Digital Signal Transmission/Decision Feedback - Revision history

Guenter at 15:26, 24 August 2022

2022-08-24T15:26:11Z

Guenter at 14:17, 22 August 2022

2022-08-22T14:17:15Z

Hwang at 11:03, 11 July 2022

2022-07-11T11:03:55Z

Hwang at 11:01, 11 July 2022

2022-07-11T11:01:58Z

Guenter at 14:59, 28 June 2022

2022-06-28T14:59:10Z

Guenter at 12:41, 27 June 2022

2022-06-27T12:41:29Z

Guenter at 16:09, 25 June 2022

2022-06-25T16:09:50Z

Guenter at 16:06, 25 June 2022

2022-06-25T16:06:09Z

Guenter at 16:04, 25 June 2022

2022-06-25T16:04:09Z

Guenter at 14:17, 25 June 2022

2022-06-25T14:17:15Z

@@ Line 96: / Line 96: @@
 The last section has already made clear that the DFE already causes an enormous SNR gain if
-[[File:EN_Dig_T_3_6_S4.png|right|frame|Eye diagrams with DFE and optimized detection time|class=fit]]
+[[File:EN_Dig_T_3_6_S3_v23png.png|right|frame|Eye diagrams with DFE and optimized detection time|class=fit]]
 * a fixed cutoff frequency&nbsp; $f_{\rm G}$&nbsp; and
 *the fixed detection time&nbsp; $T_{\rm D} = 0$

@@ Line 96: / Line 96: @@
 The last section has already made clear that the DFE already causes an enormous SNR gain if
-[[File:EN_Dig_T_3_6_S4.png|right|frame|Eye diagrams with DFE and optimized detection time '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_3_6_S4.png|right|frame|Eye diagrams with DFE and optimized detection time|class=fit]]
 * a fixed cutoff frequency&nbsp; $f_{\rm G}$&nbsp; and
 *the fixed detection time&nbsp; $T_{\rm D} = 0$

@@ Line 168: / Line 168: @@
 For a circuit realization it is sufficient if the basic corrected pulse &nbsp;$g_k(t)$&nbsp; becomes zero only at the equidistant detection times &nbsp;$T_\text{D} +\nu \cdot T$.&nbsp;
 [[File:P ID1451 Dig T 3 6 S5b version1.png|right|frame|Decision feedback with delay filter|class=fit]]
-One realization possibility is thus an asymmetric&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Mode_of_action_of_the_transversal_filter|delay filter]]&nbsp; according to the adjacent diagram,
+One realization possibility is thus an asymmetric&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Operating_principle_of_the_transversal_filter|delay filter]]&nbsp; according to the adjacent diagram,
 *whose order &nbsp;$N$&nbsp; (number of filter coefficients), and<br>

@@ Line 138: / Line 138: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp; For a transmission system over copper lines&nbsp; $($coaxial cable,&nbsp; two-wire line$)$,&nbsp; the following system variants are particularly suitable due to the achievable signal&ndash;to&ndash;noise ratio at the decision:
-*a&nbsp; '''multilevel system'''&nbsp; $($for example &nbsp;$M = 4)$&nbsp; and the optimal&nbsp; '''Nyquist equalization'''&nbsp; to compensate for the strong intersymbol interference caused by the linear channel distortions;<br>
+*a&nbsp; '''multi-level system'''&nbsp; $($for example &nbsp;$M = 4)$&nbsp; and the optimal&nbsp; '''Nyquist equalization'''&nbsp; to compensate for the strong intersymbol interference caused by the linear channel distortions;<br>
 *a&nbsp; '''binary system'''&nbsp; with relatively small bandwidth of the total frequency response &nbsp;$H_{\rm G}(f) = H_{\rm K}(f) \cdot H_{\rm E}(f)$&nbsp; and a non-linear decision with&nbsp; '''DFE'''.}}

