Difference between revisions of "Digital Signal Transmission/Causes and Effects of Intersymbol Interference"

From LNTwww
 
(64 intermediate revisions by 5 users not shown)
Line 1: Line 1:
+
 
 
{{Header
 
{{Header
|Untermenü=Impulsinterferenzen und Entzerrungsverfahren
+
|Untermenü=Intersymbol Interfering and Equalization Methods
|Vorherige Seite=Symbolweise Codierung mit Pseudoternärcodes
+
|Vorherige Seite=Symbolwise_Coding_with_Pseudo_Ternary_Codes
 
|Nächste Seite=Fehlerwahrscheinlichkeit unter Berücksichtigung von Impulsinterferenzen
 
|Nächste Seite=Fehlerwahrscheinlichkeit unter Berücksichtigung von Impulsinterferenzen
 
}}
 
}}
  
 
+
== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
== Definition des Begriffs „Impulsinterferenz” (1) ==
 
 
<br>
 
<br>
Für die beiden ersten Kapitel dieses Buches wurde vorausgesetzt, dass der Detektionsgrundimpuls <i>g<sub>d</sub></i>(<i>t</i>)
+
The third main chapter focuses on&nbsp; '''intersymbol interference'''&nbsp; $\rm (ISI)$,&nbsp; which arises,&nbsp; for example,&nbsp; from distortions of the transmission channel or is related to a realization of the receiver filter that deviates from the Nyquist condition.&nbsp;  Subsequently,&nbsp; some equalization methods are described which can be used to mitigate the system degradation due to intersymbol interference.
*entweder auf den Zeitbereich |<i>t</i>| &#8804; <i>T</i> begrenzt ist, oder<br>
 
*äquidistante Nulldurchgänge im Symbolabstand <i>T</i> aufweist.<br><br>
 
  
Bezeichnen wir die Abtastwerte von <i>g<sub>d</sub></i>(<i>t</i>) bei Vielfachen der Symboldauer <i>T</i> (Abstand der Impulse) als die Detektionsgrundimpulswerte, so wurde in den Kapiteln 1 und 2 stillschweigend vorausgesetzt:
+
The description is given throughout in the baseband.&nbsp; However,&nbsp; the results can easily be applied to the carrier frequency systems discussed in the chapter &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|"Linear Digital Modulation - Coherent Demodulation"]].&nbsp;
  
:<math>g_\nu = g_d(\nu T)   =  \left\{ \begin{array}{c} g_0    \\
+
In detail,&nbsp; this chapter deals with:
\\ 0  \\  \end{array} \right.\quad
+
#&nbsp; the&nbsp; &raquo;causes and effects&laquo;&nbsp; of intersymbol interference,
\begin{array}{*{1}c} {\rm{f\ddot{u}r}}\\  \\ {\rm{f\ddot{u}r}} \\ \end{array}
+
#&nbsp; the&nbsp; &raquo;eye diagram&laquo;&nbsp; as a suitable tool for the description of intersymbol interferences,
\begin{array}{*{20}c}\nu = 0, \\ \\  \nu \ne 0. \\
+
#&nbsp; the&nbsp; &raquo;error probability calculation&laquo;&nbsp; considering channel distortions,
\end{array}</math>
+
#&nbsp; the&nbsp; &raquo;influence of intersymbol interference in multilevel and/or coded transmission&laquo;,
 +
#&nbsp; the&nbsp; &raquo;optimal Nyquist equalizer as an example of linear channel equalization,
 +
#&nbsp; the&nbsp; &raquo;decision feedback equalization&laquo;&nbsp; $\rm (DFE)$ – an effective nonlinear decision realization,
 +
#&nbsp; the&nbsp; &raquo;correlation receiver&laquo;&nbsp; as an example of&nbsp; &raquo;maximum likelihood or maximum a-posteriori&laquo;&nbsp; $\rm (MAP)$&nbsp; decision strategy,
 +
#&nbsp; the&nbsp; &raquo;Viterbi receiver&laquo;,&nbsp; a reduced-effort MAP decision algorithm.
  
Als Konsequenz dieser Annahme hat sich daraus ergeben, dass der Nutzanteil (Index &bdquo;S&rdquo;)
 
  
:<math>d_{\rm S}(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g_d ( t - \nu \cdot T)
 
\hspace{0.3cm}{\rm mit}\hspace{0.3cm}a_\nu \in \{ -1, +1\}</math>
 
  
des Detektionssignals zu den Zeitpunkten <i>&nu;T</i> nur zwei verschiedene Werte annehmen kann, nämlich &plusmn;<i>g</i><sub>0</sub>.<br>
+
== Definition of the term "Intersymbol Interference" ==
 +
<br>
 +
For the first two main chapters of this book,&nbsp; it was assumed that the basic detection pulse''&nbsp; $g_d(t)$
 +
*either is limited to the time domain &nbsp;$|t| \le T$,&nbsp; or<br>
 +
*has equidistant zero crossings in the symbol spacing $T$.<br><br>
  
Die obere der beiden Grafiken zeigt <i>d</i><sub>S</sub>(<i>t</i>) für diesen impulsinterferenzfreien Fall mit <i>g</i><sub>0</sub> = <i>s</i><sub>0</sub> und <i>g<sub>&nu;</sub></i><sub>&ne;0</sub> = 0. Darunter gezeichnet ist der Signalverlauf für die Detektionsgrundimpulswerte
+
If we denote the samples of &nbsp;$g_d(t)$&nbsp; at multiples of the symbol duration &nbsp;$T$&nbsp; (spacing of the pulses)&nbsp; as the&nbsp; "basic detection pulse values",&nbsp; it has been tacitly assumed so far:
 +
[[File:EN_Dig_T_3_1_S1.png|right|frame|Detection signals with and without intersymbol interferences|class=fit]]
 +
:$$g_\nu = g_d(\nu T)  =  \left\{ \begin{array}{c} g_0   
 +
\\ 0 \\  \end{array} \right.\quad
 +
\begin{array}{*{1}c} {\rm{for}}\\  {\rm{for}} \\ \end{array}
 +
\begin{array}{*{20}c}\nu = 0, \\  \nu \ne 0. \\
 +
\end{array}$$
  
:<math>g_0 = 0.6 \cdot s_0, \hspace{0.2cm}g_{-1} = g_{1} =0.2 \cdot s_0, \hspace{0.2cm}g_\nu
+
As a consequence of this assumption it has resulted that in the binary case the signal component&nbsp; (index "S")&nbsp;
  =0\hspace{0.3cm}{\rm f\ddot{u}r}\hspace{0.3cm} |\nu| \ge 2 \hspace{0.05cm},</math>
+
:$$d_{\rm S}(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g_d ( t - \nu \cdot T)
 +
  \hspace{0.3cm}{\rm with}\hspace{0.3cm}a_\nu \in \{ -1, +1\}$$
  
die Impulsinterferenzen hervorrufen.
+
of the detection signal at the instants &nbsp;$\nu \cdot T$&nbsp; can take only two different values,&nbsp; namely &nbsp;$\pm g_0$.<br>
  
<br>
+
*The upper of the following two time plots shows &nbsp;$d_{\rm S}(t)$&nbsp; for this ISI-free case with &nbsp;$g_{\nu \ne 0} = 0$&nbsp; and &nbsp;$g_0 = s_0$ &nbsp; &rArr; &nbsp; the value &nbsp;$g_0$&nbsp; is equal to the maximum value &nbsp;$s_0$&nbsp; of the transmitted signal.
[[File:P_ID1362__Dig_T_3_1_S1_version1.png|Detektionssignale mit und ohne Impulsinterferenzen|class=fit]]<br><br>
 
  
In beiden Bildern ist der (jeweils dreieckförmige) Detektionsgrundimpuls <i>g<sub>d</sub></i>(<i>t</i>) rot eingezeichnet. Die Detektionszeitpunkte <i>&nu;</i><i>T</i> sind jeweils durch blaue Kreise markiert. Die Bildbeschreibung wird auf der nächsten Seite fortgesetzt.<br>
+
*Drawn below is the signal waveform for a set of basic detection pulse values that cause intersymbol interference:
 +
:$$g_0 = 0.6 \cdot s_0,\hspace{0.15cm} g_{-1} = g_{+1} =0.2 \cdot s_0,\hspace{0.15cm}g_\nu =0\hspace{0.25cm}{\rm for}\hspace{0.25cm} |\nu| \ge 2 .$$
  
 +
In both plots,&nbsp; the&nbsp; (triangular)&nbsp; basic detection pulse&nbsp; $g_d(t)$&nbsp; is drawn in red.&nbsp; The detection time points &nbsp;$\nu \cdot T$&nbsp; are marked by blue circles.
  
