Digital Signal Transmission/Viterbi Receiver - Revision history

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@@ Line 318: / Line 318: @@
 [[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]]

@@ Line 213: / Line 213: @@
 [[File:P ID1473 Dig T 3 8 S4 version1.png|right|frame|Trellis diagram&nbsp; (above)&nbsp; and simplified trellis diagram&nbsp; (below)|class=fit]]
-<br>
+<br><br>
 This&nbsp; '''simplified trellis diagram'''&nbsp; allows the following statements:
 *The decision &nbsp;$a_1= 1$&nbsp; can be made immediately at time &nbsp;$\nu = 1$,&nbsp; since under both hypotheses &ndash; the following symbol is &nbsp; $a_2= 0$&nbsp; or this symbol is &nbsp; $a_2= 1$&nbsp; &ndash; the same result  is obtained.<br>
@@ Line 222: / Line 222: @@
 *If one were to postpone the decision to time &nbsp;$\nu = 4$,&nbsp; one would not have more usable information about &nbsp; $a_1$,&nbsp; $a_2$&nbsp; and&nbsp; $a_3$.<br><br>
-All statements in this chapter can be verified with the&nbsp; (German languag) SWF  applet&nbsp; [[Applets:Viterbi|"Viterbi receiver for one precursor"]].
 == Extension to two precursors ==
@@ Line 250: / Line 250: @@
 #If one follows the continuous paths,&nbsp; one recognizes the indicated sequence.&nbsp; Since at time &nbsp; $\nu = 6$&nbsp; only blue branches arrive,&nbsp; the first six bits of the sequence are finally fixed here.<br>
 #However,&nbsp; partial sequences could also be output already at times &nbsp; $\nu = 1$,&nbsp; $\nu = 3$&nbsp; and&nbsp; $\nu = 4$,&nbsp; since at these times the same partial paths result for all four states.<br>
-#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either &nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
+#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either&nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
 == Bit error probability with maximum likelihood decision==
@@ Line 294: / Line 294: @@
   \hspace{0.05cm}.$$
-*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$ leads to the much smaller error probability
+*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$&nbsp; leads to the much smaller error probability
 :$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
   \approx {\rm  Q}(6) = 10^{-9}
@@ Line 309: / Line 309: @@
 {{BlaueBox|TEXT=
-$\text{Conclusions:}$&nbsp;
+$\text{Conclusion:}$&nbsp;
 *Maximum likelihood decision is only advantageous in the presence of intersymbol interference.&nbsp;

@@ Line 213: / Line 213: @@
 [[File:P ID1473 Dig T 3 8 S4 version1.png|right|frame|Trellis diagram&nbsp; (above)&nbsp; and simplified trellis diagram&nbsp; (below)|class=fit]]
-<br><br>
+<br>
 This&nbsp; '''simplified trellis diagram'''&nbsp; allows the following statements:
 *The decision &nbsp;$a_1= 1$&nbsp; can be made immediately at time &nbsp;$\nu = 1$,&nbsp; since under both hypotheses &ndash; the following symbol is &nbsp; $a_2= 0$&nbsp; or this symbol is &nbsp; $a_2= 1$&nbsp; &ndash; the same result  is obtained.<br>
@@ Line 222: / Line 222: @@
 *If one were to postpone the decision to time &nbsp;$\nu = 4$,&nbsp; one would not have more usable information about &nbsp; $a_1$,&nbsp; $a_2$&nbsp; and&nbsp; $a_3$.<br><br>
+All statements in this chapter can be verified with the&nbsp; (German languag) SWF  applet&nbsp; [[Applets:Viterbi|"Viterbi receiver for one precursor"]].
