Difference between revisions of "Digital Signal Transmission/Signals, Basis Functions and Vector Spaces"

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{{Header
 
{{Header
|Untermenü=Verallgemeinerte Beschreibung digitaler Modulationsverfahren
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|Untermenü=Generalized Description of Digital Modulation Methods
 
|Vorherige Seite=Viterbi–Empfänger
 
|Vorherige Seite=Viterbi–Empfänger
 
|Nächste Seite=Struktur des optimalen Empfängers
 
|Nächste Seite=Struktur des optimalen Empfängers
 
}}
 
}}
  
== # ÜBERBLICK ZUM VIERTEN HAUPTKAPITEL # ==
+
== # OVERVIEW OF THE FOURTH MAIN CHAPTER # ==
 
<br>
 
<br>
Das vierte Hauptkapitel liefert eine abstrahierte Beschreibung der Digitalsignalübertragung, die auf Basisfunktionen und Signalraumkonstellationen aufbaut. Dadurch ist es möglich, sehr unterschiedliche Konfigurationen – zum Beispiel Bandpass–Systeme und solche für das Basisband – in einheitlicher Form zu behandeln. Der jeweils optimale Empfänger besitzt in allen Fällen die gleiche Struktur.
+
The fourth main chapter provides an abstract description of digital signal transmission,&nbsp; which is based on basis functions and signal space constellations.&nbsp; This makes it possible to treat very different configurations &ndash; for example band-pass systems and those for the baseband &ndash; in a uniform way.&nbsp; The optimal receiver in each case has the same structure in all cases.
  
Im Einzelnen werden behandelt:
+
The following are dealt with in detail:
*die Bedeutung von Basisfunktionen und deren Auffinden nach dem Gram–Schmidt–Verfahren,
+
#&nbsp; The meaning of&nbsp; &raquo;basis functions&laquo;&nbsp; and finding them using the&nbsp; &raquo;Gram-Schmidt process&laquo;,
*die Struktur des optimalen Empfängers für die Basisbandübertragung,
+
#&nbsp; the&nbsp; &raquo;structure of the optimal receiver&laquo;&nbsp; for baseband transmission,
*das Theorem der Irrelevanz und dessen Bedeutung für die Herleitung optimaler Detektoren,
+
#&nbsp; the&nbsp; &raquo;theorem of irrelevance&laquo;&nbsp; and its importance for the derivation of optimal detectors,
*der optimale Empfänger für den AWGN–Kanal und Implementierungsaspekte,
+
#&nbsp; the&nbsp; &raquo;optimal receiver for the AWGN channel&laquo;&nbsp; and implementation aspects,
*die Systembeschreibung durch komplexes bzw. &nbsp;$N$–dimensionales Gaußsches Rauschen,
+
#&nbsp; the system description by&nbsp; &raquo;complex or &nbsp;$N$–dimensional Gaussian noise&laquo;,
*die Fehlerwahrscheinlichkeitsberechnung und –approximation bei sonst idealen Bedingungen,
+
#&nbsp; the&nbsp; &raquo;error probability calculation and approximation&laquo;&nbsp; under otherwise ideal conditions,
*die Anwendung der Signalraumbeschreibung auf Trägerfrequenzsysteme,
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#&nbsp; the application of the signal space description to&nbsp; &raquo;carrier frequency systems&laquo;,
*die unterschiedlichen Ergebnisse für OOK, M–ASK, M–PSK, M–QAM und M–FSK,
+
#&nbsp; the different results for&nbsp; &raquo;OOK, M-ASK, M-PSK, M-QAM and M-FSK&laquo;,
*die unterschiedlichen Ergebnisse für kohärente bzw. nichtkohärente Demodulation.
+
#&nbsp; the different results for&nbsp; &raquo;coherent and non-coherent demodulation&laquo;.
  
  
Nahezu alle Ergebnisse dieses Kapitels wurden bereits in früheren Abschnitten hergeleitet. Grundlegend neu ist jedoch die Herangehensweise:
+
Almost all results of this chapter have already been derived in previous sections.&nbsp; However,&nbsp; the approach is fundamentally new:
*Im $\rm LNTwww$&ndash;Buch &bdquo;Modulationsverfahren&rdquo; sowie in den ersten drei Kapiteln dieses Buches wurden bereits bei den Herleitungen die spezifischen Systemeigenschaften berücksichtigt &ndash; zum Beispiel, ob die Übertragung des Digitalsignals im Basisband erfolgt oder ob eine digitale Amplituden&ndash;, Frequenz&ndash; oder Phasenmodulation vorliegt.<br>
+
*In the&nbsp; $\rm LNTwww$&nbsp; book&nbsp; "Modulation Methods"&nbsp; and in the first three chapters of this book,&nbsp; the specific system properties were already taken into account in the derivations &ndash; for example,&nbsp; whether the digital signal is transmitted in baseband or whether digital amplitude,&nbsp; frequency or phase modulation is present.<br>
*Hier sollen nun die Systeme dahingehend abstrahiert werden, dass sie einheitlich behandelt werden können. Der jeweils optimale Empfänger besitzt in allen Fällen die gleiche Struktur, und die Fehlerwahrscheinlichkeit lässt sich auch für nichtgaußverteiltes Rauschen angeben.<br><br>
 
  
Anzumerken ist, dass sich durch diese eher globale Vorgehensweise gewisse Systemunzulänglichkeiten nur sehr ungenau erfassen lassen, wie zum Beispiel
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*Here the systems are to be abstracted in such a way that they can be treated uniformly.&nbsp; The optimal receiver in each case has the same structure in all cases,&nbsp; and the error probability can also be specified for non-Gaussian distributed noise.<br><br>
*der Einfluss eines  nichtoptimalen Empfangsfilters auf die Fehlerwahrscheinlichkeit,<br>
 
*ein falscher Schwellenwert (Schwellendrift), oder<br>
 
*Phasenjitter (Schwankungen der Abtastzeitpunkte).<br><br>
 
  
Insbesondere bei Vorhandensein von Impulsinterferenzen sollte also weiterhin entsprechend dem&nbsp; [[Digitalsignalübertragung/Ursachen_und_Auswirkungen_von_Impulsinterferenzen#.23_.C3.9CBERBLICK_ZUM_DRITTEN_HAUPTKAPITEL_.23|Hauptkapitel 3]]&nbsp; vorgegangen werden.<br>
+
It should be noted that this rather global approach means that certain system deficiencies can only be recorded very imprecisely,&nbsp; such as
 +
*the influence of a non-optimal receiver filter on the error probability,<br>
 +
*an incorrect threshold&nbsp; $($threshold drift$)$,&nbsp; or<br>
 +
*phase jitter&nbsp; $($fluctuations in sampling times$)$.<br><br>
  
Die Beschreibung basiert auf dem Skript [KöZ08]<ref name='KöZ08'>Kötter, R., Zeitler, G.: ''Nachrichtentechnik 2.'' Vorlesungsmanuskript, Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2008.</ref> von&nbsp; [[Biografien_und_Bibliografien/Lehrstuhlinhaber_des_LNT#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]&nbsp; und&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Georg_Zeitler_.28am_LNT_von_2007-2012.29|Georg Zeitler]], das sich stark an das Lehrbuch [WJ65]<ref name='WJ65'>Wozencraft, J. M.; Jacobs, I. M.: ''Principles of Communication Engineering.'' New York: John Wiley & Sons, 1965.</ref> anlehnt. [[Biografien_und_Bibliografien/Lehrstuhlinhaber_des_LNT#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Gerhard Kramer]], Lehrstuhlinhaber des LNT seit 2010, behandelt in seiner Vorlesung [Kra17]<ref>Kramer, G.: ''Nachrichtentechnik 2.'' Vorlesungsmanuskript, Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2017.</ref> die gleiche Thematik mit sehr ähnlicher Nomenklatur.<br>
+
In particular in the presence of intersymbol interference,&nbsp; the procedure should therefore continue in accordance with the&nbsp; [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference#.23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_.23|third main chapter]].&nbsp; <br>
  
Um unseren eigenen Studenten an der TU München das Lesen nicht unnötig zu erschweren, halten wir uns weitestgehend an diese Nomenklatur, auch wenn diese von anderen $\rm LNTwww$&ndash;Kapiteln abweicht.<br>
+
The description is based on the script&nbsp; [KöZ08]<ref name='KöZ08'>Kötter, R., Zeitler, G.:&nbsp; Lecture notes, Institute for Communications Engineering, Technical University of Munich, 2008.</ref> by&nbsp; [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]&nbsp; and&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Georg_Zeitler_.28at_LNT_from_2007-2012.29|Georg Zeitler]],&nbsp; which is closely based on the textbook [WJ65]<ref name='WJ65'>Wozencraft, J. M.; Jacobs, I. M.:&nbsp; Principles of Communication Engineering.&nbsp; New York: John Wiley & Sons, 1965.</ref>. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Gerhard Kramer]],&nbsp; who has held the chair at the LNT since 2010,&nbsp; treats the same topic with very similar nomenclature in his lecture [Kra17]<ref>Kramer, G.:&nbsp; Nachrichtentechnik 2. Lecture notes, Institute for Communications Engineering, Technical University of Munich, 2017.</ref>.&nbsp; In order not to make reading unnecessarily difficult for our own students at TU Munich,&nbsp; we stick to this nomenclature as far as possible,&nbsp; even if it deviates from other&nbsp; $\rm LNTwww$&nbsp; chapters.<br>
  
== Zur Nomenklatur im vierten Kapitel==
+
== Nomenclature in the fourth chapter==
 
<br>
 
<br>
Gegenüber den anderen Kapiteln in $\rm LNTwww$ ergeben sich hier folgende Nomenklaturunterschiede:
+
Compared to the other&nbsp;  $\rm LNTwww$&nbsp; chapters,&nbsp; the following nomenclature changes arise here:
*Die zu übertragende&nbsp; [[Signaldarstellung/Prinzip_der_Nachrichtenübertragung#Nachricht_-_Information_-_Signal|Nachricht]]&nbsp; ist ein ganzzahliger Wert&nbsp; $m \in \{m_i\}$&nbsp; mit &nbsp;$i = 0$, ... , $M-1$, wobei &nbsp;$M$&nbsp; den Symbolumfang angibt. Wenn es die Beschreibung vereinfacht, wird &nbsp;$i = 1$, ... , $M$&nbsp; &nbsp;induziert.<br>
+
*The&nbsp; [[Signal_Representation/Principles_of_Communication#Message_-_Information_-_Signal|"message"]]&nbsp; to be transmitted is an integer value&nbsp; $m \in \{m_i\}$&nbsp; with &nbsp;$i = 0$, ... , $M-1$,&nbsp; where &nbsp;$M$&nbsp; specifies the&nbsp; "symbol set size". <br>If it simplifies the description, &nbsp;$i = 1$, ... , $M$&nbsp; &nbsp; is induced.<br>
  
 +
*The result of the decision process at the receiver is also an integer with the same symbol alphabet as at the transmitter.&nbsp; <br>This result is also referred to as the&nbsp; "estimated value":
 +
:$$\hat{m} \in \{m_i \}, \hspace{0.2cm} i = 0, 1, \text{...}\hspace{0.05cm} , M-1\hspace{0.2cm} ({\rm or}\,\,i = 1, 2, \text{...}\hspace{0.05cm}, M) \hspace{0.05cm}.$$
  
*Das Ergebnis des Entscheidungsprozesses beim Empfänger ist ebenfalls ein Integerwert mit dem gleichen Symbolalphabet wie beim Sender. Man bezeichnet dieses Ergebnis auch als den ''Schätzwert'':
+
*The&nbsp; [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbol_and_bit_error_probability|"symbol error probability"]]&nbsp; $\rm Pr(symbol\hspace{0.15cm}  error)$&nbsp; or&nbsp; $p_{\rm S}$&nbsp; is usually referred to as follows in this main chapter:
:$$\hat{m} \in \{m_i \}, \hspace{0.2cm} i = 0, 1, \text{...}\hspace{0.05cm} , M-1\hspace{0.2cm} ({\rm bzw.}\,\,i = 1, 2, \text{...}\hspace{0.05cm}, M) \hspace{0.05cm}.$$
 
