Digital Signal Transmission/Viterbi Receiver - Revision history

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 2:}$  As in  [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|$\text{Example 1}$]],  we consider the unipolar source symbol sequence of length  $N=3$  with the following parameter values:		$\text{Example 2:}$  As in  [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|$\text{Example 1}$]],  we consider the unipolar source symbol sequence of length  $N=3$  with the following parameter values:
	:$$g_{0}=0.7 \hspace{0.05cm},\hspace{0.2cm}		:$$g_{0}=0.7 \hspace{0.05cm},\hspace{0.2cm}g_{-1}=0.3 \hspace{0.05cm},\hspace{0.2cm}d_{0}=0.2 \hspace{0.05cm},\hspace{0.2cm}d_{1}=0.7 \hspace{0.05cm},\hspace{0.2cm}d_{2}=0.5 \hspace{0.05cm},\hspace{0.2cm}d_{3}=0 \hspace{0.05cm}\hspace{0.05cm}.$$Then,  for the metrics  $\varepsilon_{\nu}(i)$  at time points  $\nu = 0$  to  $\nu = 3$::$$\nu = 0\text{:} \hspace{0.2cm}\varepsilon_{0}(00) = \big [0.2- (0 \cdot 0.7 + 0 \cdot0.3) \big ]^2=0.04 \hspace{0.05cm},\hspace{0.4cm} \varepsilon_{0}(01) = \big [0.2- (0 \cdot 0.7 + 1 \cdot0.3) \big ]^2=0.01 \hspace{0.05cm};$$:$$\nu = 1{:} \hspace{0.2cm}\varepsilon_{1}(00) = \big [0.7- (0 \cdot 0.7 + 0 \cdot0.3)\big ]^2=0.49 \hspace{0.05cm},\hspace{0.4cm} \varepsilon_{1}(01) = \big [0.7- (0 \cdot 0.7 + 1 \cdot0.3)\big ]^2=0.16 \hspace{0.05cm},$$:$$\hspace{1.4cm}\varepsilon_{1}(10) = \big [0.7- (1 \cdot 0.7 + 0 \cdot0.3)\big ]^2=0.00 \hspace{0.05cm}\hspace{0.4cm}\varepsilon_{1}(11) = \big [0.7- (1 \cdot 0.7 + 1 \cdot0.3)\big ]^2=0.09 \hspace{0.05cm};$$:$$\nu = 2\text{:} \hspace{0.2cm}\varepsilon_{2}(00) = \big [0.5- (0 \cdot 0.7 + 0 \cdot0.3)\big ]^2=0.25 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{2}(01) = \big [0.5- (0 \cdot 0.7 + 1 \cdot0.3)\big ]^2=0.04 \hspace{0.05cm},$$:$$\hspace{1.4cm}\varepsilon_{2}(10) = \big [0.5- (1 \cdot 0.7 + 0 \cdot0.3)\big ]^2=0.04 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{2}(11) = \big [0.5- (1 \cdot 0.7 + 1 \cdot0.3)\big ]^2=0.25 \hspace{0.05cm};$$:$$ \nu = 3\text{:} \hspace{0.2cm}\varepsilon_{3}(00) = \big [0.0- (0 \cdot 0.7 + 0 \cdot0.3)\big ]^2=0.00 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{3}(01) = \big [0.0- (0 \cdot 0.7 + 1 \cdot0.3)\big ]^2=0.09 \hspace{0.05cm},$$:$$\hspace{1.4cm}\varepsilon_{3}(10) = \big [0.0- (1 \cdot 0.7 + 0 \cdot0.3)\big ]^2=0.49 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{3}(11) = \big [0.0- (1 \cdot 0.7 + 1 \cdot0.3)\big ]^2=1.00 \hspace{0.05cm}.$$}}
	g_{-1}=0.3 \hspace{0.05cm},\hspace{0.2cm}
	d_{0}=0.2 \hspace{0.05cm},\hspace{0.2cm}
	d_{1}=0.7 \hspace{0.05cm},\hspace{0.2cm}
	d_{2}=0.5 \hspace{0.05cm},\hspace{0.2cm}
	d_{3}=0 \hspace{0.05cm}
	\hspace{0.05cm}.$$Then,  for the metrics  $\varepsilon_{\nu}(i)$  at time points  $\nu = 0$  to  $\nu = 3$::$$\nu = 0\text{:} \hspace{0.2cm}\varepsilon_{0}(00) = \big [0.2- (0 \cdot 0.7 + 0 \~~cdot~~
	0.3) \big ]^2=0.04 \hspace{0.05cm},\hspace{0.4cm} \varepsilon_{0}(01) = \big [0.2- (0 \cdot 0.7 + 1 \~~cdot~~
	0.3) \big ]^2=0.01 \hspace{0.05cm};$$:$$\nu = 1{:} \hspace{0.2cm}\varepsilon_{1}(00) = \big [0.7- (0 \cdot 0.7 + 0 \~~cdot~~
	0.3)\big ]^2=0.49 \hspace{0.05cm},\hspace{0.4cm} \varepsilon_{1}(01) = \big [0.7- (0 \cdot 0.7 + 1 \~~cdot~~
	0.3)\big ]^2=0.16 \hspace{0.05cm},$$:$$\hspace{1.4cm}\varepsilon_{1}(10) = \big [0.7- (1 \cdot 0.7 + 0 \~~cdot~~
	0.3)\big ]^2=0.00 \hspace{0.05cm}\hspace{0.4cm}\varepsilon_{1}(11) = \big [0.7- (1 \cdot 0.7 + 1 \~~cdot~~
	0.3)\big ]^2=0.09 \hspace{0.05cm};$$:$$\nu = 2\text{:} \hspace{0.2cm}\varepsilon_{2}(00) = \big [0.5- (0 \cdot 0.7 + 0 \~~cdot~~
	0.3)\big ]^2=0.25 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{2}(01) = \big [0.5- (0 \cdot 0.7 + 1 \~~cdot~~
	0.3)\big ]^2=0.04 \hspace{0.05cm},$$:$$\hspace{1.4cm}\varepsilon_{2}(10) = \big [0.5- (1 \cdot 0.7 + 0 \~~cdot~~
	0.3)\big ]^2=0.04 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{2}(11) = \big [0.5- (1 \cdot 0.7 + 1 \~~cdot~~
	0.3)\big ]^2=0.25 \hspace{0.05cm};$$:$$ \nu = 3\text{:} \hspace{0.2cm}\varepsilon_{3}(00) = \big [0.0- (0 \cdot 0.7 + 0 \~~cdot~~
	0.3)\big ]^2=0.00 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{3}(01) = \big [0.0- (0 \cdot 0.7 + 1 \~~cdot~~
	0.3)\big ]^2=0.09 \hspace{0.05cm},$$:$$\hspace{1.4cm}\varepsilon_{3}(10) = \big [0.0- (1 \cdot 0.7 + 0 \~~cdot~~
	0.3)\big ]^2=0.49 \hspace{0.05cm},\hspace{0.4cm}\varepsilon_{3}(11) = \big [0.0- (1 \cdot 0.7 + 1 \~~cdot~~
	0.3)\big ]^2=1.00 \hspace{0.05cm}.$$}}