@@ Line 62: / Line 62: @@
 == Eye opening and error probability with DFE ==
 <br>
-[[File:P ID1448 Dig T 3 6 S3 version1.png|right|frame|Eye diagrams without and with&nbsp; "Decision Feedback Equalization"&nbsp; $(f_{\rm G}\cdot T = 0.3)$ '''Korrektur'''|class=fit]] We now consider the eye diagrams
+[[File:EN_Dig_T_3_6_S3.png|right|frame|Eye diagrams without and with&nbsp; "Decision Feedback Equalization"&nbsp; $(f_{\rm G}\cdot T = 0.3)$|class=fit]] We now consider the eye diagrams
 *without DFE&nbsp; (left graph)&nbsp; and
 *with ideal DFE (right graph).
@@ Line 96: / Line 96: @@
 The last section has already made clear that the DFE already causes an enormous SNR gain if
-[[File:P ID1449 Dig T 3 6 S4 version1.png|right|frame|Eye diagrams with DFE and optimized detection time '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_3_6_S4.png|right|frame|Eye diagrams with DFE and optimized detection time '''Korrektur'''|class=fit]]
 * a fixed cutoff frequency&nbsp; $f_{\rm G}$&nbsp; and
 *the fixed detection time&nbsp; $T_{\rm D} = 0$
@@ Line 153: / Line 153: @@
 {{GraueBox|TEXT=
-[[File:P ID1450 Dig T 3 6 S5a version1.png|right|frame|Basic pulses with ideal DFE '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_3_6_S5a.png|right|frame|Basic pulses with ideal DFE|class=fit]]
 $\text{Example 2:}$&nbsp; The graph shows the basic pulse &nbsp;$g_d(t)$&nbsp;
 * for the cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.2$&nbsp; (red curve) and

@@ Line 54: / Line 54: @@
 &rArr; &nbsp; Shown on the left are the basic pulses &nbsp;$g_w(t)$&nbsp; and&nbsp; $g_k(t)$&nbsp; with ideal decision feedback,&nbsp; based on the detection time &nbsp;$T_{\rm D} = 0$&nbsp; and the delay time &nbsp;$T_{\rm V} = T/2$.&nbsp; <br>
-&rArr; &nbsp; The right pictures from [Söd01]<ref name = 'Söd01'>Söder, G.:&nbsp;Simulation digitaler Übertragungssysteme. Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2001.</ref>. &ndash; all without consideration of the noise &ndash; make clear that by the compensation of all pulse trailers by means of the correction signal &nbsp;$w(t)$,&nbsp; the distances of the useful signal samples&nbsp; $d_{\rm S}(\nu \cdot T)$&nbsp; from the decision threshold &nbsp;$E = 0$&nbsp; are changed.
+&rArr; &nbsp; The right pictures from [Söd01]<ref name = 'Söd01'>Söder, G.:&nbsp;Simulation digitaler Übertragungssysteme. Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2001.</ref> &ndash; all without consideration of the noise &ndash; make clear that by the compensation of all pulse trailers by means of the correction signal &nbsp;$w(t)$,&nbsp; the distances of the useful signal samples&nbsp; $d_{\rm S}(\nu \cdot T)$&nbsp; from the decision threshold &nbsp;$E = 0$&nbsp; are changed.
 *Particularly small distances,&nbsp;  such as at times &nbsp;$t = 6T$&nbsp; and&nbsp; $t = 7T$,&nbsp; are significantly increased and thus their error probabilities are greatly reduced&nbsp; (arrows pointing away from the threshold).<br>
@@ Line 62: / Line 62: @@
 == Eye opening and error probability with DFE ==
 <br>
-[[File:P ID1448 Dig T 3 6 S3 version1.png|right|frame|Eye diagrams without and with ''Decision Feedback Equalization''&nbsp; $(f_{\rm G}\cdot T = 0.3)$ '''Korrektur'''|class=fit]] We now consider the eye diagrams
+[[File:P ID1448 Dig T 3 6 S3 version1.png|right|frame|Eye diagrams without and with&nbsp; "Decision Feedback Equalization"&nbsp; $(f_{\rm G}\cdot T = 0.3)$ '''Korrektur'''|class=fit]] We now consider the eye diagrams
 *without DFE&nbsp; (left graph)&nbsp; and
 *with ideal DFE (right graph).

← Older revision		Revision as of 16:06, 25 June 2022
Line 27:		Line 27:
	*Since in this receiver with decision feedback,  the compensation signal  $w(t)$  is derived from the noise-free sink signal  $v(t)$,  the signal equalization is not associated with an increase in noise power as in linear equalization.  Rather,  the corrected signal  $k(t)$  has the same noise rms value  $\sigma_d$  as the signal  $d(t)$.<br><br>		*Since in this receiver with decision feedback,  the compensation signal  $w(t)$  is derived from the noise-free sink signal  $v(t)$,  the signal equalization is not associated with an increase in noise power as in linear equalization.  Rather,  the corrected signal  $k(t)$  has the same noise rms value  $\sigma_d$  as the signal  $d(t)$.<br><br>