 
 
*Die&nbsp; [[Digitalsignalübertragung/Redundanzfreie_Codierung#Symbol.E2.80.93_und_Bitfehlerwahrscheinlichkeit|Symbolfehlerwahrscheinlichkeit]]&nbsp; $\rm Pr(Symbolfehler)$&nbsp; oder auch $p_{\rm S}$&nbsp; wird in diesem Hauptkapitel  meist wie folgt bezeichnet:
 
 
:$${\rm Pr}  ({\cal E}) = {\rm Pr} ( \hat{m} \ne m) = 1 -  {\rm Pr}  ({\cal C}),
 
:$${\rm Pr}  ({\cal E}) = {\rm Pr} ( \hat{m} \ne m) = 1 -  {\rm Pr}  ({\cal C}),
\hspace{0.4cm}\text{Komplementärereignis:}\hspace{0.2cm} {\rm Pr}  ({\cal C}) = {\rm Pr} ( \hat{m} = m) \hspace{0.05cm}.$$
+
\hspace{0.4cm}\text{complementary event:}\hspace{0.2cm} {\rm Pr}  ({\cal C}) = {\rm Pr} ( \hat{m} = m) \hspace{0.05cm}.$$
 
 
*Bei einer&nbsp; [[Stochastische_Signaltheorie/Wahrscheinlichkeitsdichtefunktion|Wahrscheinlichkeitsdichtefunktion]]&nbsp; (WDF) wird nun entsprechend&nbsp; $p_r(\rho)$&nbsp;  zwischen der ''Zufallsgröße'' &nbsp; &rArr; &nbsp; $r$&nbsp; und der ''Realisierung'' &nbsp; &rArr; &nbsp; $\rho$&nbsp;  unterschieden. Bisher wurde für eine WDF die Bezeichnung &nbsp;$f_r(r)$&nbsp; verwendet.<br>
 
 
 
 
 
*Mit der Schreibweise &nbsp;$p_r(\rho)$&nbsp; sind &nbsp;$r$&nbsp; und &nbsp;$\rho$&nbsp; Skalare. Sind dagegen Zufallsgröße und Realisierung Vektoren (geeigneter Länge), so wird dies durch Fettschrift ausgedrückt: &nbsp; &nbsp; $p_{ \boldsymbol{ r}}(\boldsymbol{\rho})$&nbsp; mit den Vektoren &nbsp;$ \boldsymbol{ r}$&nbsp; und &nbsp;$\boldsymbol{\rho}$.
 
  
 +
*In a&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function|"probability density function"]]&nbsp; $\rm (PDF)$,&nbsp; a distinction is made between the&nbsp; "random variable" &nbsp; &rArr; &nbsp; $r$&nbsp; and the&nbsp; "realization" &nbsp; &rArr; &nbsp; $\rho$&nbsp; according to &nbsp; $p_r(\rho)$.&nbsp; <br>Formerly,  &nbsp;$f_r(r)$&nbsp; was used for this PDF. <br>
  
*Um Verwechslungen mit Energiewerten zu vermeiden, heißt nun der Schwellenwert &nbsp;$G$&nbsp; anstelle von &nbsp;$E$&nbsp; und dieser wird in diesem Kapitel vorwiegend als ''Entscheidungsgrenze'' bezeichnet.
+
*With the notation &nbsp;$p_r(\rho)$,&nbsp; &nbsp;$r$&nbsp; and &nbsp;$\rho$&nbsp; are scalars.&nbsp; On the other hand,&nbsp; if random variable and realization are vectors&nbsp; (of suitable length),&nbsp; this is expressed in bold type: &nbsp; &nbsp; $p_{ \boldsymbol{ r}}(\boldsymbol{\rho})$&nbsp; with the vectors &nbsp;$ \boldsymbol{ r}$&nbsp; and &nbsp;$\boldsymbol{\rho}$.
  
 +
*In order to avoid confusion with energy values,&nbsp; the&nbsp; "threshold value is"&nbsp; now called &nbsp;$G$&nbsp; instead of &nbsp;$E$.&nbsp; This is mainly referred to as the&nbsp; "decision threshold"&nbsp; in this chapter. 
  
*Ausgehend von den beiden reellen und energiebegrenzten Zeitfunktionen &nbsp;$x(t)$&nbsp; und &nbsp;$y(t)$&nbsp; erhält man für das &nbsp;[https://de.wikipedia.org/wiki/Inneres_Produkt innere Produkt]:
+
*Based on the two real and energy-limited time functions &nbsp;$x(t)$&nbsp; and &nbsp;$y(t)$,&nbsp; the &nbsp;[https://de.wikipedia.org/wiki/Inneres_Produkt "inner product"]&nbsp; is:
 
:$$<\hspace{-0.1cm}x(t), \hspace{0.05cm}y(t) \hspace{-0.1cm}> \hspace{0.15cm}= \int_{-\infty}^{+\infty}x(t) \cdot y(t)\,d \it t
 
:$$<\hspace{-0.1cm}x(t), \hspace{0.05cm}y(t) \hspace{-0.1cm}> \hspace{0.15cm}= \int_{-\infty}^{+\infty}x(t) \cdot y(t)\,d \it t
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
 
+
* This results in the&nbsp; [https://en.wikipedia.org/wiki/Euclidean_space#Euclidean_norm "Euclidean norm"]&nbsp; or&nbsp; "2&ndash;norm"&nbsp; $($or&nbsp; "norm"&nbsp; for short$)$:
* Daraus ergibt sich die&nbsp; [https://de.wikipedia.org/wiki/Euklidische_Norm Euklidische Norm]&nbsp; oder &bdquo;2&ndash;Norm&rdquo; (oder kurz &bdquo;Norm&rdquo;):
 
 
:$$||x(t) || = \sqrt{<\hspace{-0.1cm}x(t), \hspace{0.05cm}x(t) \hspace{-0.1cm}>}  
 
:$$||x(t) || = \sqrt{<\hspace{-0.1cm}x(t), \hspace{0.05cm}x(t) \hspace{-0.1cm}>}  
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
 +
*Compared to the script &nbsp;[KöZ08]<ref name='KöZ08' />,&nbsp; the naming differs as follows:
 +
#The probability of the event &nbsp;$E$&nbsp; is &nbsp;${\rm Pr}(E)$&nbsp; instead of &nbsp;$P(E)$.&nbsp; <br>This nomenclature change was also made because in some equations&nbsp; "probabilities"&nbsp; and&nbsp; "powers"&nbsp; appear together.<br>
 +
#Band&ndash;pass signals are still marked with the index "BP" and not with a tilde as in&nbsp; [KöZ08]<ref name='KöZ08' />. <br>The corresponding&nbsp; "low-pass signal"&nbsp; is&nbsp; (usually)&nbsp; provided with the index&nbsp; "TP"&nbsp; $($from German&nbsp; "Tiefpass"$)$.<br>
  
Gegenüber dem Skript &nbsp;$\rm [KöZ08]$<ref name='KöZ08' /> unterscheidet sich die Bezeichnungsweise hier wie folgt:
+
== Orthonormal basis functions ==
*Die Wahrscheinlichkeit des Ereignisses &nbsp;$E$&nbsp; ist hier &nbsp;${\rm Pr}(E)$&nbsp; anstelle von &nbsp;$P(E)$. Diese Nomenklaturänderung wurde auch deshalb vorgenommen, da in manchen Gleichungen Wahrscheinlichkeiten und Leistungen gemeinsam vorkommen.<br>
 
 
 
 
 
*Bandpass&ndash;Signale werden weiterhin mit dem Index &bdquo;BP&rdquo; gekennzeichnet und nicht wie in  [KöZ08]<ref name='KöZ08' /> mit einer Tilde. Das entsprechende Tiefpass&ndash;Signal ist (meist) mit dem Index &bdquo;TP&rdquo; versehen.<br>
 
 
 
== Orthonormale Basisfunktionen ==
 
 
<br>
 
<br>
Wir gehen in diesem Kapitel von einer Menge $\{s_i(t)\}$ möglicher Sendesignale aus, die den möglichen Nachrichten $m_i$ eineindeutig zugeordnet sind. Mit $i = 1$, ... , $M$ gilt:
+
In this chapter,&nbsp; we assume a set &nbsp;$\{s_i(t)\}$&nbsp; of possible transmitted signals that are uniquely assigned to the possible messages &nbsp;$m_i$.&nbsp; With &nbsp;$i = 1$, ... , $M$&nbsp; holds:
:$$m \in \{m_i \}, \hspace{0.2cm} s(t) \in \{s_i(t) \}\hspace{-0.1cm}: m = m_i  \hspace{0.1cm} \Leftrightarrow \hspace{0.1cm} s(t) = s_i(t) \hspace{0.05cm}.$$
+
:$$m \in \{m_i \}, \hspace{0.2cm} s(t) \in \{s_i(t) \}\hspace{-0.1cm}: \hspace{0.3cm} m = m_i  \hspace{0.1cm} \Leftrightarrow \hspace{0.1cm} s(t) = s_i(t) \hspace{0.05cm}.$$
  
Für das Folgende setzen wir weiter voraus, dass die $M$ Signale $s_i(t)$ [[Signaldarstellung/Klassifizierung_von_Signalen#Energiebegrenzte_und_leistungsbegrenzte_Signale| energiebegrenzt]] sind, was meist gleichzeitig bedeutet, dass sie nur von endlicher Dauer sind.<br>
+
For what follows,&nbsp; we further assume that the&nbsp; $M$ signals&nbsp; $s_i(t)$&nbsp; are&nbsp; [[Signal_Representation/Signal_classification#Energy.E2.80.93Limited_and_Power.E2.80.93Limited_Signals| "energy-limited"]],&nbsp; which usually means at the same time that they are of finite duration.<br>
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Satz:}$&nbsp; Eine jede Menge $\{s_1(t), \hspace{0.05cm}  \text{...} \hspace{0.05cm} , s_M(t)\}$ energiebegrenzter Signale lässt sich in $N \le M$  '''orthonormale Basisfunktionen''' $\varphi_1(t),  \hspace{0.05cm} \text{...} \hspace{0.05cm} , \varphi_N(t)$ entwickeln, wobei gilt:
+
$\text{Theorem:}$&nbsp; Any set&nbsp; $\{s_1(t), \hspace{0.05cm}  \text{...} \hspace{0.05cm} , s_M(t)\}$&nbsp; of energy-limited signals can be evolved into&nbsp; $N \le M$&nbsp; '''orthonormal basis functions'''&nbsp; $\varphi_1(t),  \hspace{0.05cm} \text{...} \hspace{0.05cm} , \varphi_N(t)$.&nbsp; It holds:
  
 
:$$s_i(t) = \sum\limits_{j = 1}^{N}s_{ij} \cdot \varphi_j(t) ,
 
:$$s_i(t) = \sum\limits_{j = 1}^{N}s_{ij} \cdot \varphi_j(t) ,
Line 88: Line 79:
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
Jeweils zwei Basisfunktionen $\varphi_j(t)$ und $\varphi_k(t)$ müssen orthonormal zueinander sein, das heißt, es muss gelten ($\delta_{jk}$ nennt man das [https://de.wikipedia.org/wiki/Kronecker-Delta Kronecker&ndash;Symbol] oder das &bdquo;Kronecker-Delta&rdquo;):
+
*In each case, two basis functions&nbsp; $\varphi_j(t)$&nbsp; and &nbsp;$\varphi_k(t)$&nbsp; must be orthonormal to each other, that is, it must hold &nbsp; <br>$(\delta_{jk}$&nbsp; is called&nbsp; [https://en.wikipedia.org/wiki/Kronecker_delta "Kronecker symbol"]&nbsp; or&nbsp; "Kronecker delta"$)$:
  
