Digital Signal Transmission/Decision Feedback - Revision history

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2026-03-16T13:46:47Z

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	$\text{Definition:}$  An  '''ideal decision feedback'''  exists when the following basic pulse is applied to the decision:		$\text{Definition:}$  An  '''ideal decision feedback'''  exists when the following basic pulse is applied to the decision:
	:$$g_k(t) = \left\{ \begin{array}{c} g_d(t)		:$$g_k(t) = \left\{ \begin{array}{c} g_d(t)\\ 0 \\ \end{array} \right.\quad\begin{array}{{1}c} \text{for} \\ \text{for} \\ \end{array}\begin{array}{{20}c} t < T_{\rm D} + T_{\rm V}, \\ t \ge T_{\rm D} + T_{\rm V}. \\\end{array}$$
	\\ 0 \\ \end{array} \right.\quad
	\begin{array}{*{1}c} \text{for} \\ \text{for} \\ \end{array}
	\begin{array}{*{20}c} t < T_{\rm D} + T_{\rm V}, \\ t \ge T_{\rm D} + T_{\rm V}. \\
	\end{array}$$

	*This means that in the ideal case the basic compensation pulse  $g_w(t)$  must exactly reproduce the linearly pre-equalized pulse  $g_d(t)$  for all times  $t > T_{\rm D} + T_{\rm V}$.		*This means that in the ideal case the basic compensation pulse  $g_w(t)$  must exactly reproduce the linearly pre-equalized pulse  $g_d(t)$  for all times  $t > T_{\rm D} + T_{\rm V}$.
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	We assume the same conditions as in the last section,  so that the following basic pulse values are present:		We assume the same conditions as in the last section,  so that the following basic pulse values are present:
	:$$g_0 = g_d(t=0) = 0.548 \cdot s_0,$$		:$$g_0 = g_d(t=0) = 0.548 \cdot s_0,$$
	:$$g_1 = g_d(t=T) = 0.214 \cdot s_0 =		:$$g_1 = g_d(t=T) = 0.214 \cdot s_0 =g_{-1} \hspace{0.05cm},$$
	g_{-1} \hspace{0.05cm},$$		:$$g_2 = g_d(t=2\hspace{0.05cm}T) = 0.012 \cdot s_0 = g_{-2}\hspace{0.05cm},$$
	:$$g_2 = g_d(t=2\hspace{0.05cm}T) = 0.012 \cdot s_0 = g_{-2}		:$$g_3 = g_{-3} = \text{...} \approx 0\hspace{0.05cm}.$$
	\hspace{0.05cm},$$
	:$$g_3 = g_{-3} = \text{...} \approx 0
	\hspace{0.05cm}.$$

	These two eye diagrams can be interpreted as follows:		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),  with binary bipolar redundancy-free coding considering symmetry:
	:$${\ddot{o}(T_{\rm D} = 0 )} = {2} \cdot \big [ g_0 - \| g_{-1}\| - \| g_{-2}\| - \| g_{1}\| - \| g_{2}\|\big ] = {2} \cdot \big [ g_0 - 2 \cdot g_{1} - 2 \cdot g_{2}\big		:$${\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}.$$
	]= 0.192 \cdot s_0 \hspace{0.05cm}.$$

	*On the other hand,  for ideal DFE,  the two trailers  $g_1$  and  $g_2$  are fully compensated and we obtain for the vertical eye opening:		*On the other hand,  for ideal DFE,  the two trailers  $g_1$  and  $g_2$  are fully compensated and we obtain for the vertical eye opening:
	:$${\ddot{o}(T_{\rm D} = 0 )} = {2} \cdot \big [ g_0 - \| g_{-1}\| - \|g_{-2}\|\big		:$${\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}.$$
	] = {2} \cdot \big [ g_0 - g_{1} - g_{2}\big ]= 0.644 \cdot s_0 \hspace{0.05cm}.$$

	*Since the correction signal  $w(t)$  is derived from the decision and thus noise-free signal  $v(t)$,  the noise rms value  $\sigma_d$  is not changed by the decision feedback.  Thus,  the SNR gain due to the DFE is		*Since the correction signal  $w(t)$  is derived from the decision and thus noise-free signal  $v(t)$,  the noise rms value  $\sigma_d$  is not changed by the decision feedback.  Thus,  the SNR gain due to the DFE is
	:$$G_{\rm DFE}=		:$$G_{\rm DFE}=20 \cdot {\rm lg}\hspace{0.1cm}\frac{0.644}{0.192} \approx 10.5\,{\rm dB} \hspace{0.05cm}.$$
	20 \cdot {\rm lg}\hspace{0.1cm}\frac{0.644}{0.192} \approx 10.5\,{\rm dB} \hspace{0.05cm}.$$

