus obtain:

:$$P_Y(Y) = \big [ {\rm Pr}( Y\hspace{-0.1cm} = 0)\hspace{0.05cm}, \ {\rm Pr}( Y \hspace{-0.1cm}= 1) \big ] = \big ( p_0\hspace{0.05cm},\ p_1 \big ) \cdot
\begin{pmatrix}
1 - \varepsilon_0 & \varepsilon_0\\
\varepsilon_1 & 1 - \varepsilon_1
\end{pmatrix} $$

:$$\begin{align*}\Rightarrow \hspace{0.3cm} {\rm Pr}( Y \hspace{-0.1cm}= 0)& =
p_0 \cdot (1 - \varepsilon_0) + p_1 \cdot \varepsilon_1 =
0.1 \cdot 0.99 + 0.9 \cdot 0.2 = 0.279\hspace{0.05cm},\\
{\rm Pr}( Y \hspace{-0.1cm}= 1) & = 1 - {\rm Pr}( Y \hspace{-0.1cm}= 0) = 0.721\end{align*}$$

:$$\Rightarrow \hspace{0.3cm}
H(Y) = H_{\rm bin} (0.279) = 0.8541 \hspace{0.12cm}{\rm bit}
\hspace{0.05cm}. $$

The joint probabilities  $p_{\mu \kappa} = \text{Pr}\big[(X = μ) ∩ (Y = κ)\big]$  between source and sink are:

:$$\begin{align*}p_{00} & = p_0 \cdot (1 - \varepsilon_0) = 0.099\hspace{0.05cm},\hspace{0.5cm}p_{01}= p_0 \cdot \varepsilon_0 = 0.001\hspace{0.05cm},\\
p_{10} & = p_1 \cdot (1 - \varepsilon_1) = 0.180\hspace{0.05cm},\hspace{0.5cm}p_{11}= p_1 \cdot \varepsilon_1 = 0.720\hspace{0.05cm}.\end{align*}$$

From this one obtains for
*the  '''joint entropy''':

:$$H(XY) = p_{00}\hspace{-0.05cm} \cdot \hspace{-0.05cm}{\rm log}_2 \hspace{0.05cm} \frac{1}{p_{00} \rm } +
p_{01} \hspace{-0.05cm} \cdot \hspace{-0.05cm}{\rm log}_2 \hspace{0.05cm} \frac{1}{p_{01} \rm } +
p_{10}\hspace{-0.05cm} \cdot \hspace{-0.05cm} {\rm log}_2 \hspace{0.05cm} \frac{1}{p_{10} \rm } +
p_{11} \hspace{-0.05cm} \cdot \hspace{-0.05cm} {\rm log}_2\hspace{0.05cm} \frac{1}{p_{11}\rm } = 1.1268\,{\rm bit} \hspace{0.05cm},$$

*the  '''mutual information''':

:$$I(X;Y) = H(X) + H(Y) - H(XY) = 0.4690 + 0.8541 - 1.1268 = 0.1963\hspace{0.12cm}{\rm bit}
\hspace{0.05cm},$$

[[File:EN_Inf_T_3_3_S3b_vers2.png|right|frame|Informationstheoretisches Modell des betrachteten Binärkanals]]
*the  '''equivocation''':

:$$H(X \vert Y) \hspace{-0.01cm} =\hspace{-0.01cm} H(X) \hspace{-0.01cm} -\hspace{-0.01cm} I(X;Y) \hspace{-0.01cm} $$
:$$\Rightarrow \hspace{0.3cm} H(X \vert Y) \hspace{-0.01cm} = \hspace{-0.01cm} 0.4690\hspace{-0.01cm} -\hspace{-0.01cm} 0.1963\hspace{-0.01cm} =\hspace{-0.01cm} 0.2727\hspace{0.12cm}{\rm bit}
\hspace{0.05cm},$$

*the '''irrelevance''':

:$$H(Y \vert X) = H(Y) - I(X;Y) $$
:$$\Rightarrow \hspace{0.3cm} H(Y \vert X) = 0.8541 - 0.1963 = 0.6578\hspace{0.12cm}{\rm bit}
\hspace{0.05cm}.$$

The results are summarised in the graph.}}

''Notes'':
*The equivocation and irrelevance could also have been calculated directly (but with extra effort) from the corresponding probability functions.
*For example, the irrelevance:

:$$H(Y|X) = \hspace{-0.2cm} \sum_{(x, y) \hspace{0.05cm}\in \hspace{0.05cm}XY} \hspace{-0.2cm} P_{XY}(x,\hspace{0.05cm}y) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{0.03cm}X}
(\hspace{0.05cm}y\hspace{0.03cm} |\hspace{0.05cm} x)}= p_{00} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{1\hspace{-0.08cm} - \hspace{-0.08cm}\varepsilon_0} +
p_{01} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{\varepsilon_0} +
p_{10} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{1\hspace{-0.08cm} - \hspace{-0.08cm}\varepsilon_1} +
p_{11} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{\varepsilon_1} = 0.6578\,{\rm bit} \hspace{0.05cm}.$$

==Definition and meaning of channel capacity ==
 
We further consider a discrete memoryless channel  $\rm (DMC)$  with a finite number of source symbols   ⇒   $|X|$  and also only finitely many sink symbols   ⇒   $|Y|$,  as shown in the first section of this chapter.
*If one calculates the mutual information  $I(X, Y)$  as explained in  $\text{Example 2}$,  it also depends on the source statistic   ⇒   $P_X(X)$.
* Ergo:   '''The mutual information'''  $I(X, Y)$ ''' is not a pure channel characteristic'''.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''channel capacity'''  introduced by  [https://en.wikipedia.org/wiki/Claude_Shannon Claude E. Shannon]  according to his standard work  [Sha48]<ref name = Sha48>Shannon, C.E.:  A Mathematical Theory of Communication. In:  Bell Syst. Techn. J. 27 (1948), S. 379-423 und S. 623-656.</ref> reads:

:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) \hspace{0.05cm}.$$

The additional unit  "bit/use"  is often added.  Since according to this definition the best possible source statistics are always the basis:
*$C$  depends only on the channel properties   ⇒   $P_{Y \vert X}(Y \vert X)$,
*but not on the source statistics   ⇒   $P_X(X)$.  }}

Shannon needed the quantity  $C$  to formulate the  "Channel Coding Theorem" – one of the highlights of the information theory he founded.

{{BlaueBox|TEXT=
$\text{Shannon's Channel Coding Theorem: }$ 
*For every transmission channel with channel capacity  $C > 0$,  there exists (at least) one  $(k,\ n)$–block code,  whose (block) error probability approaches zero  as long as the code rate  $R = k/n$  is less than or equal to the channel capacity:  
:$$R ≤ C.$$
*The prerequisite for this, however,  is that the following applies to the block length of this code:   $n → ∞.$}}

The proof of this theorem,  which is beyond the scope of our learning tutorial,  can be found for example in  [CT06]<ref name="CT06">Cover, T.M.; Thomas, J.A.:  Elements of Information Theory.  West Sussex: John Wiley & Sons, 2nd Edition, 2006.</ref>,  [Kra13]<ref name="Kra13">Kramer, G.:  Information Theory.  Lecture manuscript, Chair of Communications Engineering, Technische Universität München, 2013.</ref>  und  [Meck09]<ref name="Meck09">Mecking, M.:  Information Theory.  Lecture manuscript, Chair of Communications Engineering, Technische Universität München, 2009.</ref>.

As will be shown in  [[Aufgaben:Exercise_3.13:_Code_Rate_and_Reliability|Exercise 3.13]],  the reverse is also true.  This proof can also be found in the literature references just mentioned.

{{BlaueBox|TEXT=
$\text{Reverse of Shannon's channel coding theorem: }$ 

If the rate  $R$  of the  $(n,\ k)$–block code used is greater than the channel capacity  $C$,  then an arbitrarily small block error probability is not achievable.}}

In the chapter  [[Information_Theory/AWGN–Kanalkapazität_bei_wertdiskretem_Eingang#AWGN.E2.80.93Modell_f.C3.BCr_zeitdiskrete_bandbegrenzte_Signale|AWGN model for discrete-time band-limited signals]]  it is explained in connection with the continuous  [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|AWGN channel model]]    what phenomenally great significance Shannon's theorem has for the entire field of information technology,  not only for those interested exclusively in theory,  but also for practitioners.

==Channel capacity of a binary channel==
 
[[File:P_ID2786__Inf_T_3_3_S3a.png|right|frame|General model of the binary channel]]
The mutual information of the general  (asymmetrical)  binary channel according to this sketch was calculated in  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Transinformationsberechnung_f.C3.BCr_den_Bin.C3.A4rkanal|$\text{Example 2}$]].  In this model, the input symbols  $0$  and  $1$  are distorted to different degrees:

:$$P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{-0.01cm}X}(Y\hspace{-0.01cm} |\hspace{-0.01cm} X) =
\begin{pmatrix}
1 - \varepsilon_0 & \varepsilon_0\\
\varepsilon_1 & 1 - \varepsilon_1
\end{pmatrix} \hspace{0.05cm}.$$

The mutual information can be represented with the probability mass function  $P_X(X) = (p_0,\ p_1)$  as follows:

:$$I(X ;Y) = \sum_{\mu = 1}^{2} \hspace{0.1cm}\sum_{\kappa = 1}^{2} \hspace{0.2cm}
{\rm Pr} (\hspace{0.05cm}y_{\kappa}\hspace{0.03cm} |\hspace{0.05cm} x_{\mu}) \cdot
{\rm Pr} (\hspace{0.05cm}x_{\mu}\hspace{0.05cm})\cdot {\rm log}_2 \hspace{0.1cm} \frac{{\rm Pr}
(\hspace{0.05cm}y_{\kappa}\hspace{0.03cm} |\hspace{0.05cm} x_{\mu})}{{\rm Pr}
(\hspace{0.05cm}y_{\kappa})} $$
:$$\begin{align*}\Rightarrow \hspace{0.3cm} I(X ;Y) &= \hspace{-0.01cm} (1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_0) \cdot p_0 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_0}{(1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_0) \cdot p_0 + \varepsilon_1 \cdot p_1} +
\varepsilon_0 \cdot p_0 \cdot {\rm log}_2 \hspace{0.1cm} \frac{\varepsilon_0}{(1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_0) \cdot p_0 + \varepsilon_1 \cdot p_1} \ + \\
& + \hspace{-0.01cm} \varepsilon_1 \cdot p_1 \cdot {\rm log}_2 \hspace{0.1cm} \frac{\varepsilon_1}{\varepsilon_0 \cdot p_0 + (1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_1) \cdot p_1} + (1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_1) \cdot p_1 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_1}{\varepsilon_0 \cdot p_0 + (1 \hspace{-0.08cm}- \hspace{-0.08cm}\varepsilon_1) \cdot p_1}
\hspace{0.05cm}.\end{align*}$$

[[File:P_ID2788__Inf_T_3_3_S4a.png|right|frame|Mutual information for the "asymmetrical binary channel"]]
{{GraueBox|TEXT=
$\text{Example 3}$: 
In the following we set  $ε_0 = 0.01$  and  $ε_1 = 0.2$.

Column 4 of the adjacent table shows  (highlighted in green)  the mutual information  $I(X; Y)$  of this asymmetrical binary channel depending on the source symbol probability  $p_0 = {\rm Pr}(X = 0)$  .  One can see:
*The mutual information depends on the symbol probabilities  $p_0$  and  $p_1 = 1 - p_0$.
*Here the maximum value of  $I(X; Y)$  results in  $p_0 ≈ 0.55$    ⇒    $p_1 ≈ 0.45$.
*The result  $p_0 > p_1$  follows from the relation  $ε_0 < ε_1$  (the  "zero"  is less distorted).
*The capacity of this channel is  $C = 0.5779 \ \rm bit/use$.}}
 
In the above equation, the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|Binary Symmetric Channel]]  $\rm (BSC)$  with parameters  $ε = ε_0 = ε_1$  is also included as a special case.  Hints:
*In  [[Aufgaben:Exercise_3.10:_Mutual_Information_at_the_BSC|Exercise 3.10]]  the mutual information of the BSC is calculated for the system parameters  $ε = 0.1$  and  $p_0 = 0.2$  .
*In  [[Aufgaben:Exercise_3.10Z:_BSC_Channel_Capacity|Exercise 3.10Z]]  its channel capacity is given as follows:

:$$C_{\rm BSC} = 1 - H_{\rm bin} (\varepsilon) \hspace{0.05cm}.$$

==Properties of symmetrical channels ==
 
The capacity calculation of the (general)  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Information-theoretical_model_of_digital_signal_transmission|discrete memoryless channel]]  $\rm (DMC)$  is often complex.  It simplifies decisively if symmetries of the channel are exploited.  

[[File:EN_Inf_T_3_3_S6a_vers2.png|right|frame|Examples of symmetrical channels]]

 The diagram shows two examples:

*In the case of the  uniformly dispersive channel  all source symbols  $x ∈ X$  result in exactly the same set of transition probabilities   ⇒   $\{P_{Y\hspace{0.03cm}|\hspace{0.01cm}X}(y_κ\hspace{0.05cm}|\hspace{0.05cm}x)\}$  with  $1 ≤ κ ≤ |Y|$.  Here  $q + r + s = 1$  must always apply here  (see left graph).

*In the case of the  uniformly focusing channel , the same set of transition probabilities  ⇒   $\{P_{Y\hspace{0.03cm}|\hspace{0.01cm}X}(y\hspace{0.05cm}|\hspace{0.05cm}x_μ)\}$  with  $1 ≤ μ ≤ |X|$ results for all sink symbols  $y ∈ Y$ .   Here,  $t + u + v = 1$  need not necessarily hold  (see right graph).
 
{{BlaueBox|TEXT=
$\text{Definition:}$  If a discrete memoryless channel is both uniformly dispersive and uniformly focussing,  it is a  '''strictly symmetric channel'''. 
*With an equally distributed source alphabet, this channel has the capacity
:$$C = {\rm log}_2 \hspace{0.1cm} \vert Y \vert + \sum_{y \hspace{0.05cm}\in\hspace{0.05cm} Y} \hspace{0.1cm} P_{\hspace{0.03cm}Y \vert \hspace{0.01cm} X}(y\hspace{0.05cm} \vert \hspace{0.05cm}x) \cdot
{\rm log}_2 \hspace{0.1cm}P_{\hspace{0.01cm}Y \vert \hspace{0.01cm} X}(y\hspace{0.05cm}\vert\hspace{0.05cm} x)
\hspace{0.05cm}.$$
*Any  $x ∈ X$  can be used for this equation.}}

This definition will now be clarified by an example.

{{GraueBox|TEXT=
$\text{Example 4}$:  In the channel under consideration, there are connections between all  $ \vert X \vert = 3$  inputs and all  $ \vert Y \vert = 3$  outputs:
[[File:P_ID2794__Inf_T_3_3_S6b.png|right|frame|Strongly symmetrical channel  $\vert X \vert = \vert Y \vert= 3$]]
*A red connection stands for  $P_{Y \hspace{0.03cm}\vert\hspace{0.01cm} X}(y_κ \hspace{0.05cm} \vert \hspace{0.05cm} x_μ) = 0.7$.
*A blue connection stands for  $P_{Y \hspace{0.03cm}\vert\hspace{0.01cm} X}(y_κ \hspace{0.05cm}\vert \hspace{0.05cm} x_μ) = 0.2$.
*A green connection stands for  $P_{Y \hspace{0.03cm}\vert\hspace{0.01cm} X}(y_κ \hspace{0.05cm}\vert\hspace{0.05cm} x_μ) = 0.1$.

According to the above equation, the following applies to the channel capacity:

:$$C = {\rm log}_2 \hspace{0.1cm} (3) + 0.7 \cdot {\rm log}_2 \hspace{0.1cm} (0.7)
+ 0.2 \cdot {\rm log}_2 \hspace{0.1cm} (0.2) + 0.1 \cdot {\rm log}_2 \hspace{0.1cm} (0.1) = 0.4282 \,\,{\rm bit} \hspace{0.05cm}.$$

Notes:
*The addition of  "the same set of transition probabilities”  in the above definitions does not mean that it must apply:
:$$P_{Y \hspace{0.03cm}\vert\hspace{0.01cm} X}(y_κ \hspace{0.05cm}\vert\hspace{0.05cm} x_1) = P_{Y \hspace{0.03cm}\vert\hspace{0.01cm} X}(y_κ \hspace{0.05cm}\vert\hspace{0.05cm} x_2) = P_{Y \hspace{0.03cm}\vert\hspace{0.01cm} X}(y_κ \hspace{0.05cm}\vert\hspace{0.05cm} x_3).$$
*Rather, here a red, a blue and a green arrow leaves from each input and a red, a blue and a green arrow arrives at each output.
*The respective sequences permute:   R – G – B,     B – R – G,     G – B – R.}}

An example of a strictly symmetrical channel is the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|Binary Symmetric Channel]]  $\rm (BSC)$.  In contrast, the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_.E2.80.93_BEC|Binary Erasure Channel]]  $\rm (BEC)$  is not strictly symmetric,  because,
*although it is uniformly dispersive,
*but it is not uniformly focussing.

The following definition is less restrictive than the previous one of a strictly symmetric channel.

{{BlaueBox|TEXT=
$\text{Definition:}$  A  '''symmetric channel'''  exists,
*if it can be divided into several  $($generally $L)$  strongly symmetrical sub-channels,
*by splitting the output alphabet  $Y$  into  $L$  subsets  $Y_1$, ... , $Y_L$ .

Such a  "symmetric channel"  has the following capacity:

:$$C = \sum_{l \hspace{0.05cm}=\hspace{0.05cm} 1}^{L} \hspace{0.1cm} p_{\hspace{0.03cm}l} \cdot C_{\hspace{0.03cm}l} \hspace{0.05cm}.$$

The following designations are used here:
* $p_{\hspace{0.03cm}l}$ indicates the probability that the  $l$–th sub-channel is selected.
* $C_{\hspace{0.03cm}l}$  is the channel capacity of this  $l$–th sub-channel.}}

[[File:EN_Inf_T_3_3_S6c_v2.png|right|frame|Symmetrical channel consisting of two strongly symmetrical sub-channels  $\rm A$  and  $\rm B$]]
The diagram illustrates this definition for  $L = 2$  with the sub-channels  $\rm A$  and  $\rm B$.
*The differently drawn transitions  (dashed or dotted)  show that the two sub-channels can be different,  so that  $C_{\rm A} ≠ C_{\rm B}$  will generally apply.
*For the capacity of the total channel one thus obtains in general:

:$$C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} \hspace{0.05cm}.$$

*No statement is made here about the structure of the two sub-channels.

The following example will show that the  "Binary Erasure Channel"  $\rm (BEC)$  can also be described in principle by this diagram.   However, the two output symbols  $y_3$  and  $y_4$  must then be combined into a single symbol.
 
[[File:EN_Inf_T_3_3_S6d.png|right|frame|$\rm BEC$  in two different representations]]
{{GraueBox|TEXT=
$\text{Example 5}$:  The left figure shows the usual representation of the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_.E2.80.93_BEC|Binary Erasure Channel]]  $\rm (BEC)$  with input  $X = \{0,\ 1\}$  and output  $Y = \{0,\ 1,\ \text{E} \}$.

If one divides this according to the right grafic into
*an  "ideal channel"  $(y = x)$  for
:$$y ∈ Y_{\rm A} = \{0, 1\} \ \ ⇒ \ \ C_{\rm A} = 1 \ \rm bit/use,$$
*an  "erasure channel"  $(y = {\rm E})$  for
:$$y ∈ Y_{\rm B} = \{\rm E \} \ \ ⇒ \ \ C_{\rm B} = 0,$$

then we get with the sub-channel weights  $p_{\rm A} = 1 – λ$  and  $p_{\rm B} = λ$:

:$$C_{\rm BEC} = p_{\rm A} \cdot C_{\rm A} = 1 - \lambda \hspace{0.05cm}.$$

Both channels are strongly symmetrical.   The following applies equally for the (ideal) channel  $\rm A$ 
*for  $X = 0$  and  $X = 1$:     $\text{Pr}(Y = 0 \hspace{0.05cm}\vert \hspace{0.05cm} X) = \text{Pr}(Y = 1 \hspace{0.05cm} \vert\hspace{0.05cm} X) = 1 - λ$   ⇒   uniformly dispersive,
*for  $Y = 0$  and   $Y = 1$:     $\text{Pr}(Y \hspace{0.05cm} \vert \hspace{0.05cm} X= 0) = Pr(Y \hspace{0.05cm}\vert\hspace{0.05cm} X = 1) = 1 - λ$   ⇒   uniformly focussing.

The same applies to the erasure channel  $\rm B$.}}

In  [[Aufgaben:3.11_Streng_symmetrische_Kanäle|Exercise 3.12]]  it will be shown that the capacity of the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC|Binary Symmetric Error & Erasure Channel]]  $\rm (BSEC)$  model can be calculated in the same way.  One obtains:

:$$C_{\rm BSEC} = (1- \lambda) \cdot \left [ 1 - H_{\rm bin}(\frac{\varepsilon}{1- \lambda}) \right ]$$

*with the crossover probability  $ε$
*and the erasure probability  $λ$.

==Some basics of channel coding ==
 
In order to interpret the channel coding theorem correctly, some basics of  '''channel coding'''.  This extremely important area of Communications Engineering is covered in our learning tutorial  $\rm LNTwww$  in a separate book called  [[Channel_Coding]].

[[File:EN_Inf_T_3_3_S7a_vers2.png|center|frame|Model for binary–coded communication]]

The following description refers to the highly simplified model for  [[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes|binary block codes]]:
*The infinitely long source symbol sequence  $\underline{u}$  (not shown here)  is divided into blocks of  $k$  bits.  We denote the information block with the serial number  $j$  by  $\underline{u}_j^{(k)}$.
*Each information block  $\underline{u}_j^{(k)}$  is converted into a code word  $\underline{x}_j^{(n)}$  by the channel encoder with a yellow background, where  $n > k$  is to apply.  The ratio  $R = k/n$  is called the  '''code rate'''.
*The  "Discrete Memoryless Channel"  $\rm (DMC)$  is taken into account by transition probabilities  $P_{Y\hspace{0.03cm}|\hspace{0.03cm}X}(⋅)$ .  This block with a green background causes errors at the bit level.  The following can therefore apply:   $y_{j, \hspace{0.03cm}i} ≠ x_{j,\hspace{0.03cm} i}$.
*Thus the received blocks  $\underline{y}_j^{(n)}$  consisting of   $n$  bits can also differ from the code words  $\underline{x}_j^{(n)}$ .  Likewise, the following generally applies to the blocks after the decoder: 
:$$\underline{v}_j^{(k)} ≠ \underline{u}_j^{(k)}.$$

{{GraueBox|TEXT=
$\text{Example 6}$: 
The diagram is intended to illustrate the nomenclature used here using the example of  $k = 3$  and  $n = 4$.  The first eight blocks of the information sequence  $\underline{u}$  and the  code sequence $\underline{x}$ are shown.

[[File:EN_Inf_T_3_3_S7b_vers2.png|right|frame|Bit designation of information block and code word]]
One can see the following assignment between the blocked and the unblocked description:
*Bit 3 of the 1st information block   ⇒   $u_{1,\hspace{0.08cm} 3}$  corresponds to the symbol  $u_3$  in unblocked representation.
*Bit 1 of the 2nd information block   ⇒   $u_{2, \hspace{0.08cm}1}$  corresponds to the symbol  $u_4$  in unblocked representation.
*Bit 2 of the 6th information block   ⇒   $u_{6, \hspace{0.08cm}2}$  corresponds to the symbol  $u_{17}$  in unblocked representation.
*Bit 4 of the 1st code word   ⇒   $x_{1, \hspace{0.08cm}4}$  corresponds to the symbol  $x_4$  in unblocked representation.
*Bit 1 of the 2nd code word   ⇒   $x_{2, \hspace{0.08cm}1}$  corresponds to the symbol  $x_5$  in unblocked representation.
*Bit 2 of the 6th code word   ⇒   $x_{6, \hspace{0.08cm}2}$  corresponds to the symbol  $x_{22}$  in unblocked representation.}}

==Relationship between block errors and bit errors==
 
To interpret the channel coding theorem, we still need various definitions for error probabilities.  Various descriptive variables can be derived from the above system model:

{{BlaueBox|TEXT=
$\text{Definitions:}$
*In the present channel model, the  $\text{channel error probability}$  is given by

:$$\text{Pr(channel error)} = {\rm Pr} \left ({y}_{j,\hspace{0.05cm} i} \ne {x}_{j,\hspace{0.05cm} i}
\right )\hspace{0.05cm}.$$

:For example, in the BSC model  $\text{Pr(channel error)} = ε$  für alle  $j = 1, 2$, ...  and  $1 ≤ i ≤ n$.
*The  $\text{block error probability}$  refers to the allocated information blocks at the encoder input   ⇒   $\underline{u}_j^{(k)}$  and the decoder output   ⇒   $\underline{v}_j^{(k)}$,  each in blocks of  $k$  bits:

:$$\text{Pr(block error)} = {\rm Pr} \left (\underline{\upsilon}_j^{(k)} \ne \underline{u}_j^{(k)}
\right )\hspace{0.05cm}.$$

*The  $\text{bit error probability}$  also refers to the input and the output of the entire coding system under consideration, but at the bit level:

:$$\text{Pr(bit error)} = {\rm Pr} \left ({\upsilon}_{j,\hspace{0.05cm} i} \ne {u}_{j,\hspace{0.05cm} i}
\right )\hspace{0.05cm}.$$

:For simplicity, it is assumed here that all  $k$  bits  $u_{j,\hspace{0.08cm}i}$  $(1 ≤ i ≤ k)$  of the information block  $j$  are corrupted with equal probability.
:Otherwise, the  $k$  bits would have to be averaged.}}

There is a general relationship between the block error probability and the bit error probability:

:$${1}/{k} \cdot \text{Pr(block error)} \le \text{Pr(bit error)} \le \text{Pr(block error)}
\hspace{0.05cm}.$$

*The lower bound results when all bits are wrong in all faulty blocks.
*If there is exactly only one bit error in each faulty block, then the bit error probability is identical to the block error probability:
:$$ \text{Pr(bit error)} \equiv \text{Pr(block error)} \hspace{0.05cm}.$$

[[File:EN_Inf_T_3_3_S7c_vers2.png|frame|Definition of different error probabilities]]
{{GraueBox|TEXT=
$\text{Example 7:}$  The upper graph shows the first eight reception blocks  $\underline{y}_j^{(n)}$  with  $n = 4$  bits each.  Here channel errors are shaded green.

Below, the initial sequence  $\underline{v}$  after decoding is sketched, divided into blocks  $\underline{v}_j^{(k)}$  with  $k = 3$  bits each. Note:

*Bit errors are shaded red in the lower diagram.
*Block errors can be recognised by the blue frame.

For this, some  (due to the short sequence)  only very vague information about the error probabilities:
*Half of the reception bits are shaded green.  From this follows:  
:$$\text{Pr(channel error)} = 16/32 = 1/2.$$

*The bit error probability with the exemplary encoding and decoding is:  
:$$\text{Pr(bit error)} = 8/24 = 1/3.$$

*In contrast, with uncoded transmission would be:  
:$$\text{Pr(bit error)} = {\rm Pr(Kanalfehler)} = 1/2.$$

*Half of the decoded blocks are outlined in blue. From this follows:  
:$$\text{Pr(block error)} = 4/8 = 1/2.$$

*With  $\text{Pr(block error)}= 1/2$  and  $k = 3$  the bit error probability is in the following range:  
:$$1/6 \le \text{Pr(bit error)} \le 1/2
\hspace{0.05cm}.$$

#The upper bound with respect to bit errors is obtained when all bits in each of the four corrupted blocks are wrong:   $\text{Pr(bit error)} = 12/24 = 1/2.$
#The lower bound indicates that only one bit is wrong in each of the four corrupted blocks:   $\text{Pr(bit error)} = 4/24 = 1/6$.}}

==Rate, channel capacity and bit error probability==
 
{{BlaueBox|TEXT=
$\text{Basically:}$ 
*Channel coding increases the reliability of data transmission from the source to the sink.
*If the code rate  $R = k/n$  is reduced and the added redundancy  $(1 - R)$  is increased, the data reliability is generally improved and the bit error probability is reduced, which we will refer to as  $p_{\rm B}$  in the following:

:$$p_{\rm B} = \text{Pr(bit error)} = {\rm Pr} \left ({\upsilon}_{j,\hspace{0.05cm} i} \ne {u}_{j,\hspace{0.05cm} i}
\right )\hspace{0.05cm}.$$}}

The following theorem is based on the  "Data Processing Theorem"  and  "Fano's Lemma".  The derivation can be found in standard works on information theory, for example in  [CT06]<ref name="CT06" />:

{{BlaueBox|TEXT=
$\text{Inversion of Shannon's Channel Coding Theorem:}$ 

If one uses a channel with too small a capacity  $R$  for data transmission at rate  $C < R$, the bit error probability  $p_{\rm B}$  cannot fall below a lower bound even with the best possible channel coding:

:$$p_{\rm B} \ge H_{\rm bin}^{-1} \cdot \left ( 1 - {C}/{R}\right ) > 0\hspace{0.05cm}.$$

Here  $H_{\rm bin}(⋅)$  denotes the  [[Information_Theory/Discrete_Memoryless_Sources#Binary_entropy_function|binary entropy function]].}}

Since the probability of the block errors can never be smaller than that of the bit errors,  the block error probability „zero” is also not possible for  $R > C$ .  From the given bounds for the bit errors,

:$$ {1}/{k} \cdot {\rm Pr}({\rm block error}) \le {\rm Pr}({\rm bit error}) \le {\rm Pr}({\rm block error})\hspace{0.05cm},$$

a range for the block error probability can also be given:

:$$ {\rm Pr}({\rm bit error}) \le {\rm Pr}({\rm block error}) \le k \cdot {\rm Pr}({\rm bit error})\hspace{0.05cm}.$$

{{GraueBox|TEXT=
$\text{Example 8:}$  For a channel with capacity  $C = 1/3$  (bit), error-free data transmission   $(p_{\rm B} = 0)$  with code rate  $R < 1/3$  is possible in principle.
*However, from the channel coding theorem the special  $(k$,  $n)$–block code is not known which makes this desired result possible.   Shannon also makes no statements on this.
*All that is known is that such a best possible code works with blocks of infinite length.  For a given code rate  $R = k/n$  both  $k → ∞$  and  $n → ∞$ thus apply.
*The statement  "The bit error probability is zero"  is not the same as  "No bit errors occur":   Even with a finite number of bit errors and  $k → ∞$  ⇒   $p_{\rm B} = 0$.

With the code rate  $R = 1 > C$  (uncoded transmission) one obtains:

:$$p_{\rm B} \ge H_{\rm bin}^{-1} \cdot \left ( 1 - \frac{1/3}{1}\right )
= H_{\rm bin}^{-1}(2/3) \approx 0.174
> 0\hspace{0.05cm}.$$

With the code rate  $R = 1/2 > C$ , the bit error probability is smaller  but also different from zero:

:$$p_{\rm B} \ge H_{\rm bin}^{-1} \cdot \left ( 1 - \frac{1/3}{1/2}\right )
= H_{\rm bin}^{-1}(1/3) \approx 0.062
> 0\hspace{0.05cm}.$$}}

==Exercises for the chapter==
 
[[Aufgaben:Exercise_3.10:_Mutual_Information_at_the_BSC|Exercise 3.10: Mutual Information at the BSC]]

[[Aufgaben:Exercise_3.10Z:_BSC_Channel_Capacity|Exercise 3.10Z: BSC Channel Capacity]]

[[Aufgaben:Exercise_3.11:_Erasure_Channel|Exercise 3.11: Erasure Channel]]

[[Aufgaben:Exercise_3.11Z:_Extremely_Asymmetrical_Channel|Exercise 3.11Z: Extremely Asymmetrical Channel]]

[[Aufgaben:Exercise_3.12:_Strictly_Symmetrical_Channels|Exercise 3.12: Strictly Symmetrical Channels]]

[[Aufgaben:Exercise_3.13:_Code_Rate_and_Reliability|Exercise 3.13: Code Rate and Reliability]]

[[Aufgaben:Exercise_3.14:_Channel_Coding_Theorem|Exercise 3.14: Channel Coding Theorem]]

[[Aufgaben:Exercise_3.15:_Data_Processing_Theorem|Exercise 3.15: Data Processing Theorem]]

==References==
<references/>
{{Display}}

Linear and Time Invariant Systems/Properties of Balanced Copper Pairs

2022-02-17T11:39:07Z

Bene: Text replacement - "List of sources" to "References"

{{LastPage}}
{{Header
|Untermenü=Properties of Electrical Cables
|Vorherige Seite=Properties_of_Coaxial_Cables
|Nächste Seite=
}}
==Access network of a telecommunications system==
 
[[File:EN_LZI_T_4_3_S2.png| right|frame|Local loop area for ISDN]]
In a telecommunications system,  a distinction is made between
*the long-distance and regional network,  and
*the local loop area,

which are separated from each other by the local exchange. 

The graphic shows the network infrastructure for  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN|$\rm ISDN$]]  ("Integrated Services Digital Network").

Originally,  the entire telephone network was based on copper lines.  In the mid-1980s,  however,  the  (mainly coaxial)  copper cables were replaced by optical fiber for long-distance traffic,  as the steadily growing demand for bandwidth could only be met with optical transmission technology.
 
*Due to the immensely high installation costs,  optical fibers in the local loop area are still not economically viable today  (2009),  although plans for  "Fiber–to–the–Building"  $\rm (FttB)$  and  "Fiber–to–the–Home"  $\rm (FttH)$  have long been in the pipeline.

*Instead,  over the past twenty years,  the path has been taken to provide sufficient capacity via the conventional access network based on copper lines by developing and improving high-rate transmission systems such as  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL|$\rm DSL$]]  ("Digital Subscriber Line").

{{GraueBox|TEXT=
$\text{Example 1:}$ 
In Germany,  the so-called  "Last Mile"   (⇒   the local loop area)  is shorter than four kilometers on average in the country, and in urban areas  $90\%$  are even shorter than  $\text{2.8 km}$.  The local loop area is usually structured as follows:

[[File:EN_LZI_T_4_3_S1b.png|right|frame| Bundling and twisting of copper wires]]
*The  "main cable"  with up to  $2000$  pairs as a connection between the local exchange and the cable branch,
*the  "branch cable"  between the cable branch and the final branch,  with up to  $300$  pairs and significantly shorter than a main cable  $($maximum  $\text{500 m)}$,
*the  "house connection cable"  between the terminal box and the network termination box at the subscriber with two pairs of wires.
 
To reduce crosstalk on neighboring line pairs due to inductive and capacitive couplings and to increase the packing density,  two pairs of twisted pairs are twisted into a so-called  "star quad" .  The diagram below shows such a star quad and a bundled cable.
*Here,  five such quads each are combined to form a basic bundle,  and five basic bundles each are combined to form a main bundle.
*This thus contains  $50$  twin pairs with PE insulation  (PE:  polyethylene). }}

==Attenuation function of two-wire lines==
 
The attenuation function  $α(f)$  (per unit length) and the wave impedance  $Z_{\rm W}(f)$  of twisted pairs in real laid cables deviate to a greater or lesser extent from the theory presented in the chapter   [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory|Some Results from Line Transmission Theory]] . Reasons for this are:

*non-consideration of complex processes of eddy current formation and current displacement,  and
*inhomogeneities in cable structure in spliced cable sections.

[[File:P_ID1809__LZI_T_4_3_S2_93.png |frame| Attenuation function of two-wire lines of different diameters  '''Korrektur''' alpha, dB/km ..., auch bei '''Deutsch''']]

 Various network operators have measured  $α(f)$  and  $Z_{\rm W}(f)$  and derived empirical equations from them.   We refer here to the work of M. Pollakowski and H.W. Wellhausen of the Fernmeldetechnisches Zentralamt der Deutschen Bundespost in Darmstadt documented in  [PW95]<ref name="PW95">Pollakowski, P.; Wellhausen, H.-W.: ''Eigenschaften symmetrischer Ortsanschlusskabel im
Frequenzbereich bis 30 MHz.'' Deutsche Telekom AG, Forschungs- und Technologiezentrum
Darmstadt, 1995.</ref>.

They determined for different line diameters  $d$  among other things the empirical attenuation function per unit length from forty measurements each in the frequency range up to  $\text{30 MHz}$  according to the equation
:$$\alpha (f) = k_1 + k_2 \cdot (f/{\rm MHz})^{k_3} \hspace{0.05cm}.$$

The graph shows the measurement results:
* $d = 0.35 \ {\rm mm}$:   $k_1 = 7.9 \ {\rm dB/km}, \hspace{0.2cm}k_2 = 15.1 \ {\rm dB/km}, \hspace{0.2cm}k_3 = 0.62$,
* $d = 0.40 \ {\rm mm}$:   $k_1 = 5.1 \ {\rm dB/km}, \hspace{0.2cm}k_2 = 14.3 \ {\rm dB/km}, \hspace{0.2cm}k_3 = 0.59$,
* $d = 0.50 \ {\rm mm}$:   $k_1 = 4.4 \ {\rm dB/km}, \hspace{0.2cm}k_2 = 10.8 \ {\rm dB/km}, \hspace{0.2cm}k_3 = 0.60$,
* $d = 0.60 \ {\rm mm}$:   $k_1 = 3.8 \ {\rm dB/km}, \hspace{0.2cm}k_2 = \hspace{0.25cm}9.2 \ {\rm dB/km}, \hspace{0.2cm}k_3 = 0.61$.

One can see from this representation:
*The attenuation function per unit length   ⇒   $α(f)$  and the attenuation function   ⇒  $a_{\rm K}(f) = α(f) · l$  depend significantly on the cable diameter.  The cables laid since 1994 with  $d = 0.35 \ \rm (mm)$  and  $d = 0.5$  have about  $10\%$  larger attenuation than the older cables with  $d = 0.4$  and  $d= 0.6$.
*However,  this smaller diameter,  justified by the manufacturing and laying costs,  significantly reduces the range  $l_{\rm max}$  of the transmission systems used on these lines,  so that in the worst case expensive  "intermediate generators"  have to be used.
*The transmission methods commonly used today for copper lines, however, occupy only a relatively narrow frequency band,  e.g.   $120\ \rm kHz$  for  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN|$\rm ISDN$]]  ("Integrated Services Digital Network")  and about  $1100 \ \rm kHz$ for [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL|$\rm DSL$]]  ("Digital Subscriber Line"). 
*For  $f = 1 \ \rm MHz$,  the attenuation function per unit length for a  $0.4 \ \rm mm$  cable is about  $20 \ \rm dB/km$,  so that even with a cable length of  $l = 4 \ \rm km$  the attenuation value with these systems does not exceed  $80 \ \rm dB$.
*One exception is  [[Examples_of_Communication_Systems/xDSL–Systeme#VDSL_.E2.80.93_Very.E2.80.93high.E2.80.93speed_Digital_Subscriber_Line|$\rm VDSL$]]  ("Very High Data Rate Digital Subscriber Line"),  which is offered in Germany by Deutsche Telekom in larger cities.  Here,  the frequency range goes up to  $30 \ \rm MHz$.  For this reason,  fiber optic connections were laid right up to the cable branch in order to keep the length that still has to be bridged with copper short.  This is known as  "Fibre–to–the–Cabinet"  $\rm (FttC)$.

==Conversion between  $k$  and  $\alpha$  parameters==
To calculate the frequency response  $H_{\rm K}(f)$,  one should always start from the measured attenuation function (per unit length)
:$$\alpha (f) = k_1 + k_2 \cdot (f/f_0)^{k_3}= \alpha_{\rm I} (f) \hspace{0.05cm}, \hspace{0.2cm}{\rm with} \hspace{0.15cm} f_0 = 1\,{\rm MHz}.$$
If,  on the other hand,  one wishes to determine the associated time function in terms of the impulse response  $h_{\rm K}(t)$,  it is more convenient,  as shown in the  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs#Interpretation_and_manipulation_of_the_individual_impulse_responses|section after the next]],  if the attenuation function per unit length can be represented in the form that is also common for coaxial cables:
:$$\alpha(f) = \alpha_0 + \alpha_1 \cdot f + \alpha_2 \cdot \sqrt {f}= \alpha_{\rm II} (f).$$

As a criterion of this conversion we use that the squared deviation between both functions is minimal in the range from  $f = 0$  to  $f = B$ :
:$$\int_{0}^{B} \left [ \alpha_{\rm I} (f) - \alpha_{\rm II} (f)\right ]^2 \hspace{0.1cm}{\rm d}f \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\rm Minimum} \hspace{0.05cm} .$$
It is obvious that  $α_0 = k_1$  will hold.  The parameters  $α_1$  and  $α_2$  depend on the desired bandwidth  $B$.  According to  [[Aufgaben:Exercise_4.6:_k-parameters_and_alpha-parameters|Exercise 4.6]],  they are:
:$$\begin{align*}\alpha_1 & = 15 \cdot (B/f_0)^{k_3 -1}\cdot \frac{k_3 -0.5}{(k_3 + 1.5)(k_3 + 2)}\cdot \frac {k_2}{ {f_0} }\hspace{0.05cm} ,\\ \alpha_2 & = 10 \cdot (B/f_0)^{k_3 -0.5}\cdot \frac{1-k_3}{(k_3 + 1.5)(k_3 + 2)}\cdot \frac {k_2}{\sqrt{f_0} }\hspace{0.05cm} .\end{align*}$$
*For  $k_3 = 1$  (frequency-proportional attenuation function per unit length), the $\alpha$–parameters logically result in
:$$\alpha_1 = {k_2}/{ {f_0} }\hspace{0.05cm} ,\hspace{0.2cm} \alpha_2 = 0\hspace{0.05cm} ,$$
*while for  $k_3 = 0.5$  the following coefficients are obtained:
:$$\alpha_1 = 0\hspace{0.05cm} ,\hspace{0.2cm} \alpha_2 = {k_2}/{\sqrt{f_0}}\hspace{0.05cm}.$$

:In this case, the attenuation function per unit length  $α(f)$  increases with the square root of the frequency. This results in the same curve as for a coaxial cable, where the well-known skin effect dominates.