 == Extension to two precursors ==
@@ Line 250: / Line 250: @@
 #If one follows the continuous paths,&nbsp; one recognizes the indicated sequence.&nbsp; Since at time &nbsp; $\nu = 6$&nbsp; only blue branches arrive,&nbsp; the first six bits of the sequence are finally fixed here.<br>
 #However,&nbsp; partial sequences could also be output already at times &nbsp; $\nu = 1$,&nbsp; $\nu = 3$&nbsp; and&nbsp; $\nu = 4$,&nbsp; since at these times the same partial paths result for all four states.<br>
-#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either&nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
+#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either &nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
 == Bit error probability with maximum likelihood decision==
@@ Line 294: / Line 294: @@
   \hspace{0.05cm}.$$
-*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$&nbsp; leads to the much smaller error probability
+*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$ leads to the much smaller error probability
 :$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
   \approx {\rm  Q}(6) = 10^{-9}
@@ Line 309: / Line 309: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp;
+$\text{Conclusions:}$&nbsp;
 *Maximum likelihood decision is only advantageous in the presence of intersymbol interference.&nbsp;

@@ Line 213: / Line 213: @@
 [[File:P ID1473 Dig T 3 8 S4 version1.png|right|frame|Trellis diagram&nbsp; (above)&nbsp; and simplified trellis diagram&nbsp; (below)|class=fit]]
-<br>
+<br><br>
 This&nbsp; '''simplified trellis diagram'''&nbsp; allows the following statements:
 *The decision &nbsp;$a_1= 1$&nbsp; can be made immediately at time &nbsp;$\nu = 1$,&nbsp; since under both hypotheses &ndash; the following symbol is &nbsp; $a_2= 0$&nbsp; or this symbol is &nbsp; $a_2= 1$&nbsp; &ndash; the same result  is obtained.<br>
@@ Line 222: / Line 222: @@
 *If one were to postpone the decision to time &nbsp;$\nu = 4$,&nbsp; one would not have more usable information about &nbsp; $a_1$,&nbsp; $a_2$&nbsp; and&nbsp; $a_3$.<br><br>
-All statements in this chapter can be verified with the&nbsp; (German languag) SWF  applet&nbsp; [[Applets:Viterbi|"Viterbi receiver for one precursor"]].
 == Extension to two precursors ==
@@ Line 250: / Line 250: @@
 #If one follows the continuous paths,&nbsp; one recognizes the indicated sequence.&nbsp; Since at time &nbsp; $\nu = 6$&nbsp; only blue branches arrive,&nbsp; the first six bits of the sequence are finally fixed here.<br>
 #However,&nbsp; partial sequences could also be output already at times &nbsp; $\nu = 1$,&nbsp; $\nu = 3$&nbsp; and&nbsp; $\nu = 4$,&nbsp; since at these times the same partial paths result for all four states.<br>
-#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either &nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
+#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either&nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
 == Bit error probability with maximum likelihood decision==
@@ Line 294: / Line 294: @@
   \hspace{0.05cm}.$$
-*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$ leads to the much smaller error probability
+*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$&nbsp; leads to the much smaller error probability
 :$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
   \approx {\rm  Q}(6) = 10^{-9}

@@ Line 252: / Line 252: @@
 #In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either &nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
-== Error probability in maximum likelihood decision==
+== Bit error probability with maximum likelihood decision==
 <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&nbsp; $($e.g. with the correlation or the Viterbi receiver$)$&nbsp; is very complex.&nbsp; Thereby