 
:$$<\hspace{-0.1cm}\varphi_j(t), \hspace{0.05cm}\varphi_k(t) \hspace{-0.1cm}> = \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\,d \it t = {\rm \delta}_{jk} =
 
:$$<\hspace{-0.1cm}\varphi_j(t), \hspace{0.05cm}\varphi_k(t) \hspace{-0.1cm}> = \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\,d \it t = {\rm \delta}_{jk} =
 
\left\{ \begin{array}{c} 1 \\
 
\left\{ \begin{array}{c} 1 \\
 
  0  \end{array} \right.\quad
 
  0  \end{array} \right.\quad
\begin{array}{*{1}c} {\rm falls}\hspace{0.1cm}j = k
+
\begin{array}{*{1}c} {\rm if}\hspace{0.1cm}j = k
\\ {\rm falls}\hspace{0.1cm} j \ne k \\ \end{array}
+
\\ {\rm if}\hspace{0.1cm} j \ne k \\ \end{array}
 
  \hspace{0.05cm}.$$}}<br>
 
  \hspace{0.05cm}.$$}}<br>
  
Der Parameter $N$ gibt dabei an, wieviele Basisfunktionen $\varphi_j(t)$ benötigt werden, um die $M$ möglichen Sendesignale darzustellen. Mit anderen Worten: $N$ ist die ''Dimension des Vektorraums'', der von den $M$ Signalen aufgespannt wird. Dabei gilt:
+
Here,&nbsp; the parameter&nbsp; $N$&nbsp; indicates how many basis functions&nbsp; $\varphi_j(t)$&nbsp; are needed to represent the&nbsp; $M$&nbsp; possible transmitted signals.&nbsp; In other words: &nbsp; $N$&nbsp; is the&nbsp; "dimension of the vector space"&nbsp; spanned by the&nbsp; $M$&nbsp; signals.&nbsp; Here,&nbsp; the following holds:
*Ist $N = M$, so sind alle Sendesignale zueinander orthogonal. Sie sind nicht notwendigerweise orthonormal, das heißt, die Energien $E_i = <\hspace{-0.1cm}s_i(t), \hspace{0.05cm}s_i(t) \hspace{-0.1cm}>$ können durchaus ungleich Eins sein.<br>
+
#If&nbsp; $N = M$,&nbsp; all transmitted signals are orthogonal to each other.  
*Der Fall $N < M$ ergibt sich, wenn mindestens ein Signal $s_i(t)$ als Linearkombination von Basisfunktionen $\varphi_j(t)$ dargestellt werden kann, die sich aus anderen Signalen $s_j(t) \ne s_i(t)$ ergeben haben.<br>
+
#They are not necessarily orthonormal,&nbsp; i.e. the energies&nbsp; $E_i = <\hspace{-0.1cm}s_i(t), \hspace{0.05cm}s_i(t) \hspace{-0.1cm}>$&nbsp; may well be unequal to one.<br>
 +
#$N < M$&nbsp; arises when at least one signal&nbsp; $s_i(t)$&nbsp; can be represented as linear combination of basis functions&nbsp; $\varphi_j(t)$&nbsp; that have resulted from other signals&nbsp; $s_j(t) \ne s_i(t)$.&nbsp; <br>
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 1:}$&nbsp; Wir betrachten $M = 3$ energiebegrenzte Signale gemäß der Grafik. Man erkennt sofort:  
+
$\text{Example 1:}$&nbsp; We consider&nbsp; $M = 3$&nbsp; energy-limited signals according to the graph.&nbsp; One recognizes immediately:
*Die Signale $s_1(t)$ und $s_2(t)$ sind zueinander orthogonal.<br>
+
[[File:P ID1993 Dig T 4 1 S2 version1.png|right|frame|Representation of three transmitted signals by two basis functions|class=fit]]
 +
 
 +
*The signals&nbsp; $s_1(t)$&nbsp; and &nbsp;$s_2(t)$&nbsp; are orthogonal to each other.<br>
  
*Die Energien sind $E_1 = A^2 \cdot T = E$ und $E_2 = (A/2)^2 \cdot T = E/4$.<br>
+
*The energies are&nbsp; $E_1 = A^2 \cdot T = E$ &nbsp; and &nbsp; $E_2 = (A/2)^2 \cdot T = E/4$.<br>
  
*Die Basisfunktionen $\varphi_1(t)$ und $\varphi_2(t)$ sind jeweils formgleich mit $s_1(t)$ bzw. $s_2(t)$ und beide besitzen die Energie Eins:
+
*The basis functions&nbsp; $\varphi_1(t)$&nbsp; and &nbsp;$\varphi_2(t)$&nbsp; are equal in form to&nbsp; $s_1(t)$&nbsp; and&nbsp; $s_2(t)$,&nbsp; resp., and both have energy one:
  
:$$\varphi_1(t)=\frac{s_1(t)}{\sqrt{E_1} } = \frac{s_1(t)}{\sqrt{A^2 \cdot T} } = \frac{1}{\sqrt{ T} }  \cdot \frac{s_1(t)}{A}\hspace{0.95cm}\Rightarrow \hspace{0.1cm}s_1(t) = s_{11} \cdot \varphi_1(t)\hspace{0.05cm},\hspace{0.1cm}s_{11} = \sqrt{E}\hspace{0.05cm},$$
+
:$$\varphi_1(t)=\frac{s_1(t)}{\sqrt{E_1} } = \frac{s_1(t)}{\sqrt{A^2 \cdot T} } = \frac{1}{\sqrt{ T} }  \cdot \frac{s_1(t)}{A}$$
:$$\varphi_2(t) =\frac{s_2(t)}{\sqrt{E_2} } = \frac{s_2(t)}{\sqrt{(A/2)^2 \cdot T} } = \frac{1}{\sqrt{ T} }  \cdot \frac{s_2(t)}{A/2}\hspace{0.05cm}\hspace{0.1cm}\Rightarrow \hspace{0.1cm}s_2(t) = s_{21} \cdot \varphi_2(t)\hspace{0.05cm},\hspace{0.1cm}s_{21} = {\sqrt{E} }/{2}\hspace{0.05cm}.$$
+
:$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}s_1(t) = s_{11} \cdot \varphi_1(t)\hspace{0.05cm},\hspace{0.1cm}s_{11} = \sqrt{E}\hspace{0.05cm},$$
 +
:$$\varphi_2(t) =\frac{s_2(t)}{\sqrt{E_2} } = \frac{s_2(t)}{\sqrt{(A/2)^2 \cdot T} } = \frac{1}{\sqrt{ T} }  \cdot \frac{s_2(t)}{A/2}\hspace{0.05cm}$$
 +
:$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}s_2(t) = s_{21} \cdot \varphi_2(t)\hspace{0.05cm},\hspace{0.1cm}s_{21} = {\sqrt{E} }/{2}\hspace{0.05cm}.$$
  
*Das Signal $s_3(t)$ kann durch die Basisfunktionen $\varphi_1(t)$ und $\varphi_2(t)$ ausgedrückt werden:
+
*$s_3(t)$&nbsp; can be expressed by the previously determined basis functions&nbsp; $\varphi_1(t)$,&nbsp; $\varphi_2(t)$:&nbsp;
:$$s_3(t) =s_{31} \cdot \varphi_1(t) + s_{32} \cdot \varphi_2(t)\hspace{0.05cm},\hspace{0.3cm}
+
:$$s_3(t) =s_{31} \cdot \varphi_1(t) + s_{32} \cdot \varphi_2(t)\hspace{0.05cm},$$
 +
:$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}
 
s_{31} = {A}/{2} \cdot \sqrt {T}=  {\sqrt{E} }/{2}\hspace{0.05cm}, \hspace{0.2cm}s_{32} = - A \cdot \sqrt {T} = -\sqrt{E}  \hspace{0.05cm}.$$
 
s_{31} = {A}/{2} \cdot \sqrt {T}=  {\sqrt{E} }/{2}\hspace{0.05cm}, \hspace{0.2cm}s_{32} = - A \cdot \sqrt {T} = -\sqrt{E}  \hspace{0.05cm}.$$
  
[[File:P ID1993 Dig T 4 1 S2 version1.png|right|frame|Darstellung der drei Sendesignale durch zwei Basisfunktionen|class=fit]]
 
  
<br>Im rechten unteren Bild sind die Signale in einer 2D&ndash;Darstellung mit den Basisfunktionen $\varphi_1(t)$ und $\varphi_2(t)$ als Achsen dargestellt, wobei $E = A^2 \cdot T$ gilt und der Zusammenhang zu den anderen Grafiken durch die Farbgebung zu erkennen ist.  
+
&rArr;  &nbsp; In the lower right image,&nbsp;  the signals are shown in a two-dimensional representation
 +
*with the basis functions&nbsp; $\varphi_1(t)$&nbsp; and &nbsp;$\varphi_2(t)$&nbsp; as axes,  
 +
*where&nbsp; $E = A^2 \cdot T$&nbsp; and the relation to the other graphs can be seen by the coloring.
 +
 
  
Die vektoriellen Repräsentanten der Signale $s_1(t)$, $s_2(t)$ und $s_3(t)$ in diesem zweidimensionellen Vektorraum lassen sich daraus wie folgt ablesen:
+
&rArr;  &nbsp; The vectorial representatives of the signals&nbsp; $s_1(t)$,&nbsp; $s_2(t)$&nbsp; and&nbsp; $s_3(t)$&nbsp; in the two-dimensional vector space can be read from this sketch as follows:
 
:$$\mathbf{s}_1 = (\sqrt{ E}, \hspace{0.1cm}0), $$
 
:$$\mathbf{s}_1 = (\sqrt{ E}, \hspace{0.1cm}0), $$
 
:$$\mathbf{s}_2 = (0, \hspace{0.1cm}\sqrt{ E}/2), $$
 
:$$\mathbf{s}_2 = (0, \hspace{0.1cm}\sqrt{ E}/2), $$
Line 127: Line 126:
 
<br clear= all>
 
<br clear= all>
  
== Das Verfahren nach Gram-Schmidt==
+
== The Gram-Schmidt process==
 
<br>
 
<br>
Im letzten Beispiel auf der Seite war die Angabe der beiden orthonormalen Basisfunktionen $\varphi_1(t)$ und $\varphi_2(t)$ sehr einfach, da diese formgleich mit $s_1(t)$ bzw. $s_2(t)$ waren. Das [https://de.wikipedia.org/wiki/Gram-Schmidtsches_Orthogonalisierungsverfahren Gram&ndash;Schmidt&ndash;Verfahren] findet die Basisfunktionen $\varphi_1(t)$, ... , $\varphi_N(t)$ für beliebig vorgebbare Signale $s_1(t)$, ... , $s_M(t)$, und zwar wie folgt:
+
In &nbsp;$\text{Example 1}$&nbsp; in the last section,&nbsp; the specification of the two orthonormal basis functions&nbsp; $\varphi_1(t)$&nbsp; and&nbsp; $\varphi_2(t)$&nbsp; was very easy,&nbsp; because they were of the same form as&nbsp; $s_1(t)$&nbsp; and&nbsp; $s_2(t)$,&nbsp; respectively. The&nbsp; [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process "Gram-Schmidt process"]&nbsp; finds the basis functions&nbsp; $\varphi_1(t)$, ... , $\varphi_N(t)$&nbsp; for arbitrary predefinable signals&nbsp; $s_1(t)$, ... , $s_M(t)$, as follows:
  