== Definition des Begriffs „Impulsinterferenz” (2) ==
+
One can see from the lower signal plot:
<br>
+
*The basic detection pulse &nbsp;$g_d(t)$&nbsp; is now different from zero in the range &nbsp;$|t| \le 1.5 \cdot T$&nbsp;and thus no longer fulfills the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|"Nyquist condition in the time domain"]]&nbsp; for intersymbol interference freedom.<br>
Man erkennt aus dem unteren Signalverlauf auf der [http://en.lntwww.de/Digitalsignal%C3%BCbertragung/Ursachen_und_Auswirkungen_von_Impulsinterferenzen#Definition_des_Begriffs_.E2.80.9EImpulsinterferenz.E2.80.9D_.281.29 vorherigen Seite]:
 
*Der Detektionsgrundimpuls <i>g<sub>d</sub></i>(<i>t</i>) ist nun im Bereich |<i>t</i>| &#8804; 1.5<i>T</i> von Null verschieden und erfüllt somit nicht mehr die Nyquist&ndash;Bedingung der Impulsinterferenzfreiheit.<br>
 
  
*Dies hat zur Folge, dass zu den (mit Kreisen markierten) Detektionszeitpunkten nicht nur zwei Werte (&plusmn;<i>s</i><sub>0</sub>) möglich sind wie im oberen Bild. Vielmehr gilt für die Detektionsnutzabtastwerte:
+
*As a consequence,&nbsp; at the detection times&nbsp; (marked with circles)&nbsp; not only two values &nbsp;$(\pm s_0)$&nbsp; are possible as in the upper figure.&nbsp; Rather,&nbsp; the following applies here for the detection sampling values:
 +
:$$d_{\rm S}(\nu \cdot T)  \in \{ \pm s_0, \ \pm 0.6 s_0, \ \pm 0.2  s_0\}\hspace{0.05cm}.$$
 +
*The samples that are close to the threshold due to unfavorable neighboring pulses are more often falsified by the AWGN noise $($with noise rms value &nbsp;$\sigma_d)$&nbsp; than the samples further out.&nbsp;
  
::<math>d_{\rm S}(\nu \cdot T) \in \{ \pm s_0, \pm 0.6 s_0,\pm 0.2  s_0\}\hspace{0.05cm}.</math>
+
*Exemplarily,&nbsp; with &nbsp;$\sigma_d = 0.2 \cdot s_0$&nbsp; the blue filled points close to the threshold are falsified with probability &nbsp;$p_{\rm S} ={\rm Q} (1) \approx 16 \%$&nbsp; and the outer points&nbsp; (with white core)&nbsp; are falsified only with &nbsp;$p_{\rm S} ={\rm Q} (5) \approx 3 \cdot 10^{-7}$.&nbsp; The error probability of the red filled points&nbsp; (distance &nbsp;$0.6 \cdot s_0$&nbsp;  from zero line)&nbsp; is in between: &nbsp; $p_{\rm S} ={\rm Q} (3) \approx 0.13 \%$.
  
*Die Abtastwerte, die aufgrund ungünstiger Nachbarimpulse nahe an der Schwelle liegen, werden durch das AWGN&ndash;Rauschen (mit Rauscheffektivwert <i>&sigma;<sub>d</sub></i>) häufiger verfälscht als die weiter außen liegenden Abtastwerte.<br>
 
  
*Beispielhaft werden mit <i>&sigma;<sub>d</sub></i>/<i>s</i><sub>0</sub> = 0.2 die blau ausgefüllten Punkte nahe der Schwelle mit großer Wahrscheinlichkeit <i>p</i><sub>S</sub> = Q(1) &asymp; 16% verfälscht und die äußeren Punkte (mit weißem Kern) nur mit <i>p</i><sub>S</sub> = Q(5) &asymp; 3 &middot; 10<sup>&ndash;7</sup>. Die Fehlerwahrscheinlichkeit der gelb gefüllten Punkte (alle im Abstand 0.6 &middot; <i>s</i><sub>0</sub> von der Null&ndash;Linie) liegt dazwischen:
+
So far,&nbsp; the effects of intersymbol interference have been presented as vividly as possible.&nbsp; An exact definition is still missing.
  
::<math>p_{\rm S} ={\rm Q} (3) \approx 0.13 \% \hspace{0.05cm}.</math>
+
{{BlaueBox|TEXT= 
 +
$\text{Definition:}$&nbsp; '''Intersymbol interference'''&nbsp; $\rm (ISI)$&nbsp; is
 +
* the impairment of a symbol decision due to pulse broadening&nbsp; ("time dispersion"),&nbsp; and
 +
*the associated dependence of the error probability on the neighboring symbols.<br>
  
Bisher wurden die Auswirkungen von Impulsinterferenzen möglichst anschaulich dargelegt. Es fehlt noch eine exakte Begriffsbestimmung.<br>
 
  
{{Definition}}''':''' Unter Impulsinterferenz (englisch: <i>Intersymbol Interference</i>, ISI) versteht man die Beeinträchtigung einer Symbolentscheidung aufgrund einer Impulsverbreiterung (Zeitdispersion) und damit verbunden eine Abhängigkeit der Fehlerwahrscheinlichkeit von den Nachbarsymbolen.{{end}}<br>
+
In other words:
 +
#Falling edges of preceding pulses&nbsp; ("trailers")&nbsp; and rising edges of following pulses&nbsp; ("precursors")&nbsp; change the currently applied detection sample value.<br>
 +
#This can increase or decrease the probability of a wrong decision for the current symbol,&nbsp; depending on whether the distance to the threshold becomes smaller or larger.<br>
 +
#On statistical average &ndash; i.e. when considering an&nbsp; (infinitely)&nbsp; long symbol sequence &ndash; '''this always leads to a&nbsp; (considerable)&nbsp; increase of the&nbsp; (mean)&nbsp; symbol error probability''' &nbsp;$p_{\rm S} $.}}
  
In anderen Worten:
 
*Durch abfallende Flanken vorangegangener Impulse (&bdquo;Nachläufer&rdquo;) und ansteigende Flanken folgender Impulse (&bdquo;Vorläufer&rdquo;) wird der momentan anliegende Detektionsabtastwert verändert.<br>
 
  
*Dadurch kann die Wahrscheinlichkeit einer Fehlentscheidung für das aktuelle Symbol vergrößert oder verkleinert werden, je nachdem, ob der Abstand zur Schwelle kleiner oder größer wird.<br>
+
== Possible causes for intersymbol interference ==
 
 
*Im statistischen Mittel &ndash; also bei Betrachtung einer (unendlich) langen Symbolfolge &ndash; führt dies stets zu einer (beträchtlichen) Erhöhung der (mittleren) Symbolfehlerwahrscheinlichkeit <i>p</i><sub>S</sub>.<br>
 
 
 
 
 
== Mögliche Ursachen für Impulsinterferenzen ==
 
 
<br>
 
<br>
Die nachfolgende Grafik zeigt das [http://en.lntwww.de/Digitalsignal%C3%BCbertragung/Fehlerwahrscheinlichkeit_unter_Ber%C3%BCcksichtigung_von_Impulsinterferenzen#Definition_und_Aussagen_des_Augendiagramms Augendiagramm] für ein
+
The graphic shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram| "eye diagram"]]&nbsp; for a
*impulsinterferenzbehaftetes System ohne Rauschen (links),<br>
+
[[File:EN_Dig_T_3_1_S2.png|right|frame|Eye diagrams with and without intersymbol interference|class=fit]]  
*ein impulsinterferenzfreies System ohne Rauschen (Mitte),<br>
+
#intersymbol interference&nbsp; $\rm (ISI)$&nbsp; system without noise&nbsp; (left),<br>
*das gleiche impulsinterferenzfreie System mit Rauschen (rechts).<br><br>
+
#an ISI-free system without noise&nbsp; (middle),<br>
 +
#the same ISI-free system with noise&nbsp; (right).<br><br>
  
Auf die Definition, Bedeutung und Berechnung des Augendiagramms wird im [http://en.lntwww.de/Digitalsignal%C3%BCbertragung/Fehlerwahrscheinlichkeit_unter_Ber%C3%BCcksichtigung_von_Impulsinterferenzen#Gau.C3.9Ff.C3.B6rmiges_Empfangsfilter_.281.29 Kapitel 3.2] noch ausführlich eingegangen. Die Bilder wurden mit dem Programm &bdquo;bas&rdquo; erzeugt. Hinweis zum Download dieses Programms aus <i>LNTsim</i> finden Sie am Beginn dieses Kapitels unter Kapitelüberblick.<br>
+
The definition,&nbsp; meaning and calculation of the eye diagram will be discussed in detail in the chapter &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference|"Error Probability with Intersymbol Interference"]].&nbsp; These screenshots can be interpreted as follows:
 +
*The middle diagram is from a Nyquist system with cosine rolloff characteristic $($rolloff factor &nbsp;$r = 0.5)$.&nbsp; Thus,&nbsp; no intersymbol interference occurs.<br>
  
[[File:P_ID1364__Dig_T_3_1_S2_version1.png|Augendiagramme mit und ohne Impulsinterferenzen|class=fit]]<br><br>
+
*The right diagram is from the same ISI-free system,&nbsp; although here &nbsp;$d(t) = \pm s_0$&nbsp; does not apply.&nbsp; The deviations from the nominal values &nbsp;$\pm s_0$&nbsp; are here due to the AWGN noise.<br>
  
Diese Bilder können wie folgt interpretiert werden:
+
*From this follows the important insight: &nbsp; The question whether there is an ISI-free or  ISI-affected system can only be decided on the basis of the detection signal&nbsp; (or eye diagram)&nbsp; '''without noise'''.<br>
*Das mittlere Diagramm stammt von einem Nyquistsystem mit Cosinus&ndash;Rolloff&ndash;Charakteristik (Rolloff&ndash;Faktor <i>r</i> = 0.5). Es treten somit keine Impulsinterferenzen auf.<br>
 
  
*Auch das rechte Augendiagramm stammt von einem impulsinterferenzfreien System (genauer gesagt: vom gleichen System wie die mittlere Grafik), obwohl hier <i>d</i>(<i>&nu;</i><i>T</i>) = &plusmn;<i>s</i><sub>0</sub> nicht zutrifft. Die Abweichungen von den Sollwerten &plusmn;<i>s</i><sub>0</sub> sind hier auf das AWGN&ndash;Rauschen zurückzuführen.<br>
 
  
*Aus diesem letzten Punkt folgt die wichtige Erkenntnis: Die Frage, ob ein impulsinterferenzfreies oder ein impulsinterferenzbehaftetes System vorliegt, kann nur anhand des Detektionssignals (bzw. des Augendiagramms) ohne Rauschen entschieden werden.<br>
+
The left diagram indicates intersymbol interference,&nbsp; since no noise is taken into account here.  
 +
*The reason for this intersymbol interference could be that the overall frequency response of transmitter and receiver does not exactly fulfill the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain|"first Nyquist criterion"]]&nbsp; due to tolerances.<br>
  