In the following, three examples will illustrate how the underlying bandwidth  $B$  influences the results of this conversion.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
In the following graphs we assume the line length  $l = 1 \ \rm km$  and the diameter  $d = 0.4 \ \rm mm$,  so that the following applies:
:[[File:EN_LZI_T_4_3_S3.png|right|frame| Approximation of  $k$–parameters  by  $\alpha$–parameters]]

:$$k_1 = 5.1 \ \rm dB/km, \ k_2 = 14.3 \ \rm dB/km, \ k_3 = \ 0.59.$$
For this case the following graph shows
*the attenuation approximated  ${\rm a_K}(f)$  (in dB)  with  $α_0,  α_1$,  $α_2$  (blue curve)
*compared to the   ${\rm a_K}(f)$  curve according to  $k_1,  k_2,  k_3$  (red curve).

The three diagrams are valid for the bandwidths  $B = 10 \ \rm MHz$,  $B = 20 \ \rm MHz$  and  $B = \ \rm 30 \ MHz$.
*The determined coefficients  $α_1$  and  $α_2$  are indicated.
* $α_0 = k_1 = 5.1 \ \rm dB/km$  is always valid.

One recognizes from these representations:
*Even with the largest approximation range  $(B = 30 \ \rm MHz)$,  the blue curve  $($with  $α_0, \ α_1, \ α_2)$  approximates the measured curve  $($red curve,  described by  $k_1, \ k_2, \ k_3)$  very well.
*For smaller bandwidth  $(B = 20 \ \rm MHz$  or  $B = 10 \ \rm MHz)$  the approximation in the range  $0≤ f ≤ B$  is even better,  but then distortions occur for  $f > B$ .
*The attenuation value  $a_{\rm K}(f = 30 \ \rm MHz) ≈ 112.2 \ \rm dB$  is composed as follows for the considered two-wire cable:     $4.5\%$  is due to the coefficient  $α_0$  (ohmic losses) ,  $23.5\%$  to the  $f$–proportional component  $α_1$  and  $72\%$  to the coefficient  $α_2$.
*In comparison, the standard  $\text{2.6/9.5 mm}$  coaxial cable exhibits comparable attenuation  $(≈ 112 \ \rm dB)$  only at a length of  $l = 8.7 \ \rm km$,  with most of the attenuation  $(98.9\%)$  coming from the skin effect  $(α_2)$.}}

In the opposite direction  $(α_1, \ α_2 ⇒ k_2, \ k_3)$  the conversion rule for the exponent is:

:$$k_3 = \frac{H + 0.5} {H +1}, \hspace{0.4cm}\text{with auxiliary: }H = \frac{2} {3} \cdot \frac{\alpha_1 \cdot \sqrt{f_0}}{\alpha_2} \cdot \sqrt{B/f_0}.$$

With this result,  $k_2$  can be calculated using either of the two equations given above.

==Impulse response of a two-wire line==
 
With this coefficient conversion  $(k_1, \ k_2, \ k_3) \ \ ⇒ \ \ (α_0, \ α_1, \ α_2)$  we can now write for the total frequency response of a two-wire line:
:$$H_{\rm K}(f) = H_{\alpha 0}(f) \cdot H_{\alpha 1}(f) \cdot H_{\beta 1}(f)\cdot H_{\alpha 2}(f) \cdot H_{\beta 2}(f) \hspace{0.05cm}.$$

Here,  the following abbreviations are used:
:$$\begin{align*} H_{\alpha 0}(f) & = {\rm e}^{-\alpha_0 \hspace{0.05cm} \cdot \hspace{0.05cm} l}= {\rm e}^{-{\rm a}_0}\hspace{0.05cm},\hspace{0.2cm} {\rm a}_0= \alpha_0\hspace{0.15cm}{\rm (in \hspace{0.15cm}Np) }\cdot l,\\ H_{\alpha 1}(f) & = {\rm e}^{-\alpha_1 \cdot f \hspace{0.05cm} \cdot \hspace{0.05cm}l}= {\rm e}^{-{\rm a}_1 \cdot 2f/R}\hspace{0.05cm},\hspace{0.2cm} {\rm a}_1 = \alpha_1\hspace{0.15cm}{\rm (in \hspace{0.15cm}Np) }\cdot l \cdot {R}/{2} \hspace{0.05cm},
\\ H_{\beta 1}(f) & = {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} \beta_1 \cdot f \hspace{0.05cm} \cdot \hspace{0.05cm}l} = {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} 2 \pi \cdot f \hspace{0.05cm} \cdot \hspace{0.05cm}\tau_{\rm P}} \hspace{0.05cm},\hspace{0.2cm} \tau_{\rm P} = {\beta_1\hspace{0.15cm}{\rm (in \hspace{0.15cm}rad) }\cdot l }/({2 \pi}) \hspace{0.05cm},
\\ H_{\alpha 2}(f) & = {\rm e}^{-\alpha_2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sqrt{f} \hspace{0.05cm} \cdot \hspace{0.05cm}l}= {\rm e}^{-{\rm a}_2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sqrt{2f/R}}\hspace{0.05cm},\hspace{0.2cm} {\rm a}_2 = \alpha_2\hspace{0.15cm}{\rm (in \hspace{0.15cm}Np) }\cdot l \cdot \sqrt{R/2} \hspace{0.05cm},
\\ H_{\beta 2}(f) & = {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm}\beta_2 \hspace{0.05cm} \cdot \hspace{0.05cm}\sqrt{f} \hspace{0.05cm} \cdot \hspace{0.05cm}l}= {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} b_2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sqrt{2f/R}}\hspace{0.05cm},\hspace{0.2cm} b_2 = \beta_2\hspace{0.15cm}{\rm (in \hspace{0.15cm}rad) }\cdot l \cdot \sqrt{R/2} \hspace{0.05cm} \end{align*}$$

The meaning of the quantities implicitly defined here will be discussed a little later.

We proceed quite formally here first.  According to the  [[Signal_Representation/The_Convolution_Theorem_and_Operation|convolution theorem]],  the resulting impulse response is the  [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|Fourier retransform]]  of  $H_{\rm K}(f)$:
:$$h_{\rm K}(t) = h_{\alpha 0}(t) \star h_{\alpha 1}(t) \star h_{\beta 1}(t)\star h_{\alpha 2}(t) \star h_{\beta 2}(t) \hspace{0.05cm},$$
:$$h_{\alpha 0}(t) \quad \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\quad H_{\alpha 0}(f) \hspace{0.05cm},\hspace{0.2cm} h_{\alpha 1}(t) \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\quad H_{\alpha 1}(f) \hspace{0.05cm},\hspace{0.2cm} {\rm etc.}$$

These five components are now to be considered separately, with the numerical results referring to
*a digital transmission system with bit rate  $R = 30 \ \rm Mbit/s$  and
*a two-wire line with dimensions  $d = 0.4 \ \rm mm$  and  $l = 1 \ \rm km$. 

Thus,  the  $α$–coefficients in Neper  (Np) are:
:$$\alpha_0 = 0.59\, \frac{ {\rm Np} }{ {\rm km} } \hspace{0.05cm}, \hspace{0.2cm} \alpha_1 = 0.10\, \frac{ {\rm Np} }{ {\rm km \cdot MHz} }\hspace{0.05cm}, \hspace{0.2cm} \alpha_2 = 1.69\, \frac{ {\rm Np} }{ {\rm km \cdot \sqrt{MHz} } } \hspace{0.05cm}.$$

The phase function per unit length of this line is also given in  [PW95]<ref name="PW95">Pollakowski, P.; Wellhausen, H.-W.: ''Eigenschaften symmetrischer Ortsanschlusskabel im
Frequenzbereich bis 30 MHz.'' Deutsche Telekom AG, Forschungs- und Technologiezentrum
Darmstadt, 1995.</ref>:
:$$b_{\rm K}(f) = \beta_1 \cdot f + \beta_2 \cdot \sqrt {f}\hspace{0.05cm}, \hspace{0.2cm} \beta_1 = 32.9\, \frac{ {\rm rad} }{ {\rm km \cdot MHz} }\hspace{0.05cm}, \hspace{0.2cm} \beta_2 = 2.26\, \frac{ {\rm rad} }{ {\rm km \cdot \sqrt{MHz} } }\hspace{0.05cm}.$$
The symbol duration  $T = 1/R ≈ 33 \ \rm ns$  is suitable as a normalization quantity of time.

==Interpretation and manipulation of the individual impulse responses==
 
Now the five impulse response components  $h_{α0}(t), \ h_{α1}(t), \ h_{α2}(t), \ h_{β1}(t)$  and  $h_{β2}(t)$  are interpreted:

'''(1)'''   The first term resulting from the ohmic losses  (frequency-independent attenuation)  leads to a Dirac delta function with weight  $K$,  so that the convolution with  $h_{α0}(t)$  can be replaced by the multiplication with  $K = {\rm e}^{–0.59} ≈ 0.55$ :
:$$h_{\alpha 0}(t) = K \cdot \delta(t) \hspace{0.25cm}{\rm with}\hspace{0.25cm} K = {\rm e}^{-{\rm a}_0}\hspace{0.45cm}\Rightarrow\hspace{0.45cm} h_{\rm K}(t) = h_{\alpha 0}(t) \star h_{\rm Rest}(t) = K \cdot h_{\rm Rest}(t)\hspace{0.05cm}.$$

'''(2)'''   $H_{α1}(f)$  is a real and even function of frequency,  so the Fourier retransform is also real and symmetric at  $t =0$:
:$$H_{\alpha 1}(f) = {\rm e}^{-2\cdot{\rm a}_1 \cdot |f/R|} \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\quad h_{\alpha 1}(t)= \frac{1}{T} \cdot \frac{{\rm a}_1}{{\rm a}_1^2 + \pi \cdot (t/T)^2}\hspace{0.05cm}, \hspace{0.2cm} {\rm a}_1 \hspace{0.15cm}{\rm in \hspace{0.15cm}Np } \hspace{0.05cm}.$$
:With the exemplary numerical values  $α_1 = 0.1 \ \rm Np/(km · MHz)$,   $l = 1 \ \rm km$,   $R = 30 \ \rm MHz$   ⇒   ${\rm a}_1 = 1.5 \ \rm (Np)$,  we obtain for the maximum of this fraction:
:$$h_{α1}(t = 0) = 1/{\rm a}_1 = 2/3 · 1/T.$$

'''(3)'''   As for the coaxial cable systems,  $H_{β1}(f)$  does not lead to any signal distortion,  but only to a time delay by the  [[Linear_and_Time_Invariant_Systems/Lineare_Verzerrungen#Unterschied_zwischen_Phasen-_und_Gruppenlaufzeit|phase delay time]]:
:$$τ_{\rm P} ≈ 5.24 \ \rm µs \hspace{0.2cm} ⇒ \hspace{0.2cm} τ_{\rm P}/T ≈ 157.$$

'''(4)'''   Let us turn to the combined consideration of the  $H_{α2}(f)$  and  $H_{β2}(f)$  components,  which is described in the time domain by the partial impulse response  $h_2(t)$:
:$$H_{\alpha 2}(f) \cdot H_{\beta 2}(f) \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\quad h_{2}(t) \hspace{0.05cm}.$$

'''(5)'''   To apply the results of the chapter  [[Linear_and_Time_Invariant_Systems/Eigenschaften_von_Koaxialkabeln|Properties of Coaxial Cables]],  we replace  $β_2$  by  $α_2 · \rm rad/Np$  and  $b_2$  by  ${\rm a}_2 · \text{rad/Np}$,  so that  ${\rm a}_2$  and  $b_2$  have the same numerical value.  As an example, we substitute here:
:$$ b_2 = 8.75\, {\rm rad}\hspace{0.2cm} \Rightarrow \hspace{0.2cm} b_2 = 6.55 \,{\rm rad}\hspace{0.05cm}.$$
:One thus reduces the constant  $β_2 = 2.26 \ \rm rad/(km · \sqrt{MHz})$  to  $β_2 = 1.69 \ \rm rad/(km · \sqrt{MHz})$ .

'''(6)'''   Before we unnecessarily lead the reader to consider whether this approximation is indeed valid or not,  we freely admit right away that this assumption is the weak point of our reasoning.  A discussion of this faulty assumption follows in the  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs#Discussion_of_the_approximate_solution_found|next secction]].

'''(7)'''   Now that  ${\rm a}_2$  and  $b_2$  have the same numerical values,  we can further use the equation given in the chapter  [[Linear_and_Time_Invariant_Systems/Eigenschaften_von_Koaxialkabeln|Properties of Coaxial Cables]],  substituting  $\rm a_∗$  for  $\rm a_2$ :
:$$h_{\rm 2}(t ) = \frac {1/T \cdot {\rm a_2}}{\pi \cdot \sqrt{2 \cdot(t/T)^3}}\cdot {\rm e}^{ - {{\rm a_2}^2}/( {2\pi \hspace{0.05cm}\cdot \hspace{0.05cm} t/T})} \hspace{0.05cm}, \hspace{0.2cm} {\rm a}_2\hspace{0.15cm}{\rm in \hspace{0.15cm}Np} \hspace{0.05cm}.$$

'''(8)'''   The total impulse response without consideration of the phase delay time is thus given by
:$$h_{\rm K}(t + \tau_{\rm P}) = K \cdot h_{\alpha 1}(t) \star h_{2}(t)\hspace{0.05cm}.$$
:By shifting  $τ_{\rm P}$  to the right, the searched function  $h_{\rm K}(t)$  is obtained.  In the following example,  this procedure is illustrated by graphics.

{{GraueBox|TEXT=
$\text{Example 3:}$ 
For the following graphs,  a two-wire line with dimensions  $d = 0.4 \ \rm mm$  and  $l = 1 \ \rm km$  is still assumed. Please note the different ordinate scaling of the three diagrams in the graph.

*The bit rate is  $R = 30 \ \rm Mbit/s$   ⇒   symbol duration  $T ≈ 33\ \rm ns$.
*We assume the quantities given in the yellow box,  calculated on the last page.
*For this purpose,  the  $b_2$ value is changed from  $8.75 \ \rm rad$  to  $6.55 \ \rm rad$  to match the  ${\rm a}_2$ value.
*The effects of this measure are interpreted on the next page.
[[File:EN_LZI_T_4_3_S4.png|right|frame| For calculating the impulse response of a two-wire line]]

On the top right  $h_1(t) = h_{\rm α1}(t + τ_{\rm P})$  is shown. This component is due to the  $α_1$  and  $β_1$  coefficients. $h_1(t)$  is a symmetric function with respect to the phase delay time  $τ_{\rm P}$  with the maximum value  $(1.5T)^{–1}$, where the  $1/(1 + t^2$) decay is rapidly decreased at  $ \pm 5T$  $($right and left of  $τ_{\rm P})$.

The lower left diagram shows the signal component  $h_2(t)$,  due to the two coefficients  $α_2$  and  $β_2$.  $h_2(t)$  is identical to the  [[Linear_and_Time_Invariant_Systems/Eigenschaften_von_Koaxialkabeln#Impulsantwort_eines_Koaxialkabels|coaxial cable impulse response]]  (ignoring the delay time)  when the characteristic cable attenuation is  $6.55 \ \rm Np$  or  $56.9 \ \rm dB$.

The red curve represents the convolution product  $h_1(t) ∗ h_2(t)$.  It can be seen that the waveform is essentially fixed by  $h_2(t)$.  However,  convolution with  $h_1(t)$  leads to a  (slight)  distortion of the waveform in addition to an amplitude loss of about  $10\%$.

The resulting impulse response  $h_{\rm K}(t)$  of the 0.4mm two-wire line is shown as a blue curve in the lower right diagram.  The difference to the convolution product  $h_1(t) ∗ h_2(t)$  drawn in red results from the influence of the DC signal attenuation $($coefficient  $α_0)$. }}

You can illustrate the presented method for arbitrary parameters (diameter, length, bit rate) with the (German language) interactive SWF applet

:[[Applets:Zeitverhalten_von_Kupferkabeln|Zeitverhalten von Kupferkabeln]]  ⇒   "Time behavior of copper cables". 

==Discussion of the approximate solution found==
 
[[File:EN_LZI_T_4_3_S5.png|right|frame| Impulse response approximations of normal coaxial cable (top) and  0.4 mm  two-wire cable (bottom)]]
{{GraueBox|TEXT=
$\text{Example 4:}$ 
The following graph shows the (normalized) impulse responses  $T · h_{\rm K}(t)$  for two exemplary copper cables, namely

*for the   $\text{standard coaxial cable 2.6/9.5 mm}$  at  $\text{10.1 km}$  length (above), where:
:$$a_0 = 0.016\,{\rm Np}\hspace{0.05cm}, \hspace{0.15cm}
a_1 = 0.020\,{\rm Np}\hspace{0.05cm}, \hspace{0.15cm}
a_2 = 6.177\,{\rm Np}\hspace{0.05cm}, $$
:$$\tau_{ {\rm P} }/T = 350\hspace{0.05cm}, \hspace{0.15cm}
b_2 = 6.177\,{\rm rad}\hspace{0.05cm};$$
*for the   $\text{0.4 mm two-wire line}$ with the length $\text{1.8 km}$ (below) with the parameters
:$$a_0 = 1.057\,{\rm Np}\hspace{0.05cm}, \hspace{0.15cm}
a_1 = 0.147\,{\rm Np}\hspace{0.05cm}, \hspace{0.15cm}
a_2 = 6.177\,{\rm Np}\hspace{0.05cm}, $$
:$$\tau_{ {\rm P} }/T = 94\hspace{0.05cm}, \hspace{0.15cm}
b_2 = 8.260\,{\rm rad}\hspace{0.05cm}.$$

These values are valid for bit rate  $R = 10 \ \rm Mbit/s$   ⇒   time normalization  $T = 0.1 \ \rm µ s$.

*The two cable lengths were chosen to give exactly equal  $a_2$ parameters.
*For the two-wire cable,  the phase value  $b_2 ⇒ b_2\hspace{0.01cm}'$  was adjusted to give equal values for  $b_2\hspace{0.05cm}' = 6.177 \ \rm rad$  and  $a_2 = 6.177 \ \rm Np \ (≈ 53 \ dB)$  as for the coaxial cable.
 
The blue curves show the approximations when the  $a_0, \ a_1$ and  $b_1$ terms are neglected.  Due to the  $b_2 ⇒ b_2\hspace{0.01cm}'$  phase matching for the two-wire line,  the curves are nearly the same.  The maximum of about  $3.8\%$  is at about  $t/T = 4$  (different time scales in both diagrams!).

The red curves also take into account the  $a_0, \ a_1$ and  $b_1 $  terms.  The red curve of the coaxial cable is the actual (normalized) impulse response  $T · h_{\rm K}(t)$.

From these representations, one can further see:
*For the coaxial cable,  the  $a_0$ term and the  $a_1 $  term can be neglected.  The resulting relative error is only  $3.5\%$.
*However, the phase delay time  $τ_{\rm P}$, i.e. the  $b_1$ term,  cannot be neglected.  For the coaxial cable this gives  $τ_{\rm P}/T ≈ 350$,  while for the two-wire line it is  $τ_{\rm P}/T ≈ 94$   (note the different time scales).
*For the two-wire line (below),  do not neglect DC signal attenuation  $(a_0)$  and transverse loss  $(a_1)$ :   The red approximation  $T · h_{\rm K}\hspace{0.01cm}'(t)$  is  $70\%$  lower than the blue one and also slightly wider.}}

{{GraueBox|TEXT=
$\text{Example 5:}$  This example shows approximations of the impulse response of a two-wire line  $($length $\text{1.8 km}$,  diameter $\text{0.4 mm)}$,  so that according to  [PW95]<ref name="PW95">Pollakowski, P.; Wellhausen, H.-W.: ''Eigenschaften symmetrischer Ortsanschlusskabel im
Frequenzbereich bis 30 MHz.'' Deutsche Telekom AG, Forschungs- und Technologiezentrum
Darmstadt, 1995.</ref>  the following parameters can be assumed:
[[File:EN_LZI_T_4_3_S5c.png|frame| For the impulse response of a  $\text{0.4 mm}$  two-wire cable]]
:$$a_0 = 1.057\,{\rm Np}\hspace{0.05cm}, \hspace{0.15cm}
a_1 = 0.147\,{\rm Np}\hspace{0.05cm}, \hspace{0.15cm}
a_2 = 6.177\,{\rm Np}\hspace{0.05cm}, $$
:$$\tau_{ {\rm P} }/T = 94\hspace{0.05cm}, \hspace{0.15cm}
b_2 = 8.260\,{\rm rad}\hspace{0.05cm}.$$

The upper diagram – equal to the lower diagram in  $\text{Example 4}$ – shows two approximations
*with neglection of the  $a_0, \ a_1$  and  $b_1$ terms  (blue curve),
*with consideration of the  $a_0, \ a_1$  and  $b_1$ terms  (red curve).

For this upper diagram, we further adopted the  $a_2$ coefficient given in [PW95]<ref name="PW95"/> and lowered the mentioned coefficient  $b_2 = 8.260 \ \rm rad$  to  $b_2\hspace{0.01cm}' = 6.177 \ \rm rad$ .

''Note'':   In contrast to the coaxial cable in  [[Linear_and_Time_Invariant_Systems/Eigenschaften_von_Kupfer–Doppeladern#Interpretation_und_Manipulation_der_einzelnen_Impulsantworten|$\text{Example 3}$]],  because of  $b_2\hspace{0.01cm}' ≠ b_2$  the red curve  $T · h_{\rm K}\hspace{0.01cm}'(t)$  is also only an approximation, which is indicated in the graphic by the apostrophe.

Without the correction  $b_2\hspace{0.01cm}' = a_2 · \text{rad/Np}$,  the  [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Hilbert_transform|Hilbert Transformation]], which establishes the relationship between magnitude and phase in real and thus establishes  [https://en.wikipedia.org/wiki/Minimum_phase minimum-phase systems],  would not be satisfied.  Therefore,  it would result in a noncausal impulse response.

We therefore believe that even for a two-wire line, the two parameters  $a_2$  and  $b_2$  should have equal numerical values.

We now consider a second approach,  shown in the diagram below:
*Here,  the phase coefficient  $b_2 = 8.260 \ \rm rad$  given in  [PW95]<ref name="PW95"/>  was kept.
*Instead,  the attenuation coefficient  $a_2 = 6.177 \ \rm Np$  was adjusted to the phase coefficient (i.e., enlarged):   $a_2\hspace{0.01cm}' = 8.260 \ \rm Np$.
*The lower  (red)  impulse response   ⇒   "worst case"  is less than half as high and much wider than the upper impulse response   ⇒   "best case".
*The actual  (normalized)  impulse response  $T \cdot h_{\rm K}\hspace{0.01cm}'(t)$  will probably lie in between.  We do not allow ourselves to make more precise statements.}}

==Crosstalk on two-wire lines==
 
For transmission systems over two-wire lines,  the same  [[Linear_and_Time_Invariant_Systems/Properties_of_Coaxial_Cables#Special_features_of_coaxial_cable_systems|block diagram]]  can be assumed as for the coaxial cable systems,  where now
*for the frequency response  $H_{\rm K}(f)$  and the impulse response  $h_{\rm K}(t)$  the equations given in this section are to be used,
*white Gaussian noise  $N_0$  is no longer the dominant cause of stochastic impairments,  but  '''crosstalk'''  due to capacitive or inductive coupling of adjacent pairs now predominates.

By twisting the wire pairs of a star quad as well as the basic and main bundles according to the diagram at the end of the chapter  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs#Access_network_of_a_telecommunications_system|Access network of a telecommunications system]]  an attempt is made to achieve on average a mutual coupling as symmetrical as possible between all wire pairs.  Due to unavoidable manufacturing tolerances,  however,  a slight asymmetry always remains.  This causes
*in addition to its  "own"  useful signal,  each receiver input also receives  (usually only small)  signal components from neighboring wire pairs,
*the induced signal components represent an additional stochastic impairment for the useful signal,  which together with the thermal noise results in the resulting interfering signal  $n(t)$,
*the transmission quality cannot be improved or can only be improved to a very limited extent by increasing the transmitting power,  since this measure also increases the crosstalk interference.

[[File:EN_LZI_T_4_3_S6.png |right|frame|To clarify near-end crosstalk (NEXT) and far-end crosstalk (FEXT)]]
 
As the diagram illustrates, a distinction is made between
*'''Near-End Crosstalk'''  $\rm (NEXT)$: The interfering transmitter feeds its signal at the same end of the cable where the receiver under consideration is placed.

*'''Far–End–Crosstalk'''  $\rm (FEXT)$: The interfering transmitter and the interfering receiver are located at opposite ends of the cable.

In the case of  "FEXT",  the interference accumulates over the entire cable length,  but is also greatly attenuated by the cable attenuation.  For bundled cables in the local loop area,  "in quad crosstalk"  thus results in orders of magnitude greater interference than long-distance crosstalk,  and even near-end crosstalk interference from adjacent cores can usually be neglected.
 
{{BlaueBox|TEXT=
$\text{Without derivation:}$ 
We therefore consider only  $\text{Near-End Crosstalk (NEXT)}$  in the following.  In this case, the  [[Theory_of_Stochastic_Signals/Leistungsdichtespektrum_(LDS)#Theorem_von_Wiener-Chintchine|power-spectral density]]  $\rm (PSD)$  of the interference signal  $n(t)$  can be represented as follows, taking into account the unavoidable thermal noise  $(N_0/2)$ :
:$${\it \Phi}_n(f) = {N_0}/{2}+{\it \Phi}_{\rm NEXT}(f) \hspace{0.05cm},\hspace{0.3cm}{\rm with} \hspace{0.3cm}{\it \Phi}_{\rm NEXT}(f) = {\it \Phi}_{s}(f) \cdot\vert H_{\rm NEXT}(f)\vert ^2 \approx {\it \Phi}_{s}(f) \cdot [K_{\rm NEXT} \cdot f]^{3/2}\hspace{0.05cm}.$$}}

It should be noted about this equation:
*The equation is obtained by integrating the local couplings over the entire length of a short section,  where the couplings between all copper lines are modeled by cross capacitances and inductances.
* ${\it \Phi}_s(f)$  is the PSD of the interfering transmitter,  from which the transmission power  $P_{\rm S}$  is obtained by integration.  Assuming that the interfered transmission uses the same transmission signal and thus the same PSD  ${\it \Phi}_s(f)$  as the interferer,  it is clear that increasing  $P_{\rm S}$  only reduces the (relative) influence of thermal noise  $(N_0/2)$.
*The factor  $K_{\rm NEXT}$  quantifying the near-end crosstalk depends strongly on the core spacing,  as well as on the degree of unbalance along the cable.  In contrast,  this factor  $K_{\rm NEXT}$  is almost independent of the conductor diameter  $d$  and the cable length  $l$.
*The product  $K_{\rm NEXT} · f$  (dimensionless)  is always much smaller than  $1$  over the entire operating range of the cable,  e.g. for all frequencies  $0 ≤ f ≤ 30 \ \rm MHz$.  The crosstalk interference increases sharply $($that is,  with exponent $1.5)$  with frequency.
*In  [PW95]<ref name="PW95"/>,  the following values are given after a series of measurements over forty pairs for the frequency  $f = 10 \ \rm MHz$   $($for  $f = 30 \ \rm MHz$  these values must still be multiplied by  $3^{3/2} ≈ 5.2)$:
:* worst case:   $|H_{\rm NEXT}(f = 10 \ \rm MHz)|^2 ≈ 0.001$,
:* Averaging over 40 cores:   $|H_{\rm NEXT}(f = \ \rm 10 MHz)|^2 ≈ 0.0004$.
*This values apply to  "in-four near-end crosstalk"  (interfering transmitter and interfering receiver in the same star quad).
*Near-end crosstalk interference between more distant cores exhibits the same frequency dependence, but is smaller than  "in-four near-end crosstalk":
:* Near-end crosstalk between adjacent star quads by about  $5 \ \rm dB$,
:* Near-end crosstalk between adjacent ground bundles by about  $10 \ \rm dB$,
:* Near-end crosstalk between non-adjacent ground bundles by about  $25 \ \rm dB$.

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*In order to avoid or at least reduce such near-end crosstalk interference,  adjacent pairs of wires are often occupied by completely different signals  (analog telephony, ISDN, DSL or other broadband services)  which also use different frequency bands if possible.
*By clever selection of the twisted pairs,  adjacent cores can now be occupied with signals whose spectra overlap as little as possible,  thus reducing crosstalk interference.}}

==Exercises for the chapter==

[[Aufgaben:Exercise_4.6:_k-parameters_and_alpha-parameters| Exercise 4.6:  $k$  parameters  and  $\alpha$  parameters]]

[[Aufgaben:Exercise_4.6Z:_ISDN_Supply_Lines|Exercise 4.6Z:  ISDN Supply Lines]]

[[Aufgaben:Exercise_4.7:_Copper_Twin_Wire_0.5_mm| Exercise 4.7: Copper Twin Wire 0.5 mm]]

[[Aufgaben:Exercise_4.8:_Near-end_and_Far-end_Crosstalk_Disorders| Exercise 4.8:  Near-end and Far-end Crosstalk Disorders]]

==References==
<references/>

{{Display}}

Linear and Time Invariant Systems/Properties of Coaxial Cables

2022-02-17T11:39:06Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Properties of Electrical Cables
|Vorherige Seite=Some Results from Line Transmission Theory
|Nächste Seite=Properties of Balanced Copper Pairs
}}
==Complex propagation function of coaxial cables==
 
Coaxial cables consist of an inner conductor and – separated by a dielectric – an outer conductor.  Two different types of cable have been standardized, with the diameters of the inner and outer conductors mentioned for identification purposes:
*the  "standard coaxial cable"  whose inner conductor has a diameter of  $\text{2.6 mm}$  and whose outer diameter is  $\text{9.5 mm}$,
*the  "small coaxial cable"  with dimensions  $\text{1.2 mm}$  and  $\text{4.4 mm}$.

The cable frequency response  $H_{\rm K}(f)$  results from the cable length  $l$  and the complex propagation function (per unit length)
:$$\gamma(f) = \alpha_0 + \alpha_1 \cdot f + \alpha_2 \cdot \sqrt {f}+ {\rm j}\cdot (\beta_1 \cdot f + \beta_2 \cdot \sqrt {f})\hspace{0.05cm}\hspace{0.3cm}
\Rightarrow \hspace{0.3cm}H_{\rm K}(f) = {\rm e}^{-\gamma(f)\hspace{0.05cm} \cdot \hspace{0.05cm} l} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}|H_{\rm K}(f)| = {\rm e}^{-\alpha(f)\hspace{0.05cm} \cdot \hspace{0.05cm} l}\hspace{0.05cm}.$$

The cable specific constants for the  '''standard coaxial cable'''  $\text{(2.6/9.5 mm)}$  are:
:$$\begin{align*}\alpha_0 & = 0.00162\, \frac{ {\rm Np} }{ {\rm km} }\hspace{0.05cm}, \hspace{0.2cm} \alpha_1 = 0.000435\, \frac{ {\rm Np} }{ {\rm km \cdot MHz} }\hspace{0.05cm}, \hspace{0.2cm} \alpha_2 = 0.2722\, \frac{ {\rm Np} }{ {\rm km \cdot \sqrt{MHz} } }\hspace{0.05cm}, \\ \beta_1 & = 21.78\, \frac{ {\rm rad} }{ {\rm km \cdot MHz} }\hspace{0.05cm}, \hspace{0.2cm} \beta_2 = 0.2722\, \frac{ {\rm rad} }{ {\rm km \cdot \sqrt{MHz} } } \hspace{0.05cm}.\end{align*}$$

Accordingly,  the kilometric attenuation and phase constants for the  '''small coaxial cable'''  $\text{(1.2/4.4 mm)}$:
:$$\begin{align*}\alpha_0 & = 0.00783\, \frac{ {\rm Np} }{ {\rm km} }\hspace{0.05cm}, \hspace{0.2cm}
\alpha_1 = 0.000443\, \frac{ {\rm Np} }{ {\rm km \cdot MHz} }\hspace{0.05cm}, \hspace{0.2cm} \alpha_2 = 0.5984\, \frac{ {\rm Np} }{ {\rm km \cdot \sqrt{MHz} } }\hspace{0.05cm}, \\ \beta_1 & = 22.18\, \frac{ {\rm rad} }{ {\rm km \cdot MHz} }\hspace{0.05cm}, \hspace{0.2cm} \beta_2 = 0.5984\, \frac{ {\rm rad} }{ {\rm km \cdot \sqrt{MHz} } } \hspace{0.05cm}.\end{align*}$$

These values can be calculated from the geometric dimensions of the cables and have been confirmed by measurements at the Fernmeldetechnisches Zentralamt in Darmstadt - see [Wel77]<ref>Wellhausen, H. W.:  Dämpfung, Phase und Laufzeiten bei Weitverkehrs–Koaxialpaaren.  Frequenz 31, S. 23-28, 1977.</ref>.  They apply to a temperature of  $20^\circ\ \text{C (293 K)}$  and frequencies greater than  $\text{200 kHz}$. 

There is the following connection to the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Equivalent_circuit_diagram_of_a_short_transmission_line_section|primary line parameters]]:
*The ohmic losses originating from the frequency-independent component  $R\hspace{0.05cm}'$  are modeled by the parameter  $α_0$  and cause a  (small for coaxial cables)  frequency-independent attenuation.
*The component  $α_1 · f$  of the attenuation function (per unit length) is due to the derivation losses  $(G\hspace{0.08cm}’)$  and the frequency-proportional term  $β_1 · f$  causes only delay but no distortion.
*The components  $α_2$  and  $β_2$  are due to the  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference#Frequency_response_of_a_coaxial_cable|skin effect]],  which causes the current density inside the conductor to be lower than at the surface in the case of higher-frequency alternating current.  As a result, the serial resistance (per unit length)  $R\hspace{0.05cm}’$  of an electric line increases with the square root of the frequency.

==Characteristic cable attenuation==
 
The graph shows the frequency-dependent attenuation curve for the normal coaxial cable and the small coaxial cable.  Shown on the left is the cable attenuation per unit length of the two coaxial cable types in the frequency range up to  $\text{500 MHz}$:
[[File:EN_LZI_4_2_S2_neu.png |right|frame| Attenuation function and characteristic attenuation of coaxial cables '''Korrektur:''' alpha, Np/km]]
:$${\alpha}_{\rm K}(f) \hspace{-0.05cm} = \alpha_0 \hspace{-0.05cm}+ \hspace{-0.05cm} \alpha_1 \cdot f \hspace{-0.05cm}+ \hspace{-0.05cm} \alpha_2 \hspace{-0.05cm}\cdot \hspace{-0.05cm}\sqrt {f} \hspace{0.01cm} \hspace{0.01cm}.$$
:$$\Rightarrow \hspace{0.3cm}{\rm a}_{\rm K}(f) =\alpha_{\rm K}(f) \cdot l $$

'''Notes on the representation chosen here'''

*To make the difference between the attenuation function per unit length  "alpha"  and the function  "a"  (after multiplication by length)  more recognizable, the attenuation function is written here as  ${\rm a}_{\rm K}(f)$  and not (''italics'') as  ${a}_{\rm K}(f)$.
*The ordinate labeling is given here in &bdquo;Np/km”   Often it is also done in &bdquo;dB/km”, with the following conversion:
:$$\ln(10)/20 = 0.11513\text{...} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} 1 \ \rm dB = 0.11513\text{... Np.} $$

'''Interpretation of the left graph'''

*It can be seen from the curves shown that the error is still tolerable when neglecting the frequency-independent component  $α_0$  and the frequency-proportional term  $(α_1\cdot f)$.
*In the following, we therefore assume the following simplified attenuation function:
:$${\rm a}_{\rm K}(f) = \alpha_2 \cdot \sqrt {f} \cdot l = {\rm a}_{\rm \star}\cdot \sqrt
{ {2f}/{R}} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}|H_{\rm K}(f)| = {\rm e}^{- {\rm a}_{\rm K}(f)}\hspace{0.05cm}, \hspace{0.2cm} {\rm a}_{\rm K}(f)\hspace{0.15cm}{\rm in }\hspace{0.15cm}{\rm Np}\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Definition:}$ 
As  '''characteristic cable attenuation'''  $\rm a_∗$  we denote the attenuation of a coaxial cable at half the bit rate due to the  $α_2$ term alone   ⇒   "skin effect",  thus neglecting the  $α_0$  and the  $α_1$ term:
:$${\rm a}_{\rm \star} = {\rm a}_{\rm K}(f = {R}/{2}) = \alpha_2 \cdot \sqrt {{R}/{2}} \cdot l\hspace{0.05cm}.$$
This value is particularly suitable for comparing different conducted transmission systems with different
*coaxial cable types  (for example,  normal or small coaxial cable),  each identified by the parameter  $\alpha_2$,
*bit rates  $(R)$,  and
*cable lengths  $(l)$.}}

'''Interpretation of the right graph'''

The right diagram shows the characteristic cable attenuation  $\rm a_∗$  in "Neper"  (Np) as a function of the bit rate  $R$  and the cable length  $l$
*for the normal coaxial cable  ("2.6/9.5 mm",  left ordinate labeling)  and
*for the small coaxial cable  ("1.2/4.4 mm",  right ordinate labeling).

This diagram shows the PCM systems of hierarchy levels   $3$  to  $5$  proposed by the  [https://en.wikipedia.org/wiki/ITU-T "ITU-T"]  ("ITU Telecommunication Standardization Sector")  in the 1970s.  One recognizes:
*For all these systems for PCM speech transmission,  the characteristic cable attenuation assumes values between  $7 \ \rm Np \ \ (≈ 61 \ dB)$  and  $10.6 \ \rm Np \ \ (≈ 92 \ dB)$ .
*The system  $\text{PCM 480}$  – designed for 480 simultaneous telephone calls - with the bit rate  $R ≈ 35 \ \rm Mbit/s$  was specified for both the normal coaxial cable  $($with  $l = 9.3 \ \rm km)$  and for the small coaxial cable  $($with  $l = 4 \ \rm km)$.  The  $\rm a_∗$values  $10.4\ \rm Np$  and  $9.9\ \rm Np$  respectively are in the same order of magnitude.
*The transmission system  $\text{PCM 1920}$  of the fourth hierarchy level  (specified for the normal coaxial cable)  with  $R ≈ 140 \ \rm Mbit/s$  and  $l = 4.65 \ \rm km$  is parameterized by  $\rm a_∗ = 10.6 \ \rm Np$  or  $10.6 \ {\rm Np}· 8.688 \ \rm dB/Np ≈ 92\ \rm dB$ .
*Although the system  $\text{PCM 7680}$  in contrast has four times the capacity  $R ≈ 560 \rm Mbit/s$ ,  the characteristic cable attenuation of   $\rm a_∗ ≈ 61 \ dB$  is due to the better medium "normal coaxial cable" and the shorter cable sections by a factor of  $3$   $(l = 1. 55 \ \rm km)$  significantly lower.
*These numerical values also show that for coaxial cable systems,  the cable length  $l$  is more critical than the bit rate  $R$.  If one wants to double the cable length,  one has to reduce the bit rate by a factor  $4$ .

You can view the topic described here with the interactive HTLM 5/JS applet  [[Applets:Attenuation_of_Copper_Cables|Attenuation of Copper Cables]] .

