Digital Signal Transmission/Intersymbol Interference for Multi-Level Transmission - Revision history

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@@ Line 164: / Line 164: @@
 {{BlaueBox|TEXT=
 $\text{However, it has to be taken into account:}$&nbsp; All results in this chapter are valid under the condition &nbsp;$H_{\rm K}(f=0) = 1$.
-#If a redundancy-free signal or the duobinary-coded signal is to be transmitted over a DC-impermeable channel,&nbsp; a complex DC recovery is required,&nbsp; which is always associated with a degradation of the S/N ratio as well &nbsp;[TS87]<ref>Tröndle, K.; Söder, G.:&nbsp; Optimization of Digital Transmission Systems.&nbsp; Boston – London: Artech House, 1987,&nbsp; ISBN:&nbsp; 0-89006-225-0.</ref>.<br><br>
+#If a redundancy-free signal or the duobinary-encoded signal is to be transmitted over a DC-impermeable channel,&nbsp; a complex DC recovery is required,&nbsp; which is always associated with a degradation of the S/N ratio as well &nbsp;[TS87]<ref>Tröndle, K.; Söder, G.:&nbsp; Optimization of Digital Transmission Systems.&nbsp; Boston – London: Artech House, 1987,&nbsp; ISBN:&nbsp; 0-89006-225-0.</ref>.<br><br>
 # The AMI code is free of DC signals and can thus also be transmitted over a telephone channel &nbsp; &rArr; &nbsp; $H_{\rm K}(f=0) = 0$.&nbsp; This is the decisive reason why the AMI code is used in &nbsp;[[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN| ISDN]]&nbsp; ("Integrated Services Digital Network"),&nbsp; for example,&nbsp; despite otherwise poor properties.}}<br>

@@ Line 61: / Line 61: @@
-The graph shows the half eye opening&nbsp; $($normalized to &nbsp;$s_0)$&nbsp; as a function of the quotient &nbsp;$f_{\rm G}/R_{\rm B}$&nbsp; of the Gaussian receiver filter.&nbsp; In &nbsp;[[Aufgaben:Aufgabe_3.4Z:_Augenöffnung_und_Stufenzahl|Exercise 3.4Z]],&nbsp; this is calculated in analytical form as follows:
+The graph shows the half eye opening&nbsp; $($normalized to &nbsp;$s_0)$&nbsp; as a function of the quotient &nbsp;$f_{\rm G}/R_{\rm B}$&nbsp; of the Gaussian receiver filter.&nbsp; In &nbsp;[[Aufgaben:Exercise_3.4Z:_Eye_Opening_and_Level_Number|Exercise 3.4Z]],&nbsp; this is calculated in analytical form as follows:
 :$$\ddot{o}_{\rm norm} = \frac{\ddot{o}(T_{\rm D})}{ 2 \cdot s_0}  =  \frac{M}{ M-1}\cdot \frac{g_0}{
   s_0} -1  =  \frac{1}{ M-1}\cdot \big [1- 2 \cdot M \cdot {\rm Q} \left(

@@ Line 41: / Line 41: @@
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; Below you can see three eye diagrams&nbsp; (without noise)&nbsp; for the level numbers &nbsp;$M = 2$, &nbsp;$M = 3$&nbsp; and&nbsp; $M = 4$.
-[[File:EN_Dig_T_3_4_S1b.png|right|frame|Noiseless eye diagrams of a binary,&nbsp; ternary and quaternary system. &nbsp; Note:&nbsp; the normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.6$&nbsp; holds for all three diagrams<br><br> '''Korrektur''': Strich|class=fit]]
+[[File:EN_Dig_T_3_4_S1b_neu.png|right|frame|Noiseless eye diagrams of a binary,&nbsp; ternary and quaternary system. &nbsp; Note:&nbsp; the normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.6$&nbsp; holds for all three diagrams|class=fit]]
 *The binary eye diagram is valid for a Gaussian low-pass with the cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.6$. With the&nbsp; "main value" &nbsp;$g_0 = 0.867 \cdot s_0$,&nbsp; the&nbsp; "postcursor"&nbsp; &nbsp;$g_{1} = 0.067 \cdot s_0$&nbsp; and&nbsp; the&nbsp; "precursor"&nbsp; $g_{-1} = g_{1}$, the result in this case for the vertical eye opening&nbsp;  (rounding to one decimal place)&nbsp; is:

@@ Line 171: / Line 171: @@
 [[Aufgaben:Exercise_3.4:_Optimization_of_the_Cutoff_Frequency|Exercise 3.4: Optimization of the Cutoff Frequency]]
-[[Aufgaben:Exercise_3.4Z:_Eye_Opening_and_Number_of_Steps|Exercise 3.4Z: Eye Opening and Number of Steps]]
+[[Aufgaben:Exercise_3.4Z:_Eye_Opening_and_Level_Number|Exercise 3.4Z: Eye Opening and Level Number]]
 [[Aufgaben:Exercise_3.5:_Eye_Opening_with_Pseudoternary_Coding|Exercise 3.5: Eye Opening with Pseudoternary Coding]]

← Older revision		Revision as of 12:26, 20 June 2022
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−	⇒   Note:  The degradations due to intersymbol interference in AMI and duobinary coding can also be displayed using the HTML5/JavaScript applet  [[Applets:~~Augendiagramm~~\|"Eye Diagram and Eye Opening"]].  However,  the error probabilities apply only to the distortion-free channel  $(a_\star = 0 \ \rm dB)$.<br>	+	⇒   Note:  The degradations due to intersymbol interference in AMI and duobinary coding can also be displayed using the HTML5/JavaScript applet  [[Applets:Eye_Pattern_and_Worst-Case_Error_Probability\|"Eye Diagram and Eye Opening"]].  However,  the error probabilities apply only to the distortion-free channel  $(a_\star = 0 \ \rm dB)$.<br>

	== Cutoff frequency optimization with pseudo-ternary coding ==		== Cutoff frequency optimization with pseudo-ternary coding ==

@@ Line 110: / Line 110: @@
 [[File:EN_Dig_T_3_4_S4.png|right|frame|Eye diagrams for the pseudo-ternary codes&nbsp; (AMI code,&nbsp; duobinary code)|class=fit]]
-*the &nbsp;[[Digital_Signal_Transmission/Symbol-Wise_Coding_with_Pseudo_Ternary_Codes#Properties_of_the_AMI_code|AMI code]] &nbsp; (on the left),
+*the &nbsp;[[Digital_Signal_Transmission/Symbolwise_Coding_with_Pseudo-Ternary_Codes#Properties_of_the_AMI_code|AMI code]] &nbsp; (on the left),
-*the &nbsp;[[Digital_Signal_Transmission/Symbol-Wise_Coding_with_Pseudo_Ternary_Codes#Properties_of_the_duobinary_code|duobinary code]] &nbsp; (on the right).
+*the &nbsp;[[Digital_Signal_Transmission/Symbolwise_Coding_with_Pseudo-Ternary_Codes#Properties_of_the_duobinary_code|duobinary code]] &nbsp; (on the right).