*Das linke Diagramm weist auf Impulsinterferenzen hin, da hier kein Rauschen berücksichtigt ist. Ein Grund für diese Impulsinterferenzen könnte sein, dass der Gesamtfrequenzgang von Sender und Empfänger das [http://en.lntwww.de/Digitalsignal%C3%BCbertragung/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Frequenzbereich erste Nyquistkriterium] aufgrund von Toleranzen nicht exakt erfüllt.<br>
+
*However,&nbsp; intersymbol interference also occurs with a channel with frequency-dependent frequency response &nbsp;$H_{\rm K}(f)$,&nbsp; if the receiver does not succeed in compensating the attenuation and phase distortions of the channel completely&nbsp; (i.e. one hundred percent).<br>
  
*Impulsinterferenzen entstehen aber auch bei einem Kanal mit frequenzabhängigem Frequenzgang <i>H</i><sub>K</sub>(<i>f</i>), wenn es dem Empfänger nicht gelingt, die Dämpfungs&ndash; und Phasenverzerrungen des Kanals vollständig (hundertprozentig) zu kompensieren.<br>
+
*Finally,&nbsp; even with the middle system,&nbsp; intersymbol interference occurs if the decision is not made exactly in the center of the eye,&nbsp; but at a detection time &nbsp;$T_{\rm D} \ne 0$.&nbsp; Then the basic detection pulse values must be defined to &nbsp;$g_\nu = g_d(T_{\rm D} + \nu \cdot T)$.<br>
  
*Letztendlich kommt es auch beim mittleren System zu Impulsinterferenzen, wenn nicht exakt in Augenmitte entschieden wird, sondern zu einem Detektionszeitpunkt <i>T</i><sub>D</sub> &ne; 0. In diesem Fall müssen dann die Detektionsgrundimpulswerte zu <i>g<sub>&nu;</sub></i> = <i>g<sub>d</sub></i>(<i>T</i><sub>D</sub> + <i>&nu;</i> &middot; <i>T</i>) definiert werden.<br>
 
  
 
+
== Some remarks on the channel frequency response ==
== Einige Anmerkungen zum Kanalfrequenzgang ==
 
 
<br>
 
<br>
Für die weiteren Abschnitte von Kapitel 3 wird meist von folgendem Blockschaltbild ausgegangen.
+
For the further sections in this third main chapter the following block diagram is&nbsp;  (mostly)&nbsp; assumed.&nbsp; The main difference to the&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|block diagram of the first main chapter]]&nbsp; is the channel frequency response&nbsp; $($German:&nbsp; "Kanalfrequenzgang" &nbsp; &rArr; &nbsp; subscript&nbsp; "K"$)$,&nbsp; which is always assumed to be ideal &nbsp; &rArr; &nbsp; &nbsp;$H_{\rm K}(f) = 1$.&nbsp; <br>
  
<br>[[File:P_ID1366__Dig_T_3_1_S5_version1.png|Blockschaltbild eines Systems mit verzerrendem Kanal|class=fit]]<br><br>
+
The following applies to the&nbsp; "frequency response"&nbsp; and the&nbsp; &nbsp;"impulse response"&nbsp; of the channel &nbsp;$(\rm exp[\hspace{0.05cm} . ]$&nbsp; denotes the&nbsp; "exponential function"$)$:
 
+
:$$H_{\rm K}(f)  =  {\rm exp} \left[ - a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right] \cdot
Der wesentliche Unterschied gegenüber dem [http://en.lntwww.de/Digitalsignal%C3%BCbertragung/Systemkomponenten_eines_Basisband%C3%BCbertragungssystems#Ersatzschaltbild_und_Voraussetzungen_f.C3.BCr_Kapitel_1 Blockschaltbild zu Kapitel 1] ist der Kanalfrequenzgang <i>H</i><sub>K</sub>(<i>f</i>), der in den Kapiteln 1 und 2 stets zu <i>H</i><sub>K</sub>(<i>f</i>) = 1 und damit als ideal angenommen wurde.<br>
 
 
 
Im Folgenden gelte für den Frequenzgang und die Impulsantwort des Kanals:
 
 
 
:<math>H_{\rm K}(f)  =  {\rm exp} \left[ - a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right] \cdot
 
 
   {\rm exp} \left[ - {\rm j} \cdot a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right]
 
   {\rm exp} \left[ - {\rm j} \cdot a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right]
   \hspace{0.05cm}, </math>
+
   \hspace{0.05cm}, $$
:<math>h_{\rm K}(t)  =    \frac{ a_{{\star}\hspace{0.01cm}({\rm Np})}}{  \sqrt{2  \pi^2 \cdot R_{\rm B} \cdot t^3}}\hspace{0.1cm} \cdot
+
[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with distorting channel|class=fit]]
 +
:$$h_{\rm K}(t)  =    \frac{ a_{{\star}\hspace{0.01cm}({\rm Np})}}{  \sqrt{2  \pi^2 \cdot R_{\rm B} \cdot t^3}}\hspace{0.1cm} \cdot
 
   {\rm exp} \left[ - \frac{a_{{\star} \hspace{0.01cm}({\rm Np})}^2}{2  \pi \cdot R_{\rm B} \cdot t}\hspace{0.1cm}\right]
 
   {\rm exp} \left[ - \frac{a_{{\star} \hspace{0.01cm}({\rm Np})}^2}{2  \pi \cdot R_{\rm B} \cdot t}\hspace{0.1cm}\right]
   \hspace{0.05cm}.</math>
+
   \hspace{0.05cm}.$$
  
Hierbei gibt <i>a</i><sub>&#8727;(Np)</sub> die charakteristische Kabeldämpfung bei der halben Bitrate in Neper (Np) an:
+
Here &nbsp;$a_{{\star} \hspace{0.01cm}({\rm Np})}$&nbsp; indicates the cable attenuation at half the bit rate.&nbsp; We call this quantity the &nbsp;'''characteristic cable attenuation'''&nbsp; in Neper&nbsp; $\rm (Np)$:
 +
:$$a_{{\star} \hspace{0.01cm}({\rm Np})} = a_{\rm K}(f = {R_{\rm B}}/{2})= 0.1151 \cdot a_{{\star} \hspace{0.01cm}({\rm dB})}
 +
  \hspace{0.05cm}.$$
  
:<math>a_{{\star} \hspace{0.01cm}({\rm Np})} = a_{\rm K}(f = {R_{\rm B}}/{2})= 0.1151 \cdot a_{{\star} \hspace{0.01cm}({\rm dB})}
+
#The corresponding dB value &nbsp; &rArr; &nbsp; $a_{{\star} \hspace{0.01cm}({\rm dB})}$&nbsp; is larger by a factor of &nbsp;$1/0.1151 = 8.686$.&nbsp;
  \hspace{0.05cm}.</math>
+
#In realized systems, the characteristic cable attenuation &nbsp;$a_{{\star} \hspace{0.01cm}({\rm dB})}$&nbsp; is in the range between &nbsp;$40 \ \rm dB$&nbsp; and &nbsp;$100 \ \rm dB$.  
 +
#The addition&nbsp; "(Np)"&nbsp; or&nbsp; "(dB)"&nbsp; is omitted in the following.<br>
  
Der entsprechende dB&ndash;Wert ist um den Faktor 1/0.1151 = 8.686 größer. Bei realisierten Systemen liegt die charakteristische Kabeldämpfung <i>a</i><sub>&#8727;(dB)</sub> im Bereich zwischen 40 dB und 100 dB. Auf den Zusatz &bdquo;(Np)&rdquo; bzw. &bdquo;(dB)&rdquo; wird im Folgenden meist verzichtet.<br>
 
  
Im Kapitel 4 des Buches &bdquo;Lineare zeitinvariante Systeme&rdquo; wird gezeigt, dass diese Gleichungen die Verhältnisse bei leitungsgebundener Übertragung über Koaxialkabel mit guter Näherung wiedergeben. Bei einer Zweidrahtleitung ist die Abweichung zwischen dieser sehr einfachen, analytisch handhabbaren Formel und den tatsächlichen Gegebenheiten etwas größer.<br>
+
In the main chapter 4: &nbsp; "Properties of electrical lines" of the book &nbsp;[[Linear_and_Time_Invariant_Systems|"Linear and Time Invariant Systems"]]&nbsp; it is shown that these equations reproduce the conditions with good approximation for wireline communication systems via coaxial cable.&nbsp; For a two-wire line,&nbsp; the deviation between this very simple&nbsp; (analytically manageable)&nbsp; formula and the actual conditions is somewhat larger.<br>
  
Eine kurze Zusammenfassung der Herleitungen in Kapitel 4 des Buches &bdquo;LZI&ndash;Systeme&rdquo; folgt auf den beiden nächsten Seiten, wobei wir uns zur Vereinfachung auf ein redundanzfreies Binärsystem festlegen. Somit ist die Bitrate <i>R</i><sub>B</sub> gleich dem Kehrwert der Symboldauer <i>T</i>.<br>
+
A short summary of these derivations follows in the next two sections.
 +
*For simplicity,&nbsp; we will use a redundancy-free binary system.
 +
*Thus,&nbsp; the bit rate &nbsp;$R_{\rm B}$&nbsp; is equal to the reciprocal of the symbol duration &nbsp;$T$.<br>
  
  
== Frequenzgang eines Koaxialkabels (1) ==
+
== Frequency response of a coaxial cable==
 
<br>
 
<br>
Ein Koaxialkabel mit dem Kerndurchmesser 2.6 mm, dem Außendurchmesser 9.5 mm und der Länge <i>l</i> hat den folgenden Frequenzgang:
+
A &nbsp;'''coaxial cable'''&nbsp; with core diameter 2.6 mm,&nbsp; outer diameter 9.5 mm and length &nbsp;$l$&nbsp; has the following frequency response:
 