==Impulse response of a coaxial cable==
 
To calculate the impulse response,  the first two components of the five attenuation components of the complex propagation function (per unit length) can be neglected  (the reasoning can be found in the previous section).  So we start from the following equation:
:$$\gamma(f) = \alpha_0 + \alpha_1 \cdot f + {\rm j} \cdot \beta_1 \cdot f +\alpha_2 \cdot \sqrt {f}+ {\rm j}\cdot \beta_2 \cdot \sqrt {f} \approx {\rm j} \cdot \beta_1 \cdot f +\alpha_2 \cdot \sqrt {f}+ {\rm j}\cdot \beta_2 \cdot \sqrt {f} \hspace{0.05cm}.$$

Considering
*the cable length  $l$,
*the characteristic cable attenuation  $\rm a_∗$  and
*that  $α_2$  (in  "Np")  and  $β_2$  (in  "rad")  are numerically equal,

thus applies to the frequency response of the coaxial cable:
:$$H_{\rm K}(f) = {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm} b_1 f} \cdot {\rm e}^{-{\rm a}_{\rm \star}\hspace{0.05cm}\cdot \hspace{0.05cm} \sqrt{2f/R} }\cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}{\rm a}_{\rm \star}\hspace{0.05cm}\cdot \hspace{0.02cm} \sqrt{2f/R}}= {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm} b_1 f} \cdot {\rm e}^{-2{\rm a}_{\rm \star}\hspace{0.03cm}\cdot \hspace{0.03cm} \sqrt{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}f/R}} \hspace{0.05cm}.$$
The following abbreviations are used here:
:$$b_1\hspace{0.1cm}{(\rm in }\hspace{0.15cm}{\rm rad)}= \beta_1 \cdot l \hspace{0.05cm}, \hspace{0.8cm} {\rm a}_{\rm \star}\hspace{0.1cm}{(\rm in }\hspace{0.15cm}{\rm Np)}= \alpha_2 \cdot \sqrt {R/2} \cdot l \hspace{0.05cm}.$$

The time domain display is obtained by applying the  [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|Fourier inverse transform]]  and the  [[Signal_Representation/The_Convolution_Theorem_and_Operation|Convolution theorem]]:
:$$h_{\rm K}(t) = \mathcal{F}^{-1} \left \{ {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm} b_1
f}\right \} \star\mathcal{F}^{-1} \left \{ {\rm e}^{-2{\rm a}_{\rm \star}\hspace{0.03cm}\cdot \hspace{0.03cm} \sqrt{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}f/R} }\right \} \hspace{0.05cm}.$$

To be considered here:
*The first term yields the Dirac delta function  $δ(t - τ_{\rm P})$ shifted by the phase delay  $τ_{\rm P} = b_1/2π$ .
*The second term can be given analytically closed.  We write  $h_{\rm K}(t + τ_P)$,  so that the phase delay  $τ_{\rm P}$  need not be considered further.
:$$h_{\rm K}(t + \tau_{\rm P}) = \frac {{\rm a}_{\rm \star}}{\pi \cdot \sqrt{2 \cdot R \cdot t^3}}\cdot {\rm exp} \left [ -\frac {{\rm a}_{\rm \star}^2}{ {2\pi \cdot R\cdot t}} \right ]\hspace{0.05cm},\hspace{0.2cm}{\rm a}_{\rm \star}\hspace{0.15cm}{\rm in\hspace{0.15cm} Np}\hspace{0.05cm}.$$
*Since the bit rate  $R$  also has already been considered in the definition of the characteristic cable attenuation  $\rm a_∗$  this equation can be easily represented with the normalized time  $t\hspace{0.05cm}' = t/T$ :
:$$h_{\rm K}(t\hspace{0.05cm}' + \tau_{\rm P}\hspace{0.05cm} ') = \frac {1}{T} \cdot \frac {{\rm a}_{\rm \star}}{\pi \cdot \sqrt{2 \cdot t\hspace{0.05cm}'\hspace{0.05cm}^3}}\cdot {\rm exp} \left [ -\frac {{\rm a}_{\rm \star}^2}{ {2\pi \cdot t\hspace{0.05cm}'}} \right ]\hspace{0.05cm},\hspace{0.2cm}{\rm a}_{\rm \star}\hspace{0.15cm}{\rm in\hspace{0.15cm} Np}\hspace{0.05cm}.$$
:Here  $T = 1/R$  denotes the symbol duration of a binary system and it holds  $τ_{\rm P} \hspace{0.05cm}' = τ_{\rm P}/T$.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The results of this page are illustrated by the following graph as an example.
[[File:EN_LZI_4_2_S3_neu.png|right|frame| Impulse response of a coaxial cable with  $\rm a_∗ = 60 \ dB$ '''Korrektur:''' Rahmen]]
*The normalized impulse response  $T · h_{\rm K}(t)$  of a coaxial cable with  $\rm a_∗ = 60 \ dB \ \ (6.9\ Np)$ is shown.
*The attenuation coefficients  $α_0$  and  $α_1$  can thus be neglected,  as shown on the last page.
*For the left graph, the parameter  $β_1 = 0$  was also set.

Because of the parameterization by the coefficient  $a_∗$  suitable for this purpose and the normalization of the time to the symbol duration  $T$  the left curve is equally valid for systems with small or normal coaxial cable,  different lengths and different bit rates,  for example for a
*normal coaxial cable  $\text{2.6/9.5 mm}$,  bit rate  $R = 140 \ \rm Mbit/s$,  cable length  $l = 3 \ \rm km$   ⇒   system  $\rm A$,
*small coaxial cable  $\text{1.2/4.4 mm mm}$,  bit rate  $R = 35 \ \rm Mbit/s$, cable length  $l = 2.8 \ \rm km$   ⇒   system  $\rm B$.
 
It can be seen that even at this moderate cable attenuation  $\rm a_∗ = 60 \ \rm dB$  the impulse response already extends over more than   $200$  symbol durations due to the skin effect  $(α_2 = β_2 ≠ 0)$.  Since the integral over  $h_{\rm K}(t)$  is equal to  $H_{\rm K}(f = 0) = 1$,  the maximum value becomes very small:
:$${\rm Max}\big [h_{\rm K}(t)\big ] \approx 0.03.$$

In the diagram on the right, the effects of the phase parameter  $β_1$  can be seen.  Note the different time scales of the left and the right diagram:
*For system  $\rm A$  $(β_1 = 21.78 \ \rm rad/(km · MHz)$,  $T = 7.14\ \rm ns)$   $β_1$  leads to a phase delay of
:$$\tau_{\rm A}= \frac {\beta_1 \cdot l}{2\pi} =\frac {21.78\, { {\rm rad} }/{ {(\rm km \cdot MHz)} }\cdot 3\,{\rm km} }{2\pi} = 10.4\,{\rm \mu s}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\tau_{\rm A}\hspace{0.05cm}' = {\tau_{\rm A} }/{T} \approx 1457\hspace{0.05cm}.$$
*On the other hand,  the following can be obtained for system  $\rm B$  $(β_1 = 22.18 \ \rm rad/(km · MHz)$,  $T = 30 \ \rm ns)$:
:$$\tau_{\rm B}= \frac {\beta_1 \cdot l}{2\pi} =\frac {22.18\, { {\rm rad} }/{ {(\rm km \cdot MHz)} }\cdot 2.8\,{\rm km} }{2\pi} = 9.9\,{\rm µ s}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\tau_{\rm B}\hspace{0.05cm}' ={\tau_{\rm B} }/{T} \approx 330\hspace{0.05cm}.$$

Although  $τ_{\rm A} ≈ τ_{\rm B}$  holds,  completely different ratios result because of the time normalization to  $T = 1/R$ . }}

{{BlaueBox|TEXT=
$\text{Conclusion:}$  When simulating and optimizing communication systems, 
*one usually omits the phase term with  $b_1 = β_1 · l$, 
*since this results exclusively in a  (often not disturbing)  phase delay,  but no signal distortion.}}

==Basic reception pulse==
 
With the basic transmission pulse  $g_s(t)$   ⇒   "basic pulse of the transmitted signal  $s(t)$"  and the impulse response  $h_{\rm K}(t)$  of the channel,  the result for the basic reception pulse  $g_r(t)$   ⇒   "basic pulse of the received signal  $r(t)$"  is:
:$$g_r(t) = g_s(t) \star h_{\rm K}(t)\hspace{0.05cm}.$$
If a non-return-to-zero (NRZ) rectangular pulse  $g_s(t)$  with amplitude  $s_0$  and duration  $Δt_s = T$ is used at the transmitter,  the following results for the basic pulse at the receiver input:
:$$g_r(t) = 2 s_0 \cdot \left [ {\rm Q} \left (\frac {{\rm a}_{\rm \star}/\sqrt {\pi}}{ \sqrt{ (t/T - 0.5)}}\right ) -
{\rm Q} \left (\frac {{\rm a}_{\rm \star}/\sqrt {\pi}}{ \sqrt{ (t/T + 0.5)}}\right ) \right ]\hspace{0.05cm}.$$
Here  $\rm a_∗$  denotes the characteristic cable attenuation  (in Neper)  and  ${\rm Q}(x)$  the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceeding_probability|complementary Gaussian error function]].

[[File:EN_LZI_4_2_S4.png|right|frame|Impulse response of the coaxial cable and basic reception pulse  ]]
{{GraueBox|TEXT=
$\text{Example 2:}$ 
The figure shows for the characteristic cable attenuations  $\rm a_∗ = 40 \ \rm dB$, ... ,  $100 \ \rm dB$  (smaller  $\rm a_∗$–values   are not relevant for practice)

*the normalized coaxial cable impulse response  $T · h_{\rm K}(t)$   ⇒   solid curves,
*the basic reception pulse  ("rectangular response")  $g_r(t)$  normalized to the transmission amplitude  $s_0$   ⇒   dotted line.

One recognizes the following from this diagram:
*With  $\rm a_∗ = 40 \ \rm dB$,  the normalized rectangular response  $g_r(t)/s_0$  is slightly  (about a factor of $0.95)$  smaller at the peak than the normalized impulse response  $T · h_{\rm K}(t)$.  Here is a small difference between impulse response and basic reception pulse.
*In contrast,  for the case  $a_∗ ≥ 60 \ \rm dB$,  the rectangular response and the impulse response are indistinguishable within the drawing accuracy.
*For a return-to-zero  (RZ)  pulse,  the above equation for the basic reception impulse would still need to be multiplied by the duty cycle  $Δt_s/T$ .  In this case  $g_r(t)/s_0$  is smaller than  $T · h_{\rm K}(t)$  by at least this factor.
*The equation modified in this way is also a good approximation for other basic transmission pulses as long as  $\rm a_∗≥ 60 \ \rm dB$  is sufficiently large.  $Δt_s$  then indicates the [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse|equivalent pulse duration]]  of  $g_r(t)$.}}

We draw your attention to the  (German language)  interactive SWF applet  [[Applets:Zeitverhalten_von_Kupferkabeln|Zeitverhalten von Kupferkabeln]]  ⇒   "Time behavior of copper cables",  which deals with the topic discussed here.

==Special features of coaxial cable systems==
 
Assuming binary transmission with non-return-to-zero  (NRZ)  rectangular pulses  $($symbol duration $T)$  and a coaxial transmission channel,  the following system model is obtained.  In particular, it should be noted:

[[File:EN_LZI_4_2_S5.png |right|frame| Binary transmission system with coaxial cable ]]

*In a simulation, the phase delay time of the coaxial cable is conveniently left out of consideration.  Then the basic reception pulse  $g_r(t)$  (with ${\rm a}_{\rm \star}$ in Neper)  is approximated by
:$$g_r(t) \approx s_0 \cdot T \cdot h_{\rm K}(t) = \frac {s_0 \cdot {\rm a}_{\rm \star}/\pi}{ \sqrt{2 \cdot(t/T)^3}}\cdot {\rm e}^{ -{{\rm a}_{\rm \star}^2}/( {2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}t/T}) } \hspace{0.05cm}.$$
*Because of the good shielding of coaxial cables against other impairments,  the  [[Aufgaben:Exercise_1.3Z:_Thermal_Noise|thermal noise]]  is the dominant stochastic perturbation.  In this case,  the signal  $n(t)$  is Gaussian and white.  It can described by the (two-sided) noise power density  $N_0/2$.
*By far the largest noise component arises in the input stage of the receiver, so that it is expedient to add the noise signal  $n(t)$  at the  interface  "cable ⇒ receiver". 
*This noise addition point is also useful because the frequency response  $H_{\rm K}(f)$  decisively attenuates all noise accumulated along the cable.
*Then the received signal is with the amplitude coefficients  $a_{\nu}$:
:$$r(t) = \sum_{\nu = - \infty}^{+ \infty}a_{\nu}\cdot g_r(t - \nu \cdot T)+ n(t) \hspace{0.05cm} .$$

==Exercises for the chapter==
 
[[Aufgaben:Exercise_4.4:_Coaxial_Cable_-_Frequency_Response| Exercise 4.4: Coaxial Cable - Frequency Response]]

[[Aufgaben:Exercise_4.5:_Coaxial_Cable_-_Impulse_Response| Exercise 4.5: Coaxial Cable - Impulse Response]]

[[Aufgaben:Exercise_4.5Z:_Impulse_Response_once_again|Exercise 4.5Z: Impulse Response once again]]

==References==
<references/>

{{Display}}

Mobile Communications/Distance Dependent Attenuation and Shading

2022-02-17T11:39:06Z

Bene: Text replacement - "List of sources" to "References"

{{FirstPage}}
{{Header
|Untermenü=Time-Variant Transmission Channels
|Vorherige Seite=
|Nächste Seite=Probability Density of Rayleigh Fading
}}

== # OVERVIEW OF THE FIRST MAIN CHAPTER # ==
 
The first main chapter deals with time-variant transmission channels, a property that is of great importance for mobile communication.  The description is given throughout in the equivalent low-pass range.

It deals in detail with the following topics:

*the distance-dependent attenuation of a radio signal and various path loss models,
*the influence of shadowing , which can be modeled through ''Lognormal fading'' 
*the non-frequency selective  ''Rayleigh fading''   for channels without "Line of Sight  (LoS)",
*the consideration of the Doppler effect by the so-called  ''Jakes spectrum
*the non-frequency selective  ''Rice fading''   for channels with direct path  (Line of Sight).

== Physical description of the mobile communication channel ==
 
[[File:EN_Mob_T1_1_S1.png|right|frame|Example for a mobile radio scenario|class=fit]]
The figure shows a typical mobile radio scenario with a fixed base station and a mobile subscriber moving towards the base station at the speed  $v$ .

In this representation, the radio signal reaches the mobile station via a direct path.

However, the antenna of the mobile subscriber also receives other signal components that reach the receiver in a detour, for example
* due to reflections on houses,
* a mountain range,
* a plane,
* the ionosphere,
* the ground.
 
This scenario can be used to explain important problems in mobile communications:
*'''Path Loss''':   This measures the attenuation of the electromagnetic wave, which depends to a large extent on the distance between transmitter and receiver. 

*'''Shadowing''':   This describes a slow change in reception conditions due to the changing environment, for example when you pass a building or when you leave a wooded area. 

*'''Multipath Propagation''':   If the signal reaches the receiver on several paths with differences in propagation time, constructive or destructive superimpositions up to complete extinction occur, depending on the signal frequency.   For certain frequencies the topology is favorable, for others unfavorable.  Therefore this effect is also called Frequency Selective Fading.  

*'''Time Variance''':   The effect is caused by the movement of the transmitter and/or the receiver, because there is a different channel at each time.  The transmission quality decreases rapidly if the direct path is shadowed by an obstacle.   The received signal is then composed only of the partial signals arriving on detours, which are attenuated compared to the direct path due to scattering from trees and bushes and possibly refraction and diffraction phenomena, and which add vectorially to the total signal. 

*'''Doppler Effect''':   Depending on whether  (and also at what angle)  the mobile station is moving towards or away from the transmitter, (slight) frequency shifts occur and thus statistical links within the received signal, which cause  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference#Definition_of_the_term_.22Intersymbol_Interference.22|intersymbol interference]] . 

In this chapter we will take a closer look at path loss and shadowing effects.  The following chapters deal with time variance, also taking into account the Doppler effect.  The second main chapter describes multipath propagation, which results in echoes in mobile radio. 

== Free-space propagation ==
 
One speaks of  "free-space propagation"  when there is a line of sight between the transmitter and the receiver positioned at a distance  $d$  as in satellite communications or in space.  The radio waves propagate in  "empty space"  unhindered spherically around the transmitting antenna, but are attenuated with increasing distance due to the energy conservation law. 

Geometrically you can imagine that the radius  $R$  of the sphere and thus also the spherical surface become larger and larger and at constant total energy the energy per unit area becomes proportional to  $1/R^2$  smaller and smaller. 

We assume an unmodulated oscillation of the frequency  $f_{\rm S}$  or of the wavelength  $\lambda= c/f_{\rm S}$  where  $c = 3 \cdot 10^8\ \rm m/s$  indicates the ''speed of light'' , the signal power is  $P_{\rm S}$. 

[https://en.wikipedia.org/wiki/Harald_T._Friis Harald Friis]  gave an equation in 1944 for the reception power  $P_{\rm E}(d)$  from the distance  $d$  (this equation, however, is only valid in a vacuum):

::<math>P_{\rm E}(d) = \frac{P_{\rm S} \cdot G_{\rm S} \cdot G_{\rm E} \cdot \lambda^2}{16 \cdot \pi^2 \cdot d^2 \cdot V_{\rm add}} =
\frac{P_{\rm S} \cdot G_{\rm S} \cdot G_{\rm E} /V_{\rm add}}{K_{\rm FR}(d)} \hspace{0.05cm}.</math>

*$G_{\rm S}$  and  $G_{\rm E}$  indicate the antenna gains of transmitter and receiver, respectively.
*$V_{\rm add} > 1$  summarizes all additional losses independent of the wave propagation, e.g. through the antennas's cable feeds.
*The  '''free-space attenuation'''  $K_{\rm FR}(d)$  depends on the distance  $d$ :

::<math>K_{\rm FR}(d) = K_{\rm FR}(d_0) \cdot (d/d_0)^2 \hspace{0.2cm}{\rm with} \hspace{0.2cm}
K_{\rm FR}(d_0) = ({4 \pi d_0}/{\lambda} )^2 \hspace{0.05cm}.</math>

Usually the free-space attenuation is specified logarithmically with the pseudo unit "dB".  

Then the power loss due to free-space attenuation  $(V$  stands for "Verlust" (German)   ⇒   "loss" in dB$)$:

::<math>V_{\rm FR}(d) = 10 \cdot {\rm lg} \hspace{0.1cm} K_{\rm FR}(d) = V_{\rm 0} + 20\,\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} (d/d_0)\hspace{0.05cm},\hspace{0.5cm} V_{\rm 0} = V_{\rm FR}(d_0) = 20\,\,{\rm dB} \cdot {\rm lg} \hspace{0.2cm} ({4 \pi d_0}/{\lambda}) \hspace{0.05cm}.</math>

It should be noted about this equation:
*The equation only applies in the far field of the antenna  $(d > d_{\rm F})$.  Here  $d_{\rm F} = 2 D^2/\lambda$  the so-called  '''Fraunhofer distance'''.  For  $D$  the largest physical dimension of the transmitting antenna must be used. 

*The equation does not apply to  $d \to 0$.  This would result in the limit value  $K_{\rm FR} \to 0$, and it would result independently from  $P_{\rm S}$  always an infinite receiving power  $P_{\rm E}(d \to 0)$. 

*The free-space attenuation  $K_{\rm FR}(d)$  increases quadratically with increasing distance  $d$  and also quadratically with increasing signal frequency  $f_{\rm S}$, that is, with decreasing wavelength  $\lambda$. 

*For example, for  $\text{GSM 1800}$  $(f_{\rm S} = 1.8 \ \rm GHz$   ⇒   $\lambda \approx 17 \ \rm cm)$:  
:$$K_{\rm FR}(d = 1\ \rm km) = 1.6 \cdot 10^9.$$
:The receiver at a distance of one kilometer does not receive even one billionth of the transmitting power. 

In the  [[Aufgaben:Exercise 1.1Z: Simple Path Loss Model|Exercise 1.1Z]]  the above Friis equation is to be numerically evaluated and interpreted.   Usually, the free-space attenuation is set in relation to a suitable normalization distance  $d_0$   ⇒   $K_{\rm FR}(d/d_0)$, where   $d_0 = 1\ \rm m$  is often used. 

== Common path loss model ==
 
In contrast to satellite and radio relay links, in the case of land mobile radio in addition to free-space attenuation, other disturbing effects must be taken into account which also contribute to a reduction in reception power, namely:
*'''Reflections''':   By superimposing the transmitted signal with a signal component reflected on the ground or on other large smooth surfaces, cancellations can occur which cause a decrease in the reception power up to the fourth power of the distance  $d$  between transmitter and receiver.  For more information, see  [Zan05]<ref name = 'Zan05'>Zangl, J.:  Multi-Hop-Netze mit Kanalcodierung und Medium Access Controll (MAC).  Düsseldorf: VDI Verlag, Reihe 10, Nummer 761, 2005.</ref>  and  [PA95]<ref name = 'PA95'>Pahlavan, K.; Allen, L.:  Wireless Information Networks.  New York: John Wiley & Sons, Wiley Series in Telecommunications and Signal Processing, 1995.</ref>. 

*'''Diffraction''':   This is when the signal is not reflected but deflected from its direction of propagation, for example at the edge of a building.  A physical explanation can be found again in  [Zan05]<ref name = 'Zan05'></ref>. 

*'''Dispersion''':   If the connection  transmitter – receiver  is interrupted by several objects with irregular surfaces  (for example trees or bushes)  the signal arrives at the receiver in the form of many scattered signals with slightly different propagation times.  The size of the obstacle determines whether it is to be interpreted as a reflecting or as a scattering object. 

The effects mentioned here are responsible for the fact that mobile radio can be operated without direct line of sight  $\rm (LOS)$, and thus one of the bases for the economic success of mobile radio systems.  Negatively, these effects are caused by a lower reception power, which must be taken into account by a larger exponent than  $\gamma = 2$ .  We then no longer speak of  "free-space attenuation",  but generally of  "path attenuation factor":

::<math>K_{\rm P}(d) = K_{\rm P}(d_0) \cdot (d/d_0)^\gamma \hspace{0.05cm}.</math>

The corresponding dB–magnitude we call the  '''path loss'''   $(\rm lg$ is the logarithm to the base $10)$:

::<math>V_{\rm P}(d) = V_{\rm 0} + \gamma \cdot 10\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} (d/d_0)\hspace{0.05cm},\hspace{0.5cm}
V_{\rm 0} = V_{\rm P}(d_0) = \gamma \cdot 10\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} \frac{4 \cdot \pi \cdot d_0}{\lambda}\hspace{0.05cm}. </math>

From these equations it can be seen that the free-space attenuation  $V_{\rm FR}(d)$  is a special case of  $V_{\rm P}(d)$  with  $\gamma = 2$ .   In  [Zan05]<ref name = 'Zan05'></ref>  numerical values are given for the exponent  $\gamma$  which were determined as mean values over a large number of measurements.  Among other things
*in clear view (satellite, radio relay):   $\gamma \approx 2$, 
*in an urban setting:   $\gamma = 2.7 \ \text{...} \ 3.5$, 
*in a shaded urban setting:   $\gamma = 3.0\ \text{...} \ 5.0$, 
*inside buildings without a line of sight:   $\gamma = 4.0 \ \text{...} \ 6.0$. 

== Other, more accurate path loss models ==
 
The relatively simple path loss model shown on the last page is well suited for macro cells, but requires high base station antennas.  It was used, for example, as a reference–scenario for the standardization of  [[Mobile_Communications/General information on the LTE mobile communications standard|Long Term Evolution]]  $\rm (LTE)$. 

Of course, this very simple two–parameter model  $(V_0, \ \gamma)$  cannot reproduce all use cases with sufficient accuracy.  A large number of other models for power attenuation can be found in the literature, which are more precisely adapted to specific boundary conditions  (neighbourhood)  and also take different cell sizes into account.  Well-known are for example, see  [Gol06]<ref name='Gol06'>Goldsmith, A.:  Wireless Communications.  Cambridge University Press, Cambridge, UK, 2006.</ref>:

*the  [https://en.wikipedia.org/wiki/Hata_Model Okumura–Hata model], 

*the path loss model according to  [https://en.wikipedia.org/wiki/COST_Hata_model COST 231], 

*the  [https://ieeexplore.ieee.org/document/8597901 Dual–slope model]. 

[[File:EN_Mob_T1_1_S4.png|right|frame|Dual-slope path loss model]]
{{GraueBox|TEXT=
$\text{Example 1:}$  The Dual–slope model is often used for simulations of micro cells in urban areas.   The equation is the following, with the parameters   $d_0 = 1\ \rm m$  und  $d_{\rm BP}$  $($Breakpoint, for example  $d_{\rm BP} = 100\ \rm m)$:

::<math>V_{\rm P}(d) \hspace{-0.05cm} = \hspace{-0.05cm} V_{\rm 0} \hspace{-0.05cm}+\hspace{-0.05cm} \gamma_0 \hspace{-0.05cm}\cdot \hspace{-0.05cm} 10\,{\rm dB} \hspace{-0.05cm}\cdot \hspace{-0.05cm}{\rm lg} \hspace{0.01cm} \left ( {d}/{d_0} \right )
\hspace{-0.05cm}+ \hspace{-0.05cm}(\gamma_1 \hspace{-0.05cm}- \hspace{-0.05cm}\gamma_0) \hspace{-0.05cm}\cdot \hspace{-0.05cm}10\,{\rm dB} \hspace{-0.05cm}\cdot\hspace{-0.05cm} {\rm lg} \hspace{0.01cm} \left (1+ {d}/{d_{\rm BP} } \right )\hspace{0.05cm}.</math>

The graph shows this curve for  $V_{\rm 0} = 10 \ {\rm dB}$,  $\gamma_0 = 2$  und  $\gamma_1 = 4$  in the range from one meter to several kilometers  (thin grey curve).

To simplify matters, the asymptotic approximation shown in red in the graph is used

::<math>V_{\rm P}(d) = \left\{ \begin{array}{c} V_{\rm 0} + \gamma_0 \cdot 10\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} (d/d_0)\hspace{0.05cm},\\
V_{\rm BP} + \gamma_1 \cdot 10\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} (d/d_{\rm BP})\hspace{0.05cm}, \end{array} \right.\quad
\begin{array}{*{1}c} {\rm for} \hspace{0.15cm}d < d_{\rm BP}\hspace{0.05cm},
\\ {\rm for} \hspace{0.15cm} d \ge d_{\rm BP}\hspace{0.05cm} \\ \end{array}</math>

  The value  $V_{\rm BP} = 50 \ {\rm dB}$  is derived from the equation for the first section at the border  $d = 100\ \rm m$  of the scope. 

Note:   In the  [[Aufgaben:Exercise 1.1: Dual Slope Loss Model|Excercise 1.1]]  this model is still being examined in detail.}} 

== Additional loss due to shadowing ==
 
The disturbing influence of shadowing is explained with the help of a graphic, taken from the lecture manuscript   [Hin08]<ref name = 'Hin08'>Hindelang, T.:  Mobile Communications.  Lecture notes.  Institute for Communications Engineering. Munich: Technical University of Munich, 2008.</ref>:
[[File:EN_Mob_T1_1_S5.png|right|frame|Path loss without and with consideration of shading]]

*The previous path loss models only take into account the distance-dependent signal attenuation according to the left graph and disregard topological factors such as the influence of shading.

*In land mobile radio, shadowing causes the signal level to vary even when moving at the same distance from the base station  (on an arc of a circle) .

*This is shown in the right-hand graph, with darker areas indicating greater path loss.  The difference between the left and right images is due to shadowing. 

The effects of shadowing can be summarised as follows:
*For stationary transmitters and receivers, the shadowing is to be considered deterministic.  It causes the path loss due to the shadowing to change by a constant value  $V_{\rm S}$  (in dB):

::<math>V_{\rm P}(d) = V_{\rm 0} + \gamma \cdot 10\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} (d/d_0)+ V_{\rm S}\hspace{0.05cm}. </math>

*If the receiver  (or the sender) moves, the shadowing–loss changes according to the coordinates and therefore also with time. This means:   $V_{\rm S}$   ⇒   $V_{\rm S}(x, y)$   and   $V_{\rm S}$   ⇒   $V_{\rm S}(t)$, respectively. 

However, such channel changes are very slow due to shading.   Often the conditions remain the same for several seconds and one speaks here of  "Long Term Fading"  in contrast to fast fading like  [[Mobile_Communications/Probability density of Rayleigh fading#A very general description of the mobile communication channel|Rayleigh fading]]  and  [[Mobile_Communications/Non-frequency_selective_fading_with_direct component#Kanalmodell_und_Rice.E2.80.93WDF| Rice fading.]] 

== Lognormal channel model==
 
[[File:EN_Mob_T1_1_S6.png|right|frame|Lognormal PDF   ⇒   Shadowing loss]]
To account for the loss  $V_{\rm S}$  by shadowing, the system design must be based on statistical models that have emerged from empirical studies.

The best known is the  '''lognormal''' channel model, which uses a  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables| Gaussian PDF]]  for the random variable  $V_{\rm S}$:

::<math>f_{V_{\rm S}}(V_{\rm S}) = \frac {1}{ \sqrt{2 \pi }\cdot \sigma_{\rm S}} \cdot {\rm e }^{ - { (V_{\rm S}\hspace{0.05cm}- \hspace{0.05cm}m_{\rm S})^2}/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{\rm S}^2) } \hspace{0.05cm}.</math>

The name  "lognormal"  results from the fact that the dB–magnitude  $V_{\rm S}$, which is derived from the linear power attenuation factor via the logarithm, is normally distributed (and thus Gaussian). 

The lognormal channel model is determined by two parameters:
*The mean value  $m_{\rm S} = {\rm E}\big [V_{\rm S}\big ]$  gives the mean shadowing–loss.  
*For rural areas it is usually calculated with  $m_{\rm S} = 6 \ \rm dB$  and for urban areas it is assumed  $m_{\rm S} =14 \ \rm dB$ ... $20 \ \rm dB$. 

*Also the standard deviation  (or dispersion)   $\sigma_{\rm S}$  is different for rural areas  $(\approx 6 \ \rm dB)$  and for urban conditions  $($between  $8 \ \rm dB$  and  $12 \ \rm dB)$ . 

Note that  $V_{\rm S}$  can also take negative values when using lognormal fading   (red area in the above graphic), which actually contradicts the idea of shading.  In practice, however, this model has proven to be very good.

The  "gain by shading"  could be interpreted as follows:
*In urban canyons, reflections from buildings can cause more energy to arrive than would be expected after losing the path. 

*The path loss exponent  $\gamma$  is always fixed, for example  $\gamma = 3.76$  in urban areas.   But there are positions in the city where  $\gamma$  is smaller. 

*Such a simple model cannot reproduce all the details exactly, so one should not try to interpret all the model properties physically. 

{{BlaueBox|TEXT=
$\text{Conclusion:}$  It is useful to summarize the path loss portions in the following way:

::<math>V_{\rm P} = V_{\rm 1} + V_{\rm 2}(t) \hspace{0.25cm}{\rm with}\hspace{0.25cm} V_{\rm 1} = V_{\rm 0} + \gamma \cdot 10\,{\rm dB} \cdot {\rm lg} \hspace{0.1cm} (d/d_0)+ m_{\rm S}\hspace{0.05cm}.</math>

The second term  $V_{\rm 2}(t)$  now describes a lognormal PDF with mean value zero:

::<math>f_{V_2}(V_2) = \frac {1}{ \sqrt{2 \pi }\cdot \sigma_{\rm S} } \cdot {\rm e }^{ - V_2 ^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm} \sigma_{\rm S}^2) }\hspace{0.05cm}.</math>

The distance dependency of  $V_1$  does not play a major role and will not be further discussed here.}} 

== Time domain model for lognormal fading==
 
[[File:P ID2099 Mob T 1 1 S6b v2.png|right|frame|Path loss model with lognormal fading]]

The figure shows a time domain model, with the help of which the path loss  $V_{\rm P}$  can be simulated according to the above equation.  Please note:
*The input signal  $s(t)$  possess the power  $P_{\rm S}$.  In logarithmic representation, the power is related to  $1\ \rm mW$  and the pseudo unit  "dBm"  is added.

*The path loss  $V_1$  is generated by multiplication with  $k_1$.  The output signal  $r'(t)$  then has a power that is smaller by  $V_1$  (in dB) :

::<math>k_1 = 10^{-V_{\rm 1}/20} \hspace{0.1cm} \Rightarrow \hspace{0.1cm}
10 \cdot {\rm lg} \hspace{0.1cm} \frac{P_{\rm E}\hspace{0.05cm}' }{\rm 1\,mW}= 10 \cdot {\rm lg} \hspace{0.1cm} \frac{P_{\rm S} }{\rm 1\,mW} + 20 \cdot {\rm lg} \hspace{0.1cm} k_1 =
10 \cdot {\rm lg} \hspace{0.1cm} \frac{P_{\rm S} }{\rm 1\,mW} - V_1
\hspace{0.05cm}.</math>
[[File:P ID2102 Mob T 1 1 S6c v3.png|right|frame|Relation between Gaussian random variable  $(V_2)$  and lognormal random variable  $(z_2)$]]
*The  (mean value-free)  lognormal fading is simulated by multiplication with the random variable  $z_2(t)$.
*The PDF results from the Gaussian random quantity  $V_2$  by a  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables| nonlinear transformation]]  at the characteristic curve 
:$$z_2 = g(V_2) = 10^{-V_{\rm 2}/20}.$$

*For  $z_2< 0$  this PDF is zero, and for  $z_2\ge 0$  applies with the abbreviation  $C = \rm ln(10)/20 dB$:

:$$f_{z_{\rm 2}}(z_{\rm 2}) = \frac {{\rm e^{- {\rm ln}^2 (z_{\rm 2})
/({2 \hspace{0.05cm}\cdot \hspace{0.05cm} C^2 \hspace{0.05cm} \cdot \hspace{0.05cm} \sigma_{\rm S}^2})
} } }{ \sqrt{2 \pi }\cdot C \cdot \sigma_{\rm S} \cdot z_2} \hspace{0.05cm}.$$

The graphic illustrates the transformation.  You can see
*the Gaussian PDF of  $V_2$  (blue curve) with scatter  $\sigma_{\rm S} = 6 \ \rm dB$,
*the negative logarithmic characteristic curve (green curve), and
*the asymmetric PDF (red curve) of  $z_2(t)$  to be multiplied.

We refer here to the  [[Aufgaben:Exercise 1.2Z: Lognormal Fading Revisited|Exercise 1.2Z]].
 

== Requirements for the following chapters ==
 
The average power of all signal components arriving at the receiver can be calculated using path loss and shading models.
*The lognormal shadowing model takes into account slow changes of the reflectors due to the topology, with reception conditions changing only every five to ten meters in cities and every 30 to 100 meters in rural areas.
*In the following, the path loss and the influence of shadowing is not considered further, but normalized on  $1$. 

Paths can overlap constructively or destructively. The associated changes occur locally in the range of half the wavelength.  In mobile radio, a few centimeters are enough to find completely different reception conditions.  One speaks of   '''Fast Fading'''.  Such a channel is basically frequency–dependent and time–dependent.

[[File:P ID2100 Mob T 1 1 S7 v1.png|right|frame|Signals  $s(t)$  and  $r(t)$  for the description of the mobile radio channel in band-pass range (red) and in the equivalent low-pass range (blue); for the sketch  $ϕ(t)\equiv 0$ ]]

For the rest of this first main chapter, frequency dependence is eliminated by assuming a single fixed frequency (see graph).

The following conditions therefore apply with immediate effect:

*The input signal of the mobile radio channel is a cosine oscillation with the amplitude  $A = 1$  and the frequency  $f_{\rm T}$.  We refer to this harmonic oscillation as the  "transmission signal"  $s_{\rm BP}(t)$.  This band-pass signal is shown in red in the upper graphic. 

*The output signal  $r_{\rm BP}(t)$  of the mobile radio channel – in the following called  "reception signal"  – may differ from  $s_{\rm BP}(t)$  both in amplitude  (envelope)  and in phase   ⇒   lower graph, red. 

*Furthermore, we mostly look at the mobile radio channel in the  [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function#Motivation| equivalent low-pass range.]]  (German:  Tiefpass,  $\rm TP$).  The  "transmission signal"  is then  $s_{\rm TP}(t) = 1$  and thus real   ⇒   blue horizontal in the upper graphic. 

*The low-pass output signal  $r_{\rm TP}(t)$  is generally complex, where the envelope is given by  $a(t)$  and the phase  $\phi(t)$  is noticeable by shifts in the zero crossings   ⇒   blue envelope in the lower graph. 

{{BlaueBox|TEXT=
$\text{Conclusion:}$  For the  '''physical signal at the output of the mobile radio channel'''  ⇒   '''band-pass reception signal'''  always applies in the following:

::<math>r_{\rm BP}(t) = a(t) \cdot \cos \big [2\pi f_{\rm T} t + \phi(t)\big ]\hspace{0.3cm}\Rightarrow \hspace{0.3cm} a(t) = \vert r_{\rm BP}(t)\vert\hspace{0.05cm}, \hspace{0.2cm} \phi(t) = {\rm arc}\hspace{0.15cm} r_{\rm BP}(t)\hspace{0.05cm}.</math>}}

==Exercises for the chapter ==
 
[[Aufgaben:Exercise_1.1:_Dual_Slope_Loss_Model]]

[[Aufgaben:Exercise 1.1Z: Simple Path Loss Model]]

[[Aufgaben:Exercise 1.2: Lognormal Channel Model]]

[[Aufgaben:Exercise 1.2Z: Lognormal Fading Revisited]]

==References==

<references/>

{{Display}}

Linear and Time Invariant Systems/Conclusions from the Allocation Theorem

2022-02-17T11:39:06Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Beschreibung kausaler realisierbarer Systeme
|Vorherige Seite=Linear_Distortions
|Nächste Seite=Laplace_Transform_and_p-Transfer_Function
}}

== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
 
In the first two chapters,  filter functions with real-valued frequency responses were mostly considered for reasons of presentation so that the associated time function is symmetric about zero-time.  However,  the impulse response of a realizable system must always be causal,  that is,  $h(t)$  must be identical to zero for  $t < 0$.  This strong asymmetry of the time function  $h(t)$  implies at the same time that the frequency response  $H(f)$  of a realizable system is always complex-valued with the exception of  $H(f) = K$  where there is a fixed relation between its real part and imaginary part.

This third chapter provides a recapitulatory account of the description of causal realizable systems,  which differ also in the mathematical methods from those commonly used with non-causal systems.

In detail, the following is dealt with:

*the Hilbert transform,  which states how real and imaginary parts of  $H(f)$  are related,
*the Laplace transform,  which yields another spectral function  $H_{\rm L}(p)$  for causal  $h(t)$,
*the description of realizable systems by the pole-zero plot,  as well as
*the inverse Laplace transform using the  "theory of functions"  ("residue theorem").

For this chapter, we recommend two of our multimedia offerings:
*the  (German language)  learning video  [[Rechnen_mit_komplexen_Zahlen_(Lernvideo)|"Rechnen mit komplexen Zahlen"]]   ⇒   "Arithmetic operations involving complex numbers",
*the  (German language)  interactive SWF applet  [[Applets:Kausale_Systeme_-_Laplacetransformation|"Kausale Systeme - Laplacetransformation" ]]   ⇒   "Causal systems – Laplace transform".

==Prerequisites for the entire third main chapter==
 
In the first two chapters,  mostly real  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_–_Transfer_function|transfer functions]]   $H(f)$  were considered for which the associated impulse response  $h(t)$  is consequently always symmetric with respect to the reference time  $t = 0$.  Such transfer functions
*are suitable to explain basic relationships in a simple way,
*but unfortunately are not realizable for reasons of causality.

This becomes clear if the definition of the impulse response is considered:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
The  '''impulse response'''  $h(t)$  is equal to the output signal  $y(t)$  of the system if an infinitely short impulse with an infinitely large amplitude is applied to the input at time  $t = 0$ :   $x(t) = δ(t)$.  Such an impulse is called a  [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_delta_or_impulse|Dirac delta impulse]].}}

It is obvious that no impulse response can be realized for which  $h(t < 0) ≠ 0$  holds.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
For a  '''causal system'''  the impulse response  $h(t)$  is identical to zero for all times  $t < 0$.}}

The only real transfer function that satisfies the causality condition  "the output signal cannot start before the input signal"  is:
:$$H(f) = K \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\quad h(t) = K \cdot \delta(t).$$
All other real-valued transfer functions  $H(f)$  describe non-causal systems and thus cannot be realized by an (electrical) circuit network.