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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Without proof:}$  A simple approximation for the  $($average$)$  '''  bit error probability in maximum likelihood decision'''  is:		$\text{Without proof:}$  A simple approximation for the  $($average$)$  '''  bit error probability in maximum likelihood decision'''  is:
	:$$p_{\rm ML} = {\rm Q}\left(g_{\rm max}/{\sigma_d}\right)\hspace{0.3cm}{\rm with}\hspace{0.2cm}g_{\rm max}= {\~~rmMax~~}\hspace{0.15cm} \vert g_\nu \vert \hspace{0.05cm}.$$		:$$p_{\rm ML} = {\rm Q}\left(g_{\rm max}/{\sigma_d}\right)\hspace{0.3cm}{\rm with}\hspace{0.2cm}g_{\rm max}= {\rm Max}\hspace{0.15cm} \vert g_\nu \vert \hspace{0.05cm}.$$
	*The approximation is valid only for binary bipolar signaling.  For unipolar signaling,  the argument of the Q–function must be halved.}}<br>		*The approximation is valid only for binary bipolar signaling.  For unipolar signaling,  the argument of the Q–function must be halved.}}<br>

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	*For a binary receiver with simple threshold decision  $\rm (ThD)$  according to the chapter   [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference\|"Error Probability with Intersymbol Interference"]],  the  (average)  bit error probability for bipolar signaling is:		*For a binary receiver with simple threshold decision  $\rm (ThD)$  according to the chapter   [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference\|"Error Probability with Intersymbol Interference"]],  the  (average)  bit error probability for bipolar signaling is:
	:$$p_{\rm ThD} = {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)+{1}/{2}\cdot {\~~rmQ~~}\left(\frac{g_{0} }{\sigma_d}\right)+ {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)\approx {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right) = \frac{ {\~~rmQ~~}(2)}{4}= 0.57 \%\hspace{0.05cm}.$$		:$$p_{\rm ThD} = {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)+{1}/{2}\cdot {\rm Q}\left(\frac{g_{0} }{\sigma_d}\right)+ {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)\approx {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right) = \frac{ {\rm Q}(2)}{4}= 0.57 \%\hspace{0.05cm}.$$