−	⇒   The signal characteristics of this nonlinear equalization method  "DFE"  as well as the associated error probabilities – valid for a distortion-free channel – can be displayed with the  (German language)  SWF applet  [[Applets:Entscheidungsrückkopplung\|"~~Decision Feedback Equalization~~"]].  ~~<br>~~	+	⇒   The signal characteristics of this nonlinear equalization method  "DFE"  as well as the associated error probabilities – valid for a distortion-free channel – can be displayed with the  (German language)  SWF applet  [[Applets:Entscheidungsrückkopplung\|"Entscheidungsrückkopplung"]]   ⇒   "Decision Feedback Equalization"

@@ Line 134: / Line 134: @@
 == Realization aspects of the decision feedback ==
 <br>
-As an essential result of the last chapter&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization|"Linear Nyquist Equalization"]]&nbsp; and the current chapter "Decision Feedback" the following approach is recommended:
+As an essential result of the last chapter&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization|"Linear Nyquist Equalization"]]&nbsp; and the current chapter&nbsp; "Decision Feedback"&nbsp; the following approach is recommended:
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; For a transmission system over copper lines (coaxial cable, two-wire line), the following system variants are particularly suitable due to the achievable signal&ndash;to&ndash;noise ratio at the decision:
+$\text{Conclusion:}$&nbsp; For a transmission system over copper lines&nbsp; $($coaxial cable,&nbsp; two-wire line$)$,&nbsp; the following system variants are particularly suitable due to the achievable signal&ndash;to&ndash;noise ratio at the decision:
-*a '''multilevel system''' $($for example &nbsp;$M = 4)$&nbsp; and the optimal '''Nyquist equalization''' to compensate for the strong intersymbol interference caused by the linear channel distortions;<br>
+*a&nbsp; '''multilevel system'''&nbsp; $($for example &nbsp;$M = 4)$&nbsp; and the optimal&nbsp; '''Nyquist equalization'''&nbsp; to compensate for the strong intersymbol interference caused by the linear channel distortions;<br>
-*Since no DC signal can be transmitted via the telephone network, but &nbsp;$H_{\rm K}(f=0) = 1$&nbsp; is assumed for our calculations, <i>DC signal recovery</i>&nbsp; is required at the receiver. This statement applies in the same way to the quaternary Nyquist system.<br>
+Both system variants provide comparably good results under idealized conditions.&nbsp; But it should be noted,&nbsp; that large degradations can occur due to realization inaccuracies in both systems,&nbsp; which are mentioned here using the DFE system as an example:
-*In the DFE system, the compensation pulse &nbsp;$g_w(t)$&nbsp; must exactly replicate the pre-equalized basic pulse &nbsp;$g_d(t)$.&nbsp; This is especially difficult when &nbsp;$g_d(t)$&nbsp; is very broad $($small cutoff frequency, for example &nbsp;$f_{\rm G} \cdot T = 0.2)$&nbsp; and the optimization yields the detection time &nbsp;$T_\text{D, opt} = -T/2$.<br>
+*Since no DC signal can be transmitted via the telephone network,&nbsp; but &nbsp;$H_{\rm K}(f=0) = 1$&nbsp; is assumed for our calculations, a&nbsp; "DC signal recovery"&nbsp; is required at the receiver.&nbsp; This statement applies in the same way to the quaternary Nyquist system.<br>
-*If a wrong decision occurs due to a very large noise value, the subsequent symbols will also be falsified with a high probability. However, there are always symbol sequences which interrupt this &nbsp;[https://en.wikipedia.org/wiki/Propagation_of_uncertainty "propogation of uncertainty"].&nbsp;<br>
+*In the DFE system,&nbsp; the compensation pulse &nbsp;$g_w(t)$&nbsp; must exactly replicate the pre-equalized basic pulse &nbsp;$g_d(t)$.&nbsp; This is especially difficult when &nbsp;$g_d(t)$&nbsp; is very broad&nbsp; $($small cutoff frequency,&nbsp; e.g. &nbsp;$f_{\rm G} \cdot T = 0.2)$&nbsp; and the optimization yields the detection time &nbsp;$T_\text{D, opt} = -T/2$.<br>
 [[File:P ID1450 Dig T 3 6 S5a version1.png|right|frame|Basic pulses with ideal DFE|class=fit]]
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp; The graph shows the basic pulse &nbsp;$g_d(t)$&nbsp;
 * for the cutoff frequency &nbsp;$f_{\rm G} \cdot T = 0.2$&nbsp; (red curve) and