+
:$$H_{\rm K}(f) = {\rm e}^{-\left[ a_{\rm K}(f) + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} b_{\rm K}(f)\right] } = {\rm e}^{- \alpha_0 \hspace{0.05cm} \cdot \hspace{0.05cm} l}  \cdot
:<math>H_{\rm K}(f) = {\rm exp}\left [ a_{\rm K}(f) + {\rm j} \cdot b_{\rm K}(f) \right ] = {\rm e}^{- \alpha_0 \hspace{0.05cm} \cdot \hspace{0.05cm} l}  \cdot
 
 
   {\rm e}^{- (\alpha_1 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_1) \hspace{0.05cm}\cdot f \hspace{0.05cm}\cdot \hspace{0.05cm}l}  \cdot
 
   {\rm e}^{- (\alpha_1 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_1) \hspace{0.05cm}\cdot f \hspace{0.05cm}\cdot \hspace{0.05cm}l}  \cdot
 
   {\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l}
 
   {\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l}
     \hspace{0.05cm},</math>
+
     \hspace{0.05cm},$$
 
 
wobei bei diesen Abmessungen &ndash; man spricht vom Normalkoaxialkabel &ndash; folgende Parameter gelten:
 
  
:<math>\alpha_0 = 0.00162 \hspace{0.15cm}\frac {\rm Np}{\rm km} \hspace{0.05cm},\hspace{0.2cm}
+
where the following parameters apply for these dimensions &ndash; referred to as&nbsp; "normal coaxial cable":&nbsp;
 +
:$$\alpha_0 = 0.00162 \hspace{0.15cm}\frac {\rm Np}{\rm km} \hspace{0.05cm},\hspace{0.2cm}
 
   \alpha_1 = 0.000435 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm},
 
   \alpha_1 = 0.000435 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm},
 
   \hspace{0.2cm}
 
   \hspace{0.2cm}
   \alpha_2 = 0.2722 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot \sqrt{\rm MHz}} \hspace{0.05cm},</math>
+
   \alpha_2 = 0.2722 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot \sqrt{\rm MHz}} \hspace{0.05cm},\hspace{0.2cm}
 
+
\beta_1 = 21.78 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm},
:<math>\beta_1 = 21.78 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm},
 
 
   \hspace{0.2cm}
 
   \hspace{0.2cm}
 
   \beta_2 = 0.2722 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot \sqrt{\rm MHz}}
 
   \beta_2 = 0.2722 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot \sqrt{\rm MHz}}
   \hspace{0.05cm}.</math>
+
   \hspace{0.05cm}.$$
  
In obiger Gleichung sind die Dämpfungsparameter in &bdquo;Np&rdquo; einzusetzen, die Phasenparameter in &bdquo;rad&rdquo;.<br>
+
In the above equation the attenuation parameters&nbsp; ("alpha")&nbsp; are to be inserted in&nbsp; "Np"&nbsp; and the phase parameters&nbsp; ("beta")&nbsp;  in&nbsp; "rad".<br>
  
Die Grafik zeigt den Dämpfungsverlauf
+
[[File:EN_Dig_T_3_1_S3a.png|right|frame|Attenuation curve of a coaxial cable and approximation&nbsp; (skin effect only) |class=fit]]<br>
 +
The graph shows the exact attenuation curve for a standard coaxial cable of one kilometer length and an approximation for frequencies up to &nbsp;$f = 1000\ \rm  MHz$:
 +
:$$a_{\rm K}(f)  = \alpha_0  \cdot l \hspace{0.05cm} + \hspace{0.05cm} \alpha_1  \cdot f \cdot l
 +
+ \hspace{0.05cm} \alpha_2  \cdot \sqrt{f} \cdot l \hspace{0.05cm},$$
 +
:$$a_{\rm K}(f)  \approx \alpha_2  \cdot \sqrt{f} \cdot l.$$
 +
 +
*The axis is labeled in&nbsp; "$\rm dB$"&nbsp; on the left and&nbsp; "$\rm Np$"&nbsp; on the right.
 +
*One&nbsp; $\rm Np$&nbsp; ("Neper")&nbsp; corresponds to&nbsp; $8.686 \ \rm dB$.
  
:<math>a_{\rm K}(f)  = \alpha_0  \cdot l \hspace{0.05cm} + \hspace{0.05cm} \alpha_1  \cdot f \cdot l
 
+ \hspace{0.05cm} \alpha_2  \cdot \sqrt{f} \cdot l \approx \alpha_2  \cdot \sqrt{f} \cdot l</math>
 
  
für ein Koaxialkabel von einem Kilometer Länge für Frequenzen bis 1000 MHz. Die Achse ist sowohl in dB (links) als auch in Np (rechts) beschriftet. Ein Neper entspricht etwa 8.7 dB.<br>
+
We refer to the interactive HTML5/JavaScript applet&nbsp; [[Applets:Attenuation_of_Copper_Cables|"Attenuation of copper cables".]]<br>
  
[[File:P_ID1367__Dig_T_3_1_S3a_version1.png|Dämpfungsverlauf eines Koaxialkabels und Näherung|class=fit]]<br>
+
You can see from the diagram and the above numerical values:
 +
*The first term &nbsp;$(\alpha_0 \cdot l)$&nbsp; originating from the ohmic losses is negligible.&nbsp; Moreover,&nbsp; this term causes only frequency-independent attenuation and no signal distortion.<br>
  
Die ausführliche Bildbeschreibung folgt auf der nächsten Seite.<br><br>
+
*The second term &nbsp;$(\alpha_1 \cdot f \cdot l)$,&nbsp; due to the transverse losses,&nbsp; is proportional to frequency and therefore becomes noticeable only at very high frequencies;&nbsp; it will be neglected in the following.<br>
  
Bereits an dieser Stelle soll darauf hingewiesen werden, dass die obige Näherung
+
*The frequency-proportional phase &nbsp;$(\beta_1 \cdot f \cdot l)$&nbsp; only results in a signal delay by the transit time &nbsp;$\beta_1/(2\pi)  \cdot l$,&nbsp; but no distortion.&nbsp; This transit time will also be disregarded in the following.<br>
  
:<math>a_{\rm K}(f)  \approx \alpha_2  \cdot \sqrt{f} \cdot l</math>
+
*With these simplifications,&nbsp; the frequency response is thus determined by the skin effect alone.&nbsp; Since the numerical values for &nbsp;$\alpha_2$&nbsp; $($in Np$)$&nbsp; and &nbsp;$\beta_2$&nbsp; $($in rad$)$&nbsp; are the same,&nbsp; it follows:
 +
:$$H_{\rm K}(f) ={\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l}
 +
   \cdot {\rm e}^{- \alpha_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l \hspace{0.05cm}\cdot \sqrt{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}
 +
  2f}} \hspace{0.05cm}.$$
 +
*Often in the literature &ndash; and also in this tutorial &ndash; the attenuation measure at half the bit rate is used,&nbsp; which we call the &nbsp;"characteristic cable attenuation"&nbsp; $($in Neper$)$:
 +
:$$a_{\star} = a_{\rm K}(f ={R_{\rm B}}/{2})= a_{\rm K}(f = \frac{1}{2 \cdot T})\approx \frac{\alpha_2 \cdot
 +
  l }{ \sqrt {2\cdot T}}
 +
  \hspace{0.05cm}.$$
  
nur für Koaxialkabel zulässig ist. Bei diesen können die Koeffizienten <i>&alpha;</i><sub>0</sub> &asymp; 0 und <i>&alpha;</i><sub>1</sub> &asymp; 0 vernachlässigt werden. Für eine symmetrische Zweidrahtleitung sind diese Koeffizienten sehr viel gößer und die obige Näherung ist unzulässig.<br>
+
{{GraueBox|TEXT= 
 +
$\text{Example 1:}$&nbsp; For a binary system with half the bit rate &nbsp;$R_{\rm B}/2 = 280 \ \rm Mbit/s$&nbsp; and &nbsp;$l = 1\ \rm  km$ &nbsp;  &nbsp;$\Rightarrow \ a_{\star}  \approx 4.55 \ \rm Np$&nbsp; or &nbsp;$a_{\star} \approx 40 \ \rm dB$&nbsp; (green markings in the diagram).  
  
Genauere Informationen finden Sie im [http://en.lntwww.de/Lineare_zeitinvariante_Systeme/Kupfer%E2%80%93Doppelader#Umrechnung_zwischen_.7FUNIQ-MathJax53-QINU.7F.E2.80.93_und_.7FUNIQ-MathJax54-QINU.7F.E2.80.93_Parametern_.281.29 Kapitel 4.3] des Buches &bdquo;Lineare zeitinvariante Systeme&rdquo;.<br>
+
*But if the half bit rate is only &nbsp;$70 \ \rm Mbit/s$, &nbsp;$a_{\star} = 40 \ \rm dB$&nbsp; characterizes a transmission system with cable length &nbsp;$l = 2\ \rm  km$.
  