*Die erste Basisfunktion $\varphi_1(t)$ ist stets formgleich mit $s_1(t)$. Es gilt:
+
*The first basis function&nbsp; $\varphi_1(t)$&nbsp; is always equal in form to&nbsp; $s_1(t)$.&nbsp; It holds:
 
:$$\varphi_1(t) = \frac{s_1(t)}{\sqrt{E_1}} = \frac{s_1(t)}{|| s_1(t)||}
 
:$$\varphi_1(t) = \frac{s_1(t)}{\sqrt{E_1}} = \frac{s_1(t)}{|| s_1(t)||}
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_1(t) || = 1, \hspace{0.2cm}s_{11} =|| s_1(t)||,\hspace{0.2cm}s_{1j} = 0 \hspace{0.2cm}{\rm f{\rm \ddot{u}r }}\hspace{0.2cm} j \ge 2
+
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_1(t) || = 1, \hspace{0.2cm}s_{11} =|| s_1(t)||,\hspace{0.2cm}s_{1j} = 0 \hspace{0.2cm}{\rm f{\rm or }}\hspace{0.2cm} j \ge 2
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
*Es wird nun angenommen, dass aus den Signalen $s_1(t)$, ... , $s_{k-1}(t)$ bereits die Basisfunktionen $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$ berechnet wurden $(n \le k)$. Dann berechnen wir mittels $s_k(t)$ die Hilfsfunktion
+
*It is now assumed that from the signals&nbsp; $s_1(t)$, ... , $s_{k-1}(t)$&nbsp; the basis functions&nbsp; $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$&nbsp; have been calculated &nbsp;$(n \le k)$.&nbsp; Then,&nbsp; using&nbsp; $s_k(t)$,&nbsp; we compute the auxiliary function
:$$\theta_k(t) = s_k(t) - \sum\limits_{j = 1}^{n-1}s_{kj} \cdot \varphi_j(t) \hspace{0.4cm}{\rm mit}\hspace{0.4cm}
+
:$$\theta_k(t) = s_k(t) - \sum\limits_{j = 1}^{n-1}s_{kj} \cdot \varphi_j(t) \hspace{0.4cm}{\rm with}\hspace{0.4cm}
 
s_{kj} = \hspace{0.1cm} < \hspace{-0.1cm} s_k(t), \hspace{0.05cm}\varphi_j(t) \hspace{-0.1cm} >, \hspace{0.2cm} j = 1, \hspace{0.05cm} \text{...}\hspace{0.05cm}, n-1\hspace{0.05cm}.$$
 
s_{kj} = \hspace{0.1cm} < \hspace{-0.1cm} s_k(t), \hspace{0.05cm}\varphi_j(t) \hspace{-0.1cm} >, \hspace{0.2cm} j = 1, \hspace{0.05cm} \text{...}\hspace{0.05cm}, n-1\hspace{0.05cm}.$$
  
*Ist $\theta_k(t) \equiv 0$ &nbsp; &#8658; &nbsp; $||\theta_k(t)|| = 0$, so liefert $s_k(t)$ keine neue Basisfunktion. Vielmehr lässt sich dann $s_k(t)$ durch die $n-1$ bereits vorher gefundenen Basisfunktionen $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$ ausdrücken:
+
*If&nbsp; $\theta_k(t) \equiv 0$ &nbsp; &#8658; &nbsp; $||\theta_k(t)|| = 0$,&nbsp; then&nbsp; $s_k(t)$&nbsp; does not yield a new basis function.&nbsp; Rather,&nbsp; $s_k(t)$&nbsp; can then be expressed by the&nbsp; $n-1$&nbsp; basis functions &nbsp;$\varphi_1(t)$, ... , $\varphi_{n-1}(t)$&nbsp; already found before:
 
:$$s_k(t) = \sum\limits_{j = 1}^{n-1}s_{kj}\cdot \varphi_j(t)  \hspace{0.05cm}.$$
 
:$$s_k(t) = \sum\limits_{j = 1}^{n-1}s_{kj}\cdot \varphi_j(t)  \hspace{0.05cm}.$$
  
*Eine neue Basisfunktion (nämlich die $n$&ndash;te) ergibt sich, falls $||\theta_k(t)|| \ne 0$ ist:
+
*A new basis function&nbsp; $($namely,&nbsp; the &nbsp;$n$&ndash;th$)$&nbsp; results if &nbsp;$||\theta_k(t)|| \ne 0$:&nbsp;
 
 
 
:$$\varphi_n(t) =  \frac{\theta_k(t)}{|| \theta_k(t)||}
 
:$$\varphi_n(t) =  \frac{\theta_k(t)}{|| \theta_k(t)||}
 
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_n(t) || = 1\hspace{0.05cm}.$$
 
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_n(t) || = 1\hspace{0.05cm}.$$
  
 +
This process is continued until all&nbsp; $M$&nbsp; signals have been considered.&nbsp; Then all&nbsp; $N \le M$&nbsp; orthonormal basis functions&nbsp; $\varphi_j(t)$&nbsp; have been found.&nbsp; The special case&nbsp; $N = M$&nbsp; arises only if all&nbsp; $M$&nbsp; signals are linearly independent.<br>
  
Diese Prozedur wird fortgesetzt, bis alle $M$ Signale berücksichtigt wurden. Danach hat man alle $N \le M$ orthonormalen Basisfunktionen $\varphi_j(t)$ gefunden. Der Sonderfall $N = M$ ergibt sich nur dann, wenn alle $M$ Signale linear voneinander unabhängig sind.<br>
+
:*This process is now illustrated by an example.  
 +
:*We also refer to the&nbsp; (German language)&nbsp;  HTML5/JavaScript applet&nbsp; [https://www.lntwww.de/Applets:Das_Gram-Schmidt-Verfahren "Das Gram-Schmidt-Verfahren"] &nbsp; &rArr; &nbsp; "Gram–Schmidt process".
  
Dieses Verfahren wird nun an einem Beispiel verdeutlicht. Wir verweisen auch auf das interaktive Applet [[Applets:Gram-Schmidt-Verfahren|Gram&ndash;Schmidt&ndash;Verfahren]].
 
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 2:}$&nbsp; Wir betrachten die $M = 4$ energiebegrenzten Signale $s_1(t)$, ... , $s_4(t)$ entsprechend der Grafik. Zur Vereinfachung der Berechnungen ist hier sowohl die Amplitude als auch die Zeit normiert.  
+
$\text{Example 2:}$&nbsp; We consider the &nbsp;$M = 4$&nbsp; energy-limited signals &nbsp;$s_1(t)$, ... , $s_4(t)$.&nbsp; To simplify the calculations,&nbsp; both amplitude and time are normalized here.
  
[[File:P ID1990 Dig T 4 1 S3 version1.png|center|frame|Zum Gram-Schmidt-Verfahren|class=fit]]
+
[[File:P ID1990 Dig T 4 1 S3 version1.png|center|frame|Gram-Schmidt process|class=fit]]
  
Man erkennt aus diesen Skizzen:  
+
One can see from these sketches:
*Die Basisfunktion $\varphi_1(t)$ ist formgleich mit $s_1(t)$. Wegen $E_1 = \vert \vert s_1(t) \vert \vert ^3 = 3 \cdot 0.5^2 = 0.75$ ergibt sich $s_{11} = \vert \vert s_1(t) \vert \vert = 0.866$. $\varphi_1(t)$ selbst besitzt abschnittsweise die Werte $\pm 0.5/0.866 = \pm0.577$.
+
*The basis function&nbsp; $\varphi_1(t)$&nbsp; is equal in form to&nbsp; $s_1(t)$.&nbsp; Because&nbsp; $E_1 = \vert \vert s_1(t) \vert \vert ^3 = 3 \cdot 0.5^2 = 0.75$,&nbsp; we get&nbsp; $s_{11} = \vert \vert s_1(t) \vert \vert = 0.866$. $\varphi_1(t)$&nbsp; itself has section-wise values&nbsp; $\pm 0.5/0.866 = \pm0.577$.
  
*Zur Berechnung der Hilfsfunktion $\theta_2(t)$ berechnen wir
+
*To calculate the auxiliary function&nbsp; $\theta_2(t)$,&nbsp; we compute
  
 
:$$s_{21}  = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.05cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0 \cdot (+0.577) + 1 \cdot (-0.577)+ 0 \cdot (-0.577)= -0.577$$
 
:$$s_{21}  = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.05cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0 \cdot (+0.577) + 1 \cdot (-0.577)+ 0 \cdot (-0.577)= -0.577$$
Line 167: Line 166:
 
  \hspace{0.3cm}\Rightarrow  \hspace{0.3cm}\vert \vert \theta_2(t) \vert \vert^2 = (1/3)^2 + (2/3)^2 + (-1/3)^2 = 0.667$$
 
  \hspace{0.3cm}\Rightarrow  \hspace{0.3cm}\vert \vert \theta_2(t) \vert \vert^2 = (1/3)^2 + (2/3)^2 + (-1/3)^2 = 0.667$$
 
:$$ \Rightarrow  \hspace{0.3cm} s_{22} = \sqrt{0.667} = 0.816,\hspace{0.3cm}
 
:$$ \Rightarrow  \hspace{0.3cm} s_{22} = \sqrt{0.667} = 0.816,\hspace{0.3cm}
\varphi_2(t) = \theta_2(t)/s_{22} = (0.408, 0.816, -0.408)\hspace{0.05cm}. $$
+
\varphi_2(t) = \theta_2(t)/s_{22} = (0.408,\ 0.816,\ -0.408)\hspace{0.05cm}. $$
  
*Die inneren Produkte zwischen $s_1(t)$ mit $\varphi_1(t)$ bzw. $\varphi_2(t)$ liefern folgende Ergebnisse:
+
*The inner products between&nbsp; $s_3(t)$&nbsp; with&nbsp; $\varphi_1(t)$&nbsp; or &nbsp;$\varphi_2(t)$&nbsp; give the following results:
 
:$$s_{31}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.577) + 0.5 \cdot (-0.577)- 0.5 \cdot (-0.577)= 0.289$$
 
:$$s_{31}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.577) + 0.5 \cdot (-0.577)- 0.5 \cdot (-0.577)= 0.289$$
 
:$$s_{32}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.408) + 0.5 \cdot (+0.816)- 0.5 \cdot (-0.408)= 0.816$$
 
:$$s_{32}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.408) + 0.5 \cdot (+0.816)- 0.5 \cdot (-0.408)= 0.816$$
 
:$$\Rightarrow  \hspace{0.3cm}\theta_3(t) = s_3(t) - 0.289 \cdot \varphi_1(t)- 0.816 \cdot \varphi_2(t) = 0\hspace{0.05cm}.$$
 
:$$\Rightarrow  \hspace{0.3cm}\theta_3(t) = s_3(t) - 0.289 \cdot \varphi_1(t)- 0.816 \cdot \varphi_2(t) = 0\hspace{0.05cm}.$$
  
Das bedeutet: Die grüne Funktion $s_3(t)$ liefert keine neue Basisfunktion $\varphi_3(t)$, im Gegensatz zur Funktion $s_4(t)$. Die numerischen Ergebnisse hierfür können der Grafik entnommen werden.}}
+
*This means: &nbsp; The green function&nbsp; $s_3(t)$&nbsp; does not yield a new basis function&nbsp; $\varphi_3(t)$,&nbsp; in contrast to the function&nbsp; $s_4(t)$.&nbsp; The numerical results for this can be taken from the graph.}}
  
== Basisfunktionen komplexer Zeitsignale ==
+
== Basis functions of complex time signals ==
 