-* the compensation pulse &nbsp;$g_w(t)$&nbsp; for $T_\text{D} = -T/2$ (filled in blue).
+* the compensation pulse &nbsp;$g_w(t)$&nbsp; for $T_\text{D} = -T/2$&nbsp; (filled in blue).
-Here, a delay time &nbsp;$T_\text{V} = -T/2$&nbsp; between decision and start of signal correction is considered. It can be seen:
+Here,&nbsp; a delay time &nbsp;$T_\text{V} = -T/2$&nbsp; between decision and start of signal correction is considered.&nbsp; It can be seen:
 *For &nbsp;$T_\text{D} = -T/2$&nbsp; the first trailer &nbsp;$g_d(T_\text{D} +T) = g_d(T/2)$&nbsp; is exactly as large as the main value &nbsp;$g_d(T_\text{D}) = g_d(-T/2)$.
-*If it is not possible to fully compensate all the trailers, this quickly results in a closed eye and thus, in the worst&ndash;case, the error probability $p_{\rm U} \approx 50\%$.}}<br>
 == Decision feedback with delay filter==
 <br>
-For a circuit realization it is sufficient if the corrected basic pulse &nbsp;$g_k(t)$&nbsp; becomes zero only at the equidistant detection times &nbsp;$T_\text{D} +\nu \cdot T$.&nbsp;
+For a circuit realization it is sufficient if the basic corrected pulse &nbsp;$g_k(t)$&nbsp; becomes zero only at the equidistant detection times &nbsp;$T_\text{D} +\nu \cdot T$.&nbsp;
 [[File:P ID1451 Dig T 3 6 S5b version1.png|right|frame|Decision feedback with delay filter|class=fit]]
 One realization possibility is thus an asymmetric&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Mode_of_action_of_the_transversal_filter|delay filter]]&nbsp; according to the adjacent diagram,
 *whose order &nbsp;$N$&nbsp; (number of filter coefficients), and<br>
-are determined by the basic pulse &nbsp;$g_d(t)$&nbsp; and the detection time &nbsp;$T_\text{D}$.&nbsp;<br>
+*whose filter coefficients $&nbsp;k_\nu$&nbsp; $($mit $\nu = 1$, ... , $N)$&nbsp;
 This DFE realization has the following properties:
-*Since the output signal &nbsp;$v(t)$&nbsp; is rectangular, the compensation pulse &nbsp;$g_w(t)$&nbsp; is staircase shaped.<br>
+*Since the output signal &nbsp;$v(t)$&nbsp; is rectangular,&nbsp;  the compensation pulse &nbsp;$g_w(t)$&nbsp; is staircase shaped.<br>
-*With proper dimensioning of the filter coefficients &nbsp;$k_\nu$,&nbsp; for &nbsp;$\nu = 1$, ... , $N$:
+*With proper dimensioning of the filter coefficients &nbsp;$k_\nu$,&nbsp; <br>for &nbsp;$\nu = 1$, ... , $N$:
 :$$g_w(T_{\rm D} + \nu \cdot T) = g_d(T_{\rm D} + \nu \cdot T)
 \hspace{0.3cm}\Rightarrow\hspace{0.3cm}
 g_k(T_{\rm D} + \nu \cdot T) = 0 \hspace{0.05cm}.$$
-*At detection time &nbsp;$T_\text{D}$,&nbsp; the vertical eye opening is exactly the same as for ideal DFE. A disadvantage is a smaller horizontal eye opening.<br><br>
+*At detection time &nbsp;$T_\text{D}$,&nbsp; the vertical eye opening is exactly the same as for ideal DFE.&nbsp;  A disadvantage is a smaller horizontal eye opening.<br><br>
 [[File:P ID1452 Dig T 3 6 S5c version1.png|right|frame|Basic pulses for DFE with delay filter|class=fit]]
-{{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; The graph shows the basic pulses &nbsp;$g_d(t)$&nbsp; and &nbsp;$g_w(t)$&nbsp; for decision feedback with a second order delay filter. The same conditions apply as for &nbsp;$\text{Example 2}$&nbsp; in the last section: &nbsp; $f_{\rm G} \cdot T = 0.2$&nbsp; and&nbsp; $T_\text{D} = -T/2$. One can see:
+$\text{Example 3:}$&nbsp; The graph shows the basic pulses &nbsp;$g_d(t)$&nbsp; and &nbsp;$g_w(t)$&nbsp; for decision feedback with a second order delay filter.&nbsp; The same conditions apply as for the example in the last section: &nbsp; $f_{\rm G} \cdot T = 0.2$&nbsp; and&nbsp; $T_\text{D} = -T/2$.&nbsp; One can see:
-*However, because of the order &nbsp;$N = 2$,&nbsp; only the first two trailers &nbsp;$g_d(0.5T)$&nbsp; and &nbsp;$g_d(1.5T)$&nbsp; are compensated here.
+*The third trailer &nbsp;$g_d(2.5T)$&nbsp; could be compensate by a further filter coefficient &nbsp;$k_3$.
-*The third trailer &nbsp;$g_d(2.5T)$&nbsp; could be made zero by a further filter coefficient &nbsp;$k_3$.&nbsp;
-*In contrast, the pulse precursors &nbsp;$g_d(-1.5T)$&nbsp; and &nbsp;$g_d(-2.5T)$&nbsp; cannot be compensated in principle.}}<br>
+*In contrast,&nbsp; the precursors &nbsp;$g_d(-1.5T)$&nbsp; and &nbsp;$g_d(-2.5T)$&nbsp; cannot be compensated in principle.}}<br>
 ==Exercises for the chapter==