{{BlaueBox|TEXT=
$\text{In other words:}$   Except for the transfer function  $H(f) = K,$  '''any realistic transfer function is complex'''.
*If  $K=1$  holds additionally,  the transfer function is said to be ideal. 
*Then,  the output signal  $y(t)$  is identical to the input signal  $x(t)$  – even without attenuation or amplification.}}

==Real and imaginary part of a causal transfer function==
 
Any causal impulse response  $h(t)$  can be represented as the sum
*of an even  (German:  "gerade"   ⇒   "g")  part  $h_{\rm g}(t)$ 
*and an odd  (German:  "ungerade"   ⇒   "u")  part  $h_{\rm u}(t)$:

:$$\begin{align*} h_{ {\rm g}}(t) & = {1}/{2}\cdot \big[ h(t) + h(-t) \big]\hspace{0.05cm},\\ h_{ {\rm u}}(t) & = {1}/{2}\cdot \big[ h(t) - h(-t) \big] = h_{ {\rm g}}(t) \cdot {\rm sign}(t)\hspace{0.05cm} .\end{align*}$$

Here, the so-called  [https://en.wikipedia.org/wiki/Sign_function sign function]  is used:
:$${\rm sign}(t) = \left\{ \begin{array}{c} -1 \\
+1 \\ \end{array} \right.\quad \quad
\begin{array}{c} {\rm{for}} \\ {\rm{for}}
\\ \end{array}\begin{array}{*{20}c}
{ t < 0,} \\
{ t > 0.} \\
\end{array}$$

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The graph shows this splitting for a causal exponentially decreasing impulse response of a low-pass filter of first-order corresponding to  [[Aufgaben:Exercise_1.3Z:_Exponentially_Decreasing_Impulse_Response|Exercise 1.3Z]]:
[[File: P_ID1750__LZI_T_3_1_S2a_neu.png |right|frame| Splitting of the impulse response into an even part and an odd part|class=fit]]
:$$h(t) = \left\{ \begin{array}{c} 0 \\
0.5/T \\ 1/T \cdot {\rm e}^{-t/T} \end{array} \right.\quad
\begin{array}{c} {\rm{for} } \\ {\rm{for} }
\\ {\rm{for} } \end{array}\begin{array}{*{20}c}
{ t < 0\hspace{0.05cm},} \\
{ t = 0\hspace{0.05cm},} \\{ t > 0\hspace{0.05cm}.}

\end{array}$$

It can be seen that:
*$h_{\rm g}(t) = h_{\rm u}(t) = h(t)/2$  holds for positive times,
*$h_{\rm g}(t)$  and  $h_{\rm u}(t)$  differ only by the sign for negative times,
*$h(t) = h_{\rm g}(t) + h_{\rm u}(t)$  holds for all times, also at time  $t = 0$  (marked by circles). }}

Let us now consider the same issue in the spectral domain.  According to the  [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|Assignment Theorem]]  the following holds for the complex transfer function:  
:$$H(f) = {\rm Re} \left\{ H(f) \right \} + {\rm j} \cdot {\rm Im} \left\{ H(f) \right \}
,$$
where the following assignment is valid:

:$${\rm Re} \left\{ H(f) \right \} \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\quad h_{ {\rm g}}(t)\hspace{0.05cm},$$

:$${\rm j} \cdot {\rm Im} \left\{ H(f) \right\} \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\quad h_{ {\rm u}}(t)\hspace{0.05cm}.$$

First,  this relationship between the real part and the imaginary part of  $H(f)$  shall be worked out using another example.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
A low-pass filter of first-order is assumed and the following holds for its transfer function:
:$$H(f) = \frac{1}{1+{\rm j}\cdot f/f_{\rm G} } = \frac{1}{1+(f/f_{\rm G})^2}- {\rm j} \cdot \frac{f/f_{\rm G} }{1+(f/f_{\rm G})^2} \hspace{0.05cm}.$$
[[File:P_ID1754__LZI_T_3_1_S2b_neu.png|right|frame|Frequency response of a low-pass filter of first-order|class=fit]]
Here,  $f_{\rm G}$  represents the  $\rm 3dB$  cut-off frequency at which  $\vert H(f)\vert^2$  has decreased to half of its maximum  $($at  $f = 0)$.  The corresponding impulse response  $h(t)$  has already been shown in  $\text{Example 1}$  for  $f_{\rm G} = 1/(2πT)$.

The graph shows the real part (blue) and the imaginary part (red) of  $H(f)$.  In addition, the magnitude is shown dashed in green.

Since the time functions  $h_{\rm g}(t)$  and  $h_{\rm u}(t)$  are related by the sign function, there also exists a fixed relationship
* between the real part   ⇒   ${\rm Re} \{H(f)\}$ 
* and the imaginary part   ⇒  ${\rm Im} \{H(f)\}$ 

of the transfer function  $\{H(f)\}$   ⇒   '''Hilbert transform'''.

This is described below.}}

==Hilbert transform==
 
Here,  two time functions  $u(t)$  and  $w(t) = \sign(t) · u(t)$  are considered in the most general sense.
*The associated spectral functions are denoted by  $U(f)$  and  ${\rm j} · W(f)$.
*That is:   In this section  ${w(t) \, \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \, {\rm j} \cdot W(f) }$  is valid and not the usual Fourier correspondence  ${w(t) \, \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \, W(f)}.$

Using the correspondence   ${\rm sign}(t) \, \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \, {1}/({{\rm j} \, \pi f })$   the following is obtained after writing the [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_the_time domain|convolution integral]] out in full with the integration variable  $ν$ :
:$${\rm j} \cdot W(f) = \frac{1}{{\rm j} \, \pi f }\, \star \, U(f) \quad \Rightarrow \quad W(f) = -\frac{1}{\pi }\int\limits_{-\infty}^{+\infty} { \frac{U(\nu)}{f - \nu}}\hspace{0.1cm}{\rm d}\nu \hspace{0.05cm}.$$
However,  since at the same time   $u(t) = \sign(t) · w(t)$   also holds,  the following is valid in the same way:
:$$U(f) = \frac{1}{{\rm j} \, \pi f }\, \star \, {\rm j} \cdot W(f) \quad \Rightarrow \quad U(f) = \frac{1}{\pi }\int\limits_{-\infty}^{+\infty} { \frac{W(\nu)}{f - \nu}}\hspace{0.1cm}{\rm d}\nu \hspace{0.05cm}.$$

These  "integral transformations"  are named after their discoverer  [https://en.wikipedia.org/wiki/David_Hilbert David Hilbert].

{{BlaueBox|TEXT=
$\text{Definitions:}$  Both variants of the  '''Hilbert transformation'''  will be denoted by the following abbreviations in the further course:
:$$W(f) \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\hspace{-0.05cm}\rightarrow\quad U(f) \hspace{0.8cm}{\rm or}\hspace{0.8cm}W(f)= {\cal H}\left\{U(f) \right \}\hspace{0.05cm}.$$
*To calculate the spectrum marked by the arrowhead  –  here  $U(f)$   –  the equation with the positive sign is taken from the two otherwise identical upper equations:
:$$U(f) = \frac{1}{\pi }\int\limits_{-\infty}^{+\infty} { \frac{W(\nu)}{f - \nu} }\hspace{0.1cm}{\rm d}\nu \hspace{0.05cm}.$$
*The spectrum marked by the circle  –  here  $W(f)$   –  arises as a result from the equation with the negative sign:
:$$
W(f) = -\frac{1}{\pi }\int\limits_{-\infty}^{+\infty} { \frac{U(\nu)}{f - \nu} }\hspace{0.1cm}{\rm d}\nu \hspace{0.05cm}.$$}}

Applying the Hilbert transformation twice yields the original function with a change of sign,  and applying it four times yields the original function including the correct sign:
:$${\cal H}\left\{ {\cal H}\left\{ U(f) \right \} \right \} = -U(f),$$
:$${\cal H}\left\{ {\cal H}\left\{ {\cal H}\left\{ {\cal H}\left\{ U(f) \right \} \right \} \right \} \right \}= U(f)\hspace{0.05cm}.$$

{{GraueBox|TEXT=
$\text{Example 3:}$ 
In  [Mar94]<ref name ='Mar94'>Marko, H.:  Methoden der Systemtheorie.  3. Auflage. Berlin – Heidelberg: Springer, 1994.</ref>  the following Hilbert correspondence can be found:
:$$\frac{1}{1+x^2} \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\hspace{-0.05cm}\rightarrow\quad \frac{x}{1+x^2}\hspace{0.05cm}.$$
*Here,  $x$  is representative of a suitably normalized time or frequency variable.
*For example,  if we use  $x = f/f_{\rm G}$  as a normalized frequency variable,  then we obtain the correspondence:
:$$\frac{1}{1+(f/f_{\rm G})^2} \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\hspace{-0.05cm}\rightarrow\quad \frac{f/f_{\rm G} }{1+(f/f_{\rm G})^2}\hspace{0.05cm}.$$
Based on the equation
:$${\rm Im} \left\{ H(f) \right \} \quad \bullet\!\!-\!\!\!-\!\!\!-\!\!\hspace{-0.05cm}\rightarrow\quad {\rm Re} \left\{ H(f) \right \}$$
the result found in  [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Real_and_imaginary_part_of_a_causal_transfer_function|$\text{Example 2}$]]  is thus confirmed:
:$${\rm Im} \left\{ H(f) \right \} = \frac{-f/f_{\rm G} }{1+(f/f_{\rm G})^2}\hspace{0.05cm}.$$}}

==Some pairs of Hilbert correspondences==
 
A very pragmatic way is followed to derive Hilbert correspondences, namely as follows:

*The  [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function#Definition of the Laplace transformation|Laplace transform]]  $Y_{\rm L}(p)$  of the function  $y(t)$ is computed as described in chapter  [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].  This is already implicitly causal.
*$Y_{\rm L}(p)$  is converted into the associated Fourier spectrum  $Y(f)$  which is split it into real and imaginary parts.  To do this, the variable  $p$  is replaced by  ${\rm j \cdot 2}πf.$

[[File:EN_LZI_T_3_1_S4.png|right|frame|Table with Hilbert correspondences|class=fit]]

The real and imaginary parts – so  ${\rm Re} \{Y(f)\}$  and  ${\rm Im} \{Y(f)\}$ – are thus a pair of Hilbert transforms. Furthermore,
#  the frequency variable  $f$  is substituted by  $x$,
#  the real part  ${\rm Re} \{Y(f)\}$  by  $g(x)$,  and
#  the imaginary part  ${\rm Im} \{Y(f)\}$  by  ${\cal H} \{g(x)\}$.

The new variable  $x$  can describe both a  (suitably)  normalized frequency or a  (suitably)  normalized time. Hence, the  [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function#Representation with Hilbert transform|Hilbert transformation]]  is applicable to various problems.

The table shows some of such Hilbert pairs. The signs have been omitted so that both directions are valid.
 
{{GraueBox|TEXT=
$\text{Example 4:}$  For example,  if  ${\cal H} \{g(x)\} = f(x)$  holds,  then from this it also follows that  ${\cal H} \{f(x)\} = \, –g(x)$.  In particular, it also holds:
:$${\cal H}\left \{ \cos(x) \right\} = \sin(x)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} {\cal H}\left \{ \sin(x) \right\} = -\cos(x)\hspace{0.05cm}.$$}}

==Attenuation and phase of minimum-phase systems==
 
An important application of the Hilbert transformation is the relationship between attenuation and phase in the so-called  '''minimum-phase systems'''.  In anticipation of the following chapter  [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]],  it should be mentioned that these systems may have neither poles nor zeros in the right  $p$–half plane.

In general,  the following holds for the transfer function  $H(f)$  with the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Transmission_Line_Theory#Equivalent circuit diagram of a short transmission line section|complex transmission function]]  $g(f)$  and the attenuation function  $a(f)$  as well as the phase function  $b(f)$:
:$$H(f) = {\rm e}^{-g(f)} = {\rm e}^{-a(f)\hspace{0.05cm}- \hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}b(f)} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} g(f) = a(f)+ {\rm j} \cdot b(f)\hspace{0.05cm}.$$
Now in the case of minimum-phase systems,  not only does the Hilbert transformation hold
*regarding imaginary and real part as it does for all realizable systems   ⇒   ${\rm Im} \left\{ H(f) \right \} \, \bullet\!\!-\!\!\!-\!\!\!-\!\!\hspace{-0.05cm}\rightarrow \, {\rm Re} \left\{ H(f) \right \}\hspace{0.01cm},$
*but additionally also the Hilbert correspondence between the phase and attenuation functions is valid   ⇒   $b(f) \, \bullet\!\!-\!\!\!-\!\!\!-\!\!\hspace{-0.05cm}\rightarrow \, a(f)\hspace{0.05cm}.$

{{GraueBox|TEXT=
$\text{Example 5:}$  
A low-pass filter has the frequency response  $H(f) = 1$   ⇒   $a(f) =0$  Np in the  "pass band"   ⇒    $\vert f \vert < f_{\rm G}$,  while for higher frequencies the attenuation function  $a(f)$  has the constant value  $a_{\rm S}$  (in Neper).
[[File:P_ID1753__LZI_T_3_1_S5_neu.png|right|frame|Attenuation and phase functions of an exemplary minimum-phase low-pass filter|class=fit]]
*In this  "stop band"  ⇒    $\vert f \vert > f_{\rm G}$,   $H(f) = {\rm e}^{–a_{\rm S} }$  is very small but not zero.
*If the low-pass filter is to be causal and thus realizable, then the phase function  $b(f)$  must be equal to the Hilbert transform of the attenuation  $a(f)$ .
*Since the Hilbert transform of a constant is equal to zero, the function  $a(f) - a_{\rm S}$  can be assumed in the same way.
*This function shown dashed in the graph is  (negative)  rectangular between  $±f_{\rm G}$.  According to the  [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Some pairs of Hilbert correspondences|table]]  on the last page the following thus holds:
:$$b(f) = {a_{\rm S} }/{\pi} \cdot {\rm ln}\hspace{0.1cm}\left\vert \frac{f+f_{\rm G} }{f-f_{\rm G} }\right \vert \hspace{0.05cm}.$$
In contrast, any other phase response would result in a non-causal impulse response.}}

==Exercises for the chapter==

[[Aufgaben:Exercise_3.1:_Causality_Considerations|Exercise 3.1: Causality Considerations]]

[[Aufgaben:Exercise_3.1Z:_Hilbert_Transform|Exercise 3.1Z: Hilbert Transform]]

==References==
<references/>

{{Display}}

Mobile Communications/The Application of OFDMA and SC-FDMA in LTE

2022-02-17T11:39:06Z

Bene: Text replacement - "List of Sources" to "References"

{{Header
|Untermenü=LTE – Long Term Evolution
|Vorherige Seite=Technical Innovations of LTE
|Nächste Seite=Physical Layer for LTE
}}

== General information on LTE transmission technology ==
 
In contrast to its predecessor  [[Mobile_Communications/Characteristics_of_UMTS|$\rm UMTS$]],  Long Term Evolution  $\rm (LTE)$  uses a variant of the OFDM concept also used by  [https://en.wikipedia.org/wiki/Wireless_LAN WLAN]  to systematically divide the transmission resources.  The multiple access method  [[Modulation_Methods/Allgemeine_Beschreibung_von_OFDM#Das_Prinzip_von_OFDM_.E2.80.93_Systembetrachtung_im_Zeitbereich_.281.29| $\rm OFDM$]]  possesses the ability to protect the system against intermittent transmission disturbances, just like the UMTS basic technology  [[Examples_of_Communication_Systems/Telecommunications_Aspects_of_UMTS#Application_of_the_CDMA_method_to_UMTS|$\rm CDMA$]]. 

In principle, it would have been possible to adapt and expand the technologies used in the second and third generations of mobile communications in such a way that they also meet the required specifications for the fourth generation.  However, the rapidly increasing complexity of CDMA when receiving signals on multiple paths made the technical implementation appear to make little sense. 

The highly abstracted graphic shows the distribution of the complete bandwidth for individual subcarriers and explains the difference between  $\rm CDMA$  (UMTS) and  $\rm OFDM$  (LTE).

[[File:EN_Mob_T_4_3_S1.png|right|frame|Difference between OFDM and CDMA|class=fit]]

*In contrast to CDMA, OFDM has many subcarriers, typically even several hundred, with a bandwidth of only a few  "kHz"  each.
*To achieve this, the data stream is split and each of the many subcarriers is modulated individually with only a small bandwidth. 

LTE uses  $\rm OFDMA$, an OFDM based transmission technology.  Among the reasons for this are  [HT09]<ref name='HT09'>Holma, H.; Toskala, A.:  LTE for UMTS - OFDMA and SC-FDMA Based Radio Access.  Wiley & Sons, 2009.</ref>:
*High performance in frequency controlled channels, 
*the low complexity in the receiver, 
*good spectral properties and bandwidth flexibility, and 
*compatibility with the latest receiver and multi-antenna technologies. 

On the next page the differences between the multiple access methods "OFDM" and "OFDMA" are briefly explained. 

== Similarities and differences of OFDM and OFDMA ==
 
The principle of  "Orthogonal Frequency Division Multiplexing" is explained in detail in chapter  [[Examples_of_Communication_Systems/General Description of DSL#Motivation_for_xDSL|Motivation for xDSL]]  of the book "Examples of Communication Systems". 

[[File:EN_Mob_T_4_3_S2.png|right|frame|division of data blocks by frequency and time for OFDM and OFDMA|class=fit]]

The upper diagram shows the frequency assignment for  $\rm OFDM$:  This method splits the available frequency band into a large number of narrow–band subcarriers.  It is important to note:

*To ensure that the individual subcarriers exhibit as little intercarrier–interference as possible, their frequencies are selected so that they are orthogonal to each other.   This means: 

*At the center frequency of each subcarrier, all other carriers have no spectral components.  The goal is to select the currently most favorable resources for each user in order to obtain an overall optimal result. 

*In concrete terms, this also means that the available resources are allocated to the user who can currently do the most with them, adapted to the respective network situation.
*For this purpose, the base station for the downlink to the terminal device measures the connection quality with the help of reference symbols. 

The lower diagram shows the allocation at  "Orthogonal Frequency Division Multiple Access"  $\rm (OFDMA)$.  You can see:
*For OFDMA the resource allocation after channel fluctuations is not limited to the time domain as with OFDM, but also the frequency domain is optimally included.  Thus the OFDMA resource allocation is better adapted to the external circumstances than with OFDM.
*In order to make optimum use of this flexibility, however, coordination between the base station  ("eNodeB")  and the terminal equipment is necessary.  More on this in chapter  [[Examples_of_Communication_Systems/General Description of DSL|General Description of DSL]].

== Differences between OFDMA and SC-FDMA==
 
There are transmission methods such as 
[https://en.wikipedia.org/wiki/WiMAX WiMAX], which use OFDMA in both directions.  The LTE specification by the 3GPP consortium on the other hand specifies:

[[File:EN_Mob_T_4_3_S3.png|right|frame|Sender and Receiver Structure of a SC-FDMA System|class=fit]]

*In  $\rm Downlink$  (transmission from the base station to the terminal)  $\rm OFDMA$  is used. 

*In  $\rm Uplink$  (from terminal to base station)   $\rm SC–FDMA$  ("Single Carrier Frequency Division Multiple Access" ) is used. 

From the graphic you can see that the two systems are very similar.  In other words:   SC–FDMA is based on OFDMA (or vice versa).
*If you omit the components highlighted in red  ${\rm DFT} \ (K)$  and  ${\rm IDFT} \ (K)$  from SC–FDMA, you get the OFDMA–System. 

*The other blocks stand for Serial/Parallel converter (S/P), Parallel/Serial converter (P/S), D/A converter, A/D converter as well as Add/Remove Prefix. 

The signal generation for SC–FDMA works similar to OFDMA, but with small changes that are important for mobile radio:
*The main difference is the additional  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Argumente_f.C3.BCr_die_diskrete_Realisierung_der_FT|discrete Fourier transform]]  $\rm (DFT)$.  This has to be done on the transmit side directly after the serial/parallel conversion.

*Thus, it is no longer a multi-carrier procedure, but a single-carrier FDMA variant.  One speaks of  "DFT–spread OFDM"  because of the necessary DFT/IDFT operations. 

Let us summarize these statements briefly:

{{BlaueBox|TEXT=
$\text{SC–FDMA is different from OFDMA}$  in the following points  [see also the Internet articles  [https://en.wikipedia.org/wiki/Single-carrier_FDMA "Single-carrier FDMA"]  (in Wikipedia) and 
[http://www.rfwireless-world.com/Articles/difference-between-SC-FDMA-and-OFDMA.html "Difference between SC-FDMA and OFDMA.html"]  (from RF Wireless World)]:

*With SC–FDMA, the data symbols are sent in a group of simultaneously transmitted subcarriers instead of sending each symbol from a single orthogonal subcarrier as with OFDMA.  This subcarrier group can then be considered a separate frequency band that transmits the data sequentially.  This is where the name  "Single Carrier FDMA"  comes from. 

*While with OFDMA the data symbols directly create the different subcarriers, with SC–FDMA they first pass a discrete Fourier transform (DFT).  Thus the data symbols are first transformed from the time domain into the frequency domain before they pass through the OFDM procedure.}} 
[[File:P ID2301 Mob T 4 3 S3b v1.png|right|frame|Frequency band splitting for OFDMA and SC-FDMA|class=fit]]
 
One can also describe the difference between  "OFDMA"  and  "SC–FDMA"  in such a way:
*In an OFDMA transmission, each orthogonal subcarrier only contains the information of a single signal.
*In contrast, with SC–FDMA, each individual subcarrier contains information about all signals transmitted in this period.

This difference and the quasi–sequential transmission with SC–FDMA can be seen particularly well from the diagram on the right. 

This graphic is taken from a PDF document from  "Agilent–3GPP".
 
== Functionality of SC-FDMA==
 
Now the SC–FDMA transfer process shall be examined more in detail.  The information for this section comes largely from  [MG08]<ref name='MG08'>Myung, H.; Goodman, D.:  Single Carrier FDMA – A New Air Interface for Long Term Evolution.  West Sussex: John Wiley & Sons, 2008.</ref>.

[[File:P ID2304 Mob T 4 3 S4a v3.png|right|frame|Considered SC-FDMA transmitter |class=fit]]

The purpose and function of the  "Cyclic Prefix"  is not discussed here in detail.  The reasons for this unit are the same as for OFDM and can be read in the section  [[Modulation_Methods/Implementation of OFDM Systems#Cyclic Prefix|Cyclic Prefix]]  of the book "Modulation Methods".

The following description refers to the  SC–FDMA transmitter  shown here.  Note that with LTE the modulation is adapted to the channel quality:
*In highly noisy channels  $\rm 4–QAM$  (Quadrature Amplitude Modulation with only four signal space points)  is used.
* Under better conditions, the system then switches to a higher-level QAM, up to  $\rm 64–QAM$.
 
The following also applies:
*An input data block consists of  $K$  complex modulation symbols  $x_\nu$ which are generated at a rate of  $R_{\rm Q}\ \big[\rm symbols/s \big]$.  The discrete Fourier transform   $\rm (DFT)$  generates  $K$  symbols  $X_\mu$  in the frequency domain, which are modulated on  $K$  from a total of  $N$  orthogonal subcarriers:
::<math>X_\mu = \sum_{\nu = 0 }^{K-1}
x_\nu \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} { 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} \nu
\hspace{0.05cm}\cdot \hspace{0.05cm} \mu }/{K}} \hspace{0.05cm},</math>
*The subcarriers are distributed over a larger bandwidth of  $B_{\rm K} = N \cdot f_0$  where  $f_0 = 15 \ \rm kHz$  is the smallest addressable bandwidth for LTE.  Unused channels are shown as dashed lines in the example graphic. 

*The channel transmission rate is  $R_{\rm C} = J \cdot R_{\rm Q}$  with spreading factor  $J = N/K$.  This SC–FDMA system could simultaneously process  $J$  orthogonal input signals   ⇒   number of terminal devices that can be simultaneously connected to this base station.  In the case of LTE, for example, the values are  $K = 12$  (smallest addressable block) and  $N = 1024$. 

*According to the so-called  "Subcarrier Mapping"  which is the assignment of the symbols generated by the DFT to the available subcarriers, the symbols are then mapped to a certain bandwidth, for example   $K = 12$  maps to the range of  $0 \ \text{...} \ 180 \ \rm kHz$  or to the range of  $180 \ \rm kHz \ \text{...} \ 360 \ \rm kHz$. 

*The  $\rm IDFT$  (highlighted in blue) transforms the output values  $Y_\mu$  on the frequency domain in its time representation  $y_\nu$.  These samples are then transformed by the Parallel/Serial converter into a sequence suitable for transmission. 

== Different approaches for the Subcarrier Mapping==
 
The following figure illustrates three types of  "Subcarrier Mapping".  To simplify the representation, we will limit ourselves here to the (very small) parameter values  $K = 4$  and  $N = 12$. 
[[File:EN_Mob_T_4_3_S4b.png|right|frame|Various methods of Subcarrier Mapping|class=fit]]
$\rm DFDMA$  or  "Distributed Mapping": Here the modulation symbols are distributed over a certain range of the available channel bandwidth. 

$\rm IFDMA$  or  "Interleaved FDMA": Special form of DFDMA, when the modulation symbols are distributed over the entire bandwidth with equal distances between them. 

$\rm LFDMA$  or  "Localized Mapping": The  $K$  modulation symbols are assigned directly to adjacent subcarriers.  This corresponds to the current 3GPP–specification. 

It can be shown that with SC–FDMA the transmitter does not have to be run through the following steps individually
*Discrete Fourier Transform  $\rm (DFT)$, 
*Subcarrier Mapping,
*Inverse discrete Fourier transform  $\rm (IDFT)$  or inverse Fast Fourier transform  $\rm (IFFT)$. 

Instead, these three operations can be realized together as one single linear operation.  The complete and mathematically complex derivation can be found for example in  [MG08]<ref name='MG08'></ref>.  Each element  $y_\nu$  of the output sequence is then representable by a weighted sum of the input sequence elements  $x_\nu$  where the weights are complex-valued. 

Hence, instead of the comparatively complicated Fourier transform, the operation is reduced
*to a multiplication with a complex number, and 

*the  $J$–fold repetition of the input sequence  $\langle x_\nu \rangle $. 

In  [[Aufgaben:Exercise 4.3: Subcarrier Mapping|Exercise 4.3]]  the (transmit-side)  "Subcarrier Mapping"  is considered with more realistic values for  $K$  and  $N$  and its differences to the  "Subcarrier Demapping"  (at the receiver) are pointed out. 

== Advantages of SC-FDMA over OFDM==
 
The decisive advantage of SC–FDMA over OFDMA is its lower  "Peak–to–Average Power Ratio"  $\rm (PAPR)$  due to its single-carrier structure.  This is the ratio of current peak power  $P_{\rm max}$  to average power  $P_{\rm S}$.  $\rm PAPR$  can also be expressed by the  [[Digital Signal Transmission/Optimization of Baseband Transmission Systems#Systemoptimierung_bei_Spitzenwertbegrenzung|Crest factor]]  (quotient of the signal amplitudes).  However, the two quantities are not identical. 

[[File:P ID2308 Mob T 4 3 S5a v2.png|right|frame|(Complementary)  $\rm PAPR$  for OFDM]]

The graphic from the Internet document  [Wu09]<ref name ='Wu09'>Wu, B.:  Analyzing WiMAX Modulation Quality.   PDF Internet document, 2009.</ref>  shows in double–logarithmic representation the probability that with 64QAM–OFDM the current power  $P_{\rm max}$  is above the average power  $P_{\rm S}$.  You can see:
*The probability of large  "outliers"  is small.  For example, the average power is only exceeded in  $0.1\%$  of time by more than  $\text{10 dB}$      ⇒   marked in red. 

*Even if such high power peaks are very rare, they still pose a problem for the receiver's power amplifier.

The power amplifiers should be operated in the linear range, otherwise the signal is distorted.  Non-linearities arise in particular due to
*intercarrier interference  within the signal, 

*interference from adjacent channels due to spectrum expansions. 

Therefore, OFDM requires the amplifier to operate at a lower power level than its peak power most of the time, which can drastically reduce its efficiency.

*Because one can regard  SC–FDMA  quasi as single carrier transmission procedures, its PAPR is lower than the one of OFDMA.
*Thus, for example, a so-called  "pulse shaping filter" can be used which reduces the PAPR.

The lower PAPR is the main reason why SC–FDMA is used in the LTE uplink and not OFDMA.
*A low PAPR means longer battery life, an extremely important criterion for mobile phones/smartphones.
*At the same time, SC–FDMA offers similar performance and complexity to OFDMA.
*Since a long battery life is less important for the downlink, OFDMA is used here. 

{{GraueBox|TEXT=
$\text{Example 1:}$  We consider an OFDM system with  $N$  carriers, all with the same signal amplitude  $A$.

After a highly simplified calculation with the same proportionality factor we obtain:

*the maximum signal power is proportional to  $(N \cdot A)^2$, and 

*the average signal power is proportional to  $N \cdot A^2$ . 

This results in the  peak–to–average power ratio  ${\rm PAPR} = N$, since it's the quotient of these two powers.  Already with only two carriers this results in  ${\rm PAPR} = 2$  which corresponds to  $\text{3 dB}$. 

*So even with only two carriers the amplifier must always operate  $\text{3 dB}$  below the maximum power to avoid signal distortion in case of signal peaks.
*As will be shown below,  $\text{3 dB}$  already means a decrease in efficiency to  $85\%$.}} 

{{GraueBox|TEXT=
$\text{Example 2:}$  $\rm PAPR$  is directly related to the  "Transmit Amplifier Efficiency".  Maximum efficiency is achieved when the amplifier can operate in the vicinity of the saturation limit. The graphic shows an example of an amplifier's characteristic curve, i.e. the output power plotted against the input power. 

[[File:EN_Mob_T_4_3_S5b.png|right|frame|Decrease in amplifier efficiency with increasing back-off]]
At  $\rm PAPR = 1$  $(\text{0 dB})$  one could set the average power  $P_{\rm S}$  equal to the allowed peak power  $P_{\rm max}$ . According to the characteristic curve  $P_{\rm out}/P_{\rm in}$  the amplifier efficiency would be (exemplarily)  $95\%$. 

Nevertheless, for large  $\rm PAPR$  the amplifier must be operated below the saturation limit to avoid too much signal distortion.  Here are some numerical examples:
*At  $\rm PAPR = 2$  according to the rough calculation on the last example, the average transmit power would have to be chosen $\text{3 dB}$  lower than the allowed power,  so that  $P_{\rm max}$  would not be exceeded at any time.  The efficiency would then decrease to  $85\%$ . 

*A back–off from  $\text{3 dB}$  is usually not sufficient, but in practice values between  $\text{5 dB}$  and  $\text{8 dB}$,  taken from  [Hin08]<ref name='Hin08'>Hindelang, T.:  Mobile Communications.  Lecture Manuscript.  Chair of Communications Engineering, TU Munich, 2008.</ref>.  According to the given curve, however, at  $\text{5 dB}$  the efficiency already drops to only   $70\%$  $($system  $\rm S1$, green line$)$. 

*With system  $\rm S2$  all signal peaks reduced in  $\text{8 dB}$  can be transmitted by the amplifier without distortion, but the amplifier efficiency is then only  $40\%$.  As can be seen in the first graphic on this page, strong distortions still occur about  $2\%$  of the time.

*If the average transmit power is  $P_{\rm S} = 100\, \rm mW$, then with a  $\rm PAPR = 9 \ \Rightarrow \ \text{8 dB}$  the amplifier must work up to  $P_{\rm max} = 900\, \rm mW$  without distortion, with  $\rm PAPR = 2 \ \Rightarrow \ \text{3 dB}$  on the other hand, only up to  $200 \, \rm mW$.  The difference between the two amplifiers is an enormous cost factor.}} 

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*OFDM with a large back–off in the uplink would lead to problems, namely extremely short battery life of the mobile devices  ⇒   SC–FDMA is used in the LTE uplink. 

*In addition, the complexity of SC–FDMA is generally lower than other methods, which means cheaper terminals  [MLG06]<ref name='MLG06'>Myung, H.; Lim, J.; Goodman, D.:  Single Carrier FDMA for Uplink Wireless Transmission.  IEEE Vehicular Technology Magazine, Vol. 1, No. 3, 2006.</ref>.  If the CDMA used in UMTS were extended to the 4G standard, the receiver complexity would increase significantly due to the high frequency diversity in the channel  [IXIA09]<ref name='IXIA09'>SC-FDMA - Single Carrier FDMA in LTE.   PDF Internet document, 2009.</ref>. 

*However, the frequency domain equalization with SC–FDMA is more complicated than with OFDMA.  This is the main reason why SC–FDMA is only used in the uplink.  So these complicated equalizers have to be installed only in the base stations and not in the terminals.}} 

== Exercices to the Chapter ==
 
[[Aufgaben:Exercise 4.3: Subcarrier Mapping]]

[[Aufgaben:Exercise 4.3Z: Multiple-Access Methods in LTE]]

==References==

<references/>

{{Display}}

Theory of Stochastic Signals/Wiener–Kolmogorow Filter

2022-02-17T11:39:03Z

Bene: Text replacement - "List of sources" to "References"

{{LastPage}}
{{Header
|Untermenü=Filtering of Stochastic Signals
|Vorherige Seite=Matched-Filter
|Nächste Seite=
}}
==Optimization criterion of the Wiener-Kolmogorow filter==
 
As another example of optimal filtering, we now consider the task of reconstructing as well as possible the shape of an useful signal  $s(t)$  from the reception signal  $r(t)$,  which is disturbed by additive noise  $n(t)$,  in terms of the  ''mean square error''  (MSE):
:$${\rm{MSE}} = \mathop {\lim }\limits_{T_{\rm M} \to \infty } \frac{1}{{T_{\rm M} }}\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {\left| {d(t) - s(t)} \right|^2 \, {\rm{d}}t} \mathop = \limits^! {\rm{Minimum}}.$$

The filter is named after its inventors  [https://en.wikipedia.org/wiki/Norbert_Wiener Norbert Wiener]  and  [https://en.wikipedia.org/wiki/Andrey_Kolmogorov Andrei Nikolajewitsch Kolmogorow].  We denote the corresponding frequency response by  $H_{\rm WF}(f).$

[[File:EN_Sto_T_5_5_S1.png|right |frame| Derivation of the Wiener filter]]
The following conditions apply to this optimization task:
*The signal  $s(t)$  to be reconstructed is the result of a random process  $\{s(t)\}$, of which only the statistical properties are known in the form of the power-spectral density  ${\it Φ}_s(f)$. 
*The interference signal  $n(t)$  is given by the PSD  ${\it Φ}_n(f)$.  Correlations between the useful and interference signals are accounted for by the  [[Theory_of_Stochastic_Signals/Kreuzkorrelationsfunktion_und_Kreuzleistungsdichte#Kreuzleistungsdichtespektrum|cross-power density spectra]]  ${\it Φ}_{sn}(f) = \hspace{0.1cm} –{ {\it Φ}_{ns} }^∗(f)$. 
*The output signal of the sought filter is denoted by  $d(t)$,  which should differ as little as possible from  $s(t)$  according to the MSE.   $T_{\rm M}$  again denotes the measurement duration.
 
Let the signal  $s(t)$  be mean-free  $(m_s = 0)$  and power-limited.  This means:   The signal energy  $E_s$  is infinite due to the infinite extension of the signal   $s(t)$  and the signal power has a finite value:
:$$P_s = \mathop {\lim }\limits_{T_{\rm M} \to \infty } \frac{1}{{T_{\rm M} }}\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} |{s(t)|^2 \, {\rm{d}}t > 0.}$$

*A fundamental difference with the matched filter task is the stochastic and power-limited useful signal  $s(t)$.
*Let us recall:   In the matched filter, the signal  $g(t)$  to be reconstructed was deterministic, limited in time and thus also energy-limited.

==Result of the filter optimization==
 
{{BlaueBox|TEXT=
$\text{Here without proof:}$  The  '''transmission function of the optimal filter'''  can be determined by the so-called  ''Wiener-Hopf integral equation'',  and is:
:$$H_{\rm WF} (f) = \frac{{ {\it \Phi }_s (f) + {\it \Phi }_{ns} (f)} }{ { {\it \Phi }_s (f) + {\it \Phi }_{sn} (f) + {\it \Phi }_{ns} (f) + {\it \Phi }_n (f)}}.$$

*[https://en.wikipedia.org/wiki/Andrey_Kolmogorov A. Kolmogorow]  and  [https://en.wikipedia.org/wiki/Norbert_Wiener N. Wiener]  independently solved this optimization problem almost at the same time.
*The index "WF" stands for Wiener filter and unfortunately does not reveal the merits of Kolmogorov.
*The derivation of this result is not trivial and can be found for example in  [Hän97]<ref>Hänsler, E.: ''Statistische Signale: Grundlagen und Anwendungen.'' 2. Auflage. Berlin – Heidelberg: Springer, 1997.</ref>.  }}

The mathematical derivation of the equation is omitted.  Rather, this filter shall be clarified and interpreted in the following on the basis of some special cases.
*If signal and disturbance are uncorrelated   ⇒   ${\it Φ}_{sn}(f) = {\it Φ}_{ns}(f) = 0$, the above equation simplifies as follows:
:$$H_{\rm WF} (f) = \frac{{ {\it \Phi }_s (f) }}{{ {\it \Phi }_s (f) + {\it \Phi }_n (f) }} = \frac{1}{{1 + {\it \Phi }_n (f) / {\it \Phi }_s (f) }}.$$
*The filter then acts as a frequency-dependent divider, with the divider ratio determined by the power-spectral densities of the useful signal and the interference signal.
*The "passband" is predominantly at the frequencies where the useful signal has much larger components than the interference:
:$${\it Φ}_s(f) \gg {\it Φ}_n(f).$$
*The  ''mean square error''  (MSE) between the filter output signal  $d(t)$  and the input signal  $s(t)$  is
:$${\rm MSE} = \int\limits_{ - \infty }^{ + \infty } {\frac{{ {\it \Phi }_s (f) \cdot {\it \Phi }_n (f)}}{{ {\it \Phi }_s(f) + {\it \Phi }_n (f)}}\,{\rm{d}}f = \int\limits_{ - \infty }^{ + \infty } {H_{\rm WF} (f) \cdot {\it \Phi }_n (f)}\, {\rm{d}}f.}$$

==Interpretation of the Wiener filter==
 
Now we will illustrate the Wiener-Kolmogorov filter with two examples.

{{GraueBox|TEXT=
$\text{Example 1:}$  To illustrate the Wiener filter, we consider as a limiting case a transmitted signal  $s(t)$  with the power-spectral density  ${\it Φ}_s(f) = P_{\rm S} · δ(f ± f_{\rm S}).$
*Thus, it is known that  $s(t)$  is a harmonic oscillation with frequency  $f_{\rm S}$. 
*On the other hand, the amplitude and phase of the current sample function  $s(t)$ are unknown.

With white noise   ⇒   ${\it Φ}_n(f) = N_0/2$   the frequency response of the Wiener filter is thus:
:$$H_{\rm WF} (f) = \frac{1}{ {1 +({N_0 /2})/{\big[ P_{\rm S} \cdot\delta ( {f \pm f_{\rm S} } \big ]} })}.$$
*For all frequencies except  $f = ±f_{\rm S}$,    $H_{\rm WF}(f) = 0$ is obtained, since here the denominator becomes infinitely large.
*If we further consider that  $δ(f = ±f_{\rm S})$  is infinitely large at the point  $f = ±f_{\rm S}$,  we further obtain  $H_{\rm MF}(f = ±f_{\rm S} ) = 1. $
*Thus, the optimal filter is a bandpass around  $f_{\rm S}$  with infinitesimally small bandwidth.
*The mean square error between the transmitted signal  $s(t)$  and the filter output signal  $d(t)$  is
:$${\rm{MSE} } = \int_{ - \infty }^{ + \infty } {H_{\rm WF} (f) \cdot {\it \Phi_n} (f) \,{\rm{d} }f = \mathop {\lim }\limits_{\varepsilon \hspace{0.03cm} {\rm > \hspace{0.03cm}0,}\;\;\varepsilon \hspace{0.03cm} \to \hspace{0.03cm}\rm 0 } }\hspace{0.1cm} \int_{f_{\rm S} - \varepsilon }^{f_{\rm S} + \varepsilon }\hspace{-0.3cm} {N_0 }\,\,{\rm{d} }f = 0.$$
*This infinitely narrow bandpass filter would allow complete regeneration of the harmonics in terms of amplitude and phase, given the assumptions made.  Thus, regardless of the magnitude of the interference  $(N_0)$,    $d(t) = s(t)$  would apply.
*However, an infinitely narrow filter is not feasible.  With finite bandwidth  $Δf$,  the mean square error is ${\rm MSE} = N_0 · Δf$. }}

This example has dealt with a special case where the best possible result  $\rm MSE = 0$  would be possible, at least theoretically.  The following example makes more realistic assumptions and gives the result  $\rm MSE > 0$.

{{GraueBox|TEXT=
$\text{Example 2:}$  Now consider a  ''stochastic rectangular binary signal''  $s(t)$, additively overlaid by white noise  $n(t)$. 
[[File:P_ID662__Sto_T_5_5_S3_neu.png |frame| Signals at the Wiener filter | right]]
The diagram contains the following plots:
*At the top, the sum signal  $r(t) = s(t) + n(t)$  is shown in gray for  ${\it Φ}_0/N_0 = 5$,  where  ${\it Φ}_0$  denotes the energy of a single pulse and  $N_0$  indicates the power density of the white noise. The useful signal  $s(t)$  is drawn in blue.
*In the center of the figure, the power-spectral densities  ${\it Φ}_s(f)$  and  ${\it Φ}_n(f)$  are sketched in blue and red, respectively, and given in terms of formulas.  The resulting frequency response  $H_{\rm WF}(f)$ is drawn in green.
*The lower figure shows the output signal  $d(t)$  of the Wiener filter as a gray curve in comparison to the transmitted signal  $s(t)$ drawn in blue.  Ideally,  $d(t) = s(t)$  should be valid.

The bottom plot shows:

'''(1)'''   The mean square error (MSE) is obtained by comparing the signals  $d(t)$  and  $s(t)$.

'''(2)'''   Numerical evaluation showed  $\rm MSE$  to be about  $11\%$  of the useful power  $P_{\rm S} $.

'''(3)'''   The signal  $d(t)$  predominantly lacks the higher frequency signal components  (i.e. the jumps).

'''(4)'''   These components are filtered out in favor of a better noise suppression of these frequencies.

Under these conditions, no other filter yields a smaller (mean square) error than the Wiener filter.