 *determine the difference energies between all possible symbol sequences &nbsp; $Q_i$&nbsp; and&nbsp;  $Q_{j \ne i}$,&nbsp; where the error probability is essentially determined by the minimum difference energy;<br>
-*also the influences of matched filter &nbsp; $H_{\rm MF}(f)$&nbsp; and decorrelation filter &nbsp; $H_{\rm DF}(f)$&nbsp; are taken into account and additionally the rms value &nbsp; $\sigma_d$&nbsp; of the detection noise signal is determined.<br><br>
+*also the influences of matched filter &nbsp; $H_{\rm MF}(f)$&nbsp; and decorrelation filter &nbsp; $H_{\rm DF}(f)$&nbsp; have to be taken into account and additionally the rms value &nbsp; $\sigma_d$&nbsp; of the detection noise signal is determined.<br><br>
 {{BlaueBox|TEXT=
-$\text{Without proof:}$&nbsp; A simple approximation for the (average) &nbsp;'''error probability in maximum likelihood decision'''&nbsp; is:
+$\text{Without proof:}$&nbsp; A simple approximation for the&nbsp; $($average$)$ &nbsp;'''&nbsp; bit error probability in maximum likelihood decision'''&nbsp; is:
 :$$p_{\rm ML} = {\rm Q}\left(g_{\rm max}/{\sigma_d}\right)
   \hspace{0.3cm}{\rm with}\hspace{0.2cm}g_{\rm max}= {\rm
   Max}\hspace{0.15cm} \vert g_\nu \vert \hspace{0.05cm}.$$
-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.&nbsp; For unipolar signaling,&nbsp; the argument of the Q&ndash;function must be halved.}}<br>
-For the following interpretation and example, it is assumed that &nbsp; $v + 1$&nbsp; basic pulse values (including the main value) are different from zero. Then holds:
+For the following interpretation and example,&nbsp; it is assumed that&nbsp; $v + 1$&nbsp; basic pulse values&nbsp; $($including the main value$)$&nbsp; are different from zero.&nbsp; Then holds:
-*The Viterbi decision must consider all these basic pulse values. This means that a trellis diagram with &nbsp; $2^v$&nbsp; states has to be processed.<br>
+#The Viterbi decision must consider all these basic pulse values.&nbsp; This means that a trellis diagram with &nbsp; $2^v$&nbsp; states has to be processed.<br>
+#The precondition for the validity of the above equation is the uncorrelatedness of the noise at the decision,&nbsp; 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 &nbsp; $(p_{\rm ThD})$&nbsp; or decision feedback &nbsp; $(p_{\rm DFE})$,&nbsp; the noise rms value &nbsp; $\sigma_d$&nbsp; is assumed to be constant.<br>
+#Optimization of the maximum likelihood system results in very narrowband filters,&nbsp; since all pulse tails can be factored out by the ML algorithm.<br>
-*For the comparison with threshold decision &nbsp; $(p_{\rm SE})$&nbsp; or decision feedback &nbsp; $(p_{\rm DFE})$&nbsp; the noise rms value &nbsp; $\sigma_d$&nbsp; is assumed to be constant.<br>
+#Thus,&nbsp; with constant noise power density &nbsp; $N_0$&nbsp; $($at receiver input$)$,&nbsp; the noise rms value &nbsp; $\sigma_d$&nbsp; $($at decision$)$&nbsp; is for the ML system smaller than for other variants.<br>
+#This means: &nbsp; The signal-to-noise ratio gain by the ML decision may be even larger than the following example $($with &nbsp;$\sigma_d = \rm const.)$&nbsp; expresses.<br>
 {{GraueBox|TEXT=
-$\text{Example 5:}$&nbsp; We consider the basic pulse values &nbsp; $g_{0} = 0.6$&nbsp; and&nbsp; $g_{-1} = g_{+1} = 0.2 \ ( = g_1)$. In addition, we assume the constant noise rms value &nbsp; $\sigma_d = 0.1$.&nbsp; For simplicity, all quantities are normalized.<br>
+$\text{Example 5:}$&nbsp; We consider the basic pulse values &nbsp; $g_{0} = 0.6$&nbsp; and&nbsp; $g_{-1} = g_{+1} = 0.2 \ ( = g_1)$.&nbsp; In addition,&nbsp; we assume the constant noise rms value &nbsp; $\sigma_d = 0.1$.&nbsp; For simplicity,&nbsp; these quantities are normalized.<br>