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	*With proper dimensioning of the filter coefficients  $k_\nu$,  <br>for  $\nu = 1$, ... , $N$:		*With proper dimensioning of the filter coefficients  $k_\nu$,  <br>for  $\nu = 1$, ... , $N$:
	:$$g_w(T_{\rm D} + \nu \cdot T) = g_d(T_{\rm D} + \nu \cdot T)		:$$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}.$$
	\hspace{0.3cm}\Rightarrow\hspace{0.3cm}
	g_k(T_{\rm D} + \nu \cdot T) = 0 \hspace{0.05cm}.$$

	*At detection time  $T_\text{D}$,  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  $T_\text{D}$,  the vertical eye opening is exactly the same as for ideal DFE.  A disadvantage is a smaller horizontal eye opening.<br><br>

Maintenance script: Add German interlanguage link

2026-03-16T12:26:43Z

Add German interlanguage link

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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Entscheidungsrückkopplung]]

Guenter at 15:26, 24 August 2022

2022-08-24T15:26:11Z

← Older revision		Revision as of 17:26, 24 August 2022
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	The last section has already made clear that the DFE already causes an enormous SNR gain if		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  $f_{\rm G}$  and		* a fixed cutoff frequency  $f_{\rm G}$  and
	*the fixed detection time  $T_{\rm D} = 0$		*the fixed detection time  $T_{\rm D} = 0$

Guenter at 14:17, 22 August 2022

2022-08-22T14:17:15Z

← Older revision		Revision as of 16:17, 22 August 2022
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	The last section has already made clear that the DFE already causes an enormous SNR gain if		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  $f_{\rm G}$  and		* a fixed cutoff frequency  $f_{\rm G}$  and
	*the fixed detection time  $T_{\rm D} = 0$		*the fixed detection time  $T_{\rm D} = 0$

Hwang at 11:03, 11 July 2022

2022-07-11T11:03:55Z

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	For a circuit realization it is sufficient if the basic corrected pulse  $g_k(t)$  becomes zero only at the equidistant detection times  $T_\text{D} +\nu \cdot T$.		For a circuit realization it is sufficient if the basic corrected pulse  $g_k(t)$  becomes zero only at the equidistant detection times  $T_\text{D} +\nu \cdot T$.
	[[File:P ID1451 Dig T 3 6 S5b version1.png\|right\|frame\|Decision feedback with delay filter\|class=fit]]		[[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  [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#~~Mode_of_action_of_the_transversal_filter~~\|delay filter]]  according to the adjacent diagram,		One realization possibility is thus an asymmetric  [[Digital_Signal_Transmission/Linear_Nyquist_Equalization#Operating_principle_of_the_transversal_filter\|delay filter]]  according to the adjacent diagram,

	*whose order  $N$  (number of filter coefficients), and<br>		*whose order  $N$  (number of filter coefficients), and<br>

Hwang at 11:01, 11 July 2022

2022-07-11T11:01:58Z

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	$\text{Conclusion:}$  For a transmission system over copper lines  $($coaxial cable,  two-wire line$)$,  the following system variants are particularly suitable due to the achievable signal–to–noise ratio at the decision:		$\text{Conclusion:}$  For a transmission system over copper lines  $($coaxial cable,  two-wire line$)$,  the following system variants are particularly suitable due to the achievable signal–to–noise ratio at the decision:
	*a  '''~~multilevel~~ system'''  $($for example  $M = 4)$  and the optimal  '''Nyquist equalization'''  to compensate for the strong intersymbol interference caused by the linear channel distortions;<br>		*a  '''multi-level system'''  $($for example  $M = 4)$  and the optimal  '''Nyquist equalization'''  to compensate for the strong intersymbol interference caused by the linear channel distortions;<br>

	*a  '''binary system'''  with relatively small bandwidth of the total frequency response  $H_{\rm G}(f) = H_{\rm K}(f) \cdot H_{\rm E}(f)$  and a non-linear decision with  '''DFE'''.}}		*a  '''binary system'''  with relatively small bandwidth of the total frequency response  $H_{\rm G}(f) = H_{\rm K}(f) \cdot H_{\rm E}(f)$  and a non-linear decision with  '''DFE'''.}}