<br>
 
<br>
In der Nachrichtentechnik hat man es oft mit komplexen Zeitfunktionen zu tun,
+
In Communications Engineering,&nbsp; one often has to deal with complex time functions,
*nicht etwa, weil es komplexe Signale in der Realität gibt, sondern<br>
+
*not because there are complex signals in reality,&nbsp; but<br>
*weil die Beschreibung eines BP&ndash;Signals im äquivalenten TP&ndash;Bereich zu komplexen Signalen führt.<br><br>
 
  
Die Bestimmung der $N \le M$ komplexwertigen Basisfunktionen $\xi_k(t)$ aus den$M$ komplexen Signalen $s_i(t)$ kann ebenfalls mit dem [[Digitalsignal%C3%BCbertragung/Signale,_Basisfunktionen_und_Vektorr%C3%A4ume#Das_Verfahren_nach_Gram-Schmidt_.281.29| Gram&ndash;Schmidt&ndash;Verfahren]] erfolgen, doch ist nun zu berücksichtigen, dass das innere Produkt zweier komplexer Signale $x(t)$ und $y(t)$ wie folgt zu berechnen ist:
+
*because the description of a band-pass signal in the equivalent low-pass range leads to complex signals.<br><br>
 +
 
 +
The determination of the&nbsp; $N \le M$&nbsp; '''complex-valued basis functions'''&nbsp; $\xi_k(t)$&nbsp; from the &nbsp;$M$&nbsp; complex signals&nbsp; $s_i(t)$&nbsp; can also be done using the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#The_Gram-Schmidt_process| "Gram–Schmidt process"]],&nbsp; but it must now be taken into account that the inner product of two complex signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; must be calculated as follows:
 
:$$< \hspace{-0.1cm}x(t), \hspace{0.1cm}y(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}x(t) \cdot y^{\star}(t)\,d \it t
 
:$$< \hspace{-0.1cm}x(t), \hspace{0.1cm}y(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}x(t) \cdot y^{\star}(t)\,d \it t
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
Die entsprechenden Gleichungen lauten nun mit $i = 1, \text{..}. , M$ und $k = 1, \text{..}. , N$:
+
The corresponding equations are now with&nbsp; $i = 1, \text{..}. , M$&nbsp; and &nbsp;$k = 1, \text{..}. , N$:
 
:$$s_i(t) = \sum\limits_{k = 1}^{N}s_{ik} \cdot \xi_k(t),\hspace{0.2cm}s_i(t) \in {\cal C},\hspace{0.2cm}s_{ik} \in {\cal C}
 
:$$s_i(t) = \sum\limits_{k = 1}^{N}s_{ik} \cdot \xi_k(t),\hspace{0.2cm}s_i(t) \in {\cal C},\hspace{0.2cm}s_{ik} \in {\cal C}
 
,\hspace{0.2cm}\xi_k(t) \in {\cal C} \hspace{0.05cm},$$
 
,\hspace{0.2cm}\xi_k(t) \in {\cal C} \hspace{0.05cm},$$
Line 194: Line 194:
 
\left\{ \begin{array}{c} 1 \\
 
\left\{ \begin{array}{c} 1 \\
 
  0  \end{array} \right.\quad
 
  0  \end{array} \right.\quad
\begin{array}{*{1}c}{\rm falls}\hspace{0.15cm} k = j
+
\begin{array}{*{1}c}{\rm if}\hspace{0.25cm} k = j
\\ {\rm falls}\hspace{0.15cm} k \ne j \\ \end{array}\hspace{0.05cm}.$$
+
\\ {\rm if}\hspace{0.25cm} k \ne j \\ \end{array}\hspace{0.05cm}.$$
  
Natürlich lässt sich jede komplexe Größe auch durch zwei reelle Größen ausdrücken, nämlich durch Realteil und Imaginärteil. Somit erhält man hier folgende Gleichungen:
+
Of course,&nbsp; any complex quantity can also be expressed by two real quantities,&nbsp; namely real part and imaginary part.&nbsp; Thus,&nbsp; the following equations are obtained here:
 
:$$s_{i}(t)  = s_{{\rm I}\hspace{0.02cm}i}(t) + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}i}(t),
 
:$$s_{i}(t)  = s_{{\rm I}\hspace{0.02cm}i}(t) + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}i}(t),
 
\hspace{0.2cm} s_{{\rm I}\hspace{0.02cm}i}(t) = {\rm Re}\big [s_{i}(t)\big], \hspace{0.2cm} s_{{\rm Q}\hspace{0.02cm}i}(t) = {\rm Im} \big [s_{i}(t)\big ],$$
 
\hspace{0.2cm} s_{{\rm I}\hspace{0.02cm}i}(t) = {\rm Re}\big [s_{i}(t)\big], \hspace{0.2cm} s_{{\rm Q}\hspace{0.02cm}i}(t) = {\rm Im} \big [s_{i}(t)\big ],$$
Line 210: Line 210:
 
\hspace{0.05cm}. $$
 
\hspace{0.05cm}. $$
  
Die Nomenklatur ergibt sich aus der Hauptanwendung für komplexe Basisfunktionen, nämlich der [[Modulationsverfahren/Quadratur–Amplitudenmodulation#Allgemeine_Beschreibung_und_Signalraumzuordnung|Quadratur&ndash;Amplitudenmodulation]] (QAM).  
+
The nomenclature arises from the main application for complex basis functions, namely&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#General_description_and_signal_space_allocation|"quadrature amplitude modulation"]]&nbsp; $\rm (QAM)$.  
*Der Index &bdquo;I&rdquo; steht für Inphasekomponente und gibt den Realteil an,  
+
*The subscript&nbsp; "I"&nbsp; stands for inphase component and indicates the real part,
*während die Quadraturkomponente (Imaginärteil) mit dem Index &bdquo;Q&rdquo; gekennzeichnet ist.<br>
 
  
 +
*while the quadrature component&nbsp; (imaginary part)&nbsp; is indicated by the index&nbsp; "Q".<br>
  
Um Verwechslungen mit der imaginären Einheit zu vermeiden, sind hier die komplexen Basisfunktionen $\xi_{k}(t)$ mit $k$ induziert und nicht mit $j$.<br>
 
  
== Dimension der Basisfunktionen ==
+
To avoid confusion with the imaginary unit&nbsp; "$\rm j$",&nbsp; here the complex basis functions&nbsp; $\xi_{k}(t)$&nbsp; were induced with&nbsp; $k$&nbsp; and not with&nbsp; $j$.<br>
 +
 
 +
== Dimension of the basis functions ==
 
<br>
 
<br>
Bei der Basisbandübertragung sind die möglichen Sendesignale (Betrachtung nur einer Symboldauer)  
+
In baseband transmission, the possible transmitted signals&nbsp; $($considering only one symbol duration$)$&nbsp; are
 
:$$s_i(t) = a_i \cdot g_s(t), \hspace{0.2cm} i = 0,  \text{...}\hspace{0.05cm} , M-1,$$
 
:$$s_i(t) = a_i \cdot g_s(t), \hspace{0.2cm} i = 0,  \text{...}\hspace{0.05cm} , M-1,$$
  
wobei $g_s(t)$ den ''Sendegrundimpuls'' angibt und die $a_i$ in den ersten drei Hauptkapiteln als die ''möglichen Amplitudenkoeffizienten'' bezeichnet wurden. Anzumerken ist, dass ab sofort für die Laufvariable $i$ die Werte $0$ bis $M-1$ vorausgesetzt werden.<br>
+
where&nbsp; $g_s(t)$&nbsp; indicates the&nbsp; "basic transmission pulse"&nbsp; and the&nbsp; $a_i$&nbsp; were denoted  in the first three main chapters as the possible&nbsp; "amplitude coefficients".&nbsp; It should be noted that from now on the values&nbsp; $0$&nbsp; to&nbsp; $M-1$&nbsp; are assumed for the indexing variable&nbsp; $i$.&nbsp;<br>
  
Nach der Beschreibung dieses Kapitels handelt es sich unabhängig von der Stufenzahl $M$ um ein eindimensionales Modulationsverfahren $(N = 1)$, wobei bei der Basisbandübertragung
+
According to the description of this chapter,&nbsp; this is a one-dimensional modulation process&nbsp; $(N = 1)$,&nbsp; regardless of the level number&nbsp; $M$.
*die Basisfunktion $\varphi_1(t)$ gleich dem energienormierten Sendegrundimpuls $g_s(t)$ ist:
+
 
:$$\varphi_1(t) ={g_s(t)}/{\sqrt{E_{gs}}} \hspace{0.3cm}{\rm mit}\hspace{0.3cm}
+
{{BlaueBox|TEXT= 
 +
$\text{In the case of baseband transmission:}$
 +
*The basis function&nbsp; $\varphi_1(t)$&nbsp; is equal to the energy-normalized basic transmission pulse&nbsp; $g_s(t)$:&nbsp;
 +
:$$\varphi_1(t) ={g_s(t)}/{\sqrt{E_{gs} } } \hspace{0.3cm}{\rm with}\hspace{0.3cm}
 
E_{gs} = \int_{-\infty}^{+\infty}g_s^2(t)\,d \it t   
 
E_{gs} = \int_{-\infty}^{+\infty}g_s^2(t)\,d \it t   
\hspace{0.05cm},$$
+
\hspace{0.05cm}.$$
  
*die dimensionslosen Amplitudenkoeffizienten $a_i$ in die Signalraumpunkte $s_i$ umzurechnen sind, die die Einheit &bdquo;Wurzel aus Energie&rdquo; aufweisen.<br>
+
*The dimensionless amplitude coefficients&nbsp; $a_i$&nbsp; are to be converted into the signal space points&nbsp; $s_i$&nbsp; which have the unit "root of energy".<br>}}
  
  
Die Grafik zeigt eindimensionale Signalraumkonstellationen $(N=1)$ für die Basisbandübertragung, nämlich
+
{{GraueBox|TEXT=  
[[File:P ID1991 Dig T 4 1 S5a version2.png|right|frame|Eindimensionale Modulationsverfahren|class=fit]]
+
$\text{Example 3:}$&nbsp;
(a) binär unipolar (oben) &nbsp; &rArr; &nbsp; $M = 2$,
+
The graph shows one-dimensional signal space constellations&nbsp; $(N=1)$&nbsp; for baseband transmission, viz.
 +
[[File:EN_Dig_T_4_1_S5.png|right|frame|One-dimensional modulation processes '''KORREKTUR: 4-stufiges'''|class=fit]]
 +
# &nbsp; binary unipolar (top) &nbsp; &rArr; &nbsp; $M = 2$,
 +
# &nbsp; binary bipolar (center) &nbsp; &rArr; &nbsp; $M = 2$, and
 +
# &nbsp; quaternary bipolar (bottom) &nbsp; &rArr; &nbsp; $M = 4$.
  
(b) binär bipolar (Mitte) &nbsp; &rArr; &nbsp; $M = 2$, sowie
 
  
(c) quaternär bipolare (unten) &nbsp; &rArr; &nbsp; $M = 4$.  
+
The graph simultaneously describes the one-dimensional carrier frequency systems
 +
# &nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29|"Two-level Amplitude Shift Keying"]]&nbsp; $\text{(2&ndash;ASK)}$, 
 +
# &nbsp;  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"Binary Phase Shift Keying"]]&nbsp; $\text{(BPSK)}$,
 +
#&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29|"Four-level Amplitude Shift Keying"]]&nbsp; $\text{(4&ndash;ASK)}$.<br>
  
Die Grafik beschreibt gleichzeitig die eindimensionalen Trägerfrequenzsysteme
+
<u>Note:</u>
*oben: &nbsp; [[Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#On.E2.80.93Off.E2.80.93Keying_.282.E2.80.93ASK.29|Zweistufiges Amplitude Shift Keying]] (2&ndash;ASK), auch bekannt als &bdquo;On&ndash;Off&ndash;Keying &rdquo;,
+
*The signals&nbsp; $s_i(t)$&nbsp; and the basis function &nbsp;$\varphi_1(t)$&nbsp; always refer to the equivalent low-pass range.
*in der Mitte: &nbsp; [[Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Binary_Phase_Shift_Keying_.28BPSK.29|Binary Phase Shift Keying]] (BPSK),
 
*unten: &nbsp; [[Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#M.E2.80.93stufiges_Amplitude_Shift_Keying_.28M.E2.80.93ASK.29|Vierstufiges Amplitude Shift Keying]] (4&ndash;ASK).<br>
 
  
 +
*In the band-pass region,&nbsp; $\varphi_1(t)$&nbsp; is a harmonic oscillation limited to the time domain&nbsp; $0 \le t \le T$.
  