	*For the  [[Digital_Signal_Transmission/Decision_Feedback\|receiver with decision feedback equalization]]  $\rm (DFE)$,  considering   $g_{+1} = 0$  $($full compensation of the "postcursor"),  we obtain:		*For the  [[Digital_Signal_Transmission/Decision_Feedback\|receiver with decision feedback equalization]]  $\rm (DFE)$,  considering   $g_{+1} = 0$  $($full compensation of the "postcursor"),  we obtain:
	:$$p_{\rm DFE} ={1}/{2}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)+{1}/{2}\cdot {\~~rmQ~~}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)\approx {1}/{2}\cdot {\~~rmQ~~}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)=\frac{{\~~rmQ~~}(4)}{4}= 0.16 \cdot 10^{-4}\hspace{0.05cm}.$$		:$$p_{\rm DFE} ={1}/{2}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)+{1}/{2}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)\approx {1}/{2}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)=\frac{{\rm Q}(4)}{4}= 0.16 \cdot 10^{-4}\hspace{0.05cm}.$$

	*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$  leads to the much smaller error probability		*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$  leads to the much smaller error probability

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← Older revision		Revision as of 14:26, 16 March 2026
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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Viterbi–Empfänger]]

Guenter at 11:20, 13 July 2022

2022-07-13T11:20:19Z

← Older revision		Revision as of 13:20, 13 July 2022
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	[[Aufgaben:Exercise_3.11:_Viterbi_Receiver_and_Trellis_Diagram\|Exercise 3.11: Viterbi Receiver and Trellis Diagram]]		[[Aufgaben:Exercise_3.11:_Viterbi_Receiver_and_Trellis_Diagram\|Exercise 3.11: Viterbi Receiver and Trellis Diagram]]

	[[Aufgaben:Exercise_3.11Z:_Maximum_Likelihood_Error_Variables\|Exercise 3.11Z: ~~Maximum Likelihood Error Variables~~]]		[[Aufgaben:Exercise_3.11Z:_Maximum_Likelihood_Error_Variables\|Exercise 3.11Z: Metric and Accumutated Metric]]

	[[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors\|Exercise 3.12: Trellis Diagram for Two Precursors]]		[[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors\|Exercise 3.12: Trellis Diagram for Two Precursors]]

Guenter at 15:27, 5 July 2022

2022-07-05T15:27:46Z

← Older revision		Revision as of 17:27, 5 July 2022
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	[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]		[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]
	<br>		<br><br>
	This  '''simplified trellis diagram'''  allows the following statements:		This  '''simplified trellis diagram'''  allows the following statements:
	*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>		*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>
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	*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>		*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>

	~~All statements in this chapter can be verified with the  (German languag) SWF applet  [[Applets:Viterbi\|"Viterbi receiver for one precursor"]].~~

	== Extension to two precursors ==		== Extension to two precursors ==
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	#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>		#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>
	#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>		#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>
	#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either   "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>		#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either  "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>

	== Bit error probability with maximum likelihood decision==		== Bit error probability with maximum likelihood decision==
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$ leads to the much smaller error probability		*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$  leads to the much smaller error probability
	:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)		:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
	\approx {\rm Q}(6) = 10^{-9}		\approx {\rm Q}(6) = 10^{-9}
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{~~Conclusions~~:}$		$\text{Conclusion:}$
	*Maximum likelihood decision is only advantageous in the presence of intersymbol interference.		*Maximum likelihood decision is only advantageous in the presence of intersymbol interference.

Guenter at 15:20, 5 July 2022

2022-07-05T15:20:45Z

← Older revision		Revision as of 17:20, 5 July 2022
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	[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]		[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]
	~~<br>~~<br>		<br>
	This  '''simplified trellis diagram'''  allows the following statements:		This  '''simplified trellis diagram'''  allows the following statements:
	*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>		*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>
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	*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>		*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>

			All statements in this chapter can be verified with the  (German languag) SWF applet  [[Applets:Viterbi\|"Viterbi receiver for one precursor"]].