-*For a binary receiver with simple threshold decision &nbsp;$\rm (SE)$&nbsp; according to the chapter &nbsp;  [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference|"Error Probability with Intersymbol Interference"]],&nbsp; the (average) error probability for bipolar signaling is:
+*For a binary receiver with simple threshold decision &nbsp;$\rm (ThD)$&nbsp; according to the chapter &nbsp;  [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference|"Error Probability with Intersymbol Interference"]],&nbsp; the&nbsp; (average)&nbsp; 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
   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)
@@ Line 291: / Line 286: @@
   \hspace{0.05cm}.$$
-*For the&nbsp; [[Digital_Signal_Transmission/Decision_Feedback|receiver with decision feedback equalization]]&nbsp; $\rm (DFE)$,&nbsp; considering &nbsp; $g_{+1} = 0$&nbsp; (full compensation of the postcursor), we obtain:
+*For the&nbsp; [[Digital_Signal_Transmission/Decision_Feedback|receiver with decision feedback equalization]]&nbsp; $\rm (DFE)$,&nbsp; considering &nbsp; $g_{+1} = 0$&nbsp; $($full compensation of the "postcursor"),&nbsp; we obtain:
 :$$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 {\rm
@@ Line 299: / Line 294: @@
   \hspace{0.05cm}.$$
-*In contrast, the application of the maximum likelihood decision (with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]])&nbsp; $\rm (ML)$ leads to the much smaller error probability
+*In contrast,&nbsp; the application of the maximum likelihood decision with&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receiver"]]&nbsp; or &nbsp; [[Digital_Signal_Transmission/Viterbi_Receiver#Considered_scenario_and_prerequisites|"Viterbi receiver"]]&nbsp; $\rm (ML)$ leads to the much smaller error probability
 :$$p_{\rm ML} = {\rm Q}\left(\frac{g_{0}}{\sigma_d}\right)
   \approx {\rm  Q}(6) = 10^{-9}
   \hspace{0.05cm}.$$
-*This corresponds to a signal-to-noise ratio gain of about &nbsp; $3 \ \rm dB$&nbsp; (versus $\rm DFE$) and &nbsp; $7.5 \ \rm dB$&nbsp; (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 &nbsp; $3 \ \rm dB$&nbsp; $($versus $\rm DFE)$&nbsp; and &nbsp; $7.5 \ \rm dB$&nbsp; $($versus $\rm ThD)$&nbsp; over the other two systems.&nbsp; The result of this simple approximation was essentially confirmed by simulations.}}<br>
-To apply the described Viterbi algorithm directly, the (normalized) basic pulse values &nbsp; $g_{0} =0.2$,&nbsp; $g_{-1} =0.6$&nbsp; and&nbsp; $g_{-2} =0.2$&nbsp; 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>&nbsp;
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; A maximum likelihood decision is only advantageous in the presence of intersymbol interference. In the case of Nyquist equalization (that is: &nbsp; only the basic pulse value &nbsp; $g_0$&nbsp; is non-zero) the maximum likelihood receiver also operates symbolically and with the same error probability &nbsp; ${\rm Q}(g_0/\sigma_d)$&nbsp; as a receiver with threshold decision.}}<br>
+$\text{Conclusion:}$&nbsp;
 == Exercises for the chapter==

@@ Line 233: / Line 233: @@
   \hspace{0.05cm}.$$
-&nbsp; [[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors|Exercise 3.12]]&nbsp; discusses in detail the calculation of error quantities (metrics) and the minimization of total error quantities (accumulated metrics).
+&nbsp; [[Aufgaben:Exercise_3.12:_Trellis_Diagram_for_Two_Precursors|Exercise 3.12]]&nbsp; discusses in detail the metrics calculation and the minimization of the accumulated metrics for the case of two precursors.
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp; We consider here an exemplary trellis diagram, which reflects the (error-free) detection of the following symbol sequence:
 :$$a_{1}=0 \hspace{0.05cm},\hspace{0.2cm}
 a_{2}=0
@@ Line 243: / Line 245: @@