@@ Line 154: / Line 154: @@
 == Cutoff frequency optimization with pseudo-ternary coding ==
 <br>
-Considering a coaxial transmission channel and the thus necessary channel equalization, the following statements are possible:
+Considering a coaxial transmission channel and the thus necessary channel equalization,&nbsp; the following statements are possible:
-*The AMI code always leads to a worse signal-to-noise ratio than the redundancy-free binary code if the overall frequency response is Gaussian. With the characteristic cable attenuation &nbsp;$a_\star = 80 \ \rm dB$,&nbsp; the signal-to-noise ratio loss is about &nbsp;$11 \ \rm dB$.<br>
+*The AMI code always leads to a worse signal-to-noise ratio than the redundancy-free binary code if the overall frequency response is Gaussian.&nbsp; With the characteristic cable attenuation &nbsp;$a_\star = 80 \ \rm dB$,&nbsp; the signal-to-noise ratio loss is about &nbsp;$11 \ \rm dB$.<br>
-*This loss is due to the fact that, despite ternary coding, the symbol rate is not reduced compared to the binary reference system. This has the consequence that with the AMI code already a cutoff frequency &nbsp;$f_\text{G} \cdot T  < 0.36$&nbsp; leads to a closed eye.<br>
+*In contrast,&nbsp; with the duobinary code a closed eye results only from &nbsp;$f_\text{G} \cdot T  < 0.22$.&nbsp; As a result,&nbsp; the optimal cutoff frequency is also smaller than with the binary system.&nbsp; At 80 dB cable attenuation,&nbsp; the duobinary code in combination with &nbsp;$f_\text{G} \cdot T  =0.28$&nbsp; is &nbsp;$3.3 \ \rm dB$&nbsp; better than the best value with redundancy-free binary coding.<br>
 {{BlaueBox|TEXT=
 $\text{However, it has to be taken into account:}$&nbsp; All results in this chapter are valid under the condition &nbsp;$H_{\rm K}(f=0) = 1$.
-*If a redundancy-free signal or the duobinary-coded signal is to be transmitted over a DC-impermeable channel, a complex DC recovery is required, which is always associated with a degradation of the S/N ratio as well &nbsp;[TS87]<ref>Tröndle, K.; Söder, G.:&nbsp; Optimization of Digital Transmission Systems.&nbsp; Boston – London: Artech House, 1987,&nbsp; ISBN:&nbsp; 0-89006-225-0.</ref>.<br><br>
+#If a redundancy-free signal or the duobinary-coded signal is to be transmitted over a DC-impermeable channel,&nbsp; a complex DC recovery is required,&nbsp; which is always associated with a degradation of the S/N ratio as well &nbsp;[TS87]<ref>Tröndle, K.; Söder, G.:&nbsp; Optimization of Digital Transmission Systems.&nbsp; Boston – London: Artech House, 1987,&nbsp; ISBN:&nbsp; 0-89006-225-0.</ref>.<br><br>
+# The AMI code is free of DC signals and can thus also be transmitted over a telephone channel &nbsp; &rArr; &nbsp; $H_{\rm K}(f=0) = 0$.&nbsp; This is the decisive reason why the AMI code is used in &nbsp;[[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN| ISDN]]&nbsp; ("Integrated Services Digital Network"),&nbsp; for example,&nbsp; despite otherwise poor properties.}}<br>
 ==Exercises for the chapter==