Die dargestellten Signale $s_i(t)$ und die Basisfunktion $\varphi_1(t)$ beziehen sich stets auf den äquivalenten TP&ndash;Bereich.  
+
*In the graph on the right,&nbsp; the two or four possible signals&nbsp; $s_i(t)$&nbsp; are given for the example&nbsp; "rectangular pulse".
  
Im BP&ndash;Bereich ist $\varphi_1(t)$ eine auf den Zeitbereich $0 \le t \le T$ begrenzte harmonische Schwingung.
+
*From this,&nbsp; one can see the relationship between pulse amplitude&nbsp; $A$&nbsp; and signal energy&nbsp; $E = A^2 \cdot T$.&nbsp;}}
<br clear=all>
+
<br clear =all>
Weitere Anmerkungen:
+
{{GraueBox|TEXT= 
*In der Grafik rechts sind am Beispiel &bdquo;Rechteckimpuls&rdquo; die zwei bzw. vier möglichen Sendesignale $s_i(t)$ angegeben.  
+
$\text{Example 4:}$
*Man kann daraus den Zusammenhang zwischen Impulsamplitude $A$ und Signalenergie $E = A^2 \cdot T$ erkennen.  
+
The two-dimensional modulation processes  include:
*Die jeweils linken Darstellungen auf der $\varphi_1(t)$&ndash;Achse gelten aber unabhängig von der $g_s(t)$&ndash;Form, nicht nur für Rechtecke.<br clear =all>
+
[[File:P ID1992 Dig T 4 1 S5b version1.png|right|frame|Two-dimensional signal space constellations for multi-level PSK and QAM|class=fit]]
 +
#[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Multi-level_phase.E2.80.93shift_keying_.28M.E2.80.93PSK.29|"<i>M</i>&ndash;level Phase Shift Keying"]]&nbsp; (M&ndash;PSK),<br>
 +
#[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"Quadrature amplitude modulation"]]&nbsp; (4&ndash;QAM, 16&ndash;QAM, ...),<br>
 +
#[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_frequency_shift_keying_.282.E2.80.93FSK.29|"Binary (orthogonal) frequency shift keying"]]&nbsp; (2&ndash;FSK).<br>
  
  
== Zweidimensionale  Modulationsverfahren==
+
In general,&nbsp; for orthogonal FSK&nbsp; the number&nbsp; $N$&nbsp; of basis functions&nbsp; $\varphi_k(t)$&nbsp; is equal to the number&nbsp; $M$&nbsp; of possible transmitted signals&nbsp; $s_i(t)$ &nbsp; &rArr; &nbsp; $N=2$&nbsp; is only possible for&nbsp; $M=2$.&nbsp;
<br>
 
Zu den zweidimensionalen Modulationsverfahren $(N = 2)$ gehören
 
*[[Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Mehrstufiges_Phase.E2.80.93Shift_Keying_.28M.E2.80.93PSK.29|<i>M</i>&ndash;stufiges Phase Shift Keying]] (<i>M</i>&ndash;PSK),<br>
 
*[[Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Quadraturamplitudenmodulation_.28M.E2.80.93QAM.29|Quadratur&ndash;Amplitudenmodulation]] (4&ndash;QAM, 16&ndash;QAM, 64&ndash;QAM, ...),<br>
 
*[[Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Binary_Frequency_Shift_Keying_.282.E2.80.93FSK.29|Binäres (orthogonales) Frequency Shift Keying]] (2&ndash;FSK).<br><br>
 
  
Allgemein ist bei orthogonaler FSK die Anzahl $N$ der Basisfunktionen $\varphi_k(t)$ gleich der Anzahl $M$ möglicher Sendesignale $s_i(t)$. $N=2$ ist deshalb nur für $M=2$ möglich.<br>
+
The graph describes two-dimensional modulation processes in the band-pass&nbsp; (left)&nbsp; and in the equivalent low-pass range&nbsp; (right):
 +
*The left graph shows&nbsp; "8&ndash;PSK".&nbsp; If we restrict us to the red points only &nbsp; &rArr; &nbsp;  "4&ndash;PSK"&nbsp;  is present&nbsp; ("Quaternary Phase Shift Keying",&nbsp; QPSK).<br>
  
[[File:P ID1992 Dig T 4 1 S5b version1.png|center|frame|Zweidimensionale Signalraumkonstellationen für <i>M</i>&ndash;PSK und <i>M</i>&ndash;QAM|class=fit]]
+
*The right-hand diagram refers to&nbsp; "16&ndash;QAM"&nbsp; or &ndash; if only the signal space points outlined in red are considered &ndash; to&nbsp; "4&ndash;QAM".
 +
 +
*A comparison of the two images  with appropriate axis scaling shows that&nbsp; "4&ndash;QAM"&nbsp; is identical to&nbsp; "QPSK".<br>
  
Die Grafik zeigt Beispiele von Signalraumkonstellationen für Zweidimensionale Modulationsverfahren:
+
*When considered as a band-pass system, the basis function&nbsp; $\varphi_1(t)$&nbsp;  is cosinusoidal and &nbsp; $\varphi_2(t)$&nbsp; $($minus$)$ sinusoidal &ndash; compare&nbsp; [[Aufgaben:Exercise_4.2:_AM/PM_Oscillations|"Exercise 4.2"]].<br>
*Die linke Grafik zeigt die 8&ndash;PSK&ndash;Konstellation. Beschränkt man sich auf die rot umrandeten Punkte, so liegt eine 4&ndash;PSK (<i>Quaternary Phase Shift Keying</i>, QPSK) vor.<br>
 
  
*Die rechte Grafik bezieht sich auf die 16&ndash;QAM beziehungsweise &ndash; wenn man nur die rot umrandeten Signalraumpunkte betrachtet &ndash; auf die 4&ndash;QAM. Ein Vergleich der beiden Bilder zeigt, dass die 4&ndash;QAM mit der QPSK bei entsprechender Achsenskalierung identisch ist.<br>
+
*On the other hand,&nbsp; after transforming the QAM systems into the equivalent low-pass range, &nbsp; $\varphi_1(t)$&nbsp; is equal to the energy-normalized&nbsp; $($i.e., with energy "1"$)$&nbsp; basic transmission pulse&nbsp; $g_s(t)$,&nbsp; while &nbsp; $\varphi_2(t)={\rm  j} \cdot \varphi_1(t)$.&nbsp; For more details,&nbsp; please refer to &nbsp;[[Aufgaben:Exercise_4.2Z:_Eight-step_Phase_Shift_Keying|"Exercise 4.2Z"]].}}<br>
 
+
<br>
 
 
Die Grafiken beschreiben die Modulationsverfahren sowohl im Bandpass&ndash; als auch im äquivalenten Tiefpassbereich:
 
*Bei der Betrachtung als Bandpass&ndash;System ist die Basisfunktion $\varphi_1(t)$ cosinusförmig und $\varphi_2(t)$ (minus&ndash;)sinusförmig &ndash; vergleiche [[Aufgaben:Aufgabe_4.2:_AM/PM-Schwingungen|Aufgabe 4.2]].<br>
 
 
 
*Dagegen ist nach der Transformation der QAM&ndash;Systeme in den äquivalenten Tiefpassbereich $\varphi_1(t)$ gleich dem energienormierten  (also mit der Energie  &bdquo;1&rdquo;) Sendegrundimpuls $g_s(t)$, während $\varphi_2(t)={\rm  j} \cdot \varphi_1(t)$ zu setzen ist. Sie finden Näheres in der [[Aufgaben:Aufgabe_4.2Z:_Achtstufiges_Phase_Shift_Keying|Aufgabe 4.2Z]].<br>
 
  
== Aufgaben zum Kapitel ==
+
== Exercises for the chapter ==
 
<br>
 
<br>
[[Aufgaben:Aufgabe_4.1:_Zum_Gram-Schmidt-Verfahren|Aufgabe 4.1: Gram-Schmidt-Verfahren]]
+
[[Aufgaben:Exercise_4.1:_About_the_Gram-Schmidt_Process|Exercise 4.1: About the Gram-Schmidt Method]]
  
[[Aufgaben:Aufgabe_4.1Z:_Andere_Basisfunktionen|Aufgabe 4.1Z: Andere Basisfunktionen]]
+
[[Aufgaben:Exercise_4.1Z:_Other_Basis_Functions|Exercise 4.1Z: Other Basis Functions]]
  
[[Aufgaben:Aufgabe_4.2:_AM/PM-Schwingungen|Aufgabe 4.2: AM/PM-Schwingungen]]
+
[[Aufgaben:Aufgabe_4.2:_AM/PM-Schwingungen|Exercise 4.2: AM/PM Oscillations]]
  
[[Aufgaben:Aufgabe_4.2Z:_Achtstufiges_Phase_Shift_Keying|Aufgabe 4.2Z: Achtstufiges Phase Shift Keying]]
+
[[Aufgaben:Exercise_4.2Z:_Eight-level_Phase_Shift_Keying|Exercise 4.2Z: Eight-level Phase Shift Keying]]
  
[[Aufgaben:Aufgabe_4.3:_Unterschiedliche_Frequenzen|Aufgabe 4.3: Unterschiedliche Frequenzen]]
+
[[Aufgaben:Aufgabe_4.3:_Unterschiedliche_Frequenzen|Exercise 4.3: Different Frequencies]]
  
==Quellenverzeichnis==
+
==References==
  
 
<references/>
 
<references/>

Revision as of 13:54, 31 August 2022

# OVERVIEW OF THE FOURTH MAIN CHAPTER #


The fourth main chapter provides an abstract description of digital signal transmission,  which is based on basis functions and signal space constellations.  This makes it possible to treat very different configurations – for example band-pass systems and those for the baseband – in a uniform way.  The optimal receiver in each case has the same structure in all cases.

The following are dealt with in detail:

  1.   The meaning of  »basis functions«  and finding them using the  »Gram-Schmidt process«,
  2.   the  »structure of the optimal receiver«  for baseband transmission,
  3.   the  »theorem of irrelevance«  and its importance for the derivation of optimal detectors,
  4.   the  »optimal receiver for the AWGN channel«  and implementation aspects,
  5.   the system description by  »complex or  $N$–dimensional Gaussian noise«,
  6.   the  »error probability calculation and approximation«  under otherwise ideal conditions,
  7.   the application of the signal space description to  »carrier frequency systems«,
  8.   the different results for  »OOK, M-ASK, M-PSK, M-QAM and M-FSK«,
  9.   the different results for  »coherent and non-coherent demodulation«.


Almost all results of this chapter have already been derived in previous sections.  However,  the approach is fundamentally new:

  • In the  $\rm LNTwww$  book  "Modulation Methods"  and in the first three chapters of this book,  the specific system properties were already taken into account in the derivations – for example,  whether the digital signal is transmitted in baseband or whether digital amplitude,  frequency or phase modulation is present.
  • Here the systems are to be abstracted in such a way that they can be treated uniformly.  The optimal receiver in each case has the same structure in all cases,  and the error probability can also be specified for non-Gaussian distributed noise.