	== Extension to two precursors ==		== Extension to two precursors ==
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	#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>		#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>
	#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>		#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>
	#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either  "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>		#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either   "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>

	== Bit error probability with maximum likelihood decision==		== Bit error probability with maximum likelihood decision==
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$~~ ~~ leads to the much smaller error probability		*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$ leads to the much smaller error probability
	:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)		:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
	\approx {\rm Q}(6) = 10^{-9}		\approx {\rm Q}(6) = 10^{-9}
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{~~Conclusion~~:}$		$\text{Conclusions:}$
	*Maximum likelihood decision is only advantageous in the presence of intersymbol interference.		*Maximum likelihood decision is only advantageous in the presence of intersymbol interference.

Guenter at 13:00, 5 July 2022

2022-07-05T13:00:12Z

← Older revision		Revision as of 15:00, 5 July 2022
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	[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]		[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]
	<br>		<br><br>
	This  '''simplified trellis diagram'''  allows the following statements:		This  '''simplified trellis diagram'''  allows the following statements:
	*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>		*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>
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	*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>		*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>

	~~All statements in this chapter can be verified with the  (German languag) SWF applet  [[Applets:Viterbi\|"Viterbi receiver for one precursor"]].~~

	== Extension to two precursors ==		== Extension to two precursors ==
Line 250:		Line 250:
	#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>		#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>
	#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>		#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>
	#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either   "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>		#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either  "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>

	== Bit error probability with maximum likelihood decision==		== Bit error probability with maximum likelihood decision==
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$ leads to the much smaller error probability		*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$  leads to the much smaller error probability
	:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)		:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
	\approx {\rm Q}(6) = 10^{-9}		\approx {\rm Q}(6) = 10^{-9}

Guenter at 13:57, 3 July 2022

2022-07-03T13:57:30Z

← Older revision		Revision as of 15:57, 3 July 2022
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	#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either   "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>		#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either   "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>

	== ~~Error~~ probability in maximum likelihood decision==		== Bit error probability with maximum likelihood decision==
	<br>		<br>
	The exact error probability calculation ~~in ML~~ decision (~~for example~~ with the correlation or the Viterbi receiver) is very complex. Thereby		The exact bit error probability calculation for binary maximum likelihood decision  $($e.g. with the correlation or the Viterbi receiver$)$  is very complex.  Thereby
	*determine the difference energies between all possible symbol sequences   $Q_i$  and  $Q_{j \ne i}$,  where the error probability is essentially determined by the minimum difference energy;<br>		*determine the difference energies between all possible symbol sequences   $Q_i$  and  $Q_{j \ne i}$,  where the error probability is essentially determined by the minimum difference energy;<br>

	*also the influences of matched filter   $H_{\rm MF}(f)$  and decorrelation filter   $H_{\rm DF}(f)$  ~~are~~ taken into account and additionally the rms value   $\sigma_d$  of the detection noise signal is determined.<br><br>		*also the influences of matched filter   $H_{\rm MF}(f)$  and decorrelation filter   $H_{\rm DF}(f)$  have to be taken into account and additionally the rms value   $\sigma_d$  of the detection noise signal is determined.<br><br>

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Without proof:}$  A simple approximation for the (average)  '''error probability in maximum likelihood decision'''  is:		$\text{Without proof:}$  A simple approximation for the  $($average$)$  '''  bit error probability in maximum likelihood decision'''  is:
	:$$p_{\rm ML} = {\rm Q}\left(g_{\rm max}/{\sigma_d}\right)		:$$p_{\rm ML} = {\rm Q}\left(g_{\rm max}/{\sigma_d}\right)
	\hspace{0.3cm}{\rm with}\hspace{0.2cm}g_{\rm max}= {\rm		\hspace{0.3cm}{\rm with}\hspace{0.2cm}g_{\rm max}= {\rm
	Max}\hspace{0.15cm} \vert g_\nu \vert \hspace{0.05cm}.$$		Max}\hspace{0.15cm} \vert g_\nu \vert \hspace{0.05cm}.$$
	The approximation is valid only for binary bipolar signaling. For unipolar signaling, the argument of the Q function must be halved.}}<br>		*The approximation is valid only for binary bipolar signaling.  For unipolar signaling,  the argument of the Q–function must be halved.}}<br>