 \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{...}$$
 This simplified trellis diagram allows the following statements:
-*All branches departing from &nbsp;${\it \Gamma}_{\nu}(00)$&nbsp; or &nbsp; ${\it \Gamma}_{\nu}(01)$&nbsp; are assigned to symbol &nbsp; $\rm 0$&nbsp; and drawn in blue. The red branches departing from the two upper states denote symbol &nbsp; $\rm 1$.<br>
+#All branches departing from &nbsp;${\it \Gamma}_{\nu}(00)$&nbsp; or &nbsp; ${\it \Gamma}_{\nu}(01)$&nbsp; are assigned to symbol&nbsp; "$\rm 0$"&nbsp; and drawn in blue.&nbsp; The red branches departing from the two upper states denote symbol&nbsp; "$\rm 1$".<br>
+#If one follows the continuous paths,&nbsp; one recognizes the indicated sequence.&nbsp; Since at time &nbsp; $\nu = 6$&nbsp; only blue branches arrive,&nbsp; 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 &nbsp; $\nu = 6$&nbsp; only blue branches arrive, the first six bits of the sequence are finally fixed here.<br>
+#However,&nbsp; partial sequences could also be output already at times &nbsp; $\nu = 1$,&nbsp; $\nu = 3$&nbsp; and&nbsp; $\nu = 4$,&nbsp; since at these times the same partial paths result for all four states.<br>
+#In contrast,&nbsp; at &nbsp; $\nu = 2$&nbsp; and&nbsp; $\nu = 5$,&nbsp;decisions may not be made immediately.&nbsp; For example,&nbsp; at &nbsp; $\nu = 5$&nbsp; it is only certain that either &nbsp; "$a_5 = 0$,&nbsp; $a_6 = 1$" &nbsp; will be,&nbsp; or &nbsp; "$a_5 = 1$,&nbsp; $a_6 = 0$". }}<br>
 == Error probability in maximum likelihood decision==

@@ Line 205: / Line 205: @@
 == Simplified trellis diagram ==
 <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.&nbsp; By selecting the minimum accumulated metrics,&nbsp; 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: &nbsp; The actual transmitted amplitude coefficients are &nbsp;$a_1= 1$, &nbsp;$a_2= 1$&nbsp; and&nbsp; $a_3= 0$, and the trellis diagram drawn above was calculated assuming that due to the pulse values &nbsp; $g_{-1} = 0.3$&nbsp; and&nbsp; $g_{0} = 0.7$&nbsp; as well as the AWGN noise, the input values &nbsp;$d_0= 0.2$, &nbsp;$d_1= 0.7$, &nbsp;$d_2= 0.5$&nbsp; and&nbsp; $d_3= 0$&nbsp; are applied to the ML decision.<br>
+The trellis diagram can be further simplified by drawing in only the selected branches.&nbsp; This is illustrated in the lower part of the diagram with our numerical example: &nbsp;
-[[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&nbsp; (above)&nbsp; and simplified trellis diagram&nbsp; (below)|class=fit]]
-This ''' simplified trellis diagram''' allows the following statements:
+*At time &nbsp;$\nu = 2$&nbsp; the final decision about &nbsp; $a_2$&nbsp; cannot yet be made.&nbsp; Assuming that &nbsp; $a_3= 0$,&nbsp; one would have to decide for &nbsp; $a_2= 1$,&nbsp; while &nbsp; $a_3= 1$&nbsp; would lead to the fixed choice of &nbsp; $a_2= 0$.&nbsp; <br>
-*In contrast, at the time &nbsp;$\nu = 2$&nbsp; the final decision about &nbsp; $a_2$&nbsp; cannot yet be made. Assuming that &nbsp; $a_3= 0$,&nbsp; one would have to decide for &nbsp; $a_2= 1$,&nbsp; while the hypothesis &nbsp; $a_3= 1$&nbsp; would lead to the fixed choice of &nbsp; $a_2= 0$.&nbsp; <br>
+*At&nbsp; $\nu = 3$&nbsp; the decisions for &nbsp; $a_1= 1$, &nbsp;$a_2= 1$, &nbsp;$a_3= 0$&nbsp; are final,&nbsp;  since both continuous paths&nbsp; $($blue and red$)$&nbsp; suggest this&nbsp; $($in this case correct$)$&nbsp; sequence.
-*At time&nbsp; $\nu = 3$&nbsp; the decision for &nbsp; $a_1= 1$, &nbsp;$a_2= 1$, &nbsp;$a_3= 0$&nbsp; is final, since both continuous paths suggest this (the correct in this case) sequence.
+All statements in this chapter can be verified with the&nbsp; (German languag) SWF  applet&nbsp; [[Applets:Viterbi|"Viterbi receiver for one precursor"]].