@@ Line 8: / Line 8: @@
 == Eye opening for redundancy-free multi-level systems==
 <br>
-[[File:EN_Dig_T_3_4_S1a.png|right|frame|Block diagram for a multi-level/coded transmission system|class=fit]]
+[[File:EN_Dig_T_3_4_S1a.png|right|frame|Block diagram for a multi-level (or coded) transmission system|class=fit]]
 We further assume the following:
-*NRZ rectangular transmitted pulses,<br>
+*NRZ rectangular transmission pulses,<br>
 *coaxial cable and AWGN noise,<br>
 *ideal channel equalization, and<br>
@@ Line 17: / Line 17: @@
-In contrast to the &nbsp;[[Digital_Signal_Transmission/Berücksichtigung_von_Kanalverzerrungen_und_Entzerrung|"last chapter"]],&nbsp; the still redundancy-free transmitted signal &nbsp;$s(t)$&nbsp; is now no longer binary, but &nbsp;$M$&ndash;level, which only has an effect in the set of values of the amplitude coefficients:
+In contrast to the &nbsp;[[Digital_Signal_Transmission/Berücksichtigung_von_Kanalverzerrungen_und_Entzerrung| last chapter]],&nbsp; the still redundancy-free transmitted signal &nbsp;$s(t)$&nbsp; is now no longer binary, but of &nbsp;$M$&ndash;level,&nbsp; which only has an effect in the set of of the amplitude coefficients:
 :$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g_s ( t - \nu \cdot T)\hspace{0.3cm}{\rm with}\hspace{0.3cm}
   a_\nu \in \{ a_1,\ \text{ ...} \  , a_\mu , \ \text{ ...} \  , a_{M}\}\hspace{0.05cm}.$$
-Accordingly, the decision now has not only one, but &nbsp;$M-1$&nbsp; decision thresholds and in the eye diagram &nbsp;$M-1$&nbsp; eye openings are visible when the eye is open.<br>
+Accordingly,&nbsp; the decision unit now has not only one,&nbsp; but &nbsp;$M-1$&nbsp; decision thresholds and in the eye diagram &nbsp;$M-1$&nbsp; eye openings are visible when the eye is open.<br>
-Comparing the eye diagrams (without noise)
+Comparing the eye diagrams&nbsp; (without noise)&nbsp; of a
-*of a binary &nbsp;$(M = 2)$,<br>
+*binary &nbsp;$(M = 2)$,<br>
-*of a ternary &nbsp;$(M = 3)$, und<br>
+*ternary &nbsp;$(M = 3)$, and<br>
-*a quaternary &nbsp;$(M = 4)$
+*quaternary &nbsp;$(M = 4)$
-transmission system with the same basic transmitter pulse &nbsp;$g_d(t)$&nbsp; and the same symbol duration &nbsp;$T$, one obtains for the half vertical eye opening in general:
+transmission system with the same basic detection pulse &nbsp;$g_d(t)$&nbsp; and the same symbol duration &nbsp;$T$,&nbsp; one obtains for the half vertical eye opening in general:
 :$${\ddot{o}(T_{\rm D})}/{ 2} = \frac{g_0}{ M-1} - \sum_{\nu = 1}^{\infty} |g_{-\nu} | - \sum_{\nu = 1}^{\infty} |g_{\nu} |\hspace{0.05cm}.$$
-Here &nbsp;$g_0 = g_d(t= 0)$&nbsp; denotes the ''main value'' as in the chapter &nbsp;[[Digital_Signal_Transmission/Berücksichtigung_von_Kanalverzerrungen_und_Entzerrung|"Consideration of Channel Distortion and Equalization"]].&nbsp; The two sums in the above equation take into account
+&rArr; &nbsp;  $g_0 = g_d(t= 0)$&nbsp; denotes the&nbsp; "main value"&nbsp; as in chapter &nbsp;[[Digital_Signal_Transmission/Berücksichtigung_von_Kanalverzerrungen_und_Entzerrung|"Consideration of Channel Distortion and Equalization"]].&nbsp; The two sums in the above equation take into account
-*the ''precursors''&nbsp; $g_1$, &nbsp;$g_2$, ... of the trailing pulses (second term), and
+*the&nbsp; "precursors"&nbsp; $g_1$, &nbsp;$g_2$, ... of the trailing pulses&nbsp; (second term),&nbsp; and
-*the ''trailers''&nbsp; $g_{-1}$, $g_{-2}$, ... of the preceding pulses (last term).
+*the "trailers"&nbsp; or&nbsp; "postcursors"&nbsp;$g_{-1}$, $g_{-2}$, ... of the preceding pulses&nbsp; (last term).
-Here, &nbsp;$g_\nu = g_d(t = \nu \cdot T)$ always holds.
+Here, &nbsp;$g_\nu = g_d(t = \nu \cdot T)$&nbsp;  always holds.
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; Below you can see three eye diagrams for the level numbers &nbsp;$M = 2$, &nbsp;$M = 3$&nbsp; and&nbsp; $M = 4$.
+$\text{Example 1:}$&nbsp; Below you can see three eye diagrams&nbsp; (without noise)&nbsp; for the level numbers &nbsp;$M = 2$, &nbsp;$M = 3$&nbsp; and&nbsp; $M = 4$.
-*The binary eye diagram is valid for a Gaussian low-pass with the (normalized) cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.6$. With the main value &nbsp;$g_0 = 0.867 \cdot s_0$ and the two outliers &nbsp;$g_{1} = 0.067 \cdot s_0$&nbsp; and&nbsp; $g_{-1} = g_{1}$, the result in this case for the vertical eye opening (rounding to one decimal place) is:
+[[File:EN_Dig_T_3_4_S1b.png|right|frame|Noiseless eye diagrams of a binary,&nbsp; ternary and quaternary system. &nbsp; Note:&nbsp; the normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.6$&nbsp; holds for all three diagrams<br><br> '''Korrektur''': Strich|class=fit]]
 :$${\ddot{o}(T_{\rm D})}= 2 \cdot (g_0 - 2 \cdot g_1) \approx 1.5 \cdot s_0 \hspace{0.05cm}.$$
-*In the multi-level systems, the eye opening per se is smaller by a factor of &nbsp;$1/(M-1)$.&nbsp; Thus, the eye opening is reduced (relatively speaking) more by the (equally large) leading and trailing elements than in the binary system. One obtains with the same basic pulse values for
+*In the multi-level systems,&nbsp; the eye opening is per se smaller by a factor of &nbsp;$1/(M-1)$.&nbsp; Thus,&nbsp; the eye opening is reduced&nbsp; (relatively speaking)&nbsp; more by the&nbsp;  ISI causing pulse values&nbsp; $g_1$&nbsp; and&nbsp; $g_{-1}=g_1$&nbsp;  than in the binary system.&nbsp; One obtains with the same basic detection pulse values for
 :$$M = 3\text{:} \hspace{0.2cm}{\ddot{o}(T_{\rm D})}  =  2 \cdot (g_0/2 - 2 \cdot g_1) \approx 0.6 \cdot s_0
   \hspace{0.05cm},$$
@@ Line 51: / Line 52: @@
   \hspace{0.05cm}.$$
-It should be noted that the normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.6$&nbsp; is also the basis for these two multi-level systems.
+*But when comparing the systems,&nbsp; it should be noted that the larger level number&nbsp; $M$&nbsp; also increases the information flow.&nbsp; That is,&nbsp; the multi-level systems are better than these graphs indicate.&nbsp; More about this in the next section.}}<br>
 == Comparison between binary and quaternary system==