	For the following interpretation and example, it is assumed that   $v + 1$  basic pulse values (including the main value) are different from zero. Then holds:		For the following interpretation and example,  it is assumed that  $v + 1$  basic pulse values  $($including the main value$)$  are different from zero.  Then holds:
	*The Viterbi decision must consider all these basic pulse values. This means that a trellis diagram with   $2^v$  states has to be processed.<br>		#The Viterbi decision must consider all these basic pulse values.  This means that a trellis diagram with   $2^v$  states has to be processed.<br>
			#The precondition for the validity of the above equation is the uncorrelatedness of the noise at the decision,  which is achieved by the decorrelation filter.<br>
	*The precondition for the validity of the above equation is the uncorrelatedness of the noise at the decision, which is achieved by the decorrelation filter.<br>		#For the comparison with threshold decision   $(p_{\rm ThD})$  or decision feedback   $(p_{\rm DFE})$,  the noise rms value   $\sigma_d$  is assumed to be constant.<br>
			#Optimization of the maximum likelihood system results in very narrowband filters,  since all pulse tails can be factored out by the ML algorithm.<br>
	*For the comparison with threshold decision   $(p_{\rm SE})$  or decision feedback   $(p_{\rm DFE})$  the noise rms value   $\sigma_d$  is assumed to be constant.<br>		#Thus,  with constant noise power density   $N_0$  $($at receiver input$)$,  the noise rms value   $\sigma_d$  $($at decision$)$  is for the ML system smaller than for other variants.<br>
			#This means:   The signal-to-noise ratio gain by the ML decision may be even larger than the following example $($with  $\sigma_d = \rm const.)$  expresses.<br>
	*Optimization of the ML system results in very narrowband filters, since all pulse tails can be factored out by the ML algorithm.<br>

	*Thus, with constant noise power density   $N_0$  (at ~~the~~ receiver input), the noise rms value   $\sigma_d$  (at ~~the~~ decision) is ~~smaller~~ for the ML system than for other variants.<br>

	*This means:   The signal-to-noise ratio gain by the ML decision may be even larger than the following example $($with  $\sigma_d = \rm const.)$  expresses.<br>


	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 5:}$  We consider the basic pulse values   $g_{0} = 0.6$  and  $g_{-1} = g_{+1} = 0.2 \ ( = g_1)$. In addition, we assume the constant noise rms value   $\sigma_d = 0.1$.  For simplicity, ~~all~~ quantities are normalized.<br>		$\text{Example 5:}$  We consider the basic pulse values   $g_{0} = 0.6$  and  $g_{-1} = g_{+1} = 0.2 \ ( = g_1)$.  In addition,  we assume the constant noise rms value   $\sigma_d = 0.1$.  For simplicity,  these quantities are normalized.<br>

	*For a binary receiver with simple threshold decision  $\rm (SE)$  according to the chapter   [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference\|"Error Probability with Intersymbol Interference"]],  the (average) error probability for bipolar signaling is:		*For a binary receiver with simple threshold decision  $\rm (ThD)$  according to the chapter   [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference\|"Error Probability with Intersymbol Interference"]],  the  (average)  bit error probability for bipolar signaling is:
	:$$p_{\rm SE} = {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)		:$$p_{\rm ThD} = {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)
	+{1}/{2}\cdot {\rm		+{1}/{2}\cdot {\rm
	Q}\left(\frac{g_{0} }{\sigma_d}\right)+ {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)		Q}\left(\frac{g_{0} }{\sigma_d}\right)+ {1}/{4}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}-\hspace{-0.02cm}2 \cdot g_{1} }{\sigma_d}\right)
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*For the  [[Digital_Signal_Transmission/Decision_Feedback\|receiver with decision feedback equalization]]  $\rm (DFE)$,  considering   $g_{+1} = 0$  (full compensation of the postcursor), we obtain:		*For the  [[Digital_Signal_Transmission/Decision_Feedback\|receiver with decision feedback equalization]]  $\rm (DFE)$,  considering   $g_{+1} = 0$  $($full compensation of the "postcursor"),  we obtain:
	:$$p_{\rm DFE} ={1}/{2}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)		:$$p_{\rm DFE} ={1}/{2}\cdot {\rm Q}\left(\frac{g_{0}\hspace{-0.02cm}+\hspace{-0.02cm}g_{-1} }{\sigma_d}\right)
	+{1}/{2}\cdot {\rm		+{1}/{2}\cdot {\rm
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*In contrast, the application of the maximum likelihood decision (with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]])  $\rm (ML)$ leads to the much smaller error probability		*In contrast,  the application of the maximum likelihood decision with  [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver\|"correlation receiver"]]  or   [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites\|"Viterbi receiver"]]  $\rm (ML)$ leads to the much smaller error probability
	:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)		:$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
	\approx {\rm Q}(6) = 10^{-9}		\approx {\rm Q}(6) = 10^{-9}
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*This corresponds to a ~~signal-to-noise ratio~~ gain of about   $3 \ \rm dB$  (versus $\rm DFE$) and   $7.5 \ \rm dB$  (versus $\rm SE$) over the other two systems. The result of this simple approximation was essentially confirmed by simulations.}}<br>		*This corresponds to a SNR gain of about   $3 \ \rm dB$  $($versus $\rm DFE)$  and   $7.5 \ \rm dB$  $($versus $\rm ThD)$  over the other two systems.  The result of this simple approximation was essentially confirmed by simulations.}}<br>