It should be noted that this rather global approach means that certain system deficiencies can only be recorded very imprecisely,  such as

  • the influence of a non-optimal receiver filter on the error probability,
  • an incorrect threshold  $($threshold drift$)$,  or
  • phase jitter  $($fluctuations in sampling times$)$.

In particular in the presence of intersymbol interference,  the procedure should therefore continue in accordance with the  third main chapter

The description is based on the script  [KöZ08][1] by  Ralf Kötter  and  Georg Zeitler,  which is closely based on the textbook [WJ65][2]. Gerhard Kramer,  who has held the chair at the LNT since 2010,  treats the same topic with very similar nomenclature in his lecture [Kra17][3].  In order not to make reading unnecessarily difficult for our own students at TU Munich,  we stick to this nomenclature as far as possible,  even if it deviates from other  $\rm LNTwww$  chapters.

Nomenclature in the fourth chapter


Compared to the other  $\rm LNTwww$  chapters,  the following nomenclature changes arise here:

  • The  "message"  to be transmitted is an integer value  $m \in \{m_i\}$  with  $i = 0$, ... , $M-1$,  where  $M$  specifies the  "symbol set size".
    If it simplifies the description,  $i = 1$, ... , $M$    is induced.
  • The result of the decision process at the receiver is also an integer with the same symbol alphabet as at the transmitter. 
    This result is also referred to as the  "estimated value":
$$\hat{m} \in \{m_i \}, \hspace{0.2cm} i = 0, 1, \text{...}\hspace{0.05cm} , M-1\hspace{0.2cm} ({\rm or}\,\,i = 1, 2, \text{...}\hspace{0.05cm}, M) \hspace{0.05cm}.$$
  • The  "symbol error probability"  $\rm Pr(symbol\hspace{0.15cm} error)$  or  $p_{\rm S}$  is usually referred to as follows in this main chapter:
$${\rm Pr} ({\cal E}) = {\rm Pr} ( \hat{m} \ne m) = 1 - {\rm Pr} ({\cal C}), \hspace{0.4cm}\text{complementary event:}\hspace{0.2cm} {\rm Pr} ({\cal C}) = {\rm Pr} ( \hat{m} = m) \hspace{0.05cm}.$$
  • In a  "probability density function"  $\rm (PDF)$,  a distinction is made between the  "random variable"   ⇒   $r$  and the  "realization"   ⇒   $\rho$  according to   $p_r(\rho)$. 
    Formerly,  $f_r(r)$  was used for this PDF.
  • With the notation  $p_r(\rho)$,   $r$  and  $\rho$  are scalars.  On the other hand,  if random variable and realization are vectors  (of suitable length),  this is expressed in bold type:     $p_{ \boldsymbol{ r}}(\boldsymbol{\rho})$  with the vectors  $ \boldsymbol{ r}$  and  $\boldsymbol{\rho}$.
  • In order to avoid confusion with energy values,  the  "threshold value is"  now called  $G$  instead of  $E$.  This is mainly referred to as the  "decision threshold"  in this chapter.
  • Based on the two real and energy-limited time functions  $x(t)$  and  $y(t)$,  the  "inner product"  is:
$$<\hspace{-0.1cm}x(t), \hspace{0.05cm}y(t) \hspace{-0.1cm}> \hspace{0.15cm}= \int_{-\infty}^{+\infty}x(t) \cdot y(t)\,d \it t \hspace{0.05cm}.$$
  • This results in the  "Euclidean norm"  or  "2–norm"  $($or  "norm"  for short$)$:
$$||x(t) || = \sqrt{<\hspace{-0.1cm}x(t), \hspace{0.05cm}x(t) \hspace{-0.1cm}>} \hspace{0.05cm}.$$
  • Compared to the script  [KöZ08][1],  the naming differs as follows:
  1. The probability of the event  $E$  is  ${\rm Pr}(E)$  instead of  $P(E)$. 
    This nomenclature change was also made because in some equations  "probabilities"  and  "powers"  appear together.
  2. Band–pass signals are still marked with the index "BP" and not with a tilde as in  [KöZ08][1].
    The corresponding  "low-pass signal"  is  (usually)  provided with the index  "TP"  $($from German  "Tiefpass"$)$.

Orthonormal basis functions


In this chapter,  we assume a set  $\{s_i(t)\}$  of possible transmitted signals that are uniquely assigned to the possible messages  $m_i$.  With  $i = 1$, ... , $M$  holds:

$$m \in \{m_i \}, \hspace{0.2cm} s(t) \in \{s_i(t) \}\hspace{-0.1cm}: \hspace{0.3cm} m = m_i \hspace{0.1cm} \Leftrightarrow \hspace{0.1cm} s(t) = s_i(t) \hspace{0.05cm}.$$

For what follows,  we further assume that the  $M$ signals  $s_i(t)$  are  "energy-limited",  which usually means at the same time that they are of finite duration.

$\text{Theorem:}$  Any set  $\{s_1(t), \hspace{0.05cm} \text{...} \hspace{0.05cm} , s_M(t)\}$  of energy-limited signals can be evolved into  $N \le M$  orthonormal basis functions  $\varphi_1(t), \hspace{0.05cm} \text{...} \hspace{0.05cm} , \varphi_N(t)$.  It holds:

$$s_i(t) = \sum\limits_{j = 1}^{N}s_{ij} \cdot \varphi_j(t) , \hspace{0.3cm}i = 1,\hspace{0.05cm} \text{...}\hspace{0.1cm} , M, \hspace{0.3cm}j = 1,\hspace{0.05cm} \text{...} \hspace{0.1cm}, N \hspace{0.05cm}.$$
  • In each case, two basis functions  $\varphi_j(t)$  and  $\varphi_k(t)$  must be orthonormal to each other, that is, it must hold  
    $(\delta_{jk}$  is called  "Kronecker symbol"  or  "Kronecker delta"$)$:
$$<\hspace{-0.1cm}\varphi_j(t), \hspace{0.05cm}\varphi_k(t) \hspace{-0.1cm}> = \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\,d \it t = {\rm \delta}_{jk} = \left\{ \begin{array}{c} 1 \\ 0 \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.1cm}j = k \\ {\rm if}\hspace{0.1cm} j \ne k \\ \end{array} \hspace{0.05cm}.$$


Here,  the parameter  $N$  indicates how many basis functions  $\varphi_j(t)$  are needed to represent the  $M$  possible transmitted signals.  In other words:   $N$  is the  "dimension of the vector space"  spanned by the  $M$  signals.  Here,  the following holds:

  1. If  $N = M$,  all transmitted signals are orthogonal to each other.
  2. They are not necessarily orthonormal,  i.e. the energies  $E_i = <\hspace{-0.1cm}s_i(t), \hspace{0.05cm}s_i(t) \hspace{-0.1cm}>$  may well be unequal to one.
  3. $N < M$  arises when at least one signal  $s_i(t)$  can be represented as linear combination of basis functions  $\varphi_j(t)$  that have resulted from other signals  $s_j(t) \ne s_i(t)$. 


$\text{Example 1:}$  We consider  $M = 3$  energy-limited signals according to the graph.  One recognizes immediately:

Representation of three transmitted signals by two basis functions
  • The signals  $s_1(t)$  and  $s_2(t)$  are orthogonal to each other.
  • The energies are  $E_1 = A^2 \cdot T = E$   and   $E_2 = (A/2)^2 \cdot T = E/4$.
  • The basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$  are equal in form to  $s_1(t)$  and  $s_2(t)$,  resp., and both have energy one:
$$\varphi_1(t)=\frac{s_1(t)}{\sqrt{E_1} } = \frac{s_1(t)}{\sqrt{A^2 \cdot T} } = \frac{1}{\sqrt{ T} } \cdot \frac{s_1(t)}{A}$$
$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}s_1(t) = s_{11} \cdot \varphi_1(t)\hspace{0.05cm},\hspace{0.1cm}s_{11} = \sqrt{E}\hspace{0.05cm},$$
$$\varphi_2(t) =\frac{s_2(t)}{\sqrt{E_2} } = \frac{s_2(t)}{\sqrt{(A/2)^2 \cdot T} } = \frac{1}{\sqrt{ T} } \cdot \frac{s_2(t)}{A/2}\hspace{0.05cm}$$
$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}s_2(t) = s_{21} \cdot \varphi_2(t)\hspace{0.05cm},\hspace{0.1cm}s_{21} = {\sqrt{E} }/{2}\hspace{0.05cm}.$$
  • $s_3(t)$  can be expressed by the previously determined basis functions  $\varphi_1(t)$,  $\varphi_2(t)$: 
$$s_3(t) =s_{31} \cdot \varphi_1(t) + s_{32} \cdot \varphi_2(t)\hspace{0.05cm},$$
$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm} s_{31} = {A}/{2} \cdot \sqrt {T}= {\sqrt{E} }/{2}\hspace{0.05cm}, \hspace{0.2cm}s_{32} = - A \cdot \sqrt {T} = -\sqrt{E} \hspace{0.05cm}.$$


⇒   In the lower right image,  the signals are shown in a two-dimensional representation

  • with the basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$  as axes,
  • where  $E = A^2 \cdot T$  and the relation to the other graphs can be seen by the coloring.


⇒   The vectorial representatives of the signals  $s_1(t)$,  $s_2(t)$  and  $s_3(t)$  in the two-dimensional vector space can be read from this sketch as follows:

$$\mathbf{s}_1 = (\sqrt{ E}, \hspace{0.1cm}0), $$
$$\mathbf{s}_2 = (0, \hspace{0.1cm}\sqrt{ E}/2), $$
$$\mathbf{s}_3 = (\sqrt{ E}/2,\hspace{0.1cm}-\sqrt{ E} ) \hspace{0.05cm}.$$


The Gram-Schmidt process


In  $\text{Example 1}$  in the last section,  the specification of the two orthonormal basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$  was very easy,  because they were of the same form as  $s_1(t)$  and  $s_2(t)$,  respectively. The  "Gram-Schmidt process"  finds the basis functions  $\varphi_1(t)$, ... , $\varphi_N(t)$  for arbitrary predefinable signals  $s_1(t)$, ... , $s_M(t)$, as follows:

  • The first basis function  $\varphi_1(t)$  is always equal in form to  $s_1(t)$.  It holds:
$$\varphi_1(t) = \frac{s_1(t)}{\sqrt{E_1}} = \frac{s_1(t)}{|| s_1(t)||} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_1(t) || = 1, \hspace{0.2cm}s_{11} =|| s_1(t)||,\hspace{0.2cm}s_{1j} = 0 \hspace{0.2cm}{\rm f{\rm or }}\hspace{0.2cm} j \ge 2 \hspace{0.05cm}.$$
  • It is now assumed that from the signals  $s_1(t)$, ... , $s_{k-1}(t)$  the basis functions  $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$  have been calculated  $(n \le k)$.  Then,  using  $s_k(t)$,  we compute the auxiliary function
$$\theta_k(t) = s_k(t) - \sum\limits_{j = 1}^{n-1}s_{kj} \cdot \varphi_j(t) \hspace{0.4cm}{\rm with}\hspace{0.4cm} s_{kj} = \hspace{0.1cm} < \hspace{-0.1cm} s_k(t), \hspace{0.05cm}\varphi_j(t) \hspace{-0.1cm} >, \hspace{0.2cm} j = 1, \hspace{0.05cm} \text{...}\hspace{0.05cm}, n-1\hspace{0.05cm}.$$
  • If  $\theta_k(t) \equiv 0$   ⇒   $||\theta_k(t)|| = 0$,  then  $s_k(t)$  does not yield a new basis function.  Rather,  $s_k(t)$  can then be expressed by the  $n-1$  basis functions  $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$  already found before:
$$s_k(t) = \sum\limits_{j = 1}^{n-1}s_{kj}\cdot \varphi_j(t) \hspace{0.05cm}.$$
  • A new basis function  $($namely,  the  $n$–th$)$  results if  $||\theta_k(t)|| \ne 0$: 
$$\varphi_n(t) = \frac{\theta_k(t)}{|| \theta_k(t)||} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_n(t) || = 1\hspace{0.05cm}.$$

This process is continued until all  $M$  signals have been considered.  Then all  $N \le M$  orthonormal basis functions  $\varphi_j(t)$  have been found.  The special case  $N = M$  arises only if all  $M$  signals are linearly independent.

