Digital Signal Transmission/Causes and Effects of Intersymbol Interference - Revision history

Hwang at 14:12, 23 January 2023

2023-01-23T14:12:02Z

Hwang at 14:11, 23 January 2023

2023-01-23T14:11:26Z

Guenter at 12:50, 31 August 2022

2022-08-31T12:50:45Z

Guenter at 11:40, 8 June 2022

2022-06-08T11:40:43Z

Guenter at 11:20, 8 June 2022

2022-06-08T11:20:52Z

Guenter at 11:19, 8 June 2022

2022-06-08T11:19:06Z

Hwang at 07:35, 6 June 2022

2022-06-06T07:35:12Z

Hwang at 07:34, 6 June 2022

2022-06-06T07:34:55Z

Hwang at 07:22, 6 June 2022

2022-06-06T07:22:43Z

Guenter at 09:09, 4 June 2022

2022-06-04T09:09:57Z

@@ Line 19: / Line 19: @@
 #&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;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.

@@ Line 123: / Line 123: @@
 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>
-A short summary of these derivations follows on the next two pages.
+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>

@@ Line 13: / Line 13: @@
 In detail,&nbsp; this chapter deals with:
-*the&nbsp; &raquo;causes and effects&laquo;&nbsp; of intersymbol interference,
+#&nbsp; the&nbsp; &raquo;causes and effects&laquo;&nbsp; of intersymbol interference,
-*the&nbsp; &raquo;eye diagram&laquo;&nbsp; as a suitable tool for the description of intersymbol interferences,
+#&nbsp; the&nbsp; &raquo;eye diagram&laquo;&nbsp; as a suitable tool for the description of intersymbol interferences,
-*the&nbsp; &raquo;error probability calculation&laquo;&nbsp; considering channel distortions,
+#&nbsp; the&nbsp; &raquo;error probability calculation&laquo;&nbsp; considering channel distortions,
-*the&nbsp; &raquo;influence of intersymbol interference in multilevel and/or coded transmission&laquo;,
+#&nbsp; the&nbsp; &raquo;influence of intersymbol interference in multilevel and/or coded transmission&laquo;,
-*the&nbsp; &raquo;optimal Nyquist equalizer as an example of linear channel equalization,
+#&nbsp; the&nbsp; &raquo;optimal Nyquist equalizer as an example of linear channel equalization,
-*the&nbsp; &raquo;decision feedback equalization&laquo;&nbsp; $\rm (DFE)$ – an effective nonlinear decision realization,
+#&nbsp; the&nbsp; &raquo;decision feedback equalization&laquo;&nbsp; $\rm (DFE)$ – an effective nonlinear decision realization,
-*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;correlation receiver&laquo;&nbsp; as an example of&nbsp; &raquo;maximum likelihood or maximum a posteriori&laquo;&nbsp; $\rm (MAP)$&nbsp; decision strategy,
-*the&nbsp; &raquo;Viterbi receiver&laquo;,&nbsp; a reduced-effort MAP decision algorithm.
+#&nbsp; the&nbsp; &raquo;Viterbi receiver&laquo;,&nbsp; a reduced-effort MAP decision algorithm.

@@ Line 195: / Line 195: @@
 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 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 withe the&nbsp; "rectangular response"]]
+[[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"]]
 One can see from this time domain plot:

@@ Line 19: / Line 19: @@
 *the&nbsp; &raquo;optimal Nyquist equalizer as an example of linear channel equalization,
 *the&nbsp; &raquo;decision feedback equalization&laquo;&nbsp; $\rm (DFE)$ – an effective nonlinear decision realization,
-*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 making,
+*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,
 *the&nbsp; &raquo;Viterbi receiver&laquo;,&nbsp; a reduced-effort MAP decision algorithm.

@@ Line 101: / Line 101: @@
 == Some remarks on the channel frequency response ==
 <br>
-For the further sections in this third main chapterthe 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>
+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>
 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"$)$:

@@ Line 31: / Line 31: @@
 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 '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_3_1_S1.png|right|frame|Detection signals with and without intersymbol interferences|class=fit]]
 :$$g_\nu = g_d(\nu T)   =   \left\{ \begin{array}{c} g_0
   \\ 0  \\  \end{array} \right.\quad
@@ Line 78: / Line 78: @@
 <br>
 The graphic shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram| "eye diagram"]]&nbsp; for a
-[[File:EN_Dig_T_3_1_S2.png|right|frame|Eye diagrams with and without intersymbol interference '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_3_1_S2.png|right|frame|Eye diagrams with and without intersymbol interference|class=fit]]
 #intersymbol interference&nbsp; $\rm (ISI)$&nbsp; system without noise&nbsp; (left),<br>
 #an ISI-free system without noise&nbsp; (middle),<br>
@@ Line 107: / Line 107: @@
    {\rm exp} \left[ - {\rm j} \cdot a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right]
    \hspace{0.05cm}, $$
-[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with distorting channel  '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with distorting channel|class=fit]]
 :$$h_{\rm K}(t)  =    \frac{ a_{{\star}\hspace{0.01cm}({\rm Np})}}{  \sqrt{2  \pi^2 \cdot R_{\rm B} \cdot t^3}}\hspace{0.1cm} \cdot
    {\rm exp} \left[ - \frac{a_{{\star} \hspace{0.01cm}({\rm Np})}^2}{2  \pi \cdot R_{\rm B} \cdot t}\hspace{0.1cm}\right]
@@ Line 148: / Line 148: @@
 In the above equation the attenuation parameters&nbsp; ("alpha")&nbsp; are to be inserted in&nbsp; "Np"&nbsp; and the phase parameters&nbsp; ("beta")&nbsp;  in&nbsp; "rad".<br>
-[[File:EN_Dig_T_3_1_S3a.png|right|frame|Attenuation curve of a coaxial cable and approximation&nbsp; (skin effect only)  '''Korrektur'''|class=fit]]<br>
+[[File:EN_Dig_T_3_1_S3a.png|right|frame|Attenuation curve of a coaxial cable and approximation&nbsp; (skin effect only) |class=fit]]<br>
 The graph shows the exact attenuation curve for a standard coaxial cable of one kilometer length and an approximation for frequencies up to &nbsp;$f = 1000\ \rm  MHz$:
 :$$a_{\rm K}(f)  = \alpha_0  \cdot l \hspace{0.05cm} + \hspace{0.05cm} \alpha_1  \cdot f \cdot l
@@ Line 195: / Line 195: @@
 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 response of the coaxial cable&nbsp; (skin effect only) &nbsp; '''Korrektur''']]
+[[File:EN_Dig_T_3_1_S3b.png|right|frame|Impulse response of the coaxial cable&nbsp; (skin effect only) ]]
 One can see from this time domain plot:
@@ Line 216: / Line 216: @@
 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:
-[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with&nbsp; (highly)&nbsp; distorting channel &nbsp; '''Korrektur''']]
+[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with&nbsp; (highly)&nbsp; distorting channel]]
 *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>