	To apply the described Viterbi algorithm directly, the (normalized) basic pulse values   $g_{0} =0.2$,  $g_{-1} =0.6$  and  $g_{-2} =0.2$  must be set in the equations. Indeed, a time shift by multiples of the symbol duration $T$ with respect to the coordinate system ~~chosen for representation reasons~~ does not change the performance ~~characteristics~~ of the Viterbi decision.<br>		<u>Note:</u>
			#To apply the described Viterbi algorithm directly,  the (normalized) basic pulse values   $g_{0} =0.2$,  $g_{-1} =0.6$  and  $g_{-2} =0.2$  must be set in the equations.
			#Indeed,  a time shift by multiples of the symbol duration  $T$  with respect to the coordinate system does not change the performance of the Viterbi decision.<br>
			#The maximum likelihood error probability according to the above equation depends solely on the largest basic pulse value.
			#It is quite possible that a "precursor" $($here:  $g_{-1} =0.6)$  is larger than the main value  $g_{0}=0.2$.  In case of threshold decision this would lead to a  "closed eye".<br>

	The maximum likelihood error probability according to the above equation depends solely on the largest basic pulse value. It is quite possible that a "precursor" $($here:  $g_{-1} =0.6)$  is larger than the main value  $g_{0}$.  In case of threshold decision this would lead to a closed eye.<br>

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$  ~~A maximum~~ likelihood decision is only advantageous in the presence of intersymbol interference. In the case of Nyquist equalization (that is:   only the basic pulse value   $g_0$  is non-zero) the maximum likelihood receiver also operates symbolically and with the same error probability   ${\rm Q}(g_0/\sigma_d)$  as ~~a receiver with threshold decision~~.}}<br>		$\text{Conclusion:}$
			*Maximum likelihood decision is only advantageous in the presence of intersymbol interference.

			*In the case of Nyquist equalization  $($that is:   only the basic pulse value   $g_0$  is non-zero$)$  the maximum likelihood receiver also operates symbolically and with the same bit error probability   ${\rm Q}(g_0/\sigma_d)$  as the other receivers.}}<br>

	== Exercises for the chapter==		== Exercises for the chapter==

Guenter at 15:05, 2 July 2022

2022-07-02T15:05:07Z

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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	[[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors\|Exercise 3.12]]  discusses in detail the calculation ~~of error quantities (metrics)~~ and the minimization of ~~total error quantities (~~accumulated metrics).		[[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors\|Exercise 3.12]]  discusses in detail the metrics calculation and the minimization of the accumulated metrics for the case of two precursors.

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 4:}$  We consider here an exemplary trellis diagram, which reflects the (error-free) detection of the following symbol sequence:		$\text{Example 4:}$  We consider here an exemplary trellis diagram, which reflects the (error-free) detection of the following symbol sequence:
			[[File:P ID1474 Dig T 3 8 S5 version1.png\|right\|frame\|Trellis diagram for two precursors\|class=fit]]

	:$$a_{1}=0 \hspace{0.05cm},\hspace{0.2cm}		:$$a_{1}=0 \hspace{0.05cm},\hspace{0.2cm}
	a_{2}=0		a_{2}=0
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	\hspace{0.05cm},\hspace{0.2cm} a_{5}=1		\hspace{0.05cm},\hspace{0.2cm} a_{5}=1
	\hspace{0.05cm},\hspace{0.2cm} a_{6}=0 \hspace{0.05cm}, \hspace{0.05cm}\text{...}$$		\hspace{0.05cm},\hspace{0.2cm} a_{6}=0 \hspace{0.05cm}, \hspace{0.05cm}\text{...}$$

	~~[[File:P ID1474 Dig T 3 8 S5 version1.png\|center\|frame\|Trellis diagram for two precursors\|class=fit]]~~