Its frequency response (green curve) is as follows:
:$$H_{\rm WF} (f) = \frac{1}{ {1 + ({N_0 /2})/( {\it \Phi}_0 \cdot {\rm si^2} ( \pi f T )})} \hspace{0.15cm} .$$

From the central plot you can see further:
*The DC signal transfer factor here results in  $H_{\rm WF}(f = 0) = {\it Φ}_0/({\it Φ}_0 + N_0/2) = 10/11.$
*For multiples of the symbol repetition rate  $1/T$, where the stochastic useful signal  $s(t)$  has no spectral components,  $H_{\rm WF}(f) = 0$.
*The more useful signal components are present at a certain frequency, the more permeable the Wiener filter is at this frequency.}}

==Exercise for the chapter==
 
[[Aufgaben:Exercise_5.9:_Minimization_of_the_MSE|Exercise 5.9: Minimization of the MSE]]

==References==
<references/>

{{Display}}

Mobile Communications/The GWSSUS Channel Model

2022-02-17T11:38:56Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Frequency-Selective Transmission Channels
|Vorherige Seite=Multipath Reception in Mobile Communications
|Nächste Seite=History and Development of Mobile Communication Systems
}}

== Generalized system functions of time variant systems ==
 
Linear time-invariant systems  $\rm (LTI)$  can be completely described with only two system functions,
*the transfer function  $H(f)$  and
*the impulse response  $h(t)$   ⇒   after renaming   $h(\tau)$.

In contrast, four different functions are possible with time-variant systems  $\rm (LTV)$ .  A formal distinction of these functions with regard to time and frequency domain representation by lowercase and uppercase letters is thus excluded.

Therefore a nomenclature change will be made, which can be formalized as follows:
*The four possible system functions are uniformly denoted by  $\boldsymbol{\eta}_{12}$ . 

*The first subindex is either a  $\boldsymbol{\rm V}$  $($because of German  $\rm V\hspace{-0.05cm}$erzögerung   ⇒   delay time  $\tau)$  or  a  $\boldsymbol{\rm F}$  $($frequency  $f)$. 

*Either a  $\boldsymbol{\rm Z}$  $($because of German  $\rm Z\hspace{-0.05cm}$eit   ⇒   time  $t)$  or a  $\boldsymbol{\rm D}$  $($Doppler frequency  $f_{\rm D})$  is possible as the second subindex.

[[File:EN_Mob_T_2_3_S1_neu.png|right|frame|Relation between the four system functions|class=fit]]

 Since, in contrast to line-based transmission, the system functions of mobile communications cannot be described deterministically, but are statistical variables, the corresponding correlation functions must be considered later on. 

In the following, we will refer to these as  $\boldsymbol{\varphi}_{12}$,  and use the same indices as for the system functions  $\boldsymbol{\eta}_{12}$. 

*These formalized designations are inscribed in the graphic in blue letters.
*Additionally, the designations used in other chapters or literature are given  (grey letters).  In the other chapters these are also partly used.
 
${\rm (1)}$   At the top you can see the  '''time variant impulse response'''   ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t) \equiv h(\tau,\hspace{0.05cm} t)$  in the  "delay–time range".  The associated auto-correlation function  $\rm (ACF)$  is
:::<math>\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \big[ \eta_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1) \cdot
\eta_{\rm VZ}^{\star}(\tau_2, t_2) \big]\hspace{0.05cm}. </math>

${\rm (2)}$  For the  "frequency–time representation"  you get the  '''time-variant transfer function'''   ${\eta}_{\rm FZ}(f,\hspace{0.05cm} t) \equiv H(f,\hspace{0.05cm} t)$. 
          The Fourier transform with respect to  $\tau$  is represented in the graph by  ${\rm F}_\tau\hspace{0.05cm}[ \cdot ]$ .  The Fourier integral is written out in full:
:::<math>\eta_{\rm FZ}(f, \hspace{0.05cm} t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau,\hspace{0.05cm} t) \cdot {\rm e}^{- {\rm j}\cdot 2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau
\hspace{0.05cm}, \hspace{0.3cm} \text{kurz:} \hspace{0.2cm} \eta_{\rm FZ}(f, t)
\hspace{0.2cm} \stackrel{f, \hspace{0.05cm} \tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)
\hspace{0.05cm}.</math>

:The ACF of this time variant transfer function is general:

:::<math>\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) = {\rm E} \big [ \eta_{\rm FZ}(f_1, t_1) \cdot
\eta_{\rm FZ}^{\star}(f_2, t_2) \big ]\hspace{0.05cm}.</math>

${\rm (3)}$  The  '''Scatter–Function'''  ${\eta}_{\rm VD}(\tau,\hspace{0.05cm} f_{\rm D}) \equiv s(\tau,\hspace{0.05cm} f_{\rm D})$  corresponding to the left block describes the mobile communications channel in the  "delay–Doppler area".  
          $f_{\rm D}$  describes the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|Doppler frequency]].   The scatter function results from  ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t)$  through Fourier transformation with respect to the second parameter  $t$:

:::<math> \eta_{\rm VD}(\tau, f_{\rm D})
\hspace{0.2cm} \stackrel{f_{\rm D}, \hspace{0.05cm}t}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.3cm}
\Rightarrow \hspace{0.3cm} \varphi_{\rm VD}(\tau_1, f_{\rm D_1}, \tau_2, f_{\rm D_2}) = {\rm E} \left [ \eta_{\rm VD}(\tau_1, f_{\rm D_1}) \cdot
\eta_{\rm VD}^{\star}(\tau_2, f_{\rm D_2}) \right ]
\hspace{0.05cm}.</math>

${\rm (4)}$  Finally, we consider the so-called  '''frequency-variant transfer function'''  $\eta_{\rm FD}(f, f_{\rm D})$, i.e. the  "frequency–Doppler representation". 
          According to the graph, this can be reached in two ways:

::$$\eta_{\rm FD}(f, f_{\rm D})
\hspace{0.2cm} \stackrel{f_{\rm D}, \hspace{0.05cm}t}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm FZ}(f, t)\hspace{0.05cm},\hspace{0.5cm}\eta_{\rm FD}(f, f_{\rm D})
\hspace{0.2cm} \stackrel{f, \hspace{0.05cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VD}(\tau, f_{\rm D})\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Hints:}$ 
*The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with   ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$ ,  the index  $p$  indicates to which parameter  $\tau$,  $f$,  $t$  or  $f_{\rm D}$  does the Fourier transformation refer.

*The inner  (lighter)  arrows indicate the links via the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]].   For this we use the notation  ${ {\rm F}_p}^{-1}\hspace{0.05cm}[ \hspace{0.05cm} \cdot \hspace{0.05cm} ]$.

*The applet  [[Applets:Impulses and Spectra]] illustrates the connection between the time and frequency domain, which can be described by formulas using Fourier transformation and Fourier inverse transformation.}}

== Simplifications due to the GWSSUS requirements ==
 
The general relationship between the four system functions is very complicated due to non-stationary effects.

[[File:EN_Mob_T_2_3_S2_neu.png|right|frame|Connections between the description functions of the GWSSUS model|class=fit]]
Compared to the general model, some limitations have to be made in order to arrive at a suitable model for the mobile communications channel from which relevant statements for practical applications can be derived. 

The following definitions lead to the  $\rm GWSSUS$ model  $( \rm G$aussian  $\rm W$ide  $\rm S$ense  $\rm S$tationary  $\rm U$ncorrelated  $\rm S$cattering$)$:
*The random process of the channel impulse response  $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  is generally assumed to be complex  (i.e., description in the equivalent low-pass range),  Gaussian  $($identifier  $\rm G)$  and zero-mean  (Rayleigh, not Rice, that means, no line of sight) . 

*The random process is weakly stationary  ⇒   its characteristics change only slightly with time, and the ACF  $ {\varphi}_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1,\hspace{0.05cm}\tau_2,\hspace{0.05cm} t_2)$  of the time variant impulse response does not depend on the absolute times  $t_1$  and  $t_2$  but only on the time difference  $\Delta t = t_2 - t_1$.   This is indicated by the identifier  $\rm WSS$    ⇒   $\rm W$ide $\rm S$ense $\rm S$tationary. 

*The individual echoes due to multipath propagation are uncorrelated, which is expressed by the identifier  $\rm US$   ⇒   $\rm U$ncorrelated $\rm S$cattering.
 
The mobile communications channel can be described in full according to this graph.  The individual power density spectra  (labeled blue)  and the correlation function  (labeled red)  is explained in detail in the following pages. 

== Autocorrelation function of the time-variant impulse response==
 
We now consider the  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|auto-correlation function]]  $\rm (ACF)$  of the time variant impulse response   ⇒   $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  more closely.  It shows:

*Based on the  $\rm WSS$ property, the auto-correlation function can be written with  $\Delta t = t_2 - t_1$ :

::<math>\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = \varphi_{\rm VZ}(\tau_1, \tau_2, \Delta t)\hspace{0.05cm}.</math>

*Since the echoes were assumed to be independent of each other  $\rm (US$ property$)$, the impulse response can be assumed to be uncorrelated with respect to the delays  $\tau_1$  and  $\tau_2$.  Then:

::<math>\varphi_{\rm VZ}(\tau_1, \tau_2, \Delta t) = 0 \hspace{0.35cm}{\rm f\ddot{u}r}\hspace{0.35cm} \tau_1 \ne \tau_2\hspace{0.05cm}. </math>

*If one now replaces  $\tau_1$  with  $\tau$  and  $\tau_2$  with  $\tau + \Delta \tau$, this auto-correlation function can be represented in the following way:

::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}. </math>

*Because of the convolution property of the Dirac function, the ACF for  $\tau_1 \ne \tau_2$   ⇒   $\Delta \tau \ne 0$ disappears.

*$ {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.1cm}$  is the  '''delay–time's cross power-spectral density''', which depends on the delay  $\tau \ (= \tau_1 =\tau_2)$  and on the time difference  $\Delta t = t_2 - t_1$ .
 

{{BlaueBox|TEXT=
$\text{Please note:}$ 
*With this approach, the auto-correlation function  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  and the power-spectral density   ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  are not connected via the Fourier transform as usual, but are linked via a Dirac function:
::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}. </math>
*Not all symmetry properties that follow from the  [[Theory_of_Stochastic_Signals/Power_Density_Spectrum_(PDS)#Wiener-Khintchine_Theorem| Wiener–Khintchine theorem]]  are thus given here.  In particular it is quite possible and even very likely that such a power-spectral density is an odd function.}} 

In the overview on the last page, the  '''delay–time's cross power-spectral density'''  ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  can be seen in the top middle.
*Since  $\eta_{\rm VZ}(\tau, t) $,  like any  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response|impulse response]],  has the unit  $\rm [1/s]$ , the auto-correlation function has the unit  $\rm [1/s^2]$:

::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = {\rm E} \left [ \eta_{\rm VZ}(\tau, t) \cdot
\eta_{\rm VZ}^{\star}(\tau + \Delta \tau, t + \Delta t) \right ].</math>

*But since the Dirac function with the time argument   $\delta(\Delta \tau)$ also has the unit  $\rm [1/s]$  the delay–time's cross power-spectral density  ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  also has the unit $\rm [1/s]$:

::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.</math>

== Power density spectrum of the time-variant impulse response==
 
[[File:P ID2170 Mob T 2 3 S3a v2.png|right|frame|Delay's power-spectral density |class=fit]]
One obtains the   '''delay's power-spectral density'''   ${\it \Phi}_{\rm V}(\Delta \tau)$  by setting the second parameter of  ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$  to  $\Delta t = 0$ .  The graphic on the right shows an exemplary curve. 

The delay's power-spectral density is a central quantity for the description of the mobile communications channel.  This has the following characteristics:
*${\it \Phi}_{\rm V}(\Delta \tau_0)$  is a measure for the "power" of those signal components which are delayed by  $\tau_0$ .  For this purpose, an implicit averaging over all Doppler frequencies  $(f_{\rm D})$  is carried out. 

*The delay's power-spectral density  ${\it \Phi}_{\rm V}(\Delta \tau)$  has, like  ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$ , the unit  $\rm [1/s]$.   It characterizes the power distribution over all possible delays  $\tau$. 

*In the graphic, the power  $ P_0 \approx {\it \Phi}_{\rm V}(\Delta \tau_0)\cdot \Delta \tau$  of those signal components that arrive at the receiver via any path with a delay between  $\tau_0 \pm \Delta \tau/2$.  

*Normalizing the power-spectral density  ${\it \Phi}_{\rm V}(\Delta \tau)$  in such a way that the area is  $1$  results in the probability density function  $\rm (PDF)$ of the delay time:

::<math>{\rm PDF}_{\rm V}(\tau) = \frac{{\it \Phi}_{\rm V}(\tau)}{\int_{0 }^{\infty}{\it \Phi}_{\rm V}(\tau)\hspace{0.15cm}{\rm d}\tau} \hspace{0.05cm}.</math>

''Note on nomenclature'':
*In the book  "Stochastic Signal Theory"  we would have denoted this  [[Theory_of_Stochastic_Signals/Wahrscheinlichkeitsdichtefunktion_(WDF)#Definition_der_Wahrscheinlichkeitsdichtefunktion|probability density function]]  with $f_\tau(\tau)$.
*To make the relation between  ${\it \Phi}_{\rm V}(\Delta \tau)$  and the PDF clear and to avoid confusion with the frequency $f$  we use the nomenclature given here. 

{{GraueBox|TEXT=
$\text{Example 1: Delay models according to COST 207}$
 

In the 1990s, the European Union founded the working group COST 207 with the aim to provide standardized channel models for cellular mobile communications.  where "COST" stands for  "European Cooperation in Science and Technology". 

In this international committee profiles for the delay time  $\tau$  have been developed, based on measurements and valid for different application scenarios.   In the following, four different delay's power density spectra are given, where the normalization factor  ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$  is always used.  The graph shows the PDSs of these profiles in logarithmic representation:

[[File:P ID2175 Mob T 2 3 S4a v1.png|right|frame|Delay's power density spectra according to COST|class=fit]]

'''(1)'''  profile  $\rm RA$  ("Rural Area"):
::<math>{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0}
\hspace{0.3cm}\text{for}\hspace{0.2cm} 0 < \tau < 0.7\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 0.109\,{\rm µ s}\hspace{0.05cm}.</math>

'''(2)'''  profile  $\rm TU$  ("Typical Urban")   ⇒   cities and suburbs:
::<math>{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0}
\hspace{0.3cm}\text{for}\hspace{0.2cm} 0 < \tau < 7\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s}\hspace{0.05cm}.</math>

'''(3)'''  profile  $\rm BU$  ("Bad Urban")   ⇒   unfavourable conditions in cities:
::<math>{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0}
= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0}\\
0.5 \cdot {\rm e}^{ (5\,{\rm µ s}-\tau) / \tau_0} \end{array} \right.\quad
\begin{array}{*{1}l} \hspace{0.1cm} {\rm for}\hspace{0.3cm} 0 < \tau < 5\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s}\hspace{0.05cm},
\\ \hspace{0.1cm} {\rm for}\hspace{0.3cm} 5\,{\rm µ s} < \tau < 10\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}. \\ \end{array}</math>

'''(4)'''  profile  $\rm HT$  ("Hilly Terrain")   ⇒   hilly and mountainous regions:
::<math>{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0}
= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0}\\
0.04 \cdot {\rm e}^{ (15\,{\rm µ s}-\tau) / \tau_0} \end{array} \right.\quad
\begin{array}{*{1}l} \hspace{-0.25cm} {\rm for}\hspace{0.3cm} 0 < \tau < 2\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 0.286\,{\rm µ s}\hspace{0.05cm},
\\ \hspace{-0.25cm} {\rm for}\hspace{0.3cm} 15\,{\rm µ s} < \tau < 20\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}. \\ \end{array}</math>

One can see from the graphics:
*The exponential functions in linear representation now become straight lines. 

*For logarithmic display, you can read the PSD parameter  $\tau_0$  for  $\rm 10 \cdot lg \ (1/e) = -4.34 \ dB$  as shown in the graph for the  $\rm TU$ profile.

*These four COST profiles are described in the  [[Aufgaben:Exercise 2.8: COST Delay Models|Exercise 2.8]]  in more detail.}}
 

== ACF and PSD of the frequency-variant transfer function==
 
The system function   $\eta_{\rm FD}(f, f_{\rm D})$  described in the  [[Mobile_Communications/The_GWSSUS channel model#Generalized system functions_of time variant_systems|overview on the first page of this chapter]]  is also known as the  "Frequency-variant Transfer Function"  where the adjective  "frequency-variant"  refers to the Doppler frequency $f_{\rm D}$.

The associated auto-correlation function  $\rm (ACF)$  is defined as follows:

::<math>\varphi_{\rm FD}(f_1, f_{\rm D_1}, f_2, f_{\rm D_2}) = {\rm E} \left [ \eta_{\rm FD}(f_1, f_{\rm D_1}) \cdot
\eta_{\rm FZ}^{\star}(f_2, f_{\rm D_2}) \right ]\hspace{0.05cm}. </math>

By similar considerations as on the  [[Mobile_Communications/The_GWSSUS_Channel_Model#Power_density_spectrum_of_the_time-variant_impulse_response|previous page]]  this auto-correlation function can be represented under GWSSUS conditions as follows

::<math>\varphi_{\rm FD}(\Delta f, \Delta f_{\rm D}) = \delta(\Delta f_{\rm D}) \cdot {\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) \hspace{0.05cm}.</math>

The following applies:
*${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  is the so-called  '''frequency–Doppler's cross power-spectral density''', which is highlighted in the graphic by a yellow background. 

*The first argument  $\Delta f = f_2 - f_1$  takes into account that ACF and PSD depend only on the frequency difference due to the  "stationarity" .

*The factor  $\delta (\Delta f_{\rm D})$  with  $\Delta f_{\rm D} = f_{\rm D_2} - f_{\rm D_1}$  expresses the  "uncorrelation of the PDS"  with respect to the Doppler shift. 

*You get from  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  to  '''Doppler's power-spectral density'''   ${\it \Phi}_{\rm D}(f_{\rm D})$  if you set  $\Delta f= 0$. 

*The Doppler's power-spectral density  ${\it \Phi}_{\rm D}(f_{\rm D})$  indicates the power with which individual Doppler frequencies occur. 

*The  probability density function  $\rm (PDF)$  of the Doppler frequency is obtained from  ${\it \Phi}_{\rm D}(f_{\rm D})$  by suitable normalization.   The PDF has like  ${\it \Phi}_{\rm D}(f_{\rm D})$  the unit  $\rm [1/Hz]$. 
[[File:P ID2173 Mob T 2 3 S5 v1.png|right|frame|To calculate the Doppler's power-spectral density|class=fit]]
::<math>{\rm PDF}_{\rm D}(f_{\rm D}) = \frac{{\it \Phi}_{\rm D}(f_{\rm D})}{\int_{-\infty }^{+\infty}{\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.15cm}{\rm d}f_{\rm D}} \hspace{0.05cm}.</math>

*Often, for example for a vertical monopulse antenna in an isotropically scattered field, the  ${\it \Phi}_{\rm D}(f_{\rm D})$  is given by the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution| Jakes spectrum]] . 

The frequency–Doppler's cross power-spectral density  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  is highlighted in yellow.  The Fourier connections to the neighboring GWSSUS system description functions  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  and  ${\it \varphi}_{\rm FZ}(\Delta f, \Delta t)$ are also marked.

We refer here to the interactive applet  [[Applets:The_Doppler_Effect|The Doppler Effect]].
 
== ACF and PSD of the delay-Doppler function ==
 
The system function shown in the  [[Mobile_Communications/The_GWSSUS channel model#Generalized_system functions_of time variant_systems|Overview on the first page of this chapter]]  on the left hand side was named  $\eta_{\rm VD}(\tau, f_{\rm D})$ .  

The auto-correlation function  $\rm (ACF)$  of this delay–Doppler function can be written with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  taking into account the GWSSUS properties with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  as follows

::<math>\varphi_{\rm VD}(\tau_1, f_{\rm D_1}, \tau_2, f_{\rm D_2}) = \varphi_{\rm VD}(\Delta \tau, \Delta f_{\rm D}) =
\delta(\Delta \tau) \cdot {\rm \delta}(\Delta f_{\rm D}) \cdot {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \hspace{0.05cm}.</math>

It should be noted about this equation:
*The first Dirac function  $\delta (\Delta \tau)$  takes into account that the delays are uncorrelated  ("Uncorrelated Scattering").

*The second Dirac function  $\delta (\Delta f_{\rm D})$  follows from the stationarity  ("Wide Sense Stationary"). 

*The delay–Doppler's power-spectral density  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$ , also called  '''Scatter Function'''  can be calculated from  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  or ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  as follows:

::<math>{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) ={\rm F}_{\Delta t} \big [ {\it \Phi}_{\rm VZ}(\tau, \Delta t) \big ]
= \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VZ}(\tau, \Delta t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm D} \hspace{0.05cm}\cdot \hspace{0.05cm}\Delta t}\hspace{0.15cm}{\rm d}\Delta t \hspace{0.05cm},</math>
::<math>{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\rm F}_{f_{\rm D}}^{-1} \big [ {\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) \big ]
= \int_{-\infty}^{+\infty} {\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) \cdot {\rm e}^{+{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} \tau \hspace{0.05cm}\cdot \hspace{0.05cm} \Delta f}\hspace{0.15cm}{\rm d}\Delta f \hspace{0.05cm}. </math>

*Both, the system function  $\eta_{\rm VD}(\tau, f_{\rm D})$  and the derived functions  $\varphi _{\rm VD}(\Delta \tau, \Delta f_{\rm D})$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$,  are dimensionless.   For more information on this,  see the specification for  [[Aufgaben:Exercise 2.6: Dimensions in GWSSUS|Exercise 2.6]].

*Furthermore, if the GWSSUS requirements are met, the scatter function is equal to the product of the delay's and Doppler's PDSs:

::<math>{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.</math>

[[File:P ID2171 Mob T 2 3 S6 v1.png|right|frame|One-dimensional description functions of the GWSSUS model ]]
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 

The figure summarizes the results of this chapter so far.  It should be noted:

'''(1)'''   The influence of the delay time  $\tau$  and the Doppler frequency  $f_{\rm D}$  can be separated
*into the blue power-spectral density ${\it \Phi}_{\rm V}(\tau)$
*and the red power-spectral density ${\it \Phi}_{\rm D}(f_{\rm D})$. 

'''(2)'''   The two–dimensional  delay–Doppler's power-spectral density  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  is equal to the product of these two functions.}}
 
== ACF and PSD of the time-variant transfer function ==
 
The following diagram shows all the relationships between the individual power density spectra once again in compact form.
[[File:P ID2176 Mob T 2 3 S7 v1.png|right|frame|Compact summary of all GWSSUS description quantities |class=fit]]
This has already been discussed on the last pages:
*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#Autocorrelation_function_of_the_time-variant_impulse_response|delay–time's cross-power-spectral density]]:
:$${\it \Phi}_{\rm VZ}(\tau, \Delta t)\hspace{0.55cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta t = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau),$$
*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PDS_of_the_frequency-variant_transfer_function|frequency–Doppler's cross-power-spectral density]]:
:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta f = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm D}( f_{\rm D}),$$
*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PDS_of_the_delay-Doppler_function |delay–Doppler's cross-power-spectral density ]]:
:$${\it \Phi}_{\rm VD}(\tau, f_{\rm D})= {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.$$
 
The  '''frequency–time's correlation function'''  (marked yellow in the adjacent graph)  has not yet been considered:

::<math>\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) = {\rm E} \left [ \eta_{\rm FZ}(f_1, t_1) \cdot
\eta_{\rm FZ}^{\star}(f_2, t_2) \right ]\hspace{0.05cm}.</math>

Considering again the GWSSUS simplifications and the identity  $\eta_{\rm FZ}(f, \hspace{0.05cm}t) \equiv H(f, \hspace{0.05cm}t)$, the ACF can be also written with  $\Delta f = f_2 - f_1$  and  $\Delta t = t_2 - t_1$  as follows:

::<math>\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\varphi_{\rm FZ}(\Delta f, \Delta t)
= {\rm E} \big [ H(f, t) \cdot
H^{\star}(f + \Delta f, t + \Delta t) \big ]\hspace{0.05cm}.</math>

It should be noted in this respect:
*You already can see from the name that  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  is a correlation function and not a PSD like the functions  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$,  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$. 

*The Fourier relationships with the neighboring functions are:

::<math>{\it \Phi}_{\rm VZ}(\tau, \Delta t)
\hspace{0.2cm} \stackrel{\tau, \hspace{0.05cm}\Delta f}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} \varphi_{\rm FZ}(\Delta f, \hspace{0.05cm}\Delta t)
\hspace{0.2cm} \stackrel{\Delta t,\hspace{0.05cm} f_{\rm D}}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} {\it \Phi}_{\rm FD}(\Delta f,\hspace{0.05cm} f_{\rm D})
\hspace{0.05cm}.</math>

*If you set the parameters  $\Delta t = 0$  or  $\Delta f = 0$  in this two–dimensional function, the separate correlation functions for the frequency domain or the time domain result:

:$$\varphi_{\rm F}(\Delta f) = \varphi_{\rm FZ}(\Delta f, \Delta t = 0) \hspace{0.05cm},\hspace{0.5cm}
\varphi_{\rm Z}(\Delta t) = \varphi_{\rm FZ}(\Delta f = 0, \Delta t ) \hspace{0.05cm}.$$

*From the graph it is also clear that these correlation functions correspond to the derived power spectral densities via the Fourier transform:

::<math>\varphi_{\rm F}(\Delta f) \hspace{0.2cm} {\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau)\hspace{0.05cm},
\hspace{0.4cm}\varphi_{\rm Z}(\Delta t) \hspace{0.2cm} {\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.</math> 

== Parameters of the GWSSUS model==
 
According to the results on the last page, the mobile channel is replaced by
*the delay's power-spectral density  ${\it \Phi}_{\rm V}(\tau)$  and 

*the Doppler's power-spectral density  ${\it \Phi}_{\rm D}(f_{\rm D})$. 

By suitable normalization to the respective area  $1$  the density functions result with respect to the delay time  $\tau$  or the Doppler frequency $f_{\rm D}$. 

Characteristic values can be derived from the power spectral densities or the corresponding correlation functions.  The most important ones are listed here:

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''Multipath Spread'''  or  '''Time Delay Spread'''  $T_{\rm V}$  specifies the spread that a Dirac delta experiences through the channel on statistical average.   $T_{\rm V}$  is defined as the standard deviation  $(\sigma_{\rm V})$  the random variable  $\tau$:

::<math>T_{\rm V} = \sigma_{\rm V} = \sqrt{ {\rm E} \big [ \tau^2 \big ] - m_{\rm V}^2}
\hspace{0.05cm}.</math>

*The mean value  $m_{\rm V} = {\rm E}\big[\tau \big]$  is an  "Average Excess Delay"  for all signal components.

*${\rm E} \big [ \tau^2 \big ] $  is to be calculated as the root mean square value.}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''Coherence Bandwidth'''  $B_{\rm K}$   is the  $\Delta f$  value at which the frequency's correlation function has dropped to half of its value for the first time.

::<math>\vert \varphi_{\rm F}(\Delta f = B_{\rm K})\vert \stackrel {!}{=} {1}/{2} \cdot \vert \varphi_{\rm F}(\Delta f = 0)\vert \hspace{0.05cm}.</math>

*$B_{\rm K}$  is a measure of the minimum frequency difference by which two harmonic oscillations must differ in order to have completely different channel transmission characteristics.

*If the signal bandwidth is  $B_{\rm S} <B_{\rm K}$, then all spectral components are changed in approximately the same way by the channel. This means:   '''Precisely then there is a non-frequency selective fading'''.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  The graphic on the left side shows the delay's power-spectral density  ${\it \Phi}_{\rm V}(\tau)$
[[File:P ID2177 Mob T 2 3 S8 v3.png|right|frame|Multipath spread and coherence bandwidth|class=fit]]
*with  $T_{\rm V} = 1 \ \rm µs$  (red curve),
*with  $T_{\rm V} = 2 \ \rm µs$  (blue curve).

In the right–hand  $\varphi_{\rm F}(\Delta f)$  representation, the coherence bandwidths are drawn in:
*$B_{\rm K} = 276 \ \rm kHz$  (red curve),
*$B_{\rm K} = 138 \ \rm kHz$  (blue curve).
 
You can see from these numerical values:
*The multipath spread  $T_{\rm V}$,  obtainable from   ${\it \Phi}_{\rm V}(\tau)$ , and the coherence bandwidth  $B_{\rm K}$,  determined by  $\varphi_{\rm F}(\Delta f)$,  stand in a fixed relation to each other:  
:$$B_{\rm K} \approx 0.276/T_{\rm V}.$$
*The often  [[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications#Coherence_bandwidth_as_a_function_of_M|used approximation]]  $B_{\rm K}\hspace{0.02cm}' \approx 1/T_{\rm V}$  is very inaccurate at exponential  ${\it \Phi}_{\rm V}(\tau)$ .}}

Let us now consider the time variance characteristics derived from the time correlation function  $\varphi_{\rm Z}(\Delta t)$  or from the Doppler's power-spectral density  ${\it \Phi}_{\rm D}(f_{\rm D})$ :

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''Correlation Time'''  $T_{\rm D}$  specifies the average time that must elapse until the channel has completely changed its transmission properties due to the time variance.  Its definition is similar to the definition of the coherence bandwidth:

::<math>\vert \varphi_{\rm Z}(\Delta t = T_{\rm D})\vert \stackrel {!}{=} {1}/{2} \cdot \vert \varphi_{\rm Z}(\Delta t = 0)\vert \hspace{0.05cm}.</math>}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''Doppler Spread'''  $B_{\rm D}$  is the average frequency broadening that the individual spectral signal components experience.   The calculation is similar to multipath broadening in that the Doppler spread  $B_{\rm D}$  is calculated as the standard deviation of the random quantity $f_{\rm D}$ :

::<math>B_{\rm D} = \sigma_{\rm D} = \sqrt{ {\rm E} \left [ f_{\rm D}^2 \right ] - m_{\rm D}^2}
\hspace{0.05cm}.</math>

*First of all, the Doppler PDF is to be determined from  ${\it \Phi}_{\rm D}(f_{\rm D})$  through area normalization to  $1$. 

*This results in the mean Doppler shift  $m_{\rm D} = {\rm E}[f_{\rm D}]$  and the standard deviation  $\sigma_{\rm D}$.}} 

{{GraueBox|TEXT=
$\text{Example 3:}$  The diagram is valid for a time–variant channel without direct component.  Shown on the left is the  [[Mobile_Communications/Statistical_bonds_within_the Rayleigh process#ACF_and_PDS_with_Rayleigh_fading|Jakes spectrum]]  ${\it \Phi}_{\rm D}(f_{\rm D})$.
[[File:P ID2181 Mob T 2 3 S8b v1.png|right|frame|Doppler spread and correlation time|class=fit]]
The Doppler spread  $B_{\rm D}$  can be determined from this:

::<math>f_{\rm D,\hspace{0.05cm}max} = 50\,{\rm Hz}\hspace{-0.1cm}: \hspace{-0.1cm}\hspace{0.45cm} B_{\rm D} \approx 35\,{\rm Hz} \hspace{0.05cm},</math>
::<math>f_{\rm D,\hspace{0.05cm}max} = 100\,{\rm Hz}\hspace{-0.1cm}: \hspace{-0.1cm}\hspace{0.2cm} B_{\rm D} \approx 70\,{\rm Hz} \hspace{0.05cm}.</math>

The time correlation function  $\varphi_{\rm Z}(\Delta t)$  is sketched on the right, as the Fourier transform of  ${\it \Phi}_{\rm D}(f_{\rm D})$ .

This can be expressed for given boundary conditions with the Bessel function  ${\rm J}_0()$  as:
::<math>\varphi_{\rm Z}(\Delta t \hspace{-0.05cm} = \hspace{-0.05cm}T_{\rm D}) \hspace{-0.05cm}= \hspace{-0.05cm} {\rm J}_0(2 \pi \hspace{-0.05cm} \cdot \hspace{-0.05cm} f_{\rm D,\hspace{0.1cm}max} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\Delta t ).</math>

*The correlation duration of the blue curve is  $T_{\rm D} = 4.84 \ \rm ms$.

*For  $f_{\rm D,\hspace{0.1cm}max} = 100\,{\rm Hz}$  the correlation duration is only half.

*In this case, it generally applies:   $B_{\rm D} \cdot T_{\rm D}\approx 0.17$.}}
 
== Simulation according to the GWSSUS model ==
 
The  '''Monte–Carlo method''',  here described for the simulation of a GWSSUS mobile communication channel, is based on work by Rice  [Ric44]<ref name='Ric44'>Rice, S.O.:  Mathematical Analysis of Random Noise.  BSTJ–23, pp. 282–332 und BSTJ–24, pp. 45–156, 1945.</ref>  and  Höher [Höh90]<ref name='Höh90'>Höher, P.:  Empfang trelliscodierter PSK–Signale auf frequenzselektiven Mobilfunkkanälen – Entzerrung, Decodierung und Kanalschätzung.  Düsseldorf: VDI–Verlag, Fortschrittsberichte, Reihe 10, Nr. 147, 1990.</ref>.

*The two–dimensional impulse response is represented by a sum of  $M$  complex exponential functions.  $M$  can be interpreted as the number of different paths:

::<math>h(\tau,\ t)= \frac{1}{\sqrt {M}} \cdot \sum_{m=1}^{M} \alpha_m \cdot \delta (t - \tau_m) \cdot {\rm e}^{{\rm j} \hspace{0.05cm} \phi_{m} }\cdot {\rm e}^{ {\rm j} \hspace{0.05cm}2 \pi f_{{\rm D},\hspace{0.05cm} m} t}
\hspace{0.05cm}. </math>

*First, the delays  $\tau_m$,  the attenuation factors  $\alpha_m$,  the equally distributed phases  $\phi_m$  and the Doppler frequencies  $f_{{\rm D},\hspace{0.1cm} m}$  will be randomly generated according to the GWSSUS specifications.  The base for the random generation of the Doppler frequencies  $f_{{\rm D},\hspace{0.1cm} m}$  is the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution| Jakes spectrum]]  ${\it \Phi}_{\rm D}(f_{\rm D})$, which, appropiately normalized, simultaneously indicates the probability density function  $\rm (PDF)$  of the Doppler frequencies. 

*Because of  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})$  the delay time  $\tau_m$  is independent of the Doppler frequency  $f_{{\rm D},\hspace{0.1cm} m}$  for all  $m$.  This is valid with good approximation for terrestrial land mobile communication.  For the random generation of the parameters  $\alpha_m$  and  $\tau_m$,  which determine the delay–power-spectral density  $ {\it \Phi}_{\rm V}(\tau)$  the following  [[Mobile_Communications/The_GWSSUS_Channel_Model#Power_density_spectrum_of_the_time-variant_impulse_response|COST profiles]]  are available:   $\rm RA$  ("Rural Area"),  $\rm TU$  ("Typical Urban"),  $\rm BU$  ("Bad Urban")  and  $\rm HT$  ("Hilly Terrain"). 

*The greater the number of different paths   $M$  is chosen for the simulation, the better a real impulse response is approximated by the above equation.  However, the higher accuracy of the simulation is at the expense of its duration.  In the literature, favorable values are given for $M$  between  $100$  and  $600$ . 

[[File:P ID2183 Mob T 2 3 S9a.png|right|frame|Time variant transfer function  $($the absolute value squared is  simulated$)$]]
{{GraueBox|TEXT=
$\text{Example 4:}$  The graphic from  [Hin08]<ref name='Hin08'>Hindelang, T.:  Mobile Communications. 
Lecture notes. Institute for Communications Engineering. Munich: Technical University of Munich, 2008.</ref>  shows a simulation result:     $20 \cdot \lg \vert H(f, \hspace{0.1cm}t)\vert$  is shown as a 2D plot, where the time-variant transfer function  $H(f, \hspace{0.1cm}t)$  is also referred to as  $\eta_{\rm FZ}(f, \hspace{0.1cm}t)$  in this tutorial. 

The simulation is based on the following parameters:
*The time variance results from a movement with  $v = 3 \ \rm km/h$.

*The carrier frequency is  $f_{\rm T} = 2 \ \rm GHz$. 

*The maximum delay time is  $\tau_{\rm max} \approx 0.4 \ \rm µ s$.

* According to the approximation we obtain the coherence bandwidth  $B_{\rm K}\hspace{0.02cm}' \approx 2.5 \ \rm MHz$. 

*The maximum Doppler frequency is  $f_\text{D, max} \approx 5.5 \ \rm Hz$.

* The Doppler spread results in  $B_{\rm D} \approx 4 \ \rm Hz$.}}
 

==Exercises to the chapter==
 
[[Aufgaben:Exercise 2.5: Scatter Function]]

[[Aufgaben:Exercise 2.5Z: Multi-Path Scenario]]

[[Aufgaben:Exercise 2.6: Dimensions in GWSSUS]]

[[Aufgaben:Exercise 2.7: Coherence Bandwidth]]

[[Aufgaben:Exercise 2.7Z: Coherence Bandwidth of the LTI Two-Path Channel]]

[[Aufgaben:Exercise 2.8: COST Delay Models]]

[[Aufgaben:Exercise 2.9: Coherence Time]]

==References==

<references/>

{{Display}}

Mobile Communications/General Information on the LTE Mobile Communications Standard

2022-02-17T11:38:56Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=LTE – Long Term Evolution
|Vorherige Seite=Characteristics of UMTS
|Nächste Seite=Technical Innovations of LTE
}}

== # OVERVIEW OF THE FOURTH MAIN CHAPTER # ==
 
This chapter provides an overview of  '''Long Term Evolution'''  $\rm (LTE)$.  From today's perspective (2011), LTE is a new mobile communications standard that should replace UMTS and will probably continue to shape the next few years of mobile voice and data transmission.

In the following, a rough overview of the motivation, functionality and characteristics of LTE is given.  This is followed by a more detailed system description of the technical processes involved in LTE.  This chapter will deal with this in detail:

* The motivation for LTE and the frequency band allocation,
*the development of mobile communications standards towards LTE,
*some technical details about voice and data transmission,
*the transmission method  "SC–FDMA"  used in the uplink and its differences to  "OFDMA",
*the description and function of the different logical channels in the bit transmission layer,
*an outlook on the successor system LTE–Advanced.

''Addendum'':   The LTE chapter was written in 2011, i.e. at the time when LTE had just been introduced.  During the last editorial revision in autumn 2017, some earlier statements were revised which no longer corresponded to the facts after six years of intensive use by many customers.  However, most of the chapter remained unchanged compared to 2011, as the LTE principle has not changed in the meantime.

== Development of mobile users until 2010 ==
 
[[File:EN_LTE_T_4_1_S3.png|right|frame|Absolute and relative number of mobile devices in the years 2004 - 2010|class=fit]]
Since the turn of the millennium, the number of mobile connections has increased dramatically.
*The graph shows for the years 2004 to 2010 an increase from 1.8 to approx. 5 billion mobile devices worldwide in absolute numbers  (red bars, left scale).
*The blue bars (left scale) show the development of the world population in the same period.

*The (percentage) number of cell phones  (green curve, right scale)  in relation to the world population increased from just under 30% to over 70% between 2004 and 2010.
*The statistics include users with more than one cell phone.  2010 possessed thus by no means 70% of the world population a mobile telephone. 

*The use of mobile data services has sharply increased, especially since the introduction of flatrate tarifs.
 
The following statements refer to the year 2010:
*Global mobile data traffic grew by 159% in 2010, a much stronger increase than expected.  Since then, mobile data transmission has caused more network load than voice transmission in the mobile network. 

*Mobile data traffic alone was three times as large in 2010 as the entire traffic volume in 2000  (at that time mainly voice traffic). 

*Although smartphones accounted for only 13% of all mobile devices in 2010, they were responsible for 78% of data and voice transmission. 

*To this development also 94 million laptop users contributed, who used the Internet on the way over UMTS modems.

*Such a laptop user causes thereby on the average 22 times the data quantity of an average smartphone user.

== Essential properties of LTE ==
 
The abbreviation  $\rm LTE$  stands for  "Long Term Evolution"  and refers to the mobile communications standard that follows UMTS.  The new conceptual development of LTE was intended to satisfy the ever-increasing demand for bandwidth and higher speeds over the long time  ("Long Term").

The LTE standard was first defined in 2008 as  "UMTS Release 8"  by the  [[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard#3GPP_-_Third_Generation_Partnership_Project| $\rm 3GPP$]]  (Third Generation Partnership Project), a conglomerate of various international telecommunication associations, and has since been continuously developed further by so-called "Releases".  The commitment of the world's largest mobile communications providers has made LTE the first (largely) uniform standard for mobile communications technology. 

According to  "UMTS Release 8", LTE is also called  $\rm 3.9G$  because it initially did not fully meet the conditions specified by the ITU  (International Telecommunication Union)  for fourth generation  $\rm (4G)$  mobile communications.

In contrast, the subsequent Release 10  (dated July 2011)  complies with the  $\rm 4G$ standard.  The chapter  [[Mobile_Communications/LTE-Advanced - a further development of LTE|LTE Advanced]]  lists the features of this LTE enhancement.  This technology is also referred to as  $\rm LTE–A$. 

Here is a summary of the important system features of LTE from the page  [http://www.itwissen.info/definition/lexikon/long-term-evolution-LTE.html ITWissen] :

*LTE is based on the multiple access methods  $\rm OFDMA$  ("Orthogonal Frequency Division Multiple Access")  in the downlink and  $\rm SC–FDMA$  ("Single Carrier Frequency Division Multiple Access") in the uplink.  The detailed description of OFDMA and especially its differences to  [[Modulation_Methods/Allgemeine_Beschreibung_von_OFDM#Das_Prinzip_von_OFDM_.E2.80.93_Systembetrachtung_im_Zeitbereich_.281.29|$\rm OFDM$]]  can be found in chapter  [[Mobile_Communications/The_application_of_OFDMA_and_SC-FDMA_in_LTE|"The application of OFDMA and SC–FDMA in LTE"]].