@@ Line 8: / Line 8: @@
 == Eye opening for redundancy-free multi-level systems==
 <br>
-[[File:P ID1411 Dig T 3 4 S1a version1.png|right|frame|Block diagram for a multi-level/coded transmission system|class=fit]]
+[[File:EN_Dig_T_3_4_S1a.png|right|frame|Block diagram for a multi-level/coded transmission system|class=fit]]
 We further assume the following:
@@ Line 39: / Line 39: @@
 Here, &nbsp;$g_\nu = g_d(t = \nu \cdot T)$ always holds.
-[[File:P ID1412 Dig T 3 4 S1b version1.png|right|frame|Eye diagrams of a binary, ternary and quaternary system|class=fit]]
+[[File:EN_Dig_T_3_4_S1b.png|right|frame|Eye diagrams of a binary, ternary and quaternary system|class=fit]]
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; Below you can see three eye diagrams for the level numbers &nbsp;$M = 2$, &nbsp;$M = 3$&nbsp; and&nbsp; $M = 4$.
@@ Line 100: / Line 100: @@
 == Eye opening for the pseudo-ternary codes==
 <br>
-[[File:P ID1416 Dig T 3 4 S4 version1.png|right|frame|Eye diagrams for the pseudo-ternary codes (AMI code, duobinary code)|class=fit]]
+[[File:EN_Dig_T_3_4_S4.png|right|frame|Eye diagrams for the pseudo-ternary codes (AMI code, duobinary code)|class=fit]]
 In the chapter &nbsp;[[Digital_Signal_Transmission/Symbolwise_Coding_with_Pseudo_Ternary_Codes|"Symbolwise Coding"]]&nbsp; the &nbsp;'''pseudo-ternary codes'''&nbsp; were described in general and the eye diagrams with Nyquist pulse shaping were given for them.