  • This process is now illustrated by an example.
  • We also refer to the  (German language)  HTML5/JavaScript applet  "Das Gram-Schmidt-Verfahren"   ⇒   "Gram–Schmidt process".


$\text{Example 2:}$  We consider the  $M = 4$  energy-limited signals  $s_1(t)$, ... , $s_4(t)$.  To simplify the calculations,  both amplitude and time are normalized here.

Gram-Schmidt process

One can see from these sketches:

  • The basis function  $\varphi_1(t)$  is equal in form to  $s_1(t)$.  Because  $E_1 = \vert \vert s_1(t) \vert \vert ^3 = 3 \cdot 0.5^2 = 0.75$,  we get  $s_{11} = \vert \vert s_1(t) \vert \vert = 0.866$. $\varphi_1(t)$  itself has section-wise values  $\pm 0.5/0.866 = \pm0.577$.
  • To calculate the auxiliary function  $\theta_2(t)$,  we compute
$$s_{21} = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.05cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0 \cdot (+0.577) + 1 \cdot (-0.577)+ 0 \cdot (-0.577)= -0.577$$
$$ \Rightarrow \hspace{0.3cm}\theta_2(t) = s_2(t) - s_{21} \cdot \varphi_1(t) = (0.333, 0.667, -0.333) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\vert \vert \theta_2(t) \vert \vert^2 = (1/3)^2 + (2/3)^2 + (-1/3)^2 = 0.667$$
$$ \Rightarrow \hspace{0.3cm} s_{22} = \sqrt{0.667} = 0.816,\hspace{0.3cm} \varphi_2(t) = \theta_2(t)/s_{22} = (0.408,\ 0.816,\ -0.408)\hspace{0.05cm}. $$
  • The inner products between  $s_3(t)$  with  $\varphi_1(t)$  or  $\varphi_2(t)$  give the following results:
$$s_{31} \hspace{0.1cm} = \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.577) + 0.5 \cdot (-0.577)- 0.5 \cdot (-0.577)= 0.289$$
$$s_{32} \hspace{0.1cm} = \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.408) + 0.5 \cdot (+0.816)- 0.5 \cdot (-0.408)= 0.816$$
$$\Rightarrow \hspace{0.3cm}\theta_3(t) = s_3(t) - 0.289 \cdot \varphi_1(t)- 0.816 \cdot \varphi_2(t) = 0\hspace{0.05cm}.$$
  • This means:   The green function  $s_3(t)$  does not yield a new basis function  $\varphi_3(t)$,  in contrast to the function  $s_4(t)$.  The numerical results for this can be taken from the graph.

Basis functions of complex time signals


In Communications Engineering,  one often has to deal with complex time functions,

  • not because there are complex signals in reality,  but
  • because the description of a band-pass signal in the equivalent low-pass range leads to complex signals.

The determination of the  $N \le M$  complex-valued basis functions  $\xi_k(t)$  from the  $M$  complex signals  $s_i(t)$  can also be done using the  "Gram–Schmidt process",  but it must now be taken into account that the inner product of two complex signals  $x(t)$  and  $y(t)$  must be calculated as follows:

$$< \hspace{-0.1cm}x(t), \hspace{0.1cm}y(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}x(t) \cdot y^{\star}(t)\,d \it t \hspace{0.05cm}.$$

The corresponding equations are now with  $i = 1, \text{..}. , M$  and  $k = 1, \text{..}. , N$:

$$s_i(t) = \sum\limits_{k = 1}^{N}s_{ik} \cdot \xi_k(t),\hspace{0.2cm}s_i(t) \in {\cal C},\hspace{0.2cm}s_{ik} \in {\cal C} ,\hspace{0.2cm}\xi_k(t) \in {\cal C} \hspace{0.05cm},$$
$$< \hspace{-0.1cm}\xi_k(t),\hspace{0.1cm} \xi_j(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}\xi_k(t) \cdot \xi_j^{\star}(t)\,d \it t = {\rm \delta}_{ik} = \left\{ \begin{array}{c} 1 \\ 0 \end{array} \right.\quad \begin{array}{*{1}c}{\rm if}\hspace{0.25cm} k = j \\ {\rm if}\hspace{0.25cm} k \ne j \\ \end{array}\hspace{0.05cm}.$$

Of course,  any complex quantity can also be expressed by two real quantities,  namely real part and imaginary part.  Thus,  the following equations are obtained here:

$$s_{i}(t) = s_{{\rm I}\hspace{0.02cm}i}(t) + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}i}(t), \hspace{0.2cm} s_{{\rm I}\hspace{0.02cm}i}(t) = {\rm Re}\big [s_{i}(t)\big], \hspace{0.2cm} s_{{\rm Q}\hspace{0.02cm}i}(t) = {\rm Im} \big [s_{i}(t)\big ],$$
$$\xi_{k}(t) = \varphi_k(t) + {\rm j} \cdot \psi_k(t), \hspace{0.2cm} \varphi_k(t) = {\rm Re}\big [\xi_{k}(t)\big ], \hspace{0.2cm} \psi_k(t) = {\rm Im} \big [\xi_{k}(t)\big ],$$
$$\hspace{0.35cm} s_{ik} = s_{{\rm I}\hspace{0.02cm}ik} + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}ik}, \hspace{0.2cm} s_{{\rm I}ik} = {\rm Re} \big [s_{ik}\big ], \hspace{0.2cm} s_{{\rm Q}ik} = {\rm Im} \big [s_{ik}\big ],$$
$$ \hspace{0.35cm} s_{{\rm I}\hspace{0.02cm}ik} ={\rm Re}\big [\hspace{0.01cm} < \hspace{-0.1cm} s_i(t), \hspace{0.15cm}\varphi_k(t) \hspace{-0.1cm} > \hspace{0.1cm}\big ], \hspace{0.2cm}s_{{\rm Q}\hspace{0.02cm}ik} = {\rm Re}\big [\hspace{0.01cm} < \hspace{-0.1cm} s_i(t), \hspace{0.15cm}{\rm j} \cdot \psi_k(t) \hspace{-0.1cm} > \hspace{0.1cm}\big ] \hspace{0.05cm}. $$

The nomenclature arises from the main application for complex basis functions, namely  "quadrature amplitude modulation"  $\rm (QAM)$.

  • The subscript  "I"  stands for inphase component and indicates the real part,
  • while the quadrature component  (imaginary part)  is indicated by the index  "Q".


To avoid confusion with the imaginary unit  "$\rm j$",  here the complex basis functions  $\xi_{k}(t)$  were induced with  $k$  and not with  $j$.

Dimension of the basis functions


In baseband transmission, the possible transmitted signals  $($considering only one symbol duration$)$  are

$$s_i(t) = a_i \cdot g_s(t), \hspace{0.2cm} i = 0, \text{...}\hspace{0.05cm} , M-1,$$

where  $g_s(t)$  indicates the  "basic transmission pulse"  and the  $a_i$  were denoted in the first three main chapters as the possible  "amplitude coefficients".  It should be noted that from now on the values  $0$  to  $M-1$  are assumed for the indexing variable  $i$. 

According to the description of this chapter,  this is a one-dimensional modulation process  $(N = 1)$,  regardless of the level number  $M$.

$\text{In the case of baseband transmission:}$

  • The basis function  $\varphi_1(t)$  is equal to the energy-normalized basic transmission pulse  $g_s(t)$: 
$$\varphi_1(t) ={g_s(t)}/{\sqrt{E_{gs} } } \hspace{0.3cm}{\rm with}\hspace{0.3cm} E_{gs} = \int_{-\infty}^{+\infty}g_s^2(t)\,d \it t \hspace{0.05cm}.$$
  • The dimensionless amplitude coefficients  $a_i$  are to be converted into the signal space points  $s_i$  which have the unit "root of energy".


$\text{Example 3:}$  The graph shows one-dimensional signal space constellations  $(N=1)$  for baseband transmission, viz.

One-dimensional modulation processes KORREKTUR: 4-stufiges
  1.   binary unipolar (top)   ⇒   $M = 2$,
  2.   binary bipolar (center)   ⇒   $M = 2$, and
  3.   quaternary bipolar (bottom)   ⇒   $M = 4$.


The graph simultaneously describes the one-dimensional carrier frequency systems

  1.   "Two-level Amplitude Shift Keying"  $\text{(2–ASK)}$,
  2.   "Binary Phase Shift Keying"  $\text{(BPSK)}$,
  3.   "Four-level Amplitude Shift Keying"  $\text{(4–ASK)}$.

Note:

  • The signals  $s_i(t)$  and the basis function  $\varphi_1(t)$  always refer to the equivalent low-pass range.
  • In the band-pass region,  $\varphi_1(t)$  is a harmonic oscillation limited to the time domain  $0 \le t \le T$.
  • In the graph on the right,  the two or four possible signals  $s_i(t)$  are given for the example  "rectangular pulse".
  • From this,  one can see the relationship between pulse amplitude  $A$  and signal energy  $E = A^2 \cdot T$. 


$\text{Example 4:}$ The two-dimensional modulation processes include:

Two-dimensional signal space constellations for multi-level PSK and QAM
  1. "M–level Phase Shift Keying"  (M–PSK),
  2. "Quadrature amplitude modulation"  (4–QAM, 16–QAM, ...),
  3. "Binary (orthogonal) frequency shift keying"  (2–FSK).


In general,  for orthogonal FSK  the number  $N$  of basis functions  $\varphi_k(t)$  is equal to the number  $M$  of possible transmitted signals  $s_i(t)$   ⇒   $N=2$  is only possible for  $M=2$. 

The graph describes two-dimensional modulation processes in the band-pass  (left)  and in the equivalent low-pass range  (right):

  • The left graph shows  "8–PSK".  If we restrict us to the red points only   ⇒   "4–PSK"  is present  ("Quaternary Phase Shift Keying",  QPSK).
  • The right-hand diagram refers to  "16–QAM"  or – if only the signal space points outlined in red are considered – to  "4–QAM".
  • A comparison of the two images with appropriate axis scaling shows that  "4–QAM"  is identical to  "QPSK".
  • When considered as a band-pass system, the basis function  $\varphi_1(t)$  is cosinusoidal and   $\varphi_2(t)$  $($minus$)$ sinusoidal – compare  "Exercise 4.2".
  • On the other hand,  after transforming the QAM systems into the equivalent low-pass range,   $\varphi_1(t)$  is equal to the energy-normalized  $($i.e., with energy "1"$)$  basic transmission pulse  $g_s(t)$,  while   $\varphi_2(t)={\rm j} \cdot \varphi_1(t)$.  For more details,  please refer to  "Exercise 4.2Z".



Exercises for the chapter


Exercise 4.1: About the Gram-Schmidt Method

Exercise 4.1Z: Other Basis Functions

Exercise 4.2: AM/PM Oscillations

Exercise 4.2Z: Eight-level Phase Shift Keying

Exercise 4.3: Different Frequencies

References

  1. 1.0 1.1 1.2 Kötter, R., Zeitler, G.:  Lecture notes, Institute for Communications Engineering, Technical University of Munich, 2008.
  2. Wozencraft, J. M.; Jacobs, I. M.:  Principles of Communication Engineering.  New York: John Wiley & Sons, 1965.
  3. Kramer, G.:  Nachrichtentechnik 2. Lecture notes, Institute for Communications Engineering, Technical University of Munich, 2017.