*The use of this modulation method enables orthogonality between individual users, resulting in an increased network capacity  [HT09]<ref name ='HT09'>Holma, H.; Toskala, A.:  LTE for UMTS – OFDMA and SC–FDMA Based Radio Access.  Wiley & Sons, 2009.</ref>.  In conjunction with "Multiple Input Multiple Output" $\rm (MIMO)$, this technology currently (2011) enables peak data rates of 100 Mbit/s in the downlink.

*In addition to the significantly higher data rate compared to the  $\rm 3G$  system UMTS, LTE technology makes more efficient use of the available bandwidth.  By combining the latest state-of-the-art technology with the existing experience of GSM and UMTS, the new standard is not only much faster, but also simpler and more flexible.

== Motivation and goals of LTE ==
 
In 2010, the American telecommunications company  "Cisco Systems"  published a  [http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-c11-520862.html White Paper]  which assumes that in 2015
[[File:EN_LTE_T_4_1_S2.png|right|frame|Graphic from the Ericsson Mobility Report 2015|class=fit]]
*the use of mobile data will be twenty-six times higher than in 2010, 
*this usage is increasing by a further 92% per year, and
*the gigantic amount of 6.3 Exabyte   ⇒   $\rm6.3 \cdot 10^{18}$  byte per month is reached. 

It has also been predicted that five billion people will be connected to the Internet in 2015  [HT09]<ref name ='HT09'></ref>.  In addition, other wireless transmission technologies are being developed at the same time, which promise equally high data transmission rates.  All these factors called for further development of the 3GPP mobile communications standard "UMTS".

The  [http://www.lte-anbieter.info/lte-news/ericsson-mobility-report-weltweite-prognose-zur-lte-abdeckung Ericsson Mobility Report]  of 2015 shows that the 2010 forecast has been exceeded.  In 2014 there were already 7.1 billion mobile users with Internet access, in 2020 there should be 9.2 billion.

The 3GPP consortium started early to define the LTE targets to keep up with the rapid development of line-based connections.  The exact targets were then set out in the  "LTE Release 6"  compared to  $\rm HSPA$  technology  ("High–Speed Packet Access")  at the end of 2004.

The main goals were mentioned:
*A purely packet-oriented transmission and a high degree of mobility and security, 
*reduced complexity, cost reduction and optimized battery life of the end devices, 
*bandwidth flexibility between 1.5 MHz and 20 MHz, 
*a spectral efficiency  (data rate per one Hertz bandwidth)  as high as possible, 
*maximum possible data rates of 100 Mbit/s in downlink and 50 Mbit/s in uplink, 
*signal processing times less than 10 milliseconds. 

Compared to HSPA, this means an increase in spectral efficiency by a factor of  $2 \ \text{...}\ 4$  and a reduction in latency by half and a tenfold increase in the maximum data rate.  The individual points, which represent a large part of the LTE specific technical characteristics, are described in more detail in the chapter  [[Mobile_Communications/Technical innovations of LTE|"Technical Innovations of LTE"]].

== Development of the UMTS mobile phone standards towards LTE ==
 
The development of third generation mobile communications standards was already discussed in detail in the third chapter of this book.nbsp; For this reason, only the more recent developments are discussed in detail here.

First of all, a brief overview of UMTS releases before LTE from  [Hin08]<ref name='Hin08'>Hindelang, T.:  Mobile Communications.  Lecture notes.  Institute for Communications Engineering.  Munich: Technical University of Munich, 2008.</ref>:

*Release 99   (December 1999):   UMTS 3G FDD and TDD;   3.84 Mchip/s;   CDMA air interface. 

*Release 4   (July 2001):   Lower chip rate (1.28 Mchip/s) for TDD;   some fixes and minor improvements. 

*Release 5   (March 2002):   [https://en.wikipedia.org/wiki/IP_Multimedia_Subsystem IP Multimedia Subsystem]  $\rm (IMS)$;   [[Examples_of_Communication_Systems/Weiterentwicklungen_von_UMTS#High.E2.80.93Speed_Downlink_Packet_Access| High-Speed Downlink Packet Access]]  $\rm (HSDPA)$. 

*Release 6   (March 2005):   [[Examples_of_Communication_Systems/Weiterentwicklungen_von_UMTS#High.E2.80.93Speed_Uplink_Packet_Access| High-Speed Uplink Packet Access]]  $\rm (HSUPA)$;   Multimedia Broadcast & Multicast Services  $\rm (MBMS)$;       Cooperation with Wireless LAN;   Push–to–Talk;   [https://en.wikipedia.org/wiki/Generic_Access_Network Generic Access Network]  $\rm (GAN)$. 

*Release 7   (December 2007):   Reduced latency;   improved Quality of Service  $\rm (QoS)$;   MIMO for UMTS;   TDD option  $7.68$  Mchip/s,   real-time applications,  e.g. 
**[https://en.wikipedia.org/wiki/Voice_over_IP Voice over IP]  $\rm (VoIP)$, 
** [https://en.wikipedia.org/wiki/Enhanced_Data_Rates_for_GSM_Evolution Enhanced Data Rates for GSM Evolution EDGE Evolution]  $\rm (EDGE \ Evolution)$. 

The  Release 8  of December 2008 was synonymous with the introduction of  "Long Term Evolution"  $\rm (LTE)$ and the basis for the first generation of LTE capable terminals.  The main innovations and characteristics of Release 8, summarized by  [http://www.3gpp.org/about-3gpp 3gpp]  (Third Generation Partnership Project), were:
*High spectral efficiency and very short latency, 

*the support of different bandwidths, 

*a simple protocol and system architecture, 

*backwards compatibility and compatibility to other systems like  [[Mobile_Communications/Characteristics_of_UMTS#The_IMT.E2.80.932000 Standard |cdma2000]], 

*$\rm FDD$  (Frequency Division Duplex) and  $\rm TDD$  (Time Division Duplex) are optionally usable, 

*self-organizing networks  $\rm (SON)$  support. 

These features (and some others more) are discussed in detail in the section  [[Mobile_Communications/Technical innovations of LTE|Technical innovations of LTE]] .

*The  '''Release 9''', on the other hand, contains only minor improvements and will not be discussed in detail here.
*The  '''Release 10'''  from July 2011 describes the further development of LTE   ⇒   [[Mobile_Communications/LTE-Advanced - a further development of LTE|LTE Advanced]]  $\rm (LTE–A)$. 

== LTE frequency band splitting ==
 
[[File:EN_LTE_T_4_1_S4.png|right|frame|LTE frequencies around  $\text{800 MHz}$  and  $\text{2.6 GHz}$]]
New frequencies were needed for LTE. In Germany, there was an auction of two frequency ranges in 2010, in which all German mobile network operators took part.

The chart illustrates the results of this auction of frequencies in

*$\text{Range around 800 MHz}$  $\text{(791 ... 862 MHz)}$: Here only paired spectra were assigned for FDD:   Telekom, O2 and Vodafone; got two times 10 MHz each 

*$\text{Range around 2.6 GHz}$  $\text{(2.5 ... 2.69 GHz)}$: Here, paired spectra for FDD (140 MHz in total) and unpaired spectra for TDD (50 MHz) were assigned.

More about the difference between FDD and TDD can be found in section  [[Examples_of_Communication_Systems/General_Description_of_DSL#Motivation_for_xDSL| Motivation for xDSL]]  in the book "Examples of communication systems".

The two auctioned frequency ranges have different system characteristics, which make them interesting for different applications.
*The lower frequency range (around 800 MHz) is also called  "digital dividend"  because it was freed up by the conversion of (terrestrial) TV transmission from  $\rm PAL$  to  $\rm DVB–T$  ("Digitization").

*By agreement of the Federal Government with the (German) network operators this range must be used to help poorly supplied regions to get "Fast Internet".  Four levels were defined for the level of broadband Internet coverage in a region.  Only when 90% of one level has been covered throughout Germany the developement of the next level may begin.

*The choice for this project was the comparatively low frequency range around 800 MHz with better propagation characteristics than 2600 MHz, which is both sensible and necessary for cost-effective coverage of rural areas.  A LTE–800 base station reaches a maximum transmission radius of about 10 km.  However, the ratio of users per area is lower than with LTE–2600, which means that LTE–800 is more suitable for sparsely populated regions. 

*The frequency range from 821 MHz to 832 MHz is kept free to avoid interference between the uplink and the downlink.  One speaks of the  "Duplex Gap".  In addition, this frequency range can be used for event technology, since the frequency range around 800 MHz was already common for radio microphones before the introduction of LTE.  In areas where LTE is available everywhere, radio microphones must be able to use the duplex gap in the future. 

The difference in importance of the frequency ranges from the operator's perspective is clearly illustrated by the results of the 2010 frequency auction:
*The 60 MHz around 800 MHz generated almost EUR 3.6 billion  $\text{(60 €/Hz)}$,  the 190 MHz around 2.6 GHz only EUR 344 million  $\text{ (1.80 €/Hz)}$.
*For comparison:   The UMTS auction in 2000 resulted in the astronomical sum of 50 billion euros for 60 MHz   ⇒   $\text{833 €/Hz}$. 

== 3GPP - Third Generation Partnership Project ==
 
On the last pages the  "Third Generation Partnership Project"  $($or short $\rm 3GPP)$  has been mentioned several times.  Here is a short overview of the self-image of this group, its structure and its activities.  The information is taken directly from the  [http://www.3gpp.org/about-3gpp 3GPP Website].

3GPP is a group of various international standardization organizations that have joined forces for the purpose of standardizing mobile radio systems.  It was founded in December 1998 by five partners:
*ARIB   ('''A'''ssociation of '''R'''adio '''I'''ndustries and '''B'''usinesses, Japan) 
*ETSI   ('''E'''uropean '''T'''elecommunication '''S'''tandards '''I'''nstitute) 
*ATIS   ('''A'''lliance for '''T'''elecommunications '''I'''ndustry '''S'''olutions, USA) 
*TTA   ('''T'''elecommunications '''T'''echnology '''A'''ssociation, Korea) 
*TTC   ('''T'''elecommunications '''T'''echnology '''C'''ommittee, Japan) 

The 3GPP develops, accepts and maintains a globally applicable standard in mobile communications.  The conferences, which are held regularly and frequently, are the most important bodies in the updating of the standardization of the technical specifications of LTE.
*Change requests go through a fixed standardization process with three stages, which enables high quality and good structuring of 3GPP's work.
*When a release has reached the last stage and is completed, it is passed on to the market by the telecommunications companies united in the partner organizations. 

In  [Gut10]<ref name ='Gut10'>Gutt, E.:  LTE – eine neue Dimension mobiler Breitbandnutzung.  [http://www.ltemobile.de/uploads/media/LTE_Einfuehrung_V1.pdf PDF document] on the Internet, 2010.</ref>  the following assessment can be found: "The goal of 3GPP standardization is the creation of technical specifications  $\rm (TS)$  that describe all technical details of a mobile communications technology in detail.  The specifications for LTE are extremely comprehensive.  The level of detail is chosen so high that mobile devices from different manufacturers can function without problems in all networks".

== Exercise to chapter==

[[Aufgaben:Exercise 4.1: General Questions about LTE]]

==References==

<references/>

{{Display}}

Linear and Time Invariant Systems/Classification of the Distortions

2022-02-17T11:38:55Z

Bene: Text replacement - "List of sources" to "References"

{{Header|
Untermenü=Signal Distortion and Equalization
|Vorherige Seite=Some_Low-Pass_Functions_in_Systems_Theory
|Nächste Seite=Nonlinear_Distortions
}}

== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 
{{BlaueBox|TEXT=
$\text{Definition:}$ 
In general,  $\text{distortion}$  is understood to be undesirable deterministic changes in a message signal caused by a transmission system. }}

In addition to stochastic interferences  (noise, crosstalk, etc.), such deterministic distortions are a critical limitation on the transmission quality and rate for many communication systems.

This chapter presents these distortions in a summarising way, in particular:

*The quantitative detection of such signal falsifications via the distortion power,
*the distinguishing features between nonlinear and linear distortions,
*the meaning and computation of the distortion factor in nonlinear systems,  and
*the effects of linear damping and phase distortions.

==Prerequisites for the second main chapter==
 
[[File:P_ID873__LZI_T_2_1_S1_neu.png|frame| Description of a linear system|class=fit]]

In the following, we consider a system whose input is the signal  $x(t)$  with the corresponding spectrum  $X(f)$.  The output signal is denoted by  $y(t)$  and its spectrum by  $Y(f).$

The block labelled  "system"  can be a part of an electrical circuit or a complete communicarion system consisting of transmitter, channel and receiver.
 
For the whole main chapter  "Signal Distortions and Equalization"  the following shall apply:
*The system be '''time-invariant'''.  If the input signal  $x(t)$  results in the output signal  $y(t)$, then a later input signal of the same form – in particular  $x(t - t_0)$  – will result in the signal  $y(t - t_0)$.
*In the following,  '''no noise'''  is considered, which is always present in real systems.  For the description of these phenomena we refer to the book  [[Theory_of_Stochastic_Signals|Theory of Stochastic Signals]].
*About the system  '''no detailed knowledge'''  is assumed.  In the following, all system properties are derived from the signals  $x(t)$  and  $y(t)$  or their spectra alone.
*In particular, no specifications are made for the time being with regard to  '''linearity'''.  The "system" can be linear (prerequisite for the application of the superposition principle) or non-linear.
*Not all system properties are discernible from a single test signal  $x(t)$  and its response  $y(t)$ . Therefore,  '''sufficiently many test signals'''  must be used for evaluation.

In the following, we will classify communication systems in more detail in this respect.

==Ideal and distortion-free system==
 
{{BlaueBox|TEXT=
$\text{Definition:}$ 
One deals with an  '''ideal system'''  if the output signal  $y(t)$  is identical with the input signal  $x(t)$:
:$$y(t) \equiv x(t).$$}}

*It should be noted that such an ideal system does not exist in reality even if statistical disturbances and noise processes (that always exist but are not considered in this book) are disregarded. 
*Every transmission medium exhibits losses  (damping)  and transit times.  Even if these physical phenomena are very small, they are never zero.  Therefore it is necessary to introduce a somewhat less strict quality characteristic.

{{BlaueBox|TEXT=
$\text{Definition:}$ 
A  '''distortion-free system'''  exists if the following condition is fulfilled:
:$$y(t) = \alpha \cdot x(t - \tau).$$
*Here,  $α$  describes the damping factor and  $τ$  the transit time.
*If this condition is not met, the system is said to be  ''' distortive'''.}}

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The following diagram shows the input signal  $x(t)$  and the output signal  $y(t)$  of a nonideal but distortion-free system.  The system parameters are  $α = 0.8$  and  $τ = 0.25 \ \rm ms$.

[[File:P_ID874__LZI_T_2_1_S2_neu.png|frame|Exemplary signals of a distortion-free system|class=fit]]

Note:
*The damping factor  $α$  can be completely reversed by a receiver-side gain of  $1/α = 1.25$, but it must be taken into account that this also increases any noise.
*However, the transit time  $τ$  cannot be compensated due to  [[Signal_Representation/Signal_classification#Causal_and_Non-Causal_Signals|causality reasons]].  It now depends on the application whether such a transit time is subjectively perceived as disturbing or not. }}

*For example, even with a transit time of one second the  (unidirectional)  TV broadcast of an event is still described as "live". 
*In contrast to this, transit times of  $\text{300 ms}$  are already perceived as very disturbing in bidirectional communication – e.g. a telephone call.  You either wait for the other person to react or both participants interrupt each other.

==Quantitative measure for the signal distortions==
 
[[File:P_ID875__LZI_T_2_1_S3_neu.png|right|frame|Input and output of a distortive system and error signal (below)|class=fit]]

We now consider a distortive system on the basis of the input and output signal. 
*In doing so, we assume that apart from the signal distortions there is no additional damping factor   $α$  which is constant for all frequencies and no additional transit time  $τ$  that is constant for all frequencies.
*These conditions are fulfilled for the signal sections sketched on the right.

*In addition to the signals  $x(t)$  and  $y(t)$ , the difference signal is shown in the diagram:
:$$\varepsilon(t) = y(t) - x(t).$$
As a quantitative measure of the strength of distortions, for example, the  '''squared mean of this difference signal''' is applicable:
:$$\overline{\varepsilon^2(t)} = \frac{1}{T_{\rm M}} \cdot \int_{ 0 }^{ T_{\rm M}} {\varepsilon^2(t) }\hspace{0.1cm}{\rm d}t\hspace{0.4cm} \left( = P_{\rm V} \right).$$

The following should be noted about this equation:
*The measuring time  $T_{\rm M}$  must be chosen sufficiently large.  Actually, this equation should be formulated as a limit process.
*The squared mean mentioned above is called  "mean squared error"  $\rm (MSE)$  or   '''distortion power'''  $P_{\rm V}$  $($because of  "distortion"   ⇒   German:  "Verzerrung"   ⇒   $\rm V)$.
*If  $x(t)$  and  $y(t)$  are voltage signals, then  $P_{\rm V}$  has the unit of  ${\rm V}^2$, meaning the power is related to the resistance  $R = 1 \ Ω$  according to the above definition.

{{BlaueBox|TEXT=
$\text{Definition:}$  Making use of the power  $P_x$   $($based on  $R = 1 \ Ω)$  of the input signal  $x(t)$  the   $\text{signal–to–distortion (power) ratio}$  can be given as:
:$$\rho_{\rm V} = \frac{ P_{x} }{P_{\rm V} } \hspace{0.3cm} \Rightarrow \hspace{0.3cm} 10 \cdot \lg \hspace{0.1cm}\rho_{\rm V} =
10 \cdot \lg \hspace{0.1cm}\frac{ P_{x} }{P_{\rm V} }\hspace{0.3cm} \left( {\rm in \hspace{0.15cm} dB} \right).$$

For the signals shown in the diagram above  $P_x = 4 \ {\rm V}^2$,  $P_{\rm V} = 0.04 \ {\rm V}^2$  and thus  $10 \cdot {\rm lg} \ ρ_{\rm V} = 20 \ \rm dB$ hold.}}

We reference the interactive applet  [[Applets:Linear_Distortions_of_Periodic_Signals|Linear Distortions of Periodic Signals]].

==Elimination of damping factor and transit time==
 
The equations given on the last page do not result in applicable statements if the system is additionanlly affected by an attenuation factor  $α$  and/or a transit time  $τ$.  The diagram shows the attenuated, delayed and distorted signal

[[File:P_ID876__LZI_T_2_1_S4_neu.png|right|frame|Elimination of damping factor $α$ 
 and transit time  $τ$|class=fit]]

:$$y(t) = \alpha \cdot x(t - \tau) + \varepsilon_1(t).$$
The term  $ε_1(t)$  summarises all distortions.  It can be seen from the green area that the error signal  $ε_1(t)$  is relatively small.

In contrast to this, if the attenuation factor  $α$  and the transit time  $τ$  are unknwon, the following should be noted:
*The error signal  $ε_2(t) = y(t) - x(t)$  determined in this way is relatively large despite small distortions  $ε_1(t)$.
*Here, instead of the distortion power the distortion energy must be considered because  $x(t)$  and  $y(t)$  are energy-limited signals.
*The distortion energy is obtained by varying the unknown quantities  $α$  and  $τ$  and thus finding the minimum of the mean squared error:
:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}
{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$
*The energy of the damped and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion (power) ratio the following is applicable:
:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm D}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$
*The first of these two equations applies to time-limited and thus energy-limited signals, the second one to time-unlimited and thus power-limited signals according to the page  [[Signal_Representation/Signal_classification#Energy.E2.80.93Limited_and_Power.E2.80.93Limited_Signals|Energy-limited and power-limited signals]]  in the book "Signal Representation".

==Linear and non-linear distortions==
 
A distinction is made between linear and nonlinear distortions:

If the system is linear and time-invariant  $(\rm LTI)$, then it is fully characterised by its  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|frequency response]]  $H(f)$ and the following can be stated:
*According to the  $H(f)$  definition the following holds for the output spectrum:   $Y(f)=X(f) · H(f)$.   As a consequence according to the calculation rules of multiplication,  $Y(f)$  cannot contain any frequency components that are not already contained in  $X(f)$ .
*The inverse implies:   The output signal  $y(t)$  can include any frequency  $f_0$  already contained in the input  $x(t)$ . The prerequisite is therefore that  $X(f_0) ≠ 0$  holds.
*For an LTI system the absolute bandwidth  $B_y$  of the output signal is never greater than the bandwidth  $B_x$  of the input signal:   $B_y \le B_x .$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
The differences between linear and non-linear distortions are illustrated by the following diagram:

[[File:EN_LZI_T_2_1_S5_neu.png|frame| Linear and nonlinear distortions|class=fit]]
In the upper diagram  $B_y = B_x$  holds. There are  '''linear distortions'''  because in this frequency band  $X(f)$  and  $Y(f)$  differ.

A band limitation  $(B_y < B_x)$  is a special form of linear distortion, which will be discussed in the  [[Linear_and_Time_Invariant_Systems/Lineare_Verzerrungen|chapter after next]].

The lower diagram shows an example of  '''non-linear distortions'''  $(B_y > B_x)$.  For such a system no frequency response  $H(f)$  can be given.

Descriptive quantities applicable for non-linear systems will be explained in the next chapter  [[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen|Non-linear Distortions]] .
 
In most real transmission channels both linear and non-linear distortions occur.  However, for a whole range of problems the precise separation of the two types of distortions is essential.  In  [Kam04]<ref>Kammeyer, K.D.:  Nachrichtenübertragung.  Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>  a corresponding substitute model is shown. }}

We refer here to the  (German language)  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)|Linear and non-Linear distortions]].

==Exercises for the chapter==

[[Aufgaben:Exercise_2.1:_Linear%3F_Or_Non-Linear%3F| Exercise 2.1: Linear? Or Non-Linear?]]

[[Aufgaben:Exercise_2.1Z:_Distortion_and_Equalisation|Exercise 2.1Z: Distortion and Equalisation]]

[[Aufgaben:Exercise_2.2:_Distortion_Power|Exercise 2.2: Distortion Power]]

[[Aufgaben:Exercise_2.2Z:_Distortion_Power_again|Exercise 2.2Z: Distortion Power again]]

{{Display}}

==References==
<references/>

Signal Representation/Principles of Communication

2022-02-17T11:38:55Z

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{{FirstPage}}
{{Header|
Untermenü=Basic Terms of Communications Engineering
|Vorherige Seite=Signal Representation
|Nächste Seite=Signal Classification
}}

== OVERVIEW OF THE FIRST MAIN CHAPTER ==
 
This first chapter serves as an introduction to the whole topic, which is covered by the nine books of the series  $\rm LNTwww$.

This chapter describes in detail:

*the tasks and the basic structure of a communication system,
*the main functional units   (source, transmitter, channel, receiver and sink)  of such a system, and finally
*a classification of the signals occurring in a communication system according to several evaluation criteria:  deterministic or stochastic, energy or power limited, continuous or discrete time, continuous or discrete value, analog or digital.

At the end of the chapter follows a short summary about  "Calculating with complex numbers".

==Message - Information - Signal==
 
One distinguishes basically between the terms  "message"  and  "information", which are often used synonymously nowadays.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
An e–mail from Mr. Maier to Mrs. Miller is always a  "message".  For Mrs. Miller, however, receiving this e–mail only means an information gain if she learns something new.  The  "information"  transmitted by a message therefore depends to a great extent on the knowledge level of the recipient. In practice, the information contained in a message is rather small, especially in the field of telephony.}}

The transmission of messages   ⇒   '''communications'''  and their storage always requires an energetic or material carrier called  "signal".  Physically the representation of a message is done by signals, which can be of very different nature.

Possible appearances are:
*Electrical signals  (for example current and voltage curve),
*electromagnetic waves  (for example in radio transmission),
*progression of pressure, temperature or other physical quantities,
*acoustic signals  (for example, output signal of a loudspeaker),
*optical signals  (for example output signal of a laser).

{{BlaueBox|TEXT=
$\text{Please note:}$ 
*The signals used for  $\text{communication}$  are usually time functions.  This means that (at least) one of the signal parameters is dependent on the time parameter  $t$.  Such signal parameters are for example for a signal tone the amplitude ("volume") and the frequency ("pitch").

*For storing messages   ⇒   $\text{data storage}$  the time functions are often mapped to spatial functions of suitable physical quantities such as magnetization (magnetic band) or degree of blackening (film).}}

The set of all message signals can be cataloged by different criteria, as described in the chapter  [[Signal_Representation/Signal_classification|Signal classification]]. 

==Block diagram of a communications system==
 
In the following diagram a communication system is shown schematically.

[[File:EN_Sig_T_1_1_S2_neu.png|center|frame| General block diagram of a message transmission system   ⇒   communication system]]

The individual system components have the following tasks:
*The  '''message source'''  returns the source signal  $q(t)$, which shall be transmitted over the channel to the spatially distant sink.  The source can be for example a computer, a radio station or a telephone participant.
*In most cases the source signal  $q(t)$  itself is unsuitable for transmission and must first be converted into the signal  $s(t)$  in a suitable manner.  This process is called "modulation" and is performed by the  '''transmitter'''.  In the following the signal   $s(t)$  is called  "transmitter signal".
*During transmission over the  '''channel'''  the signal  $s(t)$  is changed in its form.  At the same time, interference and noise signals are added.  The signal at the channel output and simultaneously at the receiver input is called the  "reception Signal"  $r(t)$.
*The  '''receiver'''  must undo the conversion made by the transmitter.  If, for example, the low-frequency source signal  $q(t)$  was converted to the higher-frequency transmitter signal  $s(t)$  the receiver must also contain a demodulator and undo this conversion.
*The last block in the model above is the  '''message sink'''. The sink signal  $v(t)$  is like the source signal  $q(t)$  again low frequency.  In the ideal case, however, which in practice  (due to the unavoidable noise)  can never be reached exactly, should apply for all times  $v(t) = q(t)$.

==Message source==
 
As examples for message sources and for the source signal  $q(t)$  can be mentioned:
*Audio signals, e.g. speech or music,
*video signals, e.g. an analog television signal or an MPEG encoded streaming video,
*data signals, e.g. the data stream of a USB interface or an email on the Internet,
*measure signals, e.g. for control or regulation in a production process.

In general, a distinction is made between analog and digital sources.  The description presented in this book apply equally to analog and digital signals.  The basic differences between analog and digital signals are discussed in the chapter  [[Signal_Representation/Signal_classification|Signal classification]],  and clarified with examples throughout the (German language) learning video
:[[Analoge und digitale Signale (Lernvideo)|Analoge und digitale Signale]]   ⇒   "Analog and Digital Signals".

[[File:EN_Sig_T_1_1_S3.png|right|frame|Frequency-time representation of a speech signal]]
{{GraueBox|TEXT=
$\text{Example 2:}$ 
On the right you see the frequency-time representation of a speech signal.
*You can see the different frequency components in the kilohertz range at different times.
*By the way:   This is a male speaker.

We thank  Markus Kaindl<ref>Kaindl, M.: Kanalcodierung für Sprache und Daten in GSM-Systemen. Dissertation. Lehrstuhl für Nachrichtentechnik, TU München. VDI Fortschritt-Berichte, Reihe 10, Nr. 764, 2005.</ref>, LNT/TUM, for providing the graphic.}}

==Tasks of the transmitter==
 
The essential task of the transmitter is to convert the source signal  $q(t)$  into a transmitted signal  $s(t)$  in such a way that it is adapted as well as possible to the transmission channel while maintaining the specified performance characteristics.  For this purpose each transmitter contains corresponding  '''functional units'''  such as

*Transducer - for example a microphone for converting the physical quantity  "pressure"  (acoustic wave)  into an electrical signal,
*Signal converter - for example from  "analog"  to  "digital"  using the components  "sampling",  "quantization"  and  "PCM coding",
*Encoder for removing redundancy to data compression   (source coding)  or for systematically adding redundancy, which can be used at the receiver for  "error detection"   and/or  "error correction"   (channel coding),
*Modulator for adaptation to the transmission channel - for example a frequency conversion by means of amplitude, phase or frequency modulation or the corresponding digital methods ASK, PSK or FSK.

Depending on the application, the  '''performance characteristics''' mentioned above mean that very specific properties are required for signal transmission.

Such features are for example:

*"Power limitation"  - due to the discussions on the topic  "electrosmog"  highly topical,
*"Bandwidth efficiency"  - the UMTS auction in 2000 has shown what amounts are involved,
*"Distance"  or  "range"  - unfavorable values increase the infrastructure costs,
*"Transmission quality"  - for example a high signal-to-noise ratio or a low error rate.

==Transmission channel==
 
The  '''transmission medium'''  with its physical properties plays an essential role in the feasibility of certain transmission properties.

Examples of transmission media are:
*Electrical cables, e.g. copper wire, twisted pair,
*Coaxial cable, e.g. antenna line or cable network,
*Fiber optic cables, e.g. multimode and single mode fiber optic,
*Radio channels, e.g. broadcasting, mobile and satellite radio.

{{BlaueBox|TEXT=
$\text{Please note:}$  These transmission media are not ideal in practice and impair the transmission.

This means:   The received signal  $r(t)$  is different from the transmitted signal  $s(t)$,  possibly due to
*the channel attenuation,
*delays on the channel,
*linear and non-linear distortions.

In addition, the channel transmission properties can change significantly over time  $($'''time variance''',  example:  mobile radio$)$.}}

In addition, the  '''interfering signals'''  that occur during signal transmission must always be taken into account.  The following are examples of this:
*Noise signals - e.g. resistance and semiconductor noise,
*Pulse interference - e.g. power lines, spark interference and discharges,
*Adjacent channel interference  (crosstalk of other users, interference, cross modulation).

You will find basic information about modeling the transmission channel in general and the simple AWGN channel in the the (German language) learning video
:[[Eigenschaften_des_Übertragungskanals_(Lernvideo)|Eigenschaften des Übertragungskanals]]   ⇒   "Properties of the Transmission Channel".

==Receiver - Message sink==
 
As examples of the message sinks we can mention
*eye and ear of man,
*video recorder and call recorder,
*a smartphone that downloads a file from the Internet, or
*a control system that processes received measurement signals.

To ensure that at least in the ideal case the sink signal  $v(t)$  could coincide with the source signal  $q(t)$  – in practice however never attainable – all measures taken on the transmission side must be reversed by the receiver.

Corresponding  '''functional units'''  of a receiver are:
*Transducer -   e.g. a loudspeaker to convert an electric signal into an acoustic signal  $($counterpart of the microphone$)$,
*Signal reconstruction - e.g. the reconstruction of the analog signal from the digital samples  $($D/A converter   ⇒   counterpart to A/D converter$)$,
*Decoding - for example with the possibility of error detection and error correction  $($counterpart of the channel coder$)$.

Another important task of the receiver is to eliminate as much as possible the signal distortions and interference that occur during transmission.  The realization of such system components for transmitters and receivers is done by different electrical networks and assemblies.

Here, too, some functional units can be named as examples:
*Amplifier, filter and equalizer,
*Oscillators and nonlinear components for  (de-)modulation and synchronization,
*Digital signal processing components and signal processors.

==Signal distortions==
 
It has already been mentioned that ideally  $v(t) = q(t)$  should be valid.  However, as with any real transmission channel  $r(t) \neq s(t)$, the sink signal  $v(t)$  will be different from the source signal  $q(t)$ .  Here are some examples:

{{BlaueBox|TEXT=
$\text{Definition:}$  One speaks of  $\text{noise}$, if for the sink signal applies:
:: <math> v(t)=q(t)+n(t).</math>
*The additive noise component  $n(t)$  is always of stochastic nature and usually has no relation to the source signal  $q(t)$. 
*Such a noise term is inevitable for every transmission.}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The transmission is  $\text{distortion–free}$  if the sink signal is as follows:
:: <math>v(t)=a \cdot q(t-\tau)+n(t).</math>
In this case the sink signal differs from the source signal - except for the noise component  $n(t)$  only by
* the  attenuation factor  $\alpha$  (same for all frequencies),  and
*the  delay time  $\tau$  (also the same for all frequencies).}}

{{BlaueBox|TEXT=
$\text{Definition:}$  If the equation  $v(t)=a\cdot q(t-\tau)+n(t)$  is not fulfilled, then there are  $\text{distortions}$. 

As described in the book  [[Linear and Time Invariant Systems]]  one distinguishes between
*  [[Linear_and_Time_Invariant_Systems/Lineare_Verzerrungen|linear distortions]]  and
*  [[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen|nonlinear distortions]].

In this context we refer to the (German language) learning video 
:[[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)|Lineare und nichtlineare Verzerrungen]]  ⇒   "Linear and nonlinear distortions".
}}

Here again in slightly different words:
*The attenuation factor  $\alpha$  only causes the signal  $v(t)$  to be slightly  "quieter"  than  $q(t)$;  but both signals have the same shape.
*The delay  $\tau$  causes the signal  $v(t)$  to arrive at the receiver later than  $q(t)$  was sent.

Both effects are not particularly disturbing for a  unidirectional transmission:             For example, one can still speak of a live transmission if the television picture arrives delayed by (a fraction of) seconds.

With a  bidirectional transmission however, a long runtime can lead to problems:             During a telephone conversation, the two interlocutors then interrupt each other.

[[File:P_ID587_Sig_T_1_1_S7a.jpg|right|frame|Color template to illustrate "distortion"  and  "noise"]]
{{GraueBox|TEXT=
$\text{Example 3:}$ 
The terms used here shall now be clarified by an image signal.

On the right you see as original image a color template with  $291 × 218$  pixels and  $24$  bit color depth. Of the possible  $2^{24} = 16\hspace{0.08cm} 777\hspace{0.08cm} 216$  only a few colors are used here.

*In the lower left image, the signal is superposed with additive noise  $n(t)$  which is perceived as "snow".
*The lower right image shows the influence of (non–linear) distortions, which lead to a distortion of both brightness values and color information at the selected setting of the CCD camera.
*In the marked field of the gray staircase, the brightness corresponds approximately to the original image (above).
*On the other hand, other fields appear as too light or too dark or filled with missing colors.
*Noise effects play no role in the right image as opposed to the left image.

[[File:P_ID588_Sig_T_1_1_S7b.jpg|center|frame|Effects of  "noise"  and  "distortion"  on an image signal]] }}

==Exercises for the chapter==
 
[[Aufgaben:Exercise 1.1: Music Signals|Exercise 1.1: Music Signals]]

[[Aufgaben:Exercise_1.1Z:_ISDN-Connection|Exercise 1.1Z: ISDN Connection]]

== References ==

{{Display}}

Signal Representation/Fast Fourier Transform (FFT)

2022-02-17T11:38:55Z

Bene: Text replacement - "List of sources" to "References"

{{LastPage}}
{{Header
|Untermenü=Time and Frequency-Discrete Signal Representation
|Vorherige Seite=Spectrum Analysis
|Nächste Seite=
}}

==Complexity of DFT and IDFT==
 
A disadvantage of the direct calculation of the  (generally complex) DFT sequences

:$$\langle \hspace{0.1cm}D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm}d(\nu)\hspace{0.1cm} \rangle$$

according to the equations given in chapter  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]  $\rm (DFT)$  is the large computational cost.

We consider as an example the DFT, i.e. the calculation of the  $D(\mu)$  coefficients from the  $d(\nu)$  coefficients:

:$$N \cdot D(\mu) = \sum_{\nu = 0 }^{N-1}
d(\nu) \cdot {w}^{\hspace{0.03cm}\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu}
=
d(0) \cdot w^{\hspace{0.03cm}0} + d(1) \cdot w^{\hspace{0.03cm}\mu}+ d(2) \cdot w^{\hspace{0.03cm}2\mu}+\hspace{0.05cm}\text{ ...} \hspace{0.05cm}+ d(N-1) \cdot w^{\hspace{0.03cm}(N-1)\cdot \mu}.$$

The computational effort required for this is to be estimated, assuming that the powers of the complex rotation factor  $w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi/N}$  already exist in real and imaginary part form in a lookup table.  To calculate a single coefficient, one then needs  $N-1$  complex multiplications and as many complex additions, observing:
*Each complex addition requires two real additions:
:$$(R_1 + {\rm j} \cdot I_1) + (R_2 + {\rm j} \cdot I_2) = (R_1 +
R_2) + {\rm j} \cdot (I_1 + I_2)\hspace{0.05cm}.$$
*Each complex multiplication requires four real multiplications and two real additions  (a subtraction is treated as an addition):
:$$(R_1 + {\rm j} \cdot I_1) (R_2 + {\rm j} \cdot I_2) = (R_1 \cdot
R_2 - I_1 \cdot I_2) + {\rm j} \cdot (R_1 \cdot I_2 + R_2 \cdot
I_1)\hspace{0.05cm}.$$
*Thus, the following number of real multiplications and the number of real additions are required to calculate all $N$ coefficients in total:
:$$M = 4 \cdot N \cdot (N-1),$$
:$$A = 2 \cdot N \cdot
(N-1)+2 \cdot N \cdot (N-1)=M \hspace{0.05cm}.$$
*In today's computers, multiplications and additions/subtractions need about the same computing time.  It is sufficient to consider the total number  $\mathcal{O} = M + A$  of all operations:
:$$\mathcal{O} = 8 \cdot N \cdot (N-1) \approx 8 \cdot N^2\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*For a  Discrete Fourier Transform  (DFT) with  $N = 1000$  one already needs almost eight million arithmetic operations.  The same applies to an IDFT.
*With  $N =16$  still  $1920$  computational operations are required.}}

If the parameter  $N$  is a power to the base  $2$, more computationally efficient algorithms can be applied.  The multitude of such methods known from the literature are summarised under the collective term  $\text{Fast Fourier Transform}$  - abbreviated  $\text{FFT}$.  All these methods are based on the  "superposition theorem"  of the DFT.

==Superposition theorem of the DFT==
 
The graph illustrates the so-called  "superposition theorem"  of the DFT using the example of $N = 16$.  Shown here is the transition from the time domain to the spectral domain, i.e. the calculation of the spectral domain coefficients from the time domain coefficients:   $\langle \hspace{0.1cm}D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} d(\nu) \hspace{0.1cm}\rangle.$

[[File:EN_Sig_T_5_5_S2.png|right|frame|Superposition theorem of the DFT]]

The algorithm described thereby is characterised by the following steps:
*The sequence  $\langle \hspace{0.1cm}d(\nu)\hspace{0.1cm}\rangle$  of length  $N$  is divided into two subsequences$\langle \hspace{0.1cm} d_1(\nu)\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm} d_2(\nu)\hspace{0.1cm}\rangle$  each of half length  (highlighted in yellow and green respectively in the garafic).  With  $0 \le \nu \lt N/2$  one thus obtains the sequence elements
:$$d_1(\nu) = d(2\nu), $$
:$$d_2(\nu) = d(2\nu+1)
\hspace{0.05cm}.$$
*The initial sequences  $\langle \hspace{0.1cm}D_1(\mu )\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm}D_2(\mu )\hspace{0.1cm}\rangle$  of the two sub-blocks result from this each by its own DFT,  but now only with half length  $N/2 = 8$:
:$$\langle \hspace{0.1cm}D_1(\mu) \hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N/2)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm}d_1(\nu) \hspace{0.1cm}\rangle , $$
:$$ \langle \hspace{0.1cm}D_2(\mu)\hspace{0.1cm} \rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N/2)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm}d_2(\nu) \hspace{0.1cm}\rangle \hspace{0.05cm}.$$
*The initial values  $\langle \hspace{0.1cm} D_2(\mu )\hspace{0.1cm}\rangle$  of the lower (green) DFT $($with  $0 \le \mu \lt N/2)$  are then changed in the block outlined in red by complex rotation factors with respect to phase:
:$$D_2(\mu) \hspace{0.3cm} \Rightarrow \hspace{0.3cm}D_2(\mu) \cdot w^{\hspace{0.04cm}\mu}, \hspace{0.2cm}{\rm with}\hspace{0.1cm}w =
{\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi/N} \hspace{0.05cm}.$$
*Each single  '''Butterfly'''  in the blue bordered block (in the middle of the graph) yields two elements of the searched sequence by addition or subtraction.  With  $0 \le \mu \lt N/2$  applies:
:$$D(\mu) = {1}/{2}\cdot \big[D_1(\mu) + D_2(\mu) \cdot w^{\hspace{0.04cm}\mu}\big],$$
:$$D(\mu +{N}/{2}) = {1}/{2}\cdot \big[D_1(\mu) - D_2(\mu) \cdot w^{\hspace{0.04cm}\mu}\big]\hspace{0.05cm}.$$

'''This first application of the superposition theorem thus roughly halves the computational effort.'''

{{GraueBox|TEXT=
$\text{Example 1:}$ 
Let the DFT coefficients  $d(\nu)$  for the description of the time course be  "triangular"  according to  '''line 2'''  of the following table.  Note here the periodic continuation of the DFT, so that the linear increase for  $t \lt 0$  is given by the coefficients  $d(8), \hspace{0.05cm}\text{ ...} \hspace{0.05cm}, d(15)$.

Applying the DFT algorithm with  $N = 16$  one obtains the spectral coefficients  $D(\mu )$  given in  '''line 3'''  which would be equal  $D(\mu ) = 4 \cdot \text{si}^2(\pi \cdot \mu/2)$  if the aliasing error were neglected.  We can see that the aliasing error only affects the odd coefficients (shaded boxes). For example, $D(1) = 16/ \pi^2 \approx 1.621\neq 1.642$  should be.