	This simplified trellis diagram allows the following statements:		This simplified trellis diagram allows the following statements:
	*All branches departing from  ${\it \Gamma}_{\nu}(00)$  or   ${\it \Gamma}_{\nu}(01)$  are assigned to symbol   $\rm 0$  and drawn in blue. The red branches departing from the two upper states denote symbol   $\rm 1$.<br>		#All branches departing from  ${\it \Gamma}_{\nu}(00)$  or   ${\it \Gamma}_{\nu}(01)$  are assigned to symbol  "$\rm 0$"  and drawn in blue.  The red branches departing from the two upper states denote symbol  "$\rm 1$".<br>
			#If one follows the continuous paths,  one recognizes the indicated sequence.  Since at time   $\nu = 6$  only blue branches arrive,  the first six bits of the sequence are finally fixed here.<br>
	*If one follows the continuous paths, one recognizes the indicated sequence. Since at ~~the~~ time   $\nu = 6$  only blue branches arrive, the first six bits of the sequence are finally fixed here.<br>		#However,  partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>
			#In contrast,  at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately.  For example,  at   $\nu = 5$  it is only certain that either   "$a_5 = 0$,  $a_6 = 1$"   will be,  or   "$a_5 = 1$,  $a_6 = 0$". }}<br>
	*However, partial sequences could also be output already at times   $\nu = 1$,  $\nu = 3$  and  $\nu = 4$,  since at these times the same partial paths result for all four states.<br>

	*In contrast, at   $\nu = 2$  and  $\nu = 5$, decisions may not be made immediately. For example, at ~~time~~   $\nu = 5$  it is only certain that either   $a_5 = 0$~~  and~~  $a_6 = 1$  will be or   $a_5 = 1$~~  and~~  $a_6 = 0$~~. Other combinations are not possible~~.}}<br>

	== Error probability in maximum likelihood decision==		== Error probability in maximum likelihood decision==

Guenter at 14:50, 2 July 2022

2022-07-02T14:50:37Z

← Older revision		Revision as of 16:50, 2 July 2022
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	== Simplified trellis diagram ==		== Simplified trellis diagram ==
	<br>		<br>
	The advantage of the trellis diagram is that the number of nodes and branches does not double at each iteration step. By selecting the minimum ~~total error quantities~~, only those symbol sequences are considered further which ~~can~~ be ~~considered as~~ part of the most probable sequence ~~at all~~.<br>		The advantage of the trellis diagram is that the number of nodes and branches does not double at each iteration step.  By selecting the minimum accumulated metrics,  only those symbol sequences are considered further which could be part of the most probable sequence.<br>

	The trellis diagram can be further simplified by drawing in only the selected branches. This is illustrated in the lower part of the diagram with our numerical example~~. As a reminder~~:   The actual transmitted amplitude coefficients are  $a_1= 1$,  $a_2= 1$  and  $a_3= 0$~~, and the~~ trellis diagram drawn above was calculated assuming that due to the pulse values   $g_{-1} = 0.3$  and  $g_{0} = 0.7$  as well as ~~the~~ AWGN noise~~, the~~ input values  $d_0= 0.2$,  $d_1= 0.7$,  $d_2= 0.5$  and  $d_3= 0$  are applied to the ML decision.<br>		The trellis diagram can be further simplified by drawing in only the selected branches.  This is illustrated in the lower part of the diagram with our numerical example:
			#The actual transmitted amplitude coefficients are  $a_1= 1$,  $a_2= 1$  and  $a_3= 0$.
			#The trellis diagram drawn above was calculated assuming that due to the pulse values   $g_{-1} = 0.3$  and  $g_{0} = 0.7$  as well as AWGN noise.
			# The noisy input values  $d_0= 0.2$,  $d_1= 0.7$,  $d_2= 0.5$  and  $d_3= 0$  are applied to the maximum likelihood decision.<br>

	[[File:P ID1473 Dig T 3 8 S4 version1.png\|~~center~~\|frame\|Trellis diagram and simplified trellis diagram\|class=fit]]		[[File:P ID1473 Dig T 3 8 S4 version1.png\|right\|frame\|Trellis diagram  (above)  and simplified trellis diagram  (below)\|class=fit]]
			<br>
			This  '''simplified trellis diagram'''  allows the following statements:
			*The decision  $a_1= 1$  can be made immediately at time  $\nu = 1$,  since under both hypotheses – the following symbol is   $a_2= 0$  or this symbol is   $a_2= 1$  – the same result is obtained.<br>

	~~This ''' simplified trellis diagram''' allows the following statements:~~		*At time  $\nu = 2$  the final decision about   $a_2$  cannot yet be made.  Assuming that   $a_3= 0$,  one would have to decide for   $a_2= 1$,  while   $a_3= 1$  would lead to the fixed choice of   $a_2= 0$.  <br>
	*~~The decision about~~  $~~a_1~~= 1$  ~~can be made immediately at~~ the ~~time~~  $~~\nu = 1~~$,  ~~since under both hypotheses~~ &~~ndash~~; ~~namely the following symbol is~~   $~~a_2~~= 0$  ~~or this symbol is~~   $a_2= 1$  &~~ndash~~; the ~~same result~~   $~~a_1~~= 1$  ~~is obtained.~~<br>