== Frequenzgang eines Koaxialkabels (2) ==
+
*Note: &nbsp; The above approximation &nbsp;$a_{\rm K}(f)   \approx \alpha_2  \cdot \sqrt{f} \cdot l$&nbsp; is only admissible for coaxial cables,&nbsp; since for these the coefficients &nbsp;$\alpha_0$&nbsp; and  &nbsp;$\alpha_1$&nbsp; can be neglected.
<br>
 
Betrachten wir nochmals die Grafik von eben. Diese zeigt den Dämpfungsverlauf
 
  
:<math>a_{\rm K}(f)  = \alpha_0  \cdot l \hspace{0.05cm} + \hspace{0.05cm} \alpha_1 \cdot f \cdot l
+
*However,&nbsp; for a &nbsp;[[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs|"balanced two-wire cable"]],&nbsp; the coefficients &nbsp;$\alpha_0$&nbsp; and &nbsp;$\alpha_1$&nbsp; are much larger and the above approximation is invalid.<br>}}
+ \hspace{0.05cm} \alpha_2  \cdot \sqrt{f} \cdot l \approx \alpha_2  \cdot \sqrt{f} \cdot l</math>
 
  
für ein Koaxialkabel von einem Kilometer Länge für Frequenzen bis 1000 MHz. Die Achse ist sowohl in dB (links) als auch in Np (rechts) beschriftet. Ein Neper (Np) entspricht etwa 8.7 dB.<br>
 
  
[[File:P_ID1367__Dig_T_3_1_S3a_version1.png|Dämpfungsverlauf eines Koaxialkabels und Näherung|class=fit]]<br><br>
 
  
Man erkennt aus diesem Diagramm und den obigen Zahlenwerten:
+
== Impulse response of a coaxial cable ==
*Der von den Ohmschen Verlusten herrührende erste Term (<i>&alpha;</i><sub>0</sub> &middot; <i>l</i>) ist vernachlässigbar. Zudem bewirkt er nur eine frequenzunabhängige Dämpfung und keine Signalverzerrung.<br>
+
<br>
 +
We now consider the coaxial cable impulse response,&nbsp; which is for a binary system &nbsp;$(R_{\rm B} = 1/T)$&nbsp; as follows:
 +
:$$h_{\rm K}(t) =  \frac{ a_{\rm \star \hspace{0.01cm}(Np)}/T}{  \sqrt{2  \pi^2 \cdot (t/T)^3}}\hspace{0.1cm} \cdot
 +
  {\rm exp} \left[ - \frac{a_{\rm \star \hspace{0.01cm}(Np)}^2}{2  \pi  \cdot t/T}\hspace{0.1cm}\right]
 +
  \hspace{0.05cm}.$$
  
*Der auf die Querverluste zurückzuführende zweite Term ist proportional zur Frequenz und macht sich erst bei sehr hohen Frequenzen bemerkbar; er wird im Folgenden vernachlässigt.<br>
+
This time course is shown here for characteristic cable attenuations &nbsp;$(a_{\rm \star})$&nbsp; between &nbsp;$40 \ \rm dB$&nbsp; and &nbsp;$100 \ \rm dB$.&nbsp; Note the conversion &nbsp; $\rm  1 \ Np = 8.686 \ dB.$<br>
 +
[[File:EN_Dig_T_3_1_S3b.png|right|frame|Impulse and rectangular pulse response of the coaxial cable&nbsp; (skin effect only); &nbsp; <u>Note:</u> <br>Here,&nbsp; the basic receiver pulse &nbsp;$g_r(t)$&nbsp; is identical with the&nbsp; "rectangular pulse response"]]
  
*Die frequenzproportionale Phase <i>&beta;</i><sub>1</sub> &middot; <i>f</i> &middot; <i>l</i> hat nur eine Signalverzögerung um die Laufzeit <i>&beta;</i><sub>1</sub> &middot; <i>l</i>/2&pi; zur Folge, jedoch keine Verzerrung. Auch diese Laufzeit wird im Folgenden außer Acht gelassen.<br>
+
One can see from this time domain plot:
 +
*Even with the relatively small characteristic cable attenuation value  &nbsp;$a_{\rm \star} = 40 \ \rm dB$,&nbsp; the impulse response extends over more than &nbsp;$100$&nbsp; symbol durations &nbsp;$(T)$.<br>
  
*Mit diesen Vereinfachungen wird somit der Frequenzgang allein durch den Skineffekt bestimmt. Da die Zahlenwerte für <i>&alpha;</i><sub>2</sub> (in Np) und <i>&beta;</i><sub>2</sub> (in rad) übereinstimmen, gilt somit auch:
+
*The larger &nbsp;$a_{\rm \star}$&nbsp; is chosen,&nbsp; the broader and lower the impulse response becomes.&nbsp; The integral over &nbsp;$h_{\rm K}(t)$&nbsp; from zero to infinity is the same for all curves,&nbsp; since &nbsp;$H_{\rm K}(f=0) = 1$&nbsp; always holds&nbsp; $(\alpha_0 = 0$,&nbsp; $\alpha_1 = 0).$<br>
  
::<math>H_{\rm K}(f) ={\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l}
+
*The basic receiver pulse &nbsp;$g_r(t) = g_s(t) \star h_{\rm K}(t)$&nbsp; is nearly equal in shape to &nbsp;$h_{\rm K}(t)$.&nbsp; The right ordinate axis shows &nbsp;$g_r(t)/s_0$&nbsp; when &nbsp;$g_s(t)$&nbsp; is an NRZ rectangular pulse with height &nbsp;$s_0$&nbsp; and duration &nbsp;$T$.&nbsp; <br>
  \cdot {\rm e}^{- \alpha_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l \hspace{0.05cm}\cdot \sqrt{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}
 
  2f}} \hspace{0.05cm}.</math>
 
  
*Häufig wird in der Literatur &ndash; und auch in diesem Tutorial &ndash; das Dämpfungsmaß bei der halben Bitrate benutzt, das wir als charakteristische Kabeldämpfung (in Neper) bezeichnen:
+
*For &nbsp;$a_{\rm \star}  \ge 60 \ \rm dB$,&nbsp; &nbsp;$h_{\rm K}(t)$&nbsp; and &nbsp;$g_r(t)$&nbsp; are indistinguishable within the character accuracy when suitably normalized.&nbsp; For &nbsp;$a_{\rm \star} = 40 \ \rm dB$,&nbsp; one can see a small difference at the peak&nbsp; (yellow background); &nbsp; $g_r(t)/s_0$&nbsp; is here minimally smaller than &nbsp;$T \cdot h_{\rm K}(t)$.<br>
  
::<math>a_{\star} = a_{\rm K}(f ={R_{\rm B}}/{2})= a_{\rm K}(f = \frac{1}{2 \cdot T})\approx \frac{\alpha_2 \cdot
+
*However,&nbsp; with the characteristic attenuation &nbsp;$a_{\rm \star} = 40 \ \rm dB$,&nbsp; the pulse amplitude at the cable end is only less than &nbsp;$7\%$&nbsp; of the input amplitude.&nbsp; At &nbsp;$60 \ \rm dB$&nbsp; and &nbsp;$100 \ \rm dB$,&nbsp; this value drops to &nbsp;$3\%$&nbsp; and &nbsp;$2\%$.<br><br>
  l }{ \sqrt {2\cdot T}}
 
  \hspace{0.05cm}.</math>
 
  
*Bei einem Binärsystem mit der halben Bitrate <i>R</i><sub>B</sub>/2 = 280 Mbit/s und <i>l</i> = 1 km ergibt sich <i>a</i><sub>&#8727;</sub> zu etwa 4.55 Np bzw. 40 dB (grün eingezeichnete Markierungen). Beträgt aber die halbe Bitrate nur 70 Mbit/s, so charakterisiert <i>a</i><sub>&#8727;</sub> = 40 dB ein Übertragungssystem mit der Kabellänge <i>l</i> = 2 km.<br><br>
+
In &nbsp;[[Aufgaben:Exercise_3.1:_Impulse_response_of_the_Coaxial_Cable|Exercise 3.1]],&nbsp; the coaxial cable impulse response is analyzed in detail.&nbsp; We also refer to the interactive&nbsp; (German language)&nbsp; SWF applet &nbsp;[[Applets:Zeitverhalten_von_Kupferkabeln|"Time behavior of copper cables"]].<br>
  
Wir verweisen an dieser Stelle auf das Interaktionsmodul &nbsp; [[:File:Kabeldaempfung.swf|Dämpfung von Kupferkabeln.]]<br>
+
== Prerequisites for the entire third main chapter ==
 
 
== Impulsantwort eines Koaxialkabels ==
 
 
<br>
 
<br>
Betrachten wir nun die Impulsantwort, die bei einem Binärsystem (<i>R</i><sub>B</sub> = 1/<i>T</i>) wie folgt lautet:
+
Consider again the block diagram of a transmission system, assuming a highly distorting channel, such as is present in wireline transmission.<br>
  
:<math>h_{\rm K}(t) =  \frac{ a_{\rm \star \hspace{0.01cm}(Np)}/T}{  \sqrt{2  \pi^2 \cdot (t/T)^3}}\hspace{0.1cm} \cdot
+
Due to the channel frequency response &nbsp;$H_{\rm K}(f)$,&nbsp; which is highlighted in red in the diagram,&nbsp; there are also certain limitations for the other system components:
  {\rm exp} \left[ - \frac{a_{\rm \star \hspace{0.01cm}(Np)}^2}{2  \pi  \cdot t/T}\hspace{0.1cm}\right]
 
  \hspace{0.05cm}.</math>
 
  
Dieser Zeitverlauf ist nachfolgend für <i>a</i><sub>&#8727;</sub> = 40 dB, 60 dB, 80 dB und 100 dB dargestellt. Beachten Sie wieder die Umrechnung: 1 Np = 8.686 dB.<br>
+
[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with&nbsp; (highly)&nbsp; distorting channel]]
  
[[File:P_ID1368__Dig_T_3_1_S3b_version1.png|class=fit|Impulsantwort des Koaxialkabels]]<br>
+
*The basic receiver pulse &nbsp;$g_r(t)  = g_s(t) \star h_{\rm K}(t)$&nbsp; extends over hundreds of bits.&nbsp; Therefore,&nbsp; the receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; cannot be applied as a &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Optimal_binary_receiver_.E2.80.93_.22Matched_Filter.22_realization|"matched filter"]];&nbsp; this would approximately double the duration of the basic detection pulse &nbsp;$g_d(t)$&nbsp; compared to &nbsp;$g_r(t)$.&nbsp; <br>
  
Man erkennt aus dieser Zeitbereichsdarstellung
+
*Rather,&nbsp; the receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; must compensate for the enormous attenuation distortions &nbsp;$(\alpha_2$&ndash;term$)$ and the enormous phase distortions &nbsp;$(\beta_2$&ndash;term$)$ of the coaxial channel &nbsp;$H_{\rm K}(f)$,&nbsp; especially if a simple threshold decision is assumed.<br>
*Bereits mit der relativ kleinen charakteristischen Kabeldämpfung <i>a</i><sub>&#8727;</sub> = 40 dB erstreckt sich die Impulsantwort über mehr als 100 Symboldauern.<br>
 