[[File:Sig_T_5_5_S2b_Version2.png|right|frame|Result table for  $\text{Example 1}$  for the superposition theorem of the DFT]]

If we split the total sequence  $\langle \hspace{0.1cm}d(\nu)\hspace{0.1cm}\rangle$  into two subsequences such that the first subsequence  $\langle \hspace{0.1cm}{d_1}'(\nu)\hspace{0.1cm}\rangle$   ⇒   yellow marked has only even coefficients  $(\nu = 0, 2, \hspace{0.03cm}\text{ ...} \hspace{0.1cm}, N–2)$  and the second subsequence  $\langle \hspace{0.1cm}{d_2}'(\nu)\hspace{0.1cm}\rangle$  ⇒   green marked contains only odd coefficients  $(\nu = 1, 3, \hspace{0.03cm}\text{ ...} \hspace{0.1cm} , N-1)$  and all others are set to zero.  The corresponding sequences in the spectral domain are obtained:

:$$ \langle \hspace{0.1cm}{D_1}'(\mu)\hspace{0.1cm} \rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} {d_1}'(\nu) \hspace{0.1cm}\rangle , $$
:$$ \langle \hspace{0.1cm}{D_2}'(\mu) \hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle\hspace{0.1cm} {d_2}'(\nu) \rangle \hspace{0.1cm}\hspace{0.05cm}.$$

In the yellow or green lines  $4\hspace{0.05cm}\text{ ...} \hspace{0.05cm}7$  you can see:
*Because of  $d(\nu) = {d_1}'(\nu) + {d_2}'(\nu)$  also holds 
:$$D(\mu ) = {D_1}'(\mu ) + {D_2}'(\mu ).$$
:This can be justified, for example, with the  [[Signal_Representation/Fourier_Transform_Theorems#Multiplication_with_a_factor_-_Addition_Theorem|Addition Theorem of Linear Systems]].
*The period of the sequence  $\langle \hspace{0.1cm}{D_1}'(\mu )\hspace{0.1cm}\rangle$  due to the zeroing of every second time coefficient is now  $N/2$  unlike the period  $N$  of the sequence  $\langle \hspace{0.1cm} D(\mu )\hspace{0.1cm}\rangle$:
:$${D_1}'(\mu + {N}/{2}) ={D_1}'(\mu)\hspace{0.05cm}.$$
* The sequence  $\langle \hspace{0.1cm} {D_2}'(\mu )\hspace{0.1cm}\rangle$  additionally contains a phase factor  (shift by one sample)  which causes a sign change of two coefficients separated by  $N/2$:
:$${D_2}'(\mu + {N}/{2}) = - {D_2}'(\mu)\hspace{0.05cm}.$$
*The calculation of  $\langle \hspace{0.1cm}{D_1}'(\mu )\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm} {D_2}'(\mu )\hspace{0.1cm}\rangle$  is, however, in each case as time-consuming as the determination of  $\langle \hspace{0.1cm}D(\mu )\hspace{0.1cm}\rangle$, since  $\langle \hspace{0.1cm}{d_1}'(\nu)\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm}{d_2}'(\nu)\hspace{0.1cm}\rangle$  also consist of  $N$  elements, even if half of them are zeros.}}

{{GraueBox|TEXT=
$\text{Example 2:}$ 
To continue the first example, the previous table is now extended by the rows  $8$  to  $12$ .

[[File:Sig_T_5_5_S2c_Version2.png|right|frame|Result table for  $\text{Example 2}$  for the superposition theorem of the DFT]]

Omitting the coefficients  ${d_1}'(\nu) = 0$  with odd indices and  ${d_2}'(\nu) = 0$  with even indices, we arrive at the subsequences  $\langle \hspace{0.1cm}d_1(\nu)\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm}d_2(\nu)\hspace{0.1cm}\rangle$  corresponding to lines  $9$  and  $11$ . You can see:
*The time sequences  $\langle \hspace{0.1cm}{d_1}(\nu )\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm}{d_2}(\nu )\hspace{0.1cm}\rangle$  exhibit as well as the corresponding spectral sequences  $\langle \hspace{0.1cm}{D_1}(\mu )\hspace{0.1cm}\rangle$  and  $\langle \hspace{0.1cm}{D_2}(\mu )\hspace{0.1cm}\rangle$  only have the dimension $(N/2)$.
*A comparison of the lines  $5$,  $7$,  $10$  and  $12$  shows the following relationship for  $0 \le \mu \lt N/2$ :
:$${D_1}'(\mu) = {1}/{2}\cdot {D_1}(\mu)\hspace{0.05cm},$$
:$$ {D_2}'(\mu) = {1}/{2}\cdot {D_2}(\mu)\cdot w^{\hspace{0.04cm}\mu}\hspace{0.05cm}.$$
*Correspondingly, for  $N/2 \le \mu \lt N$:
:$${D_1}'(\mu) = {1}/{2}\cdot {D_1}(\mu - {N}/{2})\hspace{0.05cm},$$
:$$ {D_2}'(\mu) = {1}/{2}\cdot {D_2}(\mu {-} {N}/{2})\cdot w^{\hspace{0.04cm}\mu}$$
:$$ \Rightarrow \hspace{0.3cm}{D_2}'(\mu) = { - } {1}/{2}\cdot {D_2}(\mu-N/2)\cdot w^{\hspace{0.04cm}\mu {-} N/2}\hspace{0.05cm}.$$

*For example, with  $N = 16$   ⇒   $w = {\rm e}^{ - {\rm j}\hspace{0.04cm} \cdot \hspace{0.04cm}\pi/8}$  for the indices  $\mu = 1$  respectively  $\mu = 9$: 
:$${D_1}'(1) = {1.708}/{2} = 0.854,\hspace{0.8cm}
{D_2}'(1) ={1}/{2}\cdot (1.456 + {\rm j} 0.603) \cdot {\rm e}^{ - {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}
\pi/8} = 0.788$$
:$$\Rightarrow D(1) = {D_1}'(1)+ {D_2}'(1)= 1.642 \hspace{0.05cm}.$$
:$${D_9}'(1) = {1.708}/{2} = 0.854,\hspace{0.8cm}
{D_2}'(9) = - {1}/{2}\cdot (1.456 + {\rm j} 0.603) \cdot {\rm e}^{ - {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}
\pi/8} = - 0.788$$
:$$\Rightarrow D(9) = {D_1}'(9)+ {D_2}'(9)= 0.066 \hspace{0.05cm}.$$}}

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*This first application of the superposition theorem almost halves the computational effort.
*Instead of  $\mathcal{O}= 1920$  one only needs  $\mathcal{O} = 2 - 448 + 8 \cdot (4+2) + 16 \cdot 2 = 976$  real operations.
*The first summand accounts for the two DFT calculations with  $N/2 = 8$.
*The remainder is needed for the eight complex multiplications and the  $16$  complex additions and subtractions, respectively.}}

==Radix-2 algorithm according to Cooley and Tukey==
 
Like other FFT algorithms, the method presented herenbsp; [CT65]<ref name ='CT65'>Cooley, J.W.; Tukey, J.W.:  An Algorithm for the Machine Calculation of Complex Fourier Series.  In:  Mathematics of Computation, Vol. 19, No. 90. (Apr., 1965), pp. 297-301.</ref>  from   [https://en.wikipedia.org/wiki/James_Cooley James W. Cooley]  and  [https://en.wikipedia.org/wiki/John_Tukey John W. Tukey]  on the superposition theorem of the DFT.  It only works if the number of interpolation points is a power of two.

The diagram illustrates the algorithm for  $N = 8$, again showing the transformation from the time to the frequency domain.
[[File:EN_Sig_T_5_5_S3a_v2.png|right|frame|Radix-2 algorithm (flow diagram)]]

*Before the FFT algorithm, the input values  $d(0), \hspace{0.05cm}\text{...} \hspace{0.1cm}, d( N - 1)$  have to be reordered in the grey block  "Bit Reversal Operation".
*The computation is done in  $\text{log}_2 N = 3$  stages, where in each stage  $N/2 = 4$  equal computations are performed with different  $\mu$–values as exponent of the complex rotation factor.  Such a basic operation is called  $\text{butterfly}$.
*Each butterfly calculates from two  (generally complex)  input variables  $A$  and  $B$  the two output variables  $A + B \cdot w^{\mu}$  and  $A - B \cdot w^{\mu}$  according to the following sketch.

[[File:P_ID1174__Sig_T_5_5_S3b_neu.png|center|frame|Butterfly of the DFT algorithm]]
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
The complex spectral coefficients  $D(0), \hspace{0.05cm}\text{...} \hspace{0.1cm}, D( N - 1)$  are obtained at the output of the last stage after division by  $N$.
*As shown in  [[Aufgaben:Aufgabe_5.5Z:_Rechenaufwand_für_die_FFT|Exercise 5.5Z]]  compared to the DFT, this results in a much shorter computation time, for example for  $N = 1024$  by more than a factor  $150$.

*The inverse DFT for calculating the time coefficients from the spectral coefficients is done with the same algorithm and only slight modifications.}}

[[File:EN_Sig_Programm.png|right|frame|Radix-2 algorithm (C program)]]
{{GraueBox|TEXT=
$\text{Example 3:}$ 
Finally, the C program  $\text{fft(N, Re, Im)}$  according to the Radix-2 algorithm described above is given:

*The two float arrays  "Re"  and  "Im"  contain the  $N$  real and imaginary parts of the complex time coefficients  $d(0)$, ... , $d( N - 1)$.
*In the same fields  "Re"  and  "Im"  the complex coefficients  $D(0)$, ... , $D( N - 1)$  are returned to the main program.
*Due to the  "in-place"  programming,   $N$  complex memory space is thus sufficient for this algorithm but only if the input values are reordered at the beginning.
*This is done by the program  "bit-reversal", where the contents of  ${\rm Re}( \nu)$  and  ${\rm Im}( \nu)$  are entered into the elements  ${\rm Re}( \kappa)$  and  ${\rm Im}( \kappa)$.  The  $\text{Example 4}$  illustrates this procedure.

$\text{Example 4: "Bit-reversal" operation}$ 
[[File:EN_Sig_T_5_5_S3d_v2.png|100px|left|frame|Radix-2-Algorithm $($"Bit-reversal"  operation for  $N = 8)$]]

*The new index  $\kappa$  is obtained by writing the index  $\nu$  as a dual number and then representing the  $\text{log}_2 \hspace{0.05cm} N$  bits in reverse order.
*For example,  $\nu = 3$  becomes the new index  $\kappa = 6$.}}

==Exercises for the chapter==
 
[[Aufgaben:Exercise 5.5: Fast Fourier Transform|Exercise 5.5: Fast Fourier Transform]]

[[Aufgaben:Exercise 5.5Z: Complexity of The FFT|Exercise 5.5Z: Complexity of the FFT]]

==References==

{{Display}}

Information Theory/Some Preliminary Remarks on Two-Dimensional Random Variables

2022-02-17T11:38:54Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Mutual Information Between Two Discrete Random Variables
|Vorherige Seite=Weitere Quellencodierverfahren
|Nächste Seite=Verschiedene Entropien zweidimensionaler Zufallsgrößen
}}

== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
 
The focus of this third main chapter is the  '''mutual information'''  $I(X; Y)$  between two random variables  $X$  and $Y$.  With statistical dependence,  $I(X; Y)$  is smaller than the individual entropies  $H(X)$  or  $H(Y)$. 

For example, the uncertainty regarding the random variable  $X$    ⇒   entropy  $H(X)$  is reduced by the knowledge of  $Y$,  by the amount  $H(X\hspace{0.03cm}|\hspace{0.03cm}Y)$   ⇒   conditional entropy of  $X$,  if  $Y$  is known.  The remaining residue is the mutual information 
:$$I(X; Y)= H(X) - H(X\hspace{0.03cm}|\hspace{0.03cm}Y).$$

At the same time, however:
:$$I(X; Y) = H(Y) - H(Y\hspace{0.03cm}|\hspace{0.03cm}X).$$ 

The semicolon indicates that the two random variables  $X$  and  $Y$  under consideration are on an equal footing.

In detail, the third main chapter deals with

*the relationship between probability and entropy for  »2D random variables«,
*the calculation of the  »informational divergence«,  also known as the  »Kullback–Leibler distance«,
*the definition of the  »joint entropy«  $H(XY)$  and the  »conditional entropies«  $H(X\hspace{0.03cm}|\hspace{0.03cm}Y)$  and  $H(Y\hspace{0.03cm}|\hspace{0.03cm}X)$,
*the  »mutual information«  $I(X; Y)$  between two random variables,
*the  »information theory of digital signal transmission«  and the corresponding model,
*the definition and meaning of the  »channel capacity«  and its connection with the mutual information,
*the capacity calculation for  »digital memoryless channels«  such as BSC, BEC and BSEC,
*the  »Channel Coding Theorem«,  one of the highlights of Shannon's information theory.

==Introductory example on the statistical dependence of random variables ==
 
[[File:EN_Inf_T_3_1_S1.png|right|frame|Result protocol of our random experiment  "Rolling with two dice"]]

{{GraueBox|TEXT=
$\text{Example 1:}$  We start from the experiment  "Rolling with two dice", where both dice are distinguishable by colour.  The table shows the results of the first  $N = 18$  pairs of throws of this exemplary random experiment.
 
According to the nomenclature explained in the  [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Prerequisites_and_nomenclature|following section]]  $R_ν$,  $B_ν$  and  $S_ν$  are here to be understood as random variables:
*For example, the random variable  $R_3 \in \{1, \ 2, \ 3, \ 4, \ 5, \ 6\}$  indicates the number of points of the red cube on the third throw as a probability event.  The specification  $R_3 = 6$  states that in the documented realisation the red cube showed a  "6"  in the third throw.

*In line 2, the results of the red cube  $(R)$  are indicated.  The mean value of this limited sequence  $〈R_1$, ... , $R_{18}〉$  is with  $3.39$  smaller than the expected value  ${\rm E}\big[R\big] = 3.5$. 
*Line 3 shows the results of the blue cube  $(B)$.  The sequence  $〈B_1$, ... , $B_{18}〉$  has a slightly larger mean value of  $3.61$  than the unlimited sequence   ⇒   expected value ${\rm E}\big[B\big] = 3.5$. 
*Line 4 contains the sum  $S_ν = R_ν + B_ν$.  The mean value of the sequence  $〈S_1$, ... , $S_{18}〉$  is  $3.39 + 3.61 = 7$.  This is here (only by chance) equal to the expected value  $\text{E}\big[S\big] = \text{E}\big[R\big] + \text{E}\big[B\big]$.

Now the question arises between which random variables there are statistical dependencies:
*If one assumes fair dice, there are no statistical dependencies between the sequences  $〈 R\hspace{0.05cm} 〉$  and  $〈B \hspace{0.05cm}〉$  – whether bounded or unbounded:   Even if one knows  $R_ν$  for  $B_ν$  all possible results  $(1$, ... , $6)$  are equally probable.
*If one knows  $S_ν$,  however,  statements about  $R_ν$  as well as about  $B_ν$  are possible.  From  $S_{11} = 12$  follows directly  $R_{11} = B_{11} = 6$  and the sum  $S_{15} = 2$  of two dice is only possible with  $R_{15} = B_{15} = 1$.  Such dependencies are called  »deterministic«.
*From  $S_7 = 10$,  at least ranges for  $R_7$  and  $B_7$  can be given:   $R_7 ≥ 4, \ B_7 ≥ 4$.  Only three pairs are possible:  $(R_7 = 4) ∩ (B_7 = 6)$,  $(R_7 = 5) ∩ (B_7 = 5)$,  $(R_7 = 6) ∩ (B_7 = 4)$.  Here there is no deterministic relationship between the variables  $S_ν$  and  $R_ν$  $($or  $B_ν)$, but rather a so-called  [[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit#Allgemeine_Definition_von_statistischer_Abh.C3.A4ngigkeit|»statistical dependence«]].
*Such statistical dependencies exist for  $S_ν ∈ \{3, \ 4, \ 5, \ 6, \ 8, \ 9, \ 10, \ 11\}$.  On the other hand, if the sum  $S_ν = 7$, one cannot infer  $R_ν$  and  $B_ν$  from this.  For both dice, all possible numbers  $1$, ... , $6$  are equally probable.  In this case, there are also no statistical dependencies between  $S_ν$  and  $R_ν$  or between  $S_ν$  and  $B_ν$.}}

== Prerequisites and nomenclature ==
 
Throughout this chapter, we consider discrete random variables of the form  $X = \{ x_1, \ x_2, \hspace{0.05cm}$ ... $\hspace{0.05cm},\ x_{\mu},\hspace{0.05cm}$ ... $\hspace{0.05cm},\ x_M \} \hspace{0.05cm},$  and use the following nomenclature:
*The random variable itself is always denoted by a capital letter.  The lower case letter  $x$  indicates a possible realisation of the random variable  $X$.
*All realisations  $x_μ$  $($with  $μ = 1$, ... , $M)$  are real-valued.  $M$  indicates the  "symbol set size"  or  "alphabet size"  of  $X$.  Instead of  $M$,  we sometimes also use  $|X|$.

[[File:P_ID2743__Inf_T_3_1_S2.png|right|frame|Relationship between the probability space  ${\it \Omega}$  and the random variable  $X$]]
 The random variable  $X$  can, for example, be created by the transformation  ${\it \Omega} → X$ , where  ${\it \Omega}$  stands for the probability space of a random experiment. 

The diagram illustrates such a transformation:

:$${\it \Omega} = \{ \omega_1, \omega_2, \omega_3, ... \hspace{0.15cm} \}
\hspace{0.25cm} \longmapsto \hspace{0.25cm}
X = \{ x_1, \ x_2, \ x_3, \ x_4\}
\subset \cal{R}\hspace{0.05cm}.$$

*Each random event  $ω_i ∈ Ω$  is uniquely assigned to a real numerical value  $x_μ ∈ X ⊂ \cal{R}$.
*In the example considered, the running variable is  $1 ≤ μ ≤ 4$, i.e. the symbol set size is  $M = |X| = 4$.
*However, the figure is not one-to-one:   The realisation  $x_3 ∈ X$  could have resulted from the elementary event  $ω_4$  in the example, but also from  $ω_6$  $($or from some other of the infinitely many elementary events  $ω_i$ not drawn in the diagram).
 
{{BlaueBox|TEXT=
$\text{Agreement:}$  Often one refrains from indexing both the elementary events  $ω_i$  and the realisations  $x_μ$.  This results in the following shorthand notations, for example:

:$$ \{ X = x \}
\hspace{0.05cm} \equiv \hspace{0.05cm}
\{ \omega \in {\it \Omega} : \hspace{0.4cm} X(\omega) = x \}
\hspace{0.05cm},$$
:$$ \{ X \le x \}
\hspace{0.05cm} \equiv \hspace{0.05cm}
\{ \omega \in {\it \Omega} : \hspace{0.4cm} X(\omega) \le x \}
\hspace{0.05cm}.$$

With this agreement, the probabilities of the discrete random variable  $X$ are:

:$${\rm Pr}( X = x_{\mu}) = \hspace{-0.2cm} \sum_{\omega \hspace{0.1cm} \in \{ X = x_{\mu} \} }
\hspace{-0.2cm}{\rm Pr} \left ( \{ \omega \} \right )
\hspace{0.05cm}.$$}}

==Probability mass function and probability density function==
 
{{BlaueBox|TEXT=
$\text{Definition:}$  If the  $M$  probabilities of a discrete random variable  $X$   ⇒   ${\rm Pr}( X = x_{\mu})$  are combined in a similar way to a vector, we arrive at the   '''probability mass function'''  $\rm (PMF)$:

:$$P_X(X) = \big [ \hspace{0.02cm} P_X(x_1), P_X(x_2), \hspace{0.05cm}\text{...} \hspace{0.15cm}, P_X(x_{\mu}),\hspace{0.05cm} \text{...}\hspace{0.15cm}, P_X(x_M) \hspace{0.02cm} \big ] \hspace{0.05cm}.$$

The  $μ$–th element of this  "vector"  indicates the probability   $P_X(x_{\mu}) = {\rm Pr}( X = x_{\mu}) $.}}

In the book  "Theory of Stochastic Signals",  we defined a similar descriptive quantity with the  [[Theory_of_Stochastic_Signals/Wahrscheinlichkeitsdichtefunktion_(WDF)#Definition_der_Wahrscheinlichkeitsdichtefunktion|probability density function]]  $(\rm PDF)$  and designated it as  $f_X(x)$.

It should be noted, however:
*The PDF is more suitable for characterising continuous random variables, such as a  [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|Gaussian distribution]]  or a [[Theory_of_Stochastic_Signals/Gleichverteilte_Zufallsgrößen|uniform distribution]].  Only through the use of  [[Theory_of_Stochastic_Signals/Wahrscheinlichkeitsdichtefunktion#WDF-Definition_f.C3.BCr_diskrete_Zufallsgr.C3.B6.C3.9Fen| Dirac delta functions]]  does the PDF also become applicable for discrete random variables.
*The PMF provides less information about the random variable than the PDF and can also only be specified for discrete variables.  However, for the discrete information theory considered in this chapter, the PMF is sufficient.

{{GraueBox|TEXT=
$\text{Example 2:}$  We consider a probability density function  $\rm (PDF)$  without much practical relevance:

:$$f_X(x) = 0.2 \cdot \delta(x+2) + 0.3 \cdot \delta(x - 1.5)+0.5 \cdot \delta(x - {\rm \pi}) \hspace{0.05cm}. $$

Thus, for the discrete random variable  $x ∈ X = \{–2,\ +1.5,\ +\pi \} $   ⇒   symbol set size  $M = \vert X \vert = 3$, the probability function $\rm (PMF)$ is:

:$$P_X(X) = \big [ \hspace{0.1cm}0.2\hspace{0.05cm}, 0.3\hspace{0.05cm}, 0.5 \hspace{0.1cm} \big] \hspace{0.05cm}. $$

It can be seen:
*The  $\rm PMF$  only provides information about the probabilities  $\text{Pr}(x_1)$,  $\text{Pr}(x_2)$  and  $\text{Pr}(x_3)$.
*From the  $\rm PDF$,  on the other hand,  the possible realisations  $x_1$,  $x_2$  and  $x_3$  of the random variable  $X$  can also be read.
*The only requirement for the random variable is that it is real-valued.
*The possible values  $x_μ$  do not have to be positive, integer, equidistant or rational. }}

==Probability mass function and entropy==
 
In discrete information theory in contrast to transmission problems, knowledge of the probability mass function  $P_X(X)$ is sufficient, e.g. to calculate the  [[Information_Theory/Gedächtnislose_Nachrichtenquellen#Information_content_and_entropy|entropy]].

{{BlaueBox|TEXT=
$\text{Definition:}$  The  $\rm entropy$  of a discrete random variable  $X$  – i.e. its uncertainty for an observer - can be represented with the PMF  $P_X(X)$  as follows:

:$$H(X) = {\rm E} \big [ {\rm log} \hspace{0.1cm} \frac{1}{P_X(X)}\big ] \hspace{0.05cm}=\hspace{0.05cm}
- {\rm E} \big [ {\rm log} \hspace{0.1cm} {P_X(X)}\big ] \hspace{0.05cm}=\hspace{0.05cm} \sum_{\mu = 1}^{M}
P_X(x_{\mu}) \cdot {\rm log} \hspace{0.1cm} \frac{1}{P_X(x_{\mu})} \hspace{0.05cm}=\hspace{0.05cm} - \sum_{\mu = 1}^{M}
P_X(x_{\mu}) \cdot {\rm log} \hspace{0.1cm} {P_X(x_{\mu})} \hspace{0.05cm}.$$

If you use the logarithm to base  $2$, i.e.  $\log_2$ (...)   ⇒   "binary logarithm", the numerical value is provided with the pseudo-unit  "bit".  $\rm E\big[$...$\big]$ indicates the expected value. }}

For example, one obtains
*with  $P_X(X) = \big [\hspace{0.02cm}0.2, \ 0.3, \ 0.5 \hspace{0.02cm}\big ]$:
::$$H(X) = 0.2 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.2} +
0.3 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.3}
+0.5 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.5}
\approx 1.485\hspace{0.15cm}{\rm bit},$$

*with  $P_X(X) = \big [\hspace{0.02cm}1/3, \ 1/3, \ 1/3\hspace{0.02cm}\big ]$:

::$$H(X) = 3 \cdot 1/3 \cdot {\rm log}_2 \hspace{0.1cm} (3) = {\rm log}_2 \hspace{0.1cm} (3)
\approx 1.585\hspace{0.15cm}{\rm bit}.$$

The second example provides the maximum of the entropy function for the symbol set size  $M = 3$.

{{BlaueBox|TEXT=
$\text{Derivation:}$ 
For general  $M$,  this result can be derived e.g. as follows – see  [Meck]<ref>Mecking, M.: Information Theory. Lecture manuscript, Chair of Communications Engineering, Technische Universität München, 2009.</ref>:

:$$H(X) = -{\rm E} \big [ {\rm log} \hspace{0.1cm} {P_X(X)}\big ] \hspace{0.2cm} \le \hspace{0.2cm}- {\rm log} \big [ {\rm E} \hspace{0.1cm} \left [{P_X(X)}\right ] \big ] \hspace{0.05cm}.$$

This estimation  $($'''Jensens's inequality'''$)$  is admissible because the logarithm is a concave function.  According to  [[Aufgaben:3.2_Erwartungswertberechnungen|Exercise 3.2]] , the following holds:

:$$- {\rm E} \big [ {P_X(X)}\big ] \hspace{0.1cm} \le \hspace{0.1cm} M \hspace{0.3cm} \Rightarrow \hspace{0.3cm}
H(X) \le {\rm log} \hspace{0.1cm} (M) \hspace{0.05cm}.$$

The equal sign results according to the calculation above for equal probabilities, i.e. for  $P_X(x_μ) = {1}/{M}$  for all  $μ$.  In   [[Aufgaben:3.3_Entropie_von_Ternärgrößen|Exercise 3.3]],  the same situation is to be proved using the estimate    "${\rm ln} \hspace{0.1cm} (x) \le x-1$".    The equal sign applies here only for  $x = 1$.}}

If one of the  $M$  probabilities  $P_X(x_μ)$  of the PMF is equal to zero, a tighter bound can be given for the entropy:

:$$H(X) \le {\rm log} \hspace{0.1cm} (M-1) \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Agreement:}$  In the following example and on the next pages we use the following  '''nomenclature''':
*The entropy  $H(X)$  always refers to the actual probability mass function  $P_X(X)$  of the discrete random variable.  Experimentally, these quantities are obtained only after  $N → ∞$  trials.
*If the PMF is determined from a finite random sequence, we denote this probability mass function by  $Q_X(X)$  and add  „$N =$ ...” to the resulting entropy.
*This entropy approximation is not based on probabilities, but only on the  [[Theory_of_Stochastic_Signals/Wahrscheinlichkeit_und_relative_Häufigkeit#Bernoullisches_Gesetz_der_gro.C3.9Fen_Zahlen|relative frequencies]].  Only for  $N → ∞$  does this approximation agree with  $H(X)$ .}}

[[File:EN_Inf_T_3_1_S3.png|right|frame|Probability mass functions of our dice experiment]]
{{GraueBox|TEXT=
$\text{Example 3:}$  We return to our  "dice experiment". 
*The table shows the probability mass functions  $P_R(R)$  and  $P_B(B)$  for the red and blue dice as well as the approximations  $Q_R(R)$  and  $Q_B(B)$,  in each case based on the random experiment with  $N = 18$  throws. 
*The relative frequencies  $Q_R(R)$  and  $Q_B(B)$  result from the  [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen#Introductory_example_on_the_statistical_dependence_of_random_variables|exemplary random sequences]]  of  $\text{Example 1}$.

The following applies to the random variable  $R$  with the  "binary logarithm"  $($to base  $2)$:

:$$H(R) = H(R) \big \vert_{N \hspace{0.05cm}\rightarrow \hspace{0.05cm}\infty} = \sum_{\mu = 1}^{6} 1/6 \cdot {\rm log}_2 \hspace{0.1cm} (6) = {\rm log}_2 \hspace{0.1cm} (6) = 2.585\hspace{0.1cm} {\rm bit} ,$$

:$$H(R) \big \vert_{N \hspace{0.05cm} = \hspace{0.05cm}18} = 2 \cdot \frac{2}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{2} \hspace{0.1cm} +\hspace{0.1cm} 2 \cdot \frac{3}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{3} \hspace{0.1cm} +\hspace{0.1cm} 2 \cdot \frac{4}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{4} \hspace{0.1cm}= 2.530\hspace{0.1cm} {\rm bit}.$$

The blue cube of course has the same entropy:  $H(B) = H(R) = 2.585\ \rm bit$.  Here we get a slightly larger value for the approximation based on  $N = 18$ , since according to the table above  $Q_B(B)$  deviates less from the discrete uniform distribution  $P_B(B)$  than  als $Q_R(R)$  from  $P_R(R)$.

:$$H(B) \big \vert_{N \hspace{0.05cm} = \hspace{0.05cm}18} = 1 \cdot \frac{2}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{2} \hspace{0.1cm} +\hspace{0.1cm} 4 \cdot \frac{3}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{3} \hspace{0.1cm} +\hspace{0.1cm} 1 \cdot \frac{4}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{4} \hspace{0.1cm}= 2.558\hspace{0.1cm} {\rm bit} .$$

:It can be seen from the given numerical values that despite the experimental parameter  $N$,  which is here much too small, the deviation with regard to entropy is not very large.

It should be mentioned again that with finite  $N$  the following always applies:

:$$ H(R) \big \vert_{N } < H(R) = {\rm log}_2 \hspace{0.1cm} (6) \hspace{0.05cm}, \hspace{0.5cm}
H(B) \big \vert_{N } < H(B) = {\rm log}_2 \hspace{0.1cm} (6)\hspace{0.05cm}.$$}}

==Informational divergence - Kullback-Leibler distance ==
 
We consider two probability mass functions  $P_X(·)$  and  $P_Y(·)$  over the same alphabet  $X = \{ x_1, \ x_2$, ... ,  $x_M \}$,  and now define the following quantity:
{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''informational divergence'''  between the random variables defined by   $P_X(·)$  and  $P_Y(·)$  is given as follows:

:$$D(P_X \hspace{0.05cm} \vert \vert \hspace{0.05cm}P_Y) = {\rm E} \left [ {\rm log} \hspace{0.1cm} \frac{P_X(X)}{P_Y(X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{M}
P_X(x_{\mu}) \cdot {\rm log} \hspace{0.1cm} \frac{P_X(x_{\mu})}{P_Y(x_{\mu})} \hspace{0.05cm}.$$

  $D(P_X \vert \vert P_Y)$  is also called  '''Kullback–Leibler distance''' .
*This provides a measure of the  "similarity"  between the two probability functions  $P_X(·)$  and  $P_Y(·)$.

*When using the logarithm to base  $2$  the pseudo-unit  "bit"  must be added again. }}

Similarly, a second variant of the Kullback-Leibler distance can be given:
:$$D(P_Y \hspace{0.05cm} || \hspace{0.05cm}P_X) = {\rm E} \left [ {\rm log} \hspace{0.1cm} \frac{P_Y(X)}{P_X(X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{M}
P_Y(x_{\mu}) \cdot {\rm log} \hspace{0.1cm} \frac{P_Y(x_{\mu})}{P_X(x_{\mu})} \hspace{0.05cm}.$$

Compared to the first variant, each function  $P_X(·)$  is now replaced by  $P_Y(·)$  and vice versa.  Since in general  $D(P_X || P_Y)$  and  $D(P_Y || P_X)$  differ, the term  "distance"  is actually misleading.  However, we want to leave it at this naming.

If we evaluate the two equations above, we recognise the following properties:
*If the same distribution is present   ⇒   $P_Y(·) ≡ P_X(·)$,  then   $D(P_X || P_Y) = 0$.  In all other cases  $D(P_X || P_Y) > 0$.  The same applies to the variant  $D(P_Y || P_X)$.
*If  $P_X(x_μ) ≠ 0$  and  $P_Y(x_μ) = 0$  $($a single and arbitrary  $μ$  is sufficient for this$)$,  the Kullback-Leibler distance  $D(P_X || P_Y)$  has an infinitely large value.  In this case,   $D(P_Y || P_X)$  is not necessarily infinite either.
*This statement makes it clear once again that in general  $D(P_X || P_Y)$  will be unequal to  $D(P_Y || P_X)$ .

Subsequently, these two definitions are clarified with our standard example  "dice experiment".  At the same time we refer to the following exercises:
*[[Exercise_3.5:_Kullback-Leibler_Distance_and_Binomial_Distribution|Exercise 3.5: Kullback-Leibler Distance and Binomial Distribution]]
*[[Aufgaben:Exercise_3.5Z:_Kullback-Leibler_Distance_again|Exercise 3.5Z: Kullback-Leibler Distance again]]
*[[Aufgaben:Exercise_3.6:_Partitioning_Inequality|Exercise 3.6: Partitioning Inequality]]

[[File:EN_Inf_T_3_1_S3.png|right|frame|Probability mass functions of our dice experiment]]
{{GraueBox|TEXT=
$\text{Example 4:}$  For the dice experiment, we defined in  $\text{Example 3}$  the probability mass functions  $P_R(·)$  and  $P_B(·)$  and their approximations   $Q_R(·)$  and  $Q_B(·)$ .
*The random variable  $R$  with the probability mass function  $P_R(·)$  indicates the numbers of the red cube and  $B$  the numbers of the blue cube   ⇒   PMF  $P_B(·)$.
*The approximations  $Q_R(·)$  and  $Q_B(·)$  result from the former experiment with  $N = 18$  double throws  ⇒   [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Introductory_example_on_the_statistical_dependence_of_random_variables|$\text{Example 1}$]] .

Then holds:
*Since  $P_R(·)$  and  $P_B(·)$  are identical, we obtain zero for each of the Kullback-Leibler distances  $D(P_R \vert \vert P_B)$  and  $D(P_B \vert \vert P_R)$.
*The comparison of  $P_R(·)$  and  $Q_R(·)$  yields for the first variant of the Kullback-Leibler distance:

:$$\begin{align*}D(P_R \hspace{0.05cm} \vert \vert \hspace{0.05cm} Q_R) & =
{\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{P_R(\cdot)}{Q_R(\cdot)}\right ]
\hspace{0.1cm} = \sum_{\mu = 1}^{6}
P_R(r_{\mu}) \cdot {\rm log} \hspace{0.1cm} \frac{P_R(r_{\mu})}{Q_R(r_{\mu})} = \\
& = {1}/{6} \cdot \left [
2 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/6}{2/18} \hspace{0.1cm} +
2 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/6}{3/18} \hspace{0.1cm} +
2 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/6}{4/18} \hspace{0.1cm}
\right ] = 1/6 \cdot \big [
2 \cdot 0.585 + 2 \cdot 0 - 2 \cdot 0.415 \big ] \approx 0.0570\hspace{0.15cm} {\rm bit} .\end{align*}$$

:In the calculation of the expected value, the fact that  $P_R(r_1) = $  ...  $ = P_R(r_6)$,  the factor 1/6 can be excluded.  Since the logarithm to base $ 2$  was used here, the pseudo-unit  "bit”  is added.
*For the second variant of the Kullback-Leibler distance, a slightly different value results:

:$$\begin{align*}D(Q_R \hspace{0.05cm}\vert \vert \hspace{0.05cm} P_R) & =
{\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{Q_R(\cdot)}{P_R(\cdot)}\right ]
\hspace{0.1cm} = \sum_{\mu = 1}^{6}
Q_R(r_{\mu}) \cdot {\rm log} \hspace{0.1cm} \frac{Q_R(r_{\mu})}{P_R(r_{\mu})} \hspace{0.05cm} = \\
& = 2 \cdot \frac{2}{18} \cdot {\rm log}_2 \hspace{0.1cm} \frac{2/18}{1/6} \hspace{0.1cm} +
2 \cdot \frac{3}{18} \cdot {\rm log}_2 \hspace{0.1cm} \frac{3/18}{1/6} \hspace{0.1cm} +
2 \cdot \frac{4}{18} \cdot {\rm log}_2 \hspace{0.1cm} \frac{4/18}{1/6}
\approx 0.0544\hspace{0.15cm} {\rm bit} .\end{align*}$$

*For the blue cube,  one obtains  $D(P_B \vert \vert Q_B) ≈ 0.0283 \hspace{0.15cm} \rm bit$  and  $D(Q_B \vert \vert P_B) ≈ 0.0271 \hspace{0.15cm} \rm bit$, i.e. slightly smaller Kullback-Leibler distances, since the approximation  $Q_B(·)$  of  $P_B(·)$  differs less than  $Q_R(·)$  of  $P_R(·)$.
*Comparing the frequencies  $Q_R(·)$  and  $Q_B(·)$, we get  $D(Q_R \vert \vert Q_B) ≈ 0.0597 \hspace{0.15cm} \rm bit$  and  $D(Q_B \vert \vert Q_R) ≈ 0.0608 \hspace{0.15cm} \rm bit$.  Here the distances are greatest, since the differences between   $Q_B(·)$  and  $Q_R(·)$  are greater than between  $Q_R(·)$  and  $P_R(·)$  or between  $Q_B(·)$  and  $P_B(·)$.}}

==Joint probability and joint entropy ==
 
For the remainder of this third chapter, we always consider two discrete random variable  $X = \{ x_1, \ x_2$, ... ,  $x_M \}$  and  $Y = \{ y_1, \ y_2$, ... ,  $y_K \}$, whose value ranges do not necessarily have to coincide.  This means:   $K ≠ M$ $($in other notation:  $|Y| ≠ |X|)$  is quite permissible.

The probability mass function thus has a  $K×M$ matrix form with the elements

:$$P_{XY}(X = x_{\mu}\hspace{0.05cm}, \ Y = y_{\kappa}) = {\rm Pr} \big [( X = x_{\mu})\hspace{0.05cm}\cap \hspace{0.05cm} (Y = y_{\kappa}) \big ] \hspace{0.05cm}.$$

We use  $P_{XY}(X, Y)$ as a shorthand notation.  The new random variable  $XY$  contains both the properties of  $X$  and those of  $Y$.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''joint entropy''' can be represented with the two-dimensional probability mass function  $P_{XY}(X, Y)$  as an expected value as follows:

:$$H(XY) = {\rm E} \left [ {\rm log} \hspace{0.1cm} \frac{1}{P_{XY}(X, Y)}\right ] = \sum_{\mu = 1}^{M} \hspace{0.1cm} \sum_{\kappa = 1}^{K} \hspace{0.1cm}
P_{XY}(x_{\mu}\hspace{0.05cm}, y_{\kappa}) \cdot {\rm log} \hspace{0.1cm} \frac{1}{P_{XY}(x_{\mu}\hspace{0.05cm}, y_{\kappa})} \hspace{0.05cm}.$$

In the following we use throughout the logarithm to the base  $2$   ⇒   $\log(x) → \log_2(x)$.  The numerical value is thus to be assigned the pseudo-unit  "bit".

In general, the following   '''upper bound'''  can be given for the joint entropy:

:$$H(XY) \le H(X) + H(Y) \hspace{0.05cm}.$$}}

This inequality expresses the following fact:
*The equal sign only applies to the special case of statistically independent random variables, as demonstrated in the following  $\text{Example 5}$  using the random variables  $R$  and  $B$ .  Here  $R$  and  $B$  denote the numbers of the red and blue dice, respectively:
:$$H(RB) = H(R) + H(B).$$
*If, on the other hand, there are statistical dependencies as in  $\text{example 6}$  between the random variables  $R$  and  $S = R + B$, the „<” sign applies in the above inequality:
:$$H(RS) < H(R) + H(S).$$

These examples also show to what extent the joint entropies  $H(RB)$  and  $H(RS)$  change if one does not determine an infinite number of pairs of throws in the dice experiment, but only  $N = 18$.

[[File:EN_Inf_T_3_1_S5a.png|right|frame|Two-dimensional probability mass function  $P_{RB}$  and approximation  $Q_{RB}$]]
{{GraueBox|TEXT=
$\text{Example 5:}$  We return to the experiment  [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen#Introductory_example_on_the_statistical_dependence_of_random_variables|Rolling with two dice]] :

The random variables are the points of the
*red cube:   ⇒   $R = \{1, \ 2,\ 3,\ 4,\ 5,\ 6\}$,
*blue cube:  ⇒   $B = \{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$.