 == Extension to two precursors ==
 <br>
-If the basic pulse is described by the samples &nbsp; $g_{0} \ne 0$,&nbsp; $g_{-1} \ne 0$&nbsp; and&nbsp; $g_{-2} \ne 0$,&nbsp; then four metrics &nbsp; ${\it \Gamma}_{\nu}(00)$,&nbsp; ${\it \Gamma}_{\nu}(01)$,&nbsp; ${\it \Gamma}_{\nu}(10)$&nbsp; and&nbsp; ${\it \Gamma}_{\nu}(11)$&nbsp; must be determined in the trellis diagram at each time &nbsp;$\nu$.&nbsp; For example, &nbsp;${\it \Gamma}_{\nu}(01)$&nbsp; then describes the minimum total error quantity for the detection of the symbol &nbsp; $a_\nu$&nbsp; under the hypothesis that &nbsp; $a_{\nu-1} = 0$&nbsp; and&nbsp; $a_{\nu-2} = 1$,&nbsp; and it holds:
+If the basic pulse&nbsp; $g_d(t)$&nbsp; is described by the samples &nbsp; $g_{0} \ne 0$,&nbsp; $g_{-1} \ne 0$&nbsp; and&nbsp; $g_{-2} \ne 0$,&nbsp; then four metrics &nbsp; ${\it \Gamma}_{\nu}(00)$,&nbsp; ${\it \Gamma}_{\nu}(01)$,&nbsp; ${\it \Gamma}_{\nu}(10)$&nbsp; and&nbsp; ${\it \Gamma}_{\nu}(11)$&nbsp; must be determined in the trellis diagram at each time &nbsp;$\nu$.&nbsp;
 :$${\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]

@@ Line 137: / Line 137: @@
 [[File:P ID1470 Dig T 3 8 S2c version1.png|frame|right|Iterative calculation of the accumulated metrics (example)|class=fit]]
+<BR>
 *The minimum accumulated metric at time &nbsp;$\nu = 3$&nbsp; is &nbsp;$\gamma_{3}(\rm 1100) = 0.14$.&nbsp; From this,&nbsp; the coefficients of the most likely transmitted&nbsp; (unipolar)&nbsp; sequence according to the present signal values &nbsp;$d_0 = 0.2$, &nbsp;$d_1 = 0.7$, &nbsp;$d_2 = 0.5$&nbsp; and &nbsp;$d_3 = 0$&nbsp; are &nbsp;$a_1 = 1$, &nbsp;$a_2 = 1$&nbsp; and &nbsp;$a_3 = 0$&nbsp; (green path).
@@ Line 147: / Line 147: @@
-== Minimum total error quantities==
+== Minimum accumulated metric==
 <br>
 We continue from the numerical values of the last examples:
-:$$d_{0}=0.2 ,\hspace{0.1cm}
+:$$d_{0}=0.2 ,\hspace{0.3cm}
-  d_{1}=0.7 ,\hspace{0.1cm}
+  d_{1}=0.7 ,\hspace{0.3cm}
-  d_{2}=0.5 ,\hspace{0.1cm}
+  d_{2}=0.5 ,\hspace{0.3cm}
-  d_{3}=0 ,$$
+  d_{3}=0 ,\hspace{0.3cm}
-:$$g_{0}=0.7 ,\hspace{0.1cm}
+g_{0}=0.7 ,\hspace{0.3cm}
   g_{-1}=0.3 .$$
-Thus, nothing changes in the tree diagram compared to &nbsp;$\text{Example 3}$.&nbsp; Some colored markings in this diagram only indicate in which way the tree diagram can be simplified according to Viterbi's suggestions.
+Thus,&nbsp; nothing changes in the tree diagram compared to &nbsp;$\text{Example 3}$.&nbsp; Some colored markings in this diagram only indicate in which way the tree diagram can be simplified according to Viterbi's suggestions.&nbsp; Important properties of the Viterbi decision can be seen in the graph,&nbsp; e.g. at time &nbsp;$\nu = 2$:&nbsp;
-Important properties of the Viterbi decision can be seen in the above graph, for example, at time &nbsp;$\nu = 2$:&nbsp;
+*This means: &nbsp; A decision at &nbsp;$\nu = 2$&nbsp; &ndash;&nbsp; based on &nbsp;$d_0$, &nbsp;$d_1$&nbsp; and&nbsp; $d_2$&nbsp; &ndash;&nbsp; would have been in favor of sequence &nbsp;"$\rm 101$"&nbsp; instead of the transmitted sequence &nbsp;"$\rm 110$".&nbsp; <br>
-*It follows: &nbsp; A too early final determination should be avoided at all costs. However, at any time &nbsp;$\nu$&nbsp; one can already exclude several partial symbol sequences which need not be considered at later times.<br>