  
*Je größer <i>a</i><sub>&#8727;</sub> gewählt wird, desto breiter und niedriger wird die Impulsantwort. Das Integral über <i>h</i><sub>K</sub>(<i>t</i>) von 0 bis Unendlich ist für alle Kurven gleich, da stets <i>H</i><sub>K</sub>(<i>f</i> = 0) = 1 gilt.<br>
+
*Although this linear form of signal equalization can be supported by more elaborate decision strategies &ndash; e.g.  &nbsp; [[Digital_Signal_Transmission/Entscheidungsrückkopplung|"decision feedback"]], &nbsp;[[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receivers"]],&nbsp; [[Digital_Signal_Transmission/Viterbi–Empfänger|"Viterbi receivers"]].&nbsp; For wired transmission,&nbsp; a linear signal equalization &nbsp; &rArr; &nbsp; receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; cannot be completely dispensed with due to the very strong distortions.<br>
  
*Der Empfangsgrundimpuls <i>g<sub>r</sub></i>(<i>t</i>) = <i>g<sub>s</sub></i>(<i>t</i>) &#8727; <i>h</i><sub>K</sub>(<i>t</i>) ist nahezu formgleich mit <i>h</i><sub>K</sub>(<i>t</i>). Die rechte Ordinatenachse zeigt <i>g<sub>r</sub></i>(<i>t</i>)/<i>s</i><sub>0</sub>, wenn <i>g<sub>s</sub></i>(<i>t</i>) ein NRZ&ndash;Rechteckimpuls mit Höhe <i>s</i><sub>0</sub> und Dauer <i>T</i> ist.<br>
+
*The noise &nbsp;$n(t)$&nbsp; is further assumed to be additive,&nbsp; white,&nbsp; and Gaussian distributed&nbsp; $\rm (AWGN)$,&nbsp; which is justified for a coaxial cable.&nbsp; For a two-wire line,&nbsp; crosstalk from adjacent copper wires is the dominant noise,&nbsp; as explained in detail in the chapter &nbsp;[[Examples_of_Communication_Systems| "ISDN"]]&nbsp; ("Integrated Services Digital Network").<br><br>
  
*Für <i>a</i><sub>&#8727;</sub> &#8805; 60 dB sind <i>h</i><sub>K</sub>(<i>t</i>) und <i>g<sub>r</sub></i>(<i>t</i>) bei geeigneter Normierung innerhalb der Zeichengenauigkeit nicht zu unterscheiden. Für <i>a</i><sub>&#8727;</sub> = 40 dB erkennt man eine kleine Differenz an der Spitze (gelbe Hinterlegung); <i>g<sub>r</sub></i>(<i>t</i>)/<i>s</i><sub>0</sub> ist hier minimal kleiner als <i>T</i> &middot; <i>h</i><sub>K</sub>(<i>t</i>).<br>
+
{{BlaueBox|TEXT= 
 +
$\text{Conclusion:}$&nbsp; In the following chapters  we consider only the binary bipolar redundancy-free transmission &nbsp; &rArr; &nbsp; bit rate &nbsp;$R_{\rm B} = 1/T$.&nbsp; It is always assumed that:
  
*Mit der charakteristischen Dämpfung 40 dB beträgt die Impulsamplitude am Kabelende weniger als 7% der Eingangsamplitude. Bei 60 dB bzw. 100 dB sinkt dieser Wert auf 3% bzw. 1%.<br><br>
+
*The basic transmission pulse &nbsp;$g_s(t)$&nbsp; is NRZ rectangular with amplitude &nbsp;$s_0$&nbsp; and duration &nbsp;$T$.  
  
In der Aufgabe A3.1 wird die hier betrachtete Impulsantwort noch eingehend analysiert. Weiterhin verweisen wir auf das folgende Interaktionsmodul: [[:File:Zeitverhalten_von_Kupferkabeln.swf|Zeitverhalten von Kupferkabeln]]<br>
+
*Thus,&nbsp;  the transmitted signal &nbsp;$s(t)$&nbsp; is equal to &nbsp;$\pm s_0$&nbsp; at all times and the spectral function is: &nbsp; $G_s(f) = s_0 \cdot T \cdot {\rm sinc}(f T)$.<br>
  
== Blockschaltbild und Voraussetzungen für Kapitel 3 ==
+
*Splitting the equalization between transmitter and receiver according to the root&ndash;root characteristic does not make sense for conducted transmission.  
<br>
 
Betrachten wir nochmals das Blockschaltbild eines Übertragungssystems mit stark verzerrendem Kanal.<br>
 
  
[[File:P ID1369 Dig T 3 1 S5 version1.png|fit|Blockschaltbild eines  Systems mit verzerrendem Kanal]]<br>
+
*There would be too much intersymbol interference already at the transmitter.<br><br>}}
  
Aufgrund des in der Grafik rot hervorgehobenen Kanalfrequenzgangs <i>H</i><sub>K</sub>(<i>f</i>) ergeben sich auch für die anderen Systemkomponenten gewisse Einschränkungen:
 
  
*Der Empfangsgrundimpuls <i>g<sub>r</sub></i>(<i>t</i>) = <i>g<sub>s</sub></i>(<i>t</i>) &#8727; <i>h</i><sub>K</sub>(<i>t</i>) erstreckt sich über Hunderte von Bits (siehe letzte Seite). Deshalb kann das Empfangsfilter <i>H</i><sub>E</sub>(<i>f</i>) nicht als [http://en.lntwww.de/Digitalsignal%C3%BCbertragung/Fehlerwahrscheinlichkeit_bei_Basisband%C3%BCbertragung#Optimaler_Bin.C3.A4rempf.C3.A4nger_-_Realisierung_mit_Matched-Filter_.281.29 Matched&ndash;Filter] angesetzt werden, da so die Dauer des Detektionsgrundimpulses <i>g<sub>d</sub></i>(<i>t</i>) gegenüber <i>g<sub>r</sub></i>(<i>t</i>) nochmals etwa verdoppelt würde.<br>
+
== Exercises for the chapter==
 +
<br>
 +
[[Aufgaben:Exercise_3.1:_Impulse_response_of_the_Coaxial_Cable|Exercise 3.1: Impulse response of the Coaxial Cable]]
  
*Vielmehr muss <i>H</i><sub>E</sub>(<i>f</i>) die enormen Dämpfungsverzerrungen (<i>&alpha;</i><sub>2</sub>&ndash;Term) und Phasenverzerrungen (<i>&beta;</i><sub>2</sub>&ndash;Term) des koaxialen Kanals <i>H</i><sub>K</sub>(<i>f</i>) kompensieren, wenn wie in den Kapiteln 3.1 bis 3.5 von einem einfachen Schwellenwertentscheider ausgegangen wird.<br>
+
[[Aufgaben:Exercise_3.1Z:_Frequency_Response_of_the_Coaxial_Cable|Exercise 3.1Z: Frequency Response of the Coaxial Cable]]
 
 
*Diese lineare Form der Signalentzerrung kann durch aufwändigere Entscheiderstrategien &ndash; wie in den Kapiteln 3.6 bis 3.8 (Entscheidungsrückkopplung, Korrelationsempfäger,  Viterbi&ndash;Empfänger) beschrieben &ndash; unterstützt werden. Bei leitungsgebundener Übertragung kann aber aufgrund der sehr starken Verzerrungen auf eine lineare Signalentzerrung nicht vollständig verzichtet werden.<br>
 
 
 
*Das Rauschen <i>n</i>(<i>t</i>) wird weiterhin als additiv, weiß und gaußverteilt (AWGN) angesetzt, was bei einem Koaxialkabel gerechtfertigt ist. Bei einer Zweidrahtleitung ist das Nebensprechen von benachbarten Kupferadern die dominante Störung, wie im Kapitel [http://en.lntwww.de/Beispiele_von_Nachrichtensystemen ISDN] (<i>Integrated Services Digital Network</i>) im Buch &bdquo;Beispiele von Nachrichtensystemen&rdquo; ausführlich dargelegt wird.<br><br>
 
 
 
Zunächst betrachten wir in den Kapiteln 3.2 und 3.3 die binäre bipolare redundanzfreie Übertragung, wobei die Bitrate <i>R</i><sub>B</sub> = 1/<i>T</i> beträgt. Dabei wird stets vorausgesetzt:
 
 
 
*Der Sendegrundimpuls <i>g<sub>s</sub></i>(<i>t</i>) ist NRZ&ndash;rechteckförmig mit Amplitude <i>s</i><sub>0</sub> und Dauer <i>T</i>. Somit ist das Sendesignal <i>s</i>(<i>t</i>) zu allen Zeiten gleich &plusmn;<i>s</i><sub>0</sub>. Die Spektralfunktion lautet: <i>G<sub>s</sub></i>(<i>f</i>) = <i>s</i><sub>0</sub> &middot; <i>T</i> &middot; si(&pi; <i>f</i> <i>T</i>).<br>
 
 
 
*Eine Aufteilung entsprechend der Wurzel&ndash;Wurzel&ndash;Charakteristik macht bei leitungsgebundener Übertragung keinen Sinn. Es würden bereits beim Sender zu starke Impulsinterferenzen auftreten.<br><br>
 
 
 
Der Einfluss von Impulsinterferenzen bei mehrstufiger und/oder codierter Digitalsignalübertragung wird im Kapitel 3.4 behandelt.<br>
 
 
 
== Aufgaben ==
 
<br>
 
[[Aufgaben:3.1 Koaxialkabel - Impulsantwort|A3.1 Koaxialkabel - Impulsantwort]]
 
  
[[Zusatzaufgaben:3.1 Koaxialkabel - Frequenzgang]]
 
  
 
{{Display}}
 
{{Display}}

Latest revision as of 16:12, 23 January 2023

# OVERVIEW OF THE THIRD MAIN CHAPTER #


The third main chapter focuses on  intersymbol interference  $\rm (ISI)$,  which arises,  for example,  from distortions of the transmission channel or is related to a realization of the receiver filter that deviates from the Nyquist condition.  Subsequently,  some equalization methods are described which can be used to mitigate the system degradation due to intersymbol interference.