The left graph shows the probabilities 
:$$P_{RB}(r_\mu,\ b_\kappa ) ={\rm Pr}\big [(R=r_\mu) \hspace{0.05cm}\cap \hspace{0.05cm} (B=b_\kappa)\big],$$
which for all  $μ = 1$, ... , $6$  and for all  $κ = 1$, ... , $6$  equally yield the value  $1/36$.  Thus, one obtains for the joint entropy:

:$$H(RB) = H(RB) \big \vert_{N \hspace{0.05cm}\rightarrow \hspace{0.05cm}\infty} = {\rm log}_2 \hspace{0.1cm} (36) = 5.170\hspace{0.05cm} {\rm bit} .$$

One can see from the left graph and the equation given here:
*Since  $R$  and  $B$  are statistically independent of each other, the following applies.
:$$P_{RB}(R, B) = P_R(R) · P_B(B).$$
*The joint entropy is the sum of the two individual entropies:  
:$$H(RB) = H(R) + H(B).$$

The right graph shows the approximated two-dimensional probability mass function  $Q_{RB}(·)$, based on the only  $N = 18$  throws of our experiment.  Here, no quadratic form of the joint probability  $Q_{RB}(·)$  results, and the joint entropy derived from it is significantly smaller than  $H(RB)$:

:$$H(RB) \big \vert_{N \hspace{0.05cm} = \hspace{0.05cm}18} = 16 \cdot \frac{1}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{1} \hspace{0.1cm} +\hspace{0.1cm} 1 \cdot \frac{2}{18}\cdot {\rm log}_2 \hspace{0.1cm} \frac{18}{2} \hspace{0.1cm}= 4.059\hspace{0.15cm} {\rm bit} .$$}}

[[File:EN_Inf_T_3_1_S5b.png|right|frame|Two-dimensional probability mass function  $P_{RS}$  and approximation  $Q_{RS}$]]
{{GraueBox|TEXT=
$\text{Example 6:}$  In the dice experiment, in addition to the random variables  $R$  (red cube) and  $B$  (blue cube) also the sum  $S = R + B$  is considered.  The graph on the left shows that the two-dimensional probability mass function  $P_{RS}(·)$  cannot be written as a product of  $P_R(·)$  and  $P_S(·)$.

With the probability functions

:$$P_R(R) = 1/6 \cdot \big [ 1,\ 1,\ 1,\ 1,\ 1,\ 1 \big ],$$
:$$P_S(S)=1/36 \cdot \big [ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 5,\ 4,\ 3,\ 2,\ 1 \big ] $$

one obtains for the entropies:

:$$H(RS) = {\rm log}_2 \hspace{0.1cm} (36) \approx 5.170\hspace{0.15cm} {\rm bit} ,$$
:$$H(R) = {\rm log}_2 \hspace{0.1cm} (6) \approx 2.585\hspace{0.15cm} {\rm bit},$$
$$H(S) = 2 \hspace{-0.05cm}\cdot \hspace{-0.05cm}\frac{1}{36} \hspace{-0.05cm}\cdot \hspace{-0.05cm} {\rm log}_2 \hspace{0.05cm} \frac{36}{1} \hspace{0.05cm} + 2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \frac{2}{36} \hspace{-0.05cm}\cdot \hspace{-0.05cm} {\rm log}_2 \hspace{0.05cm} \frac{36}{2} \hspace{0.05cm} + 2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \frac{3}{36} \cdot {\rm log}_2 \hspace{0.05cm} \frac{36}{3} \hspace{0.05cm} + $$

::$$+ 2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \frac{4}{36} \cdot {\rm log}_2 \hspace{0.05cm} \frac{36}{4} \hspace{0.05cm} +2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \frac{5}{36} \cdot {\rm log}_2 \hspace{0.05cm} \frac{36}{5}
+ 1 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \frac{6}{36} \cdot {\rm log}_2 \hspace{0.05cm} \frac{36}{6} $$

:$$\Rightarrow \hspace{0.3cm} H(S) \approx 3.274\hspace{0.15cm} {\rm bit} . $$
 
From these numerical values one can see:
*The comparison with the  $\text{Example 5}$  shows that  $H(RS) =H(RB)$.  The reason for this is that, knowing  $R$  the random variables  $B$  and  $S$  give exactly the same information.

*Due to the statistical dependence between the red cube and the sum,   $H(RS) ≈ 5.170 \hspace{0.15cm} \rm bit$  is smaller than the sum  $H(R) + H(S) ≈ 5.877 \hspace{0.15cm} \rm bit.$
 
Shown on the right is the case where the two-dimensional probability mass function  $Q_{RS}(·)$  was determined empirically  $(N = 18)$.  Although a completely different figure emerges due to the very small  $N$  value, the approximation for  $H(RS)$  provides exactly the same value as the approximation for  $H(RB)$  in  $\text{Example 5}$:

:$$H(RS) \big \vert_{N \hspace{0.05cm} = \hspace{0.05cm}18} = H(RB) \big \vert_{N \hspace{0.05cm} = \hspace{0.05cm}18} = 4.059\,{\rm bit} \hspace{0.05cm}.$$}}

==Exercises for the chapter==
 
[[Aufgaben:Exercise_3.1:_Probabilities_when_Rolling_Dice|Exercise 3.1: Probabilities when Rolling Dice]]

[[Aufgaben:Exercise_3.1Z:_Drawing_Cards|Exercise 3.1Z: Drawing Cards]]

[[Aufgaben:Exercise_3.2:_Expected_Value_Calculations|Exercise 3.2: Expected Value Calculations]]

[[Aufgaben:Exercise_3.2Z:_Two-dimensional_Probability_Mass_Function|Exercise 3.2Z: Two-dimensional Probability Mass Function]]

[[Aufgaben:Exercise_3.3:_Entropy_of_Ternary_Quantities|Exercise 3.3: Entropy of Ternary Quantities]]

[[Aufgaben:Exercise_3.4:_Entropy_for_Different_PMF|Exercise 3.4: Entropy for Different PMF]]

[[Exercise_3.5:_Kullback-Leibler_Distance_and_Binomial_Distribution|Exercise 3.5: Kullback-Leibler Distance and Binomial Distribution]]

[[Aufgaben:Exercise_3.5Z:_Kullback-Leibler_Distance_again|Exercise 3.5Z: Kullback-Leibler Distance again]]

[[Aufgaben:Exercise_3.6:_Partitioning_Inequality|Exercise 3.6: Partitioning Inequality]]

==References==
<references />

{{Display}}

Mobile Communications/Characteristics of UMTS

2022-02-17T11:38:45Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Mobile Radio Systems of the 2nd and 3rd Generation - an Overview
|Vorherige Seite=Characteristics of GSM
|Nächste Seite=General Information on the LTE Mobile Communications Standard
}}

== Requirements for third generation mobile radio systems ==
 
The main motivation for the development of  '''Third generation mobile radio systems'''   was the realization that 2G systems could not satisfy the bandwidth requirements for the use of multimedia services.

[[File:EN_Mob_T_3_4_S1.png|right|frame|Development of the mobile radio systems|class=fit]]
The graph shows the development of mobile radio systems from 1995 to 2006 in terms of performance.  The specified data rates were still realistic for 2011 with no more than two active users in one cell.  The maximum values often stated by providers were mostly not reached in practice.

Third-generation mobile communications systems should have a greater bandwidth and sufficient reserve capacity to ensure a high quality of service  $\rm (QoS)$  even with growing requirements.

Prior to the development of the 3G systems, the "International Telecommunication Union"  (ITU) created a catalog of requirements which includes the following general conditions:
*High data rates from  $\text{144 kbit/s}$  (standard) to  $\text{2 Mbit/s}$  (in-door), 
*symmetric and asymmetric data transmission (IP services), 

*high speech quality and high spectral efficiency,
*global accessibility and distribution, 
*seamless transition from and to second generation systems, 
*applicability independent of the network used ("Virtual Home Environment"),
*provision of circuit-switched and packet-switched transmission. 

During the introduction of  $\rm UMTS$  $\rm (U\hspace{-0.02cm}$niversal $\hspace{0.05cm}\rm M\hspace{-0.03cm}$obile $\hspace{0.05cm}\rm T\hspace{-0.03cm}$elecommunication $\hspace{0.03cm}\rm S\hspace{-0.02cm}$ystem$)$  as the best known 3G standard, the expansion and diversification of the services offered was a decisive motive.  A UMTS capable terminal device must support a number of complex and multimedia applications in addition to the classic services  (voice transmission, messaging, etc.), including
*with regard to  Information:   Internet surfing (info on demand), online print media, 

*regarding  Communication:     Video and audio conference, fax, ISDN, messaging, 

*regarding  Entertainment:     Mobile TV, video on demand, Online Gaming, 

*in the  business area:     Interactive shopping, E–Commerce, 

*in the  technical area:     Online–support, distribution service (language and data), 

*in the  medical field:     Telemedicine. 

== The IMT-2000 standard ==
 
Around 1990, the  "International Telecommuncation Union"  (ITU) created the standard $\text{IMT-2000}$  $\rm (I\hspace{-0.02cm}$nternational $\hspace{0.05cm}\rm M\hspace{-0.03cm}$obile $\hspace{0.05cm}\rm T\hspace{-0.03cm}$elecommunications at the year $2000)$, which was to make the above-mentioned requirements possible.  IMT–2000 comprises a number of third-generation mobile communications systems that have been brought closer together in the course of standardization to enable the development of common terminals for all these standards. 

In order to take into account different preliminary work and to give network operators the possibility to continue to use existing network architectures in part, IMT–2000 contains several individual standards.  These can be roughly divided into four categories:
[[File:P ID2208 Mob T 3 4 S2 v1.png|right|frame|The IMT family|class=fit]]
*$\text{W – CDMA}$:   This includes the FDD component of the European UMTS standard and the American cdma2000 system.

*$\text{TD – CDMA}$:   This group includes the TDD component of UMTS as well as the Chinese TD–SCDMA, meanwhile integrated in it.

*$\text{TDMA}$:   A further development of the GSM derived EDGE and its American counterpart UWC–136, also known as D–AMPS. 

*$\text{FD – TDMA}$:   A further development of the European cordless telephony standard DECT (Digital Enhanced Cordless Telecommunication). 
 
We here concentrate on the mobile communications system UMTS developed in Europe, which supports the two standards  "W–CDMA"  and  "TD–CDMA"  of the system family IMT–2000, under the following designations:
*$\text{UTRA – FDD}$   ⇒   UMTS Terrestrial Radio Access – Frequency Division Duplex:  This consists of twelve paired uplink and downlink frequency bands each  $\text{5 MHz}$  bandwidth.  In Europe these are between  $\text{1920}$  and  $\text{1980 MHz}$  in the uplink and between  $\text{2110}$  and  $\text{2170 MHz}$  in the downlink.  In the summer of 2000, the auction of the licenses for Germany with a 20-year term brought in approx. 50 billion Euro. 

*$\text{UTRA – TDD}$   ⇒   UMTS Terrestrial Radio Access – Time Division Duplex:  Here, five bands of  $\text{5 MHz}$  bandwidth are provided in which both uplink and downlink data are to be transmitted by means of time division multiplexing.  For  "UTRA–TDD"  the frequencies between  $\text{1900}$  and  $\text{1920 MHz}$  (four channels) and between  $\text{2020}$  and  $\text{2025 MHz}$  (one channel) are reserved. 

== System architecture and basic units for UMTS ==
 
The network architecture of UMTS can be divided into two main blocks. 

⇒   The  "UMTS Terrestrial Radio Access Network"   $\text{(UTRAN)}$  ensures the wireless transmission of data between the transport level and the radio network level.  This includes the base stations and the control nodes, whose functions are described below:
:*A UMTS base station (also called  $\text{Node B}$)  comprises the antenna system and the CDMA receiver and is directly connected to the radio interfaces of all users in the cell.  The tasks of a "Node B" include data rate matching, data and channel (de)coding, interleaving, and modulation or demodulation.  Each base station can serve one or more cells (sectors). 

:*The  "Radio Network Controller"  $\rm (RNC)$  is responsible for controlling the base stations.  It is also responsible within the cells for call acceptance control, encryption and decryption, conversion to ATM ("Asynchronous Tranfer Mode"), channel assignment, handover and power control. 

⇒   The  "Core Network"  $\rm (CN)$  takes over the switching of the data within the UMTS network.  For this purpose, it contains the following hardware and software components at  "Circuit Switching" :
[[File:P ID2209 Mob T 3 4 S3 v1.png|right|frame|UMTS access level (with "Circuit Switching")|class=fit]]
:*The  "Mobile Switching Center"  $\rm (MSC)$  is responsible for localization and authentication, routing of calls, handover and encryption of user data. 

:*The  "Gateway Mobile Switching Center"  $\rm (GMSC)$  organizes the connection to other networks, for example to the landline network. 

:*Both,  MSC and GMSC,  have access to various databases like  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_GSM#Switching_and_Management_Subsystem_.28SMSS.29| Home Location Register]]  $\rm (HLR)$  and  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_GSM#Switching_and_Management_Subsystem_.28SMSS.29| Visitor Location Register]]  $\rm (VLR)$. 

The graphic shows the UMTS architecture for circuit switching, where the Core Network  $\rm (CN)$  is organized similarly to the GSM architecture.

The  [[Examples_of_Communication_Systems/UMTS%E2%80%93Netzarchitektur#Architektur_der_Zugangsebene| system architecture for packet switching]]  differs fundamentally in the following points:

*Here, the communication partners do not use the channel assigned to them exclusively, but the packets are mixed with those of other users.
*One finds there similar components as with the GSM extension  [[Examples_of_Communication_Systems/Weiterentwicklungen_des_GSM#General_Packet_Radio_Service_.28GPRS.29| General Packet Radio Service]]  $\rm (GPRS)$. 

== CDMA - Multiple access with UMTS ==
 
UMTS uses the multiple access method  [[Modulation_Methods/PN%E2%80%93Modulation#Blockschaltbild_und_.C3.A4quivalentes_Tiefpass.E2.80.93Modell| Direct Sequence Code Division Multiple Access]]  $\rm (DS–CDMA)$.  The procedure is sometimes called  "PN–Modulation". 

[[File:EN Mob T 3 4 S4.png|right|frame|Principle and signal characteristics with "DS-CDMA" for two users|class=fit]]
The graphic shows the principle using a simplified model and exemplary signals for the "user 1".  For simplification the noise signal  $n(t) \equiv 0$  is set for the displayed signals.  It is valid:
*The two source signals  $q_1(t)$  and  $q_2(t)$  use the same AWGN channel without interfering with each other. The bit duration of each data signal is  $T_{\rm B}$. 

*Each of the data signals is multiplied by an assigned spreading code,  $c_1(t)$  or   $c_2(t)$.  The sum signal is transmitted;
:$$s(t) = s_1(t) + s_2(t) = q_1(t) \cdot c_1(t) + q_2(t) \cdot c_2(t).$$

*The bandwidths of the partial signals  $s_1(t)$  and  $s_2(t)$  as well as of the resulting transmission signal  $s(t)$  are larger than the bandwidths of  $q_1(t)$  and  $q_2(t)$  by the  '''spreading factor'''   $ J = T_{\rm C}/T_{\rm B}$.  For the graphic  $J = 4$  was chosen.

*The same codes  $c_1(t)$  and   $c_2(t)$  are added multiplicatively to the receiver.  In the case of orthogonal codes and small AWGN noise  $n(t)$  the data signals can then be separated again.  This means that  $v_1(t) = q_1(t)$  and  $v_2(t) = q_2(t)$. 

*If AWGN noise is present, the output signals are different from the input signals, but the error probability is not increased by the other users as long as the spreading sequences are orthogonal. 

*In the example  $J =4$  one could thus transmit users over the same channel without interference, but only if there are   $J =4$  orthogonal spreading codes. 

== Requirements for the spreading codes==
 
The spreading codes for UMTS should
*be orthogonal to each other to avoid mutual influence of the users, and 

*allow a flexible realization of different spreading factors  $J$. 

{{GraueBox|TEXT=
$\text{Example 1:}$  An example for spreading codes are the  "Orthogonal Variable Spreading Factor"  $\rm (OVSF)$, which provide codes of length between  $J =4$  and  $J =512$.
[[File:P ID1535 Bei T 4 3 S3c v1.png|right|frame|OVSF code family and possible spreading sequences|class=fit]]

These can be created with help of a code tree, as shown in the graphic.  Thereby in each branching from a code  $C$  two new codes result:  
$(+C \ +\hspace{-0.05cm}C)$  and  $(+C \ -\hspace{-0.05cm}C)$. 

''Notes'::
*No predecessor or successor of a code may be used.
*In the example eight spreading codes with the spreading factor  $J = 8$  could be used.
*Or the four codes highlighted in yellow:   $J = 2$ once,  $J = 4$  once and the  $J = 8$ twice.
*The lower four codes with the spreading factor  $J = 8$  cannot be used here, since they all start with "$+1 \ -1$ " which is already occupied by the OVSF codes with spreading factor  $J = 2$ .}}

The situation described here is also clarified by the SWF applet  [[Applets:OVSF-Codes_(Applet)|OVSF codes]] . 

== Additional scrambling in UMTS ==
 
[[File:EN_Mob_T_3_4_S5.png|right|frame|Scrambling in UMTS|class=fit]]
In order to get more spreading codes and to be able to serve more participants, after the band spreading using  $c(t)$  the sequence is again scrambled chip by chip using  $w(t)$  without further spreading.

The use of quasi–orthogonal codes makes sense, because the amount of orthogonal codes is limited and different participants can use the same spreading codes due to the scrambling.
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*The scrambling code  $w(t)$  has the same length and rate as  $c(t)$. 
*Due to the scrambling, the codes lose their complete orthogonality; they are called "quasi–othogonal".
* In these codes, the  [[Theory_of_Stochastic_Signals/Kreuzkorrelationsfunktion_und_Kreuzleistungsdichte#Definition_der_Kreuzkorrelationsfunktion |Cross-Correlation Function]]  $\rm (CCF)$  between different spreading codes is not equal to zero.
*But they are characterized by a distinct  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)| ACF value]]  around zero, which facilitates detection at the receiver.}}

[[File:P ID1537 Bei T 4 3 S3b v2.png|right|frame|Example generator for Gold codes with  $N = 18$|class=fit]]
{{GraueBox|TEXT=
$\text{Example 2:}$  In UMTS, so-called [https://en.wikipedia.org/wiki/Gold_code Gold codes] are used for scrambling:
*The graphic from [3gpp]<ref name='3gpp'>3gpp Group: UMTS Release 6 - Technical Specification 25.213 V6.4.0., Sept. 2005.</ref> shows the block diagram for the generation of such codes.
*At first two different  [[Theory_of_Stochastic_Signals/Erzeugung_von_diskreten_Zufallsgr%C3%B6%C3%9Fen#Pseudozufallsgr.C3.B6.C3.9Fen|Pseudo–noise sequences]]  of the same length $($here:  $N = 18)$  are generated in parallel by means of shift registers and then added bitwise with "XOR gates".}}
 

[[File:EN_Mob_T_3_4_S5b.png|left|frame|Some examples and properties of suitable spreading and scrambling codes|class=fit]]
 
*In the uplink, each mobile station has its own scrambling code and the separation of the individual channels is done using the same code.

*In the downlink, on the other hand, each service area of a "Node B" has a common scrambling code. 

*The table on the left summarizes some data of the spreading and scrambling codes. 
 
== Modulation and pulse shaping for UMTS ==
 
In UMTS the following modulation methods are used in FDD–mode:
*In the downlink  [[Modulation_Methods/Quadrature_Amplitude_Modulation#Other_signal_space_constellations| Quaternary Phase Shift Keying]]  $\rm (QPSK)$ is used.  User data  (DPDCH channel)  and  control data (DPCCH channel)  are multiplexed in time. 

* In the uplink a  [https://en.wikipedia.org/wiki/Phase-shift_keying "dual channel BPSK"]  is used.  This has the same signal space as QPSK, but the $I$ and  $Q$ components transmit the information of different channels. 

[[File:EN_Mob_T_3_4_S6.png|right|frame|Modulation and pulse shaping for UMTS|class=fit]]
The graphic shows the  $I/Q$ multiplexing method  (another name for the "dual channel BPSK")  in the equivalent low-pass range. 

*The spread user data of the DPDCH channel is modulated and transmitted on the inphase component  (real part) and the control data of the DPCCH channel, also spread, is modulated on the quadrature component  (imaginary part). 

*The quadrature component is weighted with the square root of the power ratio  $G$  between  $I$  and  $Q$  to compensate power differences.  Then the sum signal  $(I + {\rm j} \cdot Q)$  is multiplied by a complex scrambling code. 

*Finally the pulse shaping is done with  $g_s(t)$  corresponding to the  [[Digital_Signal_Transmission/Optimierung_der_Basisband%C3%BCbertragungssysteme#Wurzel.E2.80.93Nyquist.E2.80.93Systeme| Root Raised Cosine]].  Since the reception filter is adapted to  $G_s(f)$  the overall frequency response thus fulfills the  [[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Frequenzbereich| first Nyquist criterion]]. 

Further information on this topic can be found in the section  [[Examples_of_Communication_Systems/Telecommunications_Aspects_of_UMTS#Frequency_responses_and_pulse_shaping_for_UMTS|
Frequency responses and pulse shaping for UMTS]]  of the book "Examples of communication systems".  There you will also find a graphic with the Nyquist frequency response  $H(f)$.  It is a  [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Cosinus-Rolloff-Tiefpass|Raised Cosine]] with the following dimensioning:
*The UMTS chip rate is  $R_{\rm C} = 3.84 \ \rm Mbit/s$.  The center of the slope must be at  $f_{\rm N} =R_{\rm C}/2 = 1.92 \ \rm MHz$  to avoid intersymbol interference   ⇒   $H(f = \pm f_{\rm N}) = 0.5$.

*For UMTS the rolloff factor  $r = 0.22$  has been defined.
*This results in the two cutoff frequencies  $f_1 = 0.78 \cdot f_{\rm N} \approx 1.498 \ \rm MHz$  and  $f_2 = 1.22 \cdot f_{\rm N} \approx 2.342 \ \rm MHz$. 

*The required absolute frequency bandwidth is thus  $B = 2 \cdot f_2 = 1.22 \cdot f_{\rm N} \approx 4.684 \ \rm MHz$, so that for each UMTS channel sufficient bandwidth  $(5 \ \rm MHz)$  is available. 

== UMTS extensions HSDPA and HSUPA ==
 
In order to meet the ever-increasing demand for higher data rates in mobile communications, the UMTS standard has been continuously developed.  The most important changes within the third generation resulted from the introduction of
*[[Examples_of_Communication_Systems/Weiterentwicklungen_von_UMTS#High.E2.80.93Speed_Downlink_Packet_Access| High Speed Downlink Packet Access]]  $\rm (HSDPA)$  (Release 5, 2002, market launch 2006) and 

*[[Examples_of_Communication_Systems/Weiterentwicklungen_von_UMTS#High.E2.80.93Speed_Uplink_Packet_Access| High Speed Uplink Packet Access]]  $\rm (HSUPA)$  (Release 6, 2005, market launch 2007). 

Together, HSDPA and HSDUPA result in the  $\rm HSPA$ standard. 

The main motivation for these further developments was to increase data rate/throughput and minimize response times in packet-switched transmission.
*For the downlink, data rates up to  $\text{7 Mbit/s}$  were quite feasible with HSDPA since 2011.
*But also (more theoretical) "Best–Case" rates of up to  $\text{28.8 Mbit/s}$  (with 64–QAM and MIMO) were given.

These increases were achieved by
*the introduction of additional  [[Examples_of_Communication_Systems/Weiterentwicklungen_von_UMTS#Zus.C3.A4tzliche_Kan.C3.A4le_in_HSDPA | shared channels]]  (for example  $\rm HS\hspace{0.05cm}–\hspace{-0.05cm}DSCH$), 

*the  [[Examples_of_Communication_Systems/Further_Developments_of_UMTS#HARQ_procedure_and_.22Node_B_Scheduling.22| Hybrid–ARQ Procedure]]  $\rm (HARQ)$  and  "Node B scheduling", 

*the use of  [[Examples_of_Communication_Systems/Further_Developments_of_UMTS#Adaptive_modulation.2C_adaptive_coding_and_adaptive_transmission_rate| adaptive M-QAM, coding and transmission rate"]]. 

In addition to the use of HARQ and "Node B scheduling", the significant improvement through HSUPA is due to the introduction of the additional upstream channel  $\rm E\hspace{0.05cm}–\hspace{-0.05cm}DCH$  (Enhanced Dedicated Channel).
*Among other things, this minimizes the influence of applications with very different and sometimes very intensive data volumes ("Bursty Traffic").  However, unlike UMTS–R99, HSUPA does not guarantee a fixed bandwidth in the upward direction. 

*This flexible and efficient bandwidth allocation depending on channel conditions increased the cell capacity enormously.  In practice, data rates of up to  $\text{3 Mbit/s}$  were achieved from 2011, even when taking into account the large number of users.  The values specified by developers for best conditions were significantly higher. 

==Exercises for the chapter==

 
[[Aufgaben:Exercise 3.6: FDMA, TDMA and CDMA]]

[[Aufgaben:Exercise 3.6Z: Concepts of 3G Mobile Communications Systems]]

[[Aufgaben:Exercise 3.7: PN Modulation]]

[[Aufgaben:Exercise 3.7Z: Spread Spectrum in UMTS]]

[[Aufgaben:Exercise 3.8: OVSF Codes]]

[[Aufgaben:Exercise 3.9: Further Developments of UMTS]]

==References==

{{Display}}

Information Theory/Natural Discrete Sources

2022-02-17T11:38:38Z

Bene: Text replacement - "List of sources" to "References"

{{Header
|Untermenü=Entropy of Discrete Sources
|Vorherige Seite=Discrete Sources with Memory
|Nächste Seite=General_Description
}}

==Difficulties with the determination of entropy ==
 
Up to now, we have been dealing exclusively with artificially generated symbol sequences.  Now we consider written texts.  Such a text can be seen as a natural discrete message source, which of course can also be analyzed information-theoretically by determining its entropy.

Even today (2011), natural texts are still often represented with the 8 bit character set according to ANSI ("American National Standard Institute"), although there are several "more modern" encodings;

The  $M = 2^8 = 256$  ANSI characters are used as follows:
* '''No.  0   to   31''':   control commands that cannot be printed or displayed,
* '''No.  32   to  127''':   identical to the characters of the 7 bit ASCII code,
* '''No.  128   to 159''':   additional control characters or alphanumeric characters for Windows,
* '''No.  160   to   255''':   identical to the Unicode charts.

Theoretically, one could also define the entropy here as the border crossing point of the entropy approximation  $H_k$  for  $k \to \infty$,  according to the procedure from the  [[Information_Theory/Sources_with_Memory#Generalization to k -tuple and boundary crossing|last chapter]].  In practice, however, insurmountable numerical limitations can be found here as well:

*Already for the entropy approximation  $H_2$  there are  $M^2 = 256^2 = 65\hspace{0.1cm}536$  possible two-tuples.  Thus, the calculation requires the same amount of memory (in bytes).   If you assume that you need for a sufficiently safe statistic  $100$  equivalents per tuple on average,  the length of the source symbol sequence should already be  $N > 6.5 · 10^6$.
*The number of possible three-tuples is  $M^3 > 16 · 10^7$  and thus the required source symbol length is already  $N > 1.6 · 10^9$.  This corresponds to a book with about  $500\hspace{0.1cm}000$  pages to  $42$  lines per page and  $80$  characters per line.
*For a natural text the statistical ties extend much further than two or three characters.  Küpfmüller gives a value of  $100$  for the German language.  To determine the 100th entropy approximation you need  $2^{800}$ ≈ $10^{240}$  frequencies and for the safe statistics  $100$  times more characters.

A justified question is therefore:   How did  [https://en.wikipedia.org/wiki/Karl_K%C3%BCpfm%C3%BCller Karl Küpfmüller]  determined the entropy of the German language in 1954?  How did  [https://en.wikipedia.org/wiki/Claude_Shannon Claude Elwood Shannon]  do the same for the English language, even before Küpfmüller?  One thing is revealed beforehand:   Not with the approach described above.

==Entropy estimation according to Küpfmüller ==
 
Karl Küpfmüller has investigated the entropy of German texts in his published assessment   [Küpf54]<ref name ='Küpf54'>Küpfmüller, K.:  Die Entropie der deutschen Sprache.  Fernmeldetechnische Zeitung 7, 1954, S. 265-272.</ref>  the following assumptions are made:
*an alphabet with  $26$  letters  (no umlauts and punctuation marks),
*not taking into account the space character,
*no distinction between upper and lower case.

The maximum average information content is therefore 
:$$H_0 = \log_2 (26) ≈ 4.7\ \rm bit/letter.$$

Küpfmueller's estimation is based on the following considerations:

'''(1)'''  The  '''first entropy approximation'''  results from the letter frequencies in German texts.  According to a study of 1939, "e" is with a frequency of   $16. 7\%$  the most frequent, the rarest is "x" with  $0.02\%$.  Averaged over all letters we obtain 
:$$H_1 \approx 4.1\,\, {\rm bit/letter}\hspace{0.05 cm}.$$

'''(2)'''  Regarding the  '''syllable frequency'''  Küpfmüller evaluates the  "Häufigkeitswörterbuch der deutschen Sprache"  (Frequency Dictionary of the German Language), published by  [https://en.wikipedia.org/wiki/Friedrich_Wilhelm_Kaeding Friedrich Wilhelm Kaeding]  in 1898.  He distinguishes between root syllables, prefixes, and final syllables and thus arrives at the average information content of all syllables:

:$$H_{\rm syllable} = \hspace{-0.1cm} H_{\rm root} + H_{\rm prefix} + H_{\rm final} + H_{\rm rest} \approx
4.15 + 0.82+1.62 + 2.0 \approx 8.6\,\, {\rm bit/syllable}
\hspace{0.05cm}.$$

:The following proportions were taken into account:
:*According to the Kaeding study of 1898, the  $400$  most common root syllables  (beginning with "de")  represent $47\%$  of a German text and contribute to the entropy with  $H_{\text{root}} ≈ 4.15 \ \rm bit/syllable$.
:*The contribution of  $242$  most common prefixes - in the first place "ge" with  $9\%$ - is numbered by Küpfmüller with  $H_{\text{prefix}} ≈ 0.82 \ \rm bit/syllable$.
:*The contribution of the  $118$  most used final syllables is  $H_{\text{final}} ≈ 1.62 \ \rm bit/syllable$.  Most often, "en" appears at the end of words with  $30\%$ .
:*The remaining  $14\%$  is distributed over syllables not yet measured.  Küpfmüller assumes that there are  $4000$  and that they are equally distributed.  He assumes  $H_{\text{rest}} ≈ 2 \ \rm bit/syllable$  for this.

'''(3)'''  As average number of letters per syllable Küpfmüller determined the value  $3.03$.  From this he deduced the  '''third entropy approximation''''  regarding the letters:
:$$H_3 \approx {8.6}/{3.03}\approx 2.8\,\, {\rm bit/letter}\hspace{0.05 cm}.$$

'''(4)'''  Küpfmueller's estimation of the entropy approximation  $H_3$  based mainly on the syllable frequencies according to  '''(2)'''  and the mean value of  $3.03$  letters per syllable. To get another entropy approximation  $H_k$  with greater  $k$  Küpfmüller additionally analyzed the words in German texts.  He came to the following results:

:*The  $322$  most common words provide an entropy contribution of  $4.5 \ \rm bit/word$.
:*The contributions of the remaining  $40\hspace{0.1cm}000$ words  were estimated.  Assuming that the frequencies of rare words are reciprocal to their ordinal number ([https://en.wikipedia.org/wiki/Zipf%27s_law Zipf's Law]).
:*With these assumptions the average information content (related to words) is about   $11 \ \rm bit/word$.

'''(5)'''  The counting "letters per word" resulted in average  $5.5$.  Analogous to point  '''(3)'''  the entropy approximation for  $k = 5.5$  was approximated.  Küpfmüller gives the value: 
:$$H_{5.5} \approx {11}/{5.5}\approx 2\,\, {\rm bit/letter}\hspace{0.05 cm}.$$
:Of course,  $k$  can only assume integer values,  according to  [[Information_Theory/Sources_with_Memory#Generalization to k-tuple and boundary crossing|its definition]].  This equation is therefore to be interpreted in such a way that for  $H_5$  a somewhat larger and for  $H_6$  a somewhat smaller value than  $2 \ {\rm bit/letter}$  will result.

'''(6)'''  Now you can try to get the final value of entropy for  $k \to \infty$  by extrapolation from these three points  $H_1$,  $H_3$  and  $H_{5.5}$ :
[[File:EN_Inf_T_1_3_S2.png|right|frame|Approximate values of the entropy of the German language according to Küpfmüller]]

:*The continuous line, taken from Küpfmüller's original work  [Küpf54]<ref name ='Küpf54'>Küpfmüller, K.:  Die Entropie der deutschen Sprache.  Fernmeldetechnische Zeitung 7, 1954, S. 265-272.</ref>, leads to the final entropy value  $H = 1.6 \ \rm bit/letter$.
:*The green curves are two extrapolation attempts (of a continuous function course through three points) of the  $\rm LNTwww$'s author.
:*These and the brown arrows are actually only meant to show that such an extrapolation is  (carefully worded)  somewhat vague.

'''(7)'''  Küpfmüller then tried to verify the final value  $H = 1.6 \ \rm bit/letter$  found by him with this first estimation with a completely different methodology - see next section. After this estimation he revised his result slightly to 
:$$H = 1.51 \ \rm bit/letter.$$

'''(8)'''  Three years earlier, after a completely different approach, Claude E. Shannon had given the entropy value  $H ≈ 1 \ \rm bit/letter$  for the English language, but taking into account the space character.  In order to be able to compare his results with Shannom, Küpfmüller subsequently included the space character in his result.

:*The correction factor is the quotient of the average word length without considering the space  $(5.5)$  and the average word length with consideration of the space  $(5.5+1 = 6.5)$.
:*This correction led to Küpfmueller's final result: 
:$$H =1.51 \cdot {5.5}/{6.5}\approx 1.3\,\, {\rm bit/letter}\hspace{0.05 cm}.$$

==A further entropy estimation by Küpfmüller ==
 
For the sake of completeness, Küpfmüller's considerations are presented here, which led him to the final result  $H = 1.51 \ \rm bit/letter$.    Since there was no documentation for the statistics of word groups or whole sentences, he estimated the entropy value of the German language as follows:
*Any contiguous German text is covered behind a certain word.  The preceding text is read and the reader should try to determine the following word from the context of the preceding text.
*For a large number of such attempts, the percentage of hits gives a measure of the relationships between words and sentences.  It can be seen that for one and the same type of text (novels, scientific writings, etc.) by one and the same author, a constant final value of this hit ratio is reached relatively quickly  (about one hundred to two hundred attempts).
*The hit ratio, however, depends quite strongly on the type of text.  For different texts, values between  $15\%$  and  $33\%$  are obtained, with the mean value at  $22\%$.  This also means:   On average,  $22\%$  of the words in a German text can be determined from the context.
*Alternatively:   The word count of a long text can be reduced with the factor  $0.78$  without a significant loss of the message content of the text.  Starting from the reference value  $H_{5. 5} = 2 \ \rm bit/letter$  $($see dot  '''(5)'''  in the last section$)$  for a word of medium length this results in the entropy  $H ≈ 0.78 · 2 = 1.56 \ \rm bit/letter$.
*Küpfmüller verified this value with a comparable empirical study regarding the syllables and thus determined the reduction factor  $0.54$  (regarding syllables).  Küpfmüller gives  $H = 0. 54 · H_3 ≈ 1.51 \ \rm bit/letter$  as the final result, where  $H_3 ≈ 2.8 \ \rm bit/letter$  corresponds to the entropy of a syllable of medium length  $($about three letters, see point  '''(3)'''  on the last page$)$ .

The remarks on this and the previous page, which may be perceived as very critical, are not intended to diminish the importance of neither Küpfmüller's entropy estimation, nor Shannon's contributions to the same topic.
*They are only meant to point out the great difficulties that arise in this task.
*This is perhaps also the reason why no one has dealt with this problem intensively since the 1950s.

==Some own simulation results==
 
Karl Küpfmüller's data regarding the entropy of the German language will now be compared with some (very simple) simulation results that were worked out by the author of this chapter (Günter Söder) at the Department of Communications Engineering at the Technical University of Munich as part of an internship for students.  The results are based on the German Bible in ASCII format with  $N \approx 4.37 \cdot 10^6$  characters. This corresponds to a book with  $1300$  pages at  $42$  lines per page and  $80$  characters per line.

The symbol set size has been reduced to  $M = 33$  and includes the characters '''a''',  '''b''',  '''c''',  ... .  '''x''',  '''y''',  '''z''',  '''ä''',  '''ö''',  '''ü''',  '''ß''',  $\rm BS$,  $\rm DI$,  $\rm PM$.   Our analysis did not differentiate between upper and lower case letters.  In contrast to Küpfmüller's analysis, we also took into account:
*the German umlauts  '''ä''',  '''ö''',  '''ü'''  and  '''ß''', which make up about  $1.2\%$  of the biblical text,
*the class  "Digits"   ⇒   $\rm DI$  with about  $1.3\%$  because of the verse numbering within the bible,
*the class  "Punctuation Marks"   ⇒   $\rm PM$  with about  $3\%$,
*the class  "Blank Space"   ⇒   $\rm BS$  as the most common character  $(17.8\%)$, even more than the "e"  $(12.8\%)$.

The following table summarizes the results:   $N$  indicates the analyzed file size in characters (bytes).   The decision content  $H_0$  as well as the entropy approximations  $H_1$,  $H_2$  and  $H_3$  were each determined from  $N$  characters and are each given in "bit/characters".

[[File:EN_Inf_T_1_3_S3_v2.png|left|frame|Entropy values  (in bit/characters)  of the German Bible]]
 
*Please do not consider these results to be scientific research.
*It is only an attempt to give students an understanding of the subject matter in an internship.
*The basis of this study was the Bible, since we had both its German and English versions available to us in the appropriate ASCII format.
 
The results of the above table can be summarized as follows:
*In all rows the entropy approximations  $H_k$  decreases monotously with increasing  $k$.  The decrease is convex, that means:   $H_1 - H_2 > H_2 - H_3$.   The extrapolation of the final value  $(k \to \infty)$  from the three entropy approximations determined in each case is not possible  (or only extremely vague).
*If the evaluation of the digits  $\rm (DI)$  and additionally the evaluation of the punctuation marks  $\rm (PM)$  is omitted, the approximations  $H_1$  $($by  $0. 114)$,  $H_2$  $($by  $0.063)$  and  $H_3$  $($by  $0.038)$  decrease.   On the final entropy   $H$  as the limit value of  $H_k$  for  $k \to \infty$  the omission of digits and punctuation will probably have little effect.
*If one leaves also the blank spaces  $(\rm BS)$  out of consideration  $($Row 4   ⇒   $M = 30)$, the result is almost the same constellation as Küpfmüller originally considered.  The only difference are the rather rare German special characters '''ä''',  '''ö''',  '''ü'''  and  '''ß'''.
*The  $H_1$–value indicated in the last row  $(4.132)$  corresponds very well with the value  $H_1 ≈ 4.1$  determined by Küpfmüller.   However, with regard to the  $H_3$–values there are clear differences:   Our analysis results in a larger value  $(H_3 ≈ 3.4)$  than Küpfmüller  $(H_3 ≈ 2.8)$.
*From the frequency of the blank spaces  $(17.8\%)$  here results an average word length of  $1/0.178 - 1 ≈ 4.6$, a smaller value than Küpfmüller  ($5.5$)  had given.  The discrepancy can be partly explained with our analysis file "Bible"  (many spaces due to verse numbering).
*Interesting is the comparison of lines 3 and 4.  If  $\rm BS$  is taken into account, then although  $H_0$  from  $\log_2 \ (30) \approx 4.907$  to  $\log_2 \ (31) \approx 4. 954$  enlarges, but thereby reduces  $H_1$  $($by the factor  $0.98)$,  $H_2$  $($by  $0.96)$  and  $H_3$  $($by  $0.93)$. Küpfmüller has intuitively taken this factor into account with  $85\%$.

Although we consider this own study to be rather insignificant, we believe that for today's texts the  $1.0 \ \rm bit/character$  given by Shannon are somewhat too low for the English language and also Küpfmüllers  $1.3 \ \rm bit/character$  for the German language, among other things because:
*The symbol set size today is larger than that considered by Shannon and Küpfmüller in the 1950s; for example, for the ASCII character set  $M = 256$.
*The multiple formatting options (underlining, bold and italics, indents, colors) further increase the information content of a document.

==Synthetically generated texts ==
 
The graphic shows artificially generated German and English texts, which are taken from  [Küpf54]<ref name ='Küpf54'>Küpfmüller, K.:  Die Entropie der deutschen Sprache.  Fernmeldetechnische Zeitung 7, 1954, S. 265-272.</ref>.  The underlying symbol set size is  $M = 27$,  that means, all letters  (without umlauts and  '''ß''')  and the space character are considered.

[[File:EN_Inf_T_1_3_S4_v4.png|right|frame|Artificially generated German and English texts]]

*The  "Zero-order Character Approximation"  assumes equally probable characters in each case.  There is therefore no difference between German (red) and English (blue).

*The  "First-order Character Approximation"  already considers the different frequencies, the higher order approximations also the preceding characters.

*In the  "Fourth-order Character Approximation"  one can already recognize meaningful words.  Here the probability for a new letter depends on the last three characters.

*The  "First-order Word Approximation"  synthesizes sentences according to the word probabilities.  The  "Second-order Word Approximation"  also considers the previous word.

Further information on the synthetic generation of German and English texts can be found in the  [[Aufgaben:1.8_Synthetisch_erzeugte_Texte|Exercise 1.8]].

==Exercises for the chapter==
 
[[Aufgaben:1.7 Entropie natürlicher Texte|Aufgabe 1.7: Entropie natürlicher Texte]]

[[Aufgaben:1.8 Synthetisch erzeugte Texte|Aufgabe 1.8: Synthetisch erzeugte Texte]]

==References==
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