	*~~In contrast, at the time~~  $\nu = 2$  the ~~final decision about~~   $~~a_2~~$  ~~cannot yet be made. Assuming that~~   $a_3= 0$,  ~~one would have to decide for~~   $~~a_2= 1~~$,  ~~while~~ the ~~hypothesis~~   $~~a_3~~= 1$  would ~~lead to the fixed choice of~~   $a_2~~= 0~~$.  <br>		*At  $\nu = 3$  the decisions for   $a_1= 1$,  $a_2= 1$,  $a_3= 0$  are final,  since both continuous paths  $($blue and red$)$  suggest this  $($in this case correct$)$  sequence.
			*If one were to postpone the decision to time  $\nu = 4$,  one would not have more usable information about   $a_1$,  $a_2$  and  $a_3$.<br><br>

	*At time  $\nu = 3$  the decision for   $a_1= 1$,  $a_2= 1$,  $a_3= 0$  is final, since both continuous paths suggest this (the correct in this case) sequence.		All statements in this chapter can be verified with the  (German languag) SWF applet  [[Applets:Viterbi\|"Viterbi receiver for one precursor"]].
	*If one were to postpone the decision to time  $\nu = 4$,  one would no longer have usable information about   $a_1$,  $a_2$  and  $a_3$  available.<br><br>

	All statements in this chapter can be verified with the ~~interactive~~ applet [[Applets:Viterbi\|"Viterbi receiver for one precursor"]].

	== Extension to two precursors ==		== Extension to two precursors ==
	<br>		<br>
	If the basic pulse is described by the samples   $g_{0} \ne 0$,  $g_{-1} \ne 0$  and  $g_{-2} \ne 0$,  then four metrics   ${\it \Gamma}_{\nu}(00)$,  ${\it \Gamma}_{\nu}(01)$,  ${\it \Gamma}_{\nu}(10)$  and  ${\it \Gamma}_{\nu}(11)$  must be determined in the trellis diagram at each time  $\nu$.  For example,  ${\it \Gamma}_{\nu}(01)$  ~~then~~ describes the minimum ~~total error quantity~~ for the detection of the symbol   $a_\nu$  under the hypothesis ~~that~~   $a_{\nu-1} = 0$  and  $a_{\nu-2} = 1$,  ~~and it~~ holds:		If the basic pulse  $g_d(t)$  is described by the samples   $g_{0} \ne 0$,  $g_{-1} \ne 0$  and  $g_{-2} \ne 0$,  then four metrics   ${\it \Gamma}_{\nu}(00)$,  ${\it \Gamma}_{\nu}(01)$,  ${\it \Gamma}_{\nu}(10)$  and  ${\it \Gamma}_{\nu}(11)$  must be determined in the trellis diagram at each time  $\nu$.

			For example,  ${\it \Gamma}_{\nu}(01)$  describes the minimum accumulated metric for the detection of the symbol   $a_\nu$  under the hypothesis   "$a_{\nu-1} = 0$  and  $a_{\nu-2} = 1$".  It holds:
	:$${\it \Gamma}_{\nu}(01) = {\rm Min}\big[{\it \Gamma}_{\nu-1}(00) + \varepsilon_{\nu}(001),		:$${\it \Gamma}_{\nu}(01) = {\rm Min}\big[{\it \Gamma}_{\nu-1}(00) + \varepsilon_{\nu}(001),
	\hspace{0.2cm}{\it \Gamma}_{\nu-1}(10) + \varepsilon_{\nu}(101)\big]		\hspace{0.2cm}{\it \Gamma}_{\nu-1}(10) + \varepsilon_{\nu}(101)\big]

← Older revision		Revision as of 13:20, 13 July 2022
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	[[Aufgaben:Exercise_3.11:_Viterbi_Receiver_and_Trellis_Diagram\|Exercise 3.11: Viterbi Receiver and Trellis Diagram]]		[[Aufgaben:Exercise_3.11:_Viterbi_Receiver_and_Trellis_Diagram\|Exercise 3.11: Viterbi Receiver and Trellis Diagram]]

	[[Aufgaben:Exercise_3.11Z:_Maximum_Likelihood_Error_Variables\|Exercise 3.11Z: ~~Maximum Likelihood Error Variables~~]]		[[Aufgaben:Exercise_3.11Z:_Maximum_Likelihood_Error_Variables\|Exercise 3.11Z: Metric and Accumutated Metric]]

	[[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors\|Exercise 3.12: Trellis Diagram for Two Precursors]]		[[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors\|Exercise 3.12: Trellis Diagram for Two Precursors]]