The description is given throughout in the baseband.  However,  the results can easily be applied to the carrier frequency systems discussed in the chapter  "Linear Digital Modulation - Coherent Demodulation"

In detail,  this chapter deals with:

  1.   the  »causes and effects«  of intersymbol interference,
  2.   the  »eye diagram«  as a suitable tool for the description of intersymbol interferences,
  3.   the  »error probability calculation«  considering channel distortions,
  4.   the  »influence of intersymbol interference in multilevel and/or coded transmission«,
  5.   the  »optimal Nyquist equalizer as an example of linear channel equalization,
  6.   the  »decision feedback equalization«  $\rm (DFE)$ – an effective nonlinear decision realization,
  7.   the  »correlation receiver«  as an example of  »maximum likelihood or maximum a-posteriori«  $\rm (MAP)$  decision strategy,
  8.   the  »Viterbi receiver«,  a reduced-effort MAP decision algorithm.


Definition of the term "Intersymbol Interference"


For the first two main chapters of this book,  it was assumed that the basic detection pulse  $g_d(t)$

  • either is limited to the time domain  $|t| \le T$,  or
  • has equidistant zero crossings in the symbol spacing $T$.

If we denote the samples of  $g_d(t)$  at multiples of the symbol duration  $T$  (spacing of the pulses)  as the  "basic detection pulse values",  it has been tacitly assumed so far:

Detection signals with and without intersymbol interferences
$$g_\nu = g_d(\nu T) = \left\{ \begin{array}{c} g_0 \\ 0 \\ \end{array} \right.\quad \begin{array}{*{1}c} {\rm{for}}\\ {\rm{for}} \\ \end{array} \begin{array}{*{20}c}\nu = 0, \\ \nu \ne 0. \\ \end{array}$$

As a consequence of this assumption it has resulted that in the binary case the signal component  (index "S") 

$$d_{\rm S}(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g_d ( t - \nu \cdot T) \hspace{0.3cm}{\rm with}\hspace{0.3cm}a_\nu \in \{ -1, +1\}$$

of the detection signal at the instants  $\nu \cdot T$  can take only two different values,  namely  $\pm g_0$.

  • The upper of the following two time plots shows  $d_{\rm S}(t)$  for this ISI-free case with  $g_{\nu \ne 0} = 0$  and  $g_0 = s_0$   ⇒   the value  $g_0$  is equal to the maximum value  $s_0$  of the transmitted signal.
  • Drawn below is the signal waveform for a set of basic detection pulse values that cause intersymbol interference:
$$g_0 = 0.6 \cdot s_0,\hspace{0.15cm} g_{-1} = g_{+1} =0.2 \cdot s_0,\hspace{0.15cm}g_\nu =0\hspace{0.25cm}{\rm for}\hspace{0.25cm} |\nu| \ge 2 .$$

In both plots,  the  (triangular)  basic detection pulse  $g_d(t)$  is drawn in red.  The detection time points  $\nu \cdot T$  are marked by blue circles.

One can see from the lower signal plot:

  • The basic detection pulse  $g_d(t)$  is now different from zero in the range  $|t| \le 1.5 \cdot T$ and thus no longer fulfills the  "Nyquist condition in the time domain"  for intersymbol interference freedom.
  • As a consequence,  at the detection times  (marked with circles)  not only two values  $(\pm s_0)$  are possible as in the upper figure.  Rather,  the following applies here for the detection sampling values:
$$d_{\rm S}(\nu \cdot T) \in \{ \pm s_0, \ \pm 0.6 s_0, \ \pm 0.2 s_0\}\hspace{0.05cm}.$$
  • The samples that are close to the threshold due to unfavorable neighboring pulses are more often falsified by the AWGN noise $($with noise rms value  $\sigma_d)$  than the samples further out. 
  • Exemplarily,  with  $\sigma_d = 0.2 \cdot s_0$  the blue filled points close to the threshold are falsified with probability  $p_{\rm S} ={\rm Q} (1) \approx 16 \%$  and the outer points  (with white core)  are falsified only with  $p_{\rm S} ={\rm Q} (5) \approx 3 \cdot 10^{-7}$.  The error probability of the red filled points  (distance  $0.6 \cdot s_0$  from zero line)  is in between:   $p_{\rm S} ={\rm Q} (3) \approx 0.13 \%$.


So far,  the effects of intersymbol interference have been presented as vividly as possible.  An exact definition is still missing.

$\text{Definition:}$  Intersymbol interference  $\rm (ISI)$  is

  • the impairment of a symbol decision due to pulse broadening  ("time dispersion"),  and
  • the associated dependence of the error probability on the neighboring symbols.


In other words:

  1. Falling edges of preceding pulses  ("trailers")  and rising edges of following pulses  ("precursors")  change the currently applied detection sample value.
  2. This can increase or decrease the probability of a wrong decision for the current symbol,  depending on whether the distance to the threshold becomes smaller or larger.
  3. On statistical average – i.e. when considering an  (infinitely)  long symbol sequence – this always leads to a  (considerable)  increase of the  (mean)  symbol error probability  $p_{\rm S} $.


Possible causes for intersymbol interference


The graphic shows the   "eye diagram"  for a

Eye diagrams with and without intersymbol interference
  1. intersymbol interference  $\rm (ISI)$  system without noise  (left),
  2. an ISI-free system without noise  (middle),
  3. the same ISI-free system with noise  (right).

The definition,  meaning and calculation of the eye diagram will be discussed in detail in the chapter  "Error Probability with Intersymbol Interference".  These screenshots can be interpreted as follows:

  • The middle diagram is from a Nyquist system with cosine rolloff characteristic $($rolloff factor  $r = 0.5)$.  Thus,  no intersymbol interference occurs.
  • The right diagram is from the same ISI-free system,  although here  $d(t) = \pm s_0$  does not apply.  The deviations from the nominal values  $\pm s_0$  are here due to the AWGN noise.
  • From this follows the important insight:   The question whether there is an ISI-free or ISI-affected system can only be decided on the basis of the detection signal  (or eye diagram)  without noise.


The left diagram indicates intersymbol interference,  since no noise is taken into account here.

  • The reason for this intersymbol interference could be that the overall frequency response of transmitter and receiver does not exactly fulfill the  "first Nyquist criterion"  due to tolerances.
  • However,  intersymbol interference also occurs with a channel with frequency-dependent frequency response  $H_{\rm K}(f)$,  if the receiver does not succeed in compensating the attenuation and phase distortions of the channel completely  (i.e. one hundred percent).
  • Finally,  even with the middle system,  intersymbol interference occurs if the decision is not made exactly in the center of the eye,  but at a detection time  $T_{\rm D} \ne 0$.  Then the basic detection pulse values must be defined to  $g_\nu = g_d(T_{\rm D} + \nu \cdot T)$.


Some remarks on the channel frequency response


For the further sections in this third main chapter the following block diagram is  (mostly)  assumed.  The main difference to the  block diagram of the first main chapter  is the channel frequency response  $($German:  "Kanalfrequenzgang"   ⇒   subscript  "K"$)$,  which is always assumed to be ideal   ⇒    $H_{\rm K}(f) = 1$. 

The following applies to the  "frequency response"  and the   "impulse response"  of the channel  $(\rm exp[\hspace{0.05cm} . ]$  denotes the  "exponential function"$)$:

$$H_{\rm K}(f) = {\rm exp} \left[ - a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right] \cdot {\rm exp} \left[ - {\rm j} \cdot a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right] \hspace{0.05cm}, $$
Block diagram of a system with distorting channel
$$h_{\rm K}(t) = \frac{ a_{{\star}\hspace{0.01cm}({\rm Np})}}{ \sqrt{2 \pi^2 \cdot R_{\rm B} \cdot t^3}}\hspace{0.1cm} \cdot {\rm exp} \left[ - \frac{a_{{\star} \hspace{0.01cm}({\rm Np})}^2}{2 \pi \cdot R_{\rm B} \cdot t}\hspace{0.1cm}\right] \hspace{0.05cm}.$$

Here  $a_{{\star} \hspace{0.01cm}({\rm Np})}$  indicates the cable attenuation at half the bit rate.  We call this quantity the  characteristic cable attenuation  in Neper  $\rm (Np)$:

$$a_{{\star} \hspace{0.01cm}({\rm Np})} = a_{\rm K}(f = {R_{\rm B}}/{2})= 0.1151 \cdot a_{{\star} \hspace{0.01cm}({\rm dB})} \hspace{0.05cm}.$$
  1. The corresponding dB value   ⇒   $a_{{\star} \hspace{0.01cm}({\rm dB})}$  is larger by a factor of  $1/0.1151 = 8.686$. 
  2. In realized systems, the characteristic cable attenuation  $a_{{\star} \hspace{0.01cm}({\rm dB})}$  is in the range between  $40 \ \rm dB$  and  $100 \ \rm dB$.
  3. The addition  "(Np)"  or  "(dB)"  is omitted in the following.


In the main chapter 4:   "Properties of electrical lines" of the book  "Linear and Time Invariant Systems"  it is shown that these equations reproduce the conditions with good approximation for wireline communication systems via coaxial cable.  For a two-wire line,  the deviation between this very simple  (analytically manageable)  formula and the actual conditions is somewhat larger.

A short summary of these derivations follows in the next two sections.

  • For simplicity,  we will use a redundancy-free binary system.
  • Thus,  the bit rate  $R_{\rm B}$  is equal to the reciprocal of the symbol duration  $T$.