@@ Line 62: / Line 62: @@
 == Eye opening and error probability with DFE ==
 <br>
-We now consider the eye diagrams
+[[File:P ID1448 Dig T 3 6 S3 version1.png|right|frame|Eye diagrams without and with ''Decision Feedback Equalization''&nbsp; $(f_{\rm G}\cdot T = 0.3)$ '''Korrektur'''|class=fit]] We now consider the eye diagrams
 *without DFE&nbsp; (left graph)&nbsp; and
 *with ideal DFE (right graph).
@@ Line 78: / Line 77: @@
 These two eye diagrams can be interpreted as follows:
-*For the conventional receiver (without DFE), with binary bipolar redundancy-free coding considering symmetry:
+*For the conventional receiver (without DFE),&nbsp;  with binary bipolar redundancy-free coding considering symmetry:
 :$${\ddot{o}(T_{\rm D} = 0 )}  =  {2} \cdot \big [  g_0 -  | g_{-1}| -  | g_{-2}| -  | g_{1}| -  | g_{2}|\big ] =  {2} \cdot \big [  g_0 -  2 \cdot g_{1} -  2 \cdot g_{2}\big
   ]= 0.192 \cdot s_0 \hspace{0.05cm}.$$
-*On the other hand, for ideal DFE, the two trailers&nbsp; $g_1$&nbsp; and&nbsp; $g_2$&nbsp; are fully compensated and we obtain for the vertical eye opening:
+*On the other hand,&nbsp; for ideal DFE,&nbsp; the two trailers&nbsp; $g_1$&nbsp; and&nbsp; $g_2$&nbsp; are fully compensated and we obtain for the vertical eye opening:
 :$${\ddot{o}(T_{\rm D} = 0 )} = {2} \cdot \big  [  g_0 -  | g_{-1}| -  |g_{-2}|\big
   ]  =  {2} \cdot \big [  g_0 -   g_{1} -   g_{2}\big ]= 0.644 \cdot s_0 \hspace{0.05cm}.$$
-*Since the correction signal&nbsp;  $w(t)$&nbsp; is derived from the decision and thus noise-free signal &nbsp;$v(t)$,&nbsp; the noise rms value &nbsp;$\sigma_d$&nbsp; is not changed by the decision feedback. Thus, the noise rms gain due to the DFE is equal to in the considered example
+*Since the correction signal&nbsp;  $w(t)$&nbsp; is derived from the decision and thus noise-free signal &nbsp;$v(t)$,&nbsp; the noise rms value &nbsp;$\sigma_d$&nbsp; is not changed by the decision feedback.&nbsp; Thus,&nbsp; the SNR gain due to the DFE is
 :$$G_{\rm DFE}=
 \cdot {\rm lg}\hspace{0.1cm}\frac{0.644}{0.192} \approx 10.5\,{\rm dB} \hspace{0.05cm}.$$
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; For a coaxial cable with characteristic cable attenuation&nbsp; $a_\star = 80 \ \rm dB$&nbsp; and&nbsp; $10 \cdot \lg \ (E_{\rm B}/N_0) = 80 \ \rm  dB$,&nbsp; for example, this signal-to-noise ratio gain means that the worst-case error probability &nbsp;$p_{\rm U}$&nbsp; is reduced by DFE from&nbsp; $7\%$&nbsp; to about&nbsp; $4 \cdot 10^{-7}$&nbsp; &ndash; a quite remarkable improvement.}}<br>
+$\text{Conclusion:}$&nbsp; For a coaxial cable with characteristic cable attenuation&nbsp; $a_\star = 80 \ \rm dB$&nbsp; and&nbsp; $10 \cdot \lg \ (E_{\rm B}/N_0) = 80 \ \rm  dB$,&nbsp; <br>this SNR gain means that the worst-case error probability &nbsp;$p_{\rm U}$&nbsp; is reduced by DFE from&nbsp; $7\%$&nbsp; to about&nbsp; $4 \cdot 10^{-7}$&nbsp; &ndash; a quite remarkable improvement.}}<br>