+*It follows: &nbsp; A too early final determination should be avoided at all costs.&nbsp; However,&nbsp; at any time &nbsp;$\nu$&nbsp; one can already exclude several partial symbol sequences which need not be considered at later times.<br>
-*To &nbsp;$\gamma_{2}(\rm 111) = 0.35$&nbsp; the same error quantities &nbsp;$\varepsilon_{3}(\rm 11)$&nbsp; and &nbsp;$\varepsilon_{3}(\rm 10)$&nbsp; are added as to &nbsp;$\gamma_{2}(\rm 101) = 0.05$. Because of&nbsp;  $\gamma_{2}(\rm 111) > \gamma_{2}(\rm 101)$&nbsp; it is therefore already certain at &nbsp;$\nu = 2$&nbsp; that &nbsp;$\rm 111$&nbsp; cannot be part of the most probable sequence. The same is true for &nbsp;$\rm 001$&nbsp; and &nbsp;$\rm 011$. Therefore, these nodes and all their successors need not be considered further (brown overlaps).<br>
+*To &nbsp;$\gamma_{2}(\rm 111) = 0.35$&nbsp; the same metrics &nbsp;$\varepsilon_{3}(\rm 11)$&nbsp; and &nbsp;$\varepsilon_{3}(\rm 10)$&nbsp; are added as to &nbsp;$\gamma_{2}(\rm 101) = 0.05$.&nbsp; Because of&nbsp;  $\gamma_{2}(\rm 111) > \gamma_{2}(\rm 101)$&nbsp; it is therefore already certain at &nbsp;$\nu = 2$&nbsp; that &nbsp;"$\rm 111$"&nbsp; cannot be part of the most probable sequence.
-*Also, the rounded nodes &nbsp;$\gamma_{2}(\rm 000) = 0.78$, &nbsp;$\gamma_{2}(\rm 010) = 0.24$&nbsp; and&nbsp; $\gamma_{2}(\rm 100) = 0.26$&nbsp; are not part of the most probable sequence, since they are larger than the node &nbsp;$\gamma_{2}(\rm 110) = 0.14$, which is marked in green. Also, these and their successors need not be considered from time &nbsp;$\nu = 3$&nbsp; (green covers).<br>
+*The same is true for &nbsp;"$\rm 001$"&nbsp; and &nbsp;"$\rm 011$".&nbsp; Therefore,&nbsp; these nodes and all their successors need not be considered further&nbsp; $($brown coverages$)$.<br>
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;
-*Of the eight nodes at &nbsp;$\nu = 2$,&nbsp; only two need to be followed up, namely the rectangle &nbsp;$\gamma_{2}(\rm 101) = 0.05$&nbsp; and the rounded rectangle &nbsp;$\gamma_{2}(\rm 110) = 0.14$.
+#Of the eight nodes at &nbsp;$\nu = 2$,&nbsp; only two need to be followed up,&nbsp; namely the rectangle &nbsp;$\gamma_{2}(\rm 101) = 0.05$&nbsp; and the rounded rectangle &nbsp;$\gamma_{2}(\rm 110) = 0.14$.
-*These describe the minimum total error quantities assuming that &nbsp;$a_3 = 0$&nbsp; and &nbsp;$a_3 = 1$,&nbsp; respectively.}}
+#These describe the minimum accumulated metrics assuming that &nbsp;$a_3 = 0$&nbsp; and &nbsp;$a_3 = 1$,&nbsp; respectively.}}
-== Representation of the minimum total error quantities &ndash; Trellis diagram ==
+== Representation with the trellis diagram ==
 <br>
-We now generalize the result of this example. Under the still valid assumption that the basic pulse has only one precursor &nbsp;$(g_{-1})$&nbsp; besides the main value &nbsp;$(g_0)$,&nbsp; the two &nbsp;'''minimum total error quantities'''&nbsp; at time &nbsp;$\nu$&nbsp; are formally given by
+We now generalize the result of this example.&nbsp; Under the still valid assumption that the basic pulse&nbsp; $g_d(t)$&nbsp; has only one precursor &nbsp;$(g_{-1})$&nbsp; besides the main value &nbsp;$(g_0)$,&nbsp; the two &nbsp;'''minimum accumulated metrics'''&nbsp; at time &nbsp;$\nu$&nbsp; are formally given by