Frequency response of a coaxial cable


A  coaxial cable  with core diameter 2.6 mm,  outer diameter 9.5 mm and length  $l$  has the following frequency response:

$$H_{\rm K}(f) = {\rm e}^{-\left[ a_{\rm K}(f) + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} b_{\rm K}(f)\right] } = {\rm e}^{- \alpha_0 \hspace{0.05cm} \cdot \hspace{0.05cm} l} \cdot {\rm e}^{- (\alpha_1 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_1) \hspace{0.05cm}\cdot f \hspace{0.05cm}\cdot \hspace{0.05cm}l} \cdot {\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l} \hspace{0.05cm},$$

where the following parameters apply for these dimensions – referred to as  "normal coaxial cable": 

$$\alpha_0 = 0.00162 \hspace{0.15cm}\frac {\rm Np}{\rm km} \hspace{0.05cm},\hspace{0.2cm} \alpha_1 = 0.000435 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm}, \hspace{0.2cm} \alpha_2 = 0.2722 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot \sqrt{\rm MHz}} \hspace{0.05cm},\hspace{0.2cm} \beta_1 = 21.78 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm}, \hspace{0.2cm} \beta_2 = 0.2722 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot \sqrt{\rm MHz}} \hspace{0.05cm}.$$

In the above equation the attenuation parameters  ("alpha")  are to be inserted in  "Np"  and the phase parameters  ("beta")  in  "rad".

Attenuation curve of a coaxial cable and approximation  (skin effect only)


The graph shows the exact attenuation curve for a standard coaxial cable of one kilometer length and an approximation for frequencies up to  $f = 1000\ \rm MHz$:

$$a_{\rm K}(f) = \alpha_0 \cdot l \hspace{0.05cm} + \hspace{0.05cm} \alpha_1 \cdot f \cdot l + \hspace{0.05cm} \alpha_2 \cdot \sqrt{f} \cdot l \hspace{0.05cm},$$
$$a_{\rm K}(f) \approx \alpha_2 \cdot \sqrt{f} \cdot l.$$
  • The axis is labeled in  "$\rm dB$"  on the left and  "$\rm Np$"  on the right.
  • One  $\rm Np$  ("Neper")  corresponds to  $8.686 \ \rm dB$.


We refer to the interactive HTML5/JavaScript applet  "Attenuation of copper cables".

You can see from the diagram and the above numerical values:

  • The first term  $(\alpha_0 \cdot l)$  originating from the ohmic losses is negligible.  Moreover,  this term causes only frequency-independent attenuation and no signal distortion.
  • The second term  $(\alpha_1 \cdot f \cdot l)$,  due to the transverse losses,  is proportional to frequency and therefore becomes noticeable only at very high frequencies;  it will be neglected in the following.
  • The frequency-proportional phase  $(\beta_1 \cdot f \cdot l)$  only results in a signal delay by the transit time  $\beta_1/(2\pi) \cdot l$,  but no distortion.  This transit time will also be disregarded in the following.
  • With these simplifications,  the frequency response is thus determined by the skin effect alone.  Since the numerical values for  $\alpha_2$  $($in Np$)$  and  $\beta_2$  $($in rad$)$  are the same,  it follows:
$$H_{\rm K}(f) ={\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l} \cdot {\rm e}^{- \alpha_2 \hspace{0.05cm}\cdot \hspace{0.05cm}l \hspace{0.05cm}\cdot \sqrt{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2f}} \hspace{0.05cm}.$$
  • Often in the literature – and also in this tutorial – the attenuation measure at half the bit rate is used,  which we call the  "characteristic cable attenuation"  $($in Neper$)$:
$$a_{\star} = a_{\rm K}(f ={R_{\rm B}}/{2})= a_{\rm K}(f = \frac{1}{2 \cdot T})\approx \frac{\alpha_2 \cdot l }{ \sqrt {2\cdot T}} \hspace{0.05cm}.$$

$\text{Example 1:}$  For a binary system with half the bit rate  $R_{\rm B}/2 = 280 \ \rm Mbit/s$  and  $l = 1\ \rm km$    $\Rightarrow \ a_{\star} \approx 4.55 \ \rm Np$  or  $a_{\star} \approx 40 \ \rm dB$  (green markings in the diagram).

  • But if the half bit rate is only  $70 \ \rm Mbit/s$,  $a_{\star} = 40 \ \rm dB$  characterizes a transmission system with cable length  $l = 2\ \rm km$.
  • Note:   The above approximation  $a_{\rm K}(f) \approx \alpha_2 \cdot \sqrt{f} \cdot l$  is only admissible for coaxial cables,  since for these the coefficients  $\alpha_0$  and  $\alpha_1$  can be neglected.
  • However,  for a  "balanced two-wire cable",  the coefficients  $\alpha_0$  and  $\alpha_1$  are much larger and the above approximation is invalid.


Impulse response of a coaxial cable


We now consider the coaxial cable impulse response,  which is for a binary system  $(R_{\rm B} = 1/T)$  as follows:

$$h_{\rm K}(t) = \frac{ a_{\rm \star \hspace{0.01cm}(Np)}/T}{ \sqrt{2 \pi^2 \cdot (t/T)^3}}\hspace{0.1cm} \cdot {\rm exp} \left[ - \frac{a_{\rm \star \hspace{0.01cm}(Np)}^2}{2 \pi \cdot t/T}\hspace{0.1cm}\right] \hspace{0.05cm}.$$

This time course is shown here for characteristic cable attenuations  $(a_{\rm \star})$  between  $40 \ \rm dB$  and  $100 \ \rm dB$.  Note the conversion   $\rm 1 \ Np = 8.686 \ dB.$

Impulse and rectangular pulse response of the coaxial cable  (skin effect only);   Note:
Here,  the basic receiver pulse  $g_r(t)$  is identical with the  "rectangular pulse response"

One can see from this time domain plot:

  • Even with the relatively small characteristic cable attenuation value  $a_{\rm \star} = 40 \ \rm dB$,  the impulse response extends over more than  $100$  symbol durations  $(T)$.
  • The larger  $a_{\rm \star}$  is chosen,  the broader and lower the impulse response becomes.  The integral over  $h_{\rm K}(t)$  from zero to infinity is the same for all curves,  since  $H_{\rm K}(f=0) = 1$  always holds  $(\alpha_0 = 0$,  $\alpha_1 = 0).$
  • The basic receiver pulse  $g_r(t) = g_s(t) \star h_{\rm K}(t)$  is nearly equal in shape to  $h_{\rm K}(t)$.  The right ordinate axis shows  $g_r(t)/s_0$  when  $g_s(t)$  is an NRZ rectangular pulse with height  $s_0$  and duration  $T$. 
  • For  $a_{\rm \star} \ge 60 \ \rm dB$,   $h_{\rm K}(t)$  and  $g_r(t)$  are indistinguishable within the character accuracy when suitably normalized.  For  $a_{\rm \star} = 40 \ \rm dB$,  one can see a small difference at the peak  (yellow background);   $g_r(t)/s_0$  is here minimally smaller than  $T \cdot h_{\rm K}(t)$.
  • However,  with the characteristic attenuation  $a_{\rm \star} = 40 \ \rm dB$,  the pulse amplitude at the cable end is only less than  $7\%$  of the input amplitude.  At  $60 \ \rm dB$  and  $100 \ \rm dB$,  this value drops to  $3\%$  and  $2\%$.

In  Exercise 3.1,  the coaxial cable impulse response is analyzed in detail.  We also refer to the interactive  (German language)  SWF applet  "Time behavior of copper cables".

Prerequisites for the entire third main chapter


Consider again the block diagram of a transmission system, assuming a highly distorting channel, such as is present in wireline transmission.

Due to the channel frequency response  $H_{\rm K}(f)$,  which is highlighted in red in the diagram,  there are also certain limitations for the other system components:

Block diagram of a system with  (highly)  distorting channel
  • The basic receiver pulse  $g_r(t) = g_s(t) \star h_{\rm K}(t)$  extends over hundreds of bits.  Therefore,  the receiver filter  $H_{\rm E}(f)$  cannot be applied as a  "matched filter";  this would approximately double the duration of the basic detection pulse  $g_d(t)$  compared to  $g_r(t)$. 
  • Rather,  the receiver filter  $H_{\rm E}(f)$  must compensate for the enormous attenuation distortions  $(\alpha_2$–term$)$ and the enormous phase distortions  $(\beta_2$–term$)$ of the coaxial channel  $H_{\rm K}(f)$,  especially if a simple threshold decision is assumed.
  • Although this linear form of signal equalization can be supported by more elaborate decision strategies – e.g.   "decision feedback",  "correlation receivers""Viterbi receivers".  For wired transmission,  a linear signal equalization   ⇒   receiver filter  $H_{\rm E}(f)$  cannot be completely dispensed with due to the very strong distortions.
  • The noise  $n(t)$  is further assumed to be additive,  white,  and Gaussian distributed  $\rm (AWGN)$,  which is justified for a coaxial cable.  For a two-wire line,  crosstalk from adjacent copper wires is the dominant noise,  as explained in detail in the chapter   "ISDN"  ("Integrated Services Digital Network").

$\text{Conclusion:}$  In the following chapters we consider only the binary bipolar redundancy-free transmission   ⇒   bit rate  $R_{\rm B} = 1/T$.  It is always assumed that:

  • The basic transmission pulse  $g_s(t)$  is NRZ rectangular with amplitude  $s_0$  and duration  $T$.
  • Thus,  the transmitted signal  $s(t)$  is equal to  $\pm s_0$  at all times and the spectral function is:   $G_s(f) = s_0 \cdot T \cdot {\rm sinc}(f T)$.
  • Splitting the equalization between transmitter and receiver according to the root–root characteristic does not make sense for conducted transmission.
  • There would be too much intersymbol interference already at the transmitter.


Exercises for the chapter


Exercise 3.1: Impulse response of the Coaxial Cable

Exercise 3.1Z: Frequency Response of the Coaxial Cable