 == Optimization of a transmission system with DFE ==
 <br>
-The last section has already made clear that the DFE already causes an enormous gain in the signal-to-noise ratio if
+The last section has already made clear that the DFE already causes an enormous SNR gain if
 [[File:P ID1449 Dig T 3 6 S4 version1.png|right|frame|Eye diagrams with DFE and optimized detection time|class=fit]]
-* of a fixed cutoff frequency&nbsp; $f_{\rm G}$&nbsp; and
+* a fixed cutoff frequency&nbsp; $f_{\rm G}$&nbsp; and
 *the fixed detection time&nbsp; $T_{\rm D} = 0$
-is assumed. However, the system can be further improved if the two parameters&nbsp; $f_{\rm G}$&nbsp; and&nbsp; $T_{\rm D}$&nbsp; are optimized together.<br>
+is assumed.&nbsp; However,&nbsp; the system can be further improved if the two parameters&nbsp; $f_{\rm G}$&nbsp; and&nbsp; $T_{\rm D}$&nbsp; are optimized together.<br>
 The graph shows the eye diagrams without noise for
-*$f_{\rm G} \cdot T = 0.3$&nbsp; (left) and
+*$f_{\rm G} \cdot T = 0.3$&nbsp; (left),
 *$f_{\rm G} \cdot T = 0.2$&nbsp; (right).
-For the diagram and the following calculations, the characteristic cable attenuation $a_\star = 80 \ \rm dB$ and the AWGN parameter $10 \cdot \lg \ (E_{\rm B}/N_0) = 80 \ \rm  dB$ (with $E_{\rm B} = s_0^2 \cdot T$) are still assumed.
+For the diagram and the following calculations  are still assumed:
-*With&nbsp; $f_{\rm G} \cdot T = 0.3$,&nbsp; by shifting the detection time to &nbsp;$T_\text{D, opt} = -0.3T$,&nbsp; the eye opening can be increased to&nbsp; $\ddot{o}(T_\text{D, opt}) =  0.779 \cdot s_0 $.&nbsp;
+With the DFE receiver,&nbsp; however,&nbsp; you can further reduce the cutoff frequency.&nbsp; The reason is the better noise behavior with a smaller cutoff frequency.&nbsp;
-*Compared to&nbsp; $T_{\rm D} = 0$&nbsp; (compare last section), this results in a further gain of&nbsp; $G_{T_\text{D, opt}}= 20 \cdot {\rm lg}\hspace{0.1cm}{0.779}/{644} \approx 1.65\,{\rm dB} \hspace{0.05cm}.$
+*For example,&nbsp; the normalized noise rms value results instead of &nbsp;$\sigma_d/s_0 = 0.065$&nbsp; $($for &nbsp;$f_{\rm G} \cdot T = 0.3)$&nbsp; to &nbsp;$\sigma_d/s_0 = 0.010$&nbsp; $($for &nbsp;$f_{\rm G} \cdot T = 0.2)$.
-With the DFE receiver, however, you can additionally reduce the cutoff frequency further. The reason is the better noise behavior with a smaller cutoff frequency. The normalized noise rms value results instead of &nbsp;$\sigma_d/s_0 = 0.065$&nbsp; $($for &nbsp;$f_{\rm G} \cdot T = 0.3)$&nbsp; for example to &nbsp;$\sigma_d/s_0 = 0.010$&nbsp; $($for &nbsp;$f_{\rm G} \cdot T = 0.2)$.
+* Thus&nbsp; (practically)&nbsp; $p_{\rm U}\approx 0$.&nbsp; However,&nbsp; this configuration is not relevant in practice: &nbsp; Already a minimal tolerance of the system parameters leads to a&nbsp; "closed eye".<br>