Guenter at 14:59, 28 June 2022

2022-06-28T14:59:10Z

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	== Eye opening and error probability with DFE ==		== Eye opening and error probability with DFE ==
	<br>		<br>
	[[File:~~P ID1448 Dig T 3 6 S3 version1~~.png\|right\|frame\|Eye diagrams without and with  "Decision Feedback Equalization"  $(f_{\rm G}\cdot T = 0.3)$ ~~'''Korrektur'''~~\|class=fit]] We now consider the eye diagrams		[[File:EN_Dig_T_3_6_S3.png\|right\|frame\|Eye diagrams without and with  "Decision Feedback Equalization"  $(f_{\rm G}\cdot T = 0.3)$\|class=fit]] We now consider the eye diagrams
	*without DFE  (left graph)  and		*without DFE  (left graph)  and
	*with ideal DFE (right graph).		*with ideal DFE (right graph).
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	The last section has already made clear that the DFE already causes an enormous SNR gain 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 '''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  $f_{\rm G}$  and		* a fixed cutoff frequency  $f_{\rm G}$  and
	*the fixed detection time  $T_{\rm D} = 0$		*the fixed detection time  $T_{\rm D} = 0$
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	[[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]]
	$\text{Example 2:}$  The graph shows the basic pulse  $g_d(t)$		$\text{Example 2:}$  The graph shows the basic pulse  $g_d(t)$
	* for the cutoff frequency  $f_{\rm G} \cdot T = 0.2$  (red curve) and		* for the cutoff frequency  $f_{\rm G} \cdot T = 0.2$  (red curve) and

Guenter at 12:41, 27 June 2022

2022-06-27T12:41:29Z

← Older revision		Revision as of 14:41, 27 June 2022
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	The last section has already made clear that the DFE already causes an enormous SNR gain 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]]		[[File:P ID1449 Dig T 3 6 S4 version1.png\|right\|frame\|Eye diagrams with DFE and optimized detection time '''Korrektur'''\|class=fit]]
	* a fixed cutoff frequency  $f_{\rm G}$  and		* a fixed cutoff frequency  $f_{\rm G}$  and
	*the fixed detection time  $T_{\rm D} = 0$		*the fixed detection time  $T_{\rm D} = 0$

Guenter at 16:09, 25 June 2022

2022-06-25T16:09:50Z

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	⇒   Shown on the left are the basic pulses  $g_w(t)$  and  $g_k(t)$  with ideal decision feedback,  based on the detection time  $T_{\rm D} = 0$  and the delay time  $T_{\rm V} = T/2$.  <br>		⇒   Shown on the left are the basic pulses  $g_w(t)$  and  $g_k(t)$  with ideal decision feedback,  based on the detection time  $T_{\rm D} = 0$  and the delay time  $T_{\rm V} = T/2$.  <br>

	⇒   The right pictures from [Söd01]<ref name = 'Söd01'>Söder, G.: Simulation digitaler Übertragungssysteme. Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2001.</ref>. – all without consideration of the noise – make clear that by the compensation of all pulse trailers by means of the correction signal  $w(t)$,  the distances of the useful signal samples  $d_{\rm S}(\nu \cdot T)$  from the decision threshold  $E = 0$  are changed.		⇒   The right pictures from [Söd01]<ref name = 'Söd01'>Söder, G.: Simulation digitaler Übertragungssysteme. Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2001.</ref> – all without consideration of the noise – make clear that by the compensation of all pulse trailers by means of the correction signal  $w(t)$,  the distances of the useful signal samples  $d_{\rm S}(\nu \cdot T)$  from the decision threshold  $E = 0$  are changed.

	*Particularly small distances,  such as at times  $t = 6T$  and  $t = 7T$,  are significantly increased and thus their error probabilities are greatly reduced  (arrows pointing away from the threshold).<br>		*Particularly small distances,  such as at times  $t = 6T$  and  $t = 7T$,  are significantly increased and thus their error probabilities are greatly reduced  (arrows pointing away from the threshold).<br>
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	== Eye opening and error probability with DFE ==		== Eye opening and error probability with DFE ==
	<br>		<br>
	[[File:P ID1448 Dig T 3 6 S3 version1.png\|right\|frame\|Eye diagrams without and with ''Decision Feedback Equalization''  $(f_{\rm G}\cdot T = 0.3)$ '''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  "Decision Feedback Equalization"  $(f_{\rm G}\cdot T = 0.3)$ '''Korrektur'''\|class=fit]] We now consider the eye diagrams
	*without DFE  (left graph)  and		*without DFE  (left graph)  and
	*with ideal DFE (right graph).		*with ideal DFE (right graph).

Guenter at 16:06, 25 June 2022

2022-06-25T16:06:09Z

← Older revision		Revision as of 18:06, 25 June 2022
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	*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"