 :$${\it \Gamma}_{\nu}(0)  =  {\rm Min}\left[{\it \Gamma}_{\nu-1}(0) + \varepsilon_{\nu}(00),
   \hspace{0.2cm}{\it \Gamma}_{\nu-1}(1) + \varepsilon_{\nu}(10)\right] \hspace{0.05cm},$$
@@ Line 187: / Line 190: @@
   \hspace{0.05cm}.$$
-The <i>total error minimization</i>&nbsp; procedure can be clearly illustrated in the &nbsp;'''trellis diagram'''.&nbsp; <br>
+The procedure&nbsp; "minimization of the accumulated metrics"&nbsp;  can be clearly illustrated in the &nbsp;'''trellis diagram'''.&nbsp; <br>
+[[File:P ID1472 Dig T 3 8 S3b version1.png|right|frame|Example trellis diagram&nbsp; $($the numerical values are adapted to the previous examples$)$|class=fit]]
 A comparison with the figures in the last sections shows:
-*The lower branch represents the minimum total error quantity &nbsp;${\it \Gamma}_{\nu}(0)$&nbsp; calculated at each time &nbsp; $\nu$&nbsp; under the hypothesis that &nbsp;$a_{\nu + 1} = 0$&nbsp; will hold (blue rounded squares).<br>
+*The lower branch represents the minimum accumulated metric &nbsp;${\it \Gamma}_{\nu}(0)$&nbsp; calculated at each time &nbsp; $\nu$&nbsp; under the hypothesis &nbsp;$a_{\nu + 1} = 0$&nbsp;  $($blue rounded squares$)$.<br>
-*In contrast, the upper branch describes the minimum total error quantities &nbsp;${\it \Gamma}_{\nu}(1)$&nbsp; under the assumption&nbsp; $a_{\nu + 1} = 1$&nbsp; (red squares). Again, the numerical values are adapted to the previous example.<br>
+*The upper branch describes the minimum accumulated metric &nbsp;${\it \Gamma}_{\nu}(1)$&nbsp; under the assumption&nbsp; $a_{\nu + 1} = 1$&nbsp; $($red squares$)$.&nbsp; <br>
-*Besides &nbsp;${\it \Gamma}_{\nu}(0)$&nbsp; and &nbsp;${\it \Gamma}_{\nu}(1)$,&nbsp; the maximum likelihood decision must also store the associated symbol sequences (paths). These branches are highlighted in red and blue, respectively, in the graph.<br>
+*Besides &nbsp;${\it \Gamma}_{\nu}(0)$&nbsp; and &nbsp;${\it \Gamma}_{\nu}(1)$,&nbsp; the maximum likelihood decision must also store the associated symbol sequences&nbsp; ("paths").&nbsp; These branches are highlighted in red and blue, resp.<br>
-*If &nbsp;${\it \Gamma}_{\nu}$&nbsp; arises from the node &nbsp;${\it \Gamma}_{\nu-1}(0)$&nbsp; &ndash; that is, if the lower of the two incoming branches is highlighted &ndash; the associated symbol is &nbsp;$\rm 0$, otherwise the symbol is &nbsp;$\rm 1$.<br>
+*If &nbsp;${\it \Gamma}_{\nu}$&nbsp; arises from the node &nbsp;${\it \Gamma}_{\nu-1}(0)$&nbsp; &ndash;&nbsp; that is,&nbsp; if the lower of the two incoming branches is highlighted &ndash;&nbsp; the associated symbol is &nbsp;"$\rm 0$",&nbsp; otherwise the symbol is &nbsp;"$\rm 1$".<br>
-*At time &nbsp;$\nu = 3$,&nbsp; for example, both &nbsp;${\it \Gamma}_{3}(0) = 0.14$&nbsp; and &nbsp;${\it \Gamma}_{3}(1) = 0.23$&nbsp; result from the precursor &nbsp;${\it \Gamma}_{2}(0)$, so that both selected paths point to the symbol &nbsp;$\rm 0$,&nbsp; respectively (blue branches).<br>
+*For &nbsp; $\nu = 3$:&nbsp; Both, &nbsp;${\it \Gamma}_{3}(0) = 0.14$&nbsp; and &nbsp;${\it \Gamma}_{3}(1) = 0.23$,&nbsp; result from the precursor &nbsp;${\it \Gamma}_{2}(0)$,&nbsp; so that both selected paths point to the symbol &nbsp;"$\rm 0$"&nbsp;  (blue branches).<br>
 == Simplified trellis diagram ==

@@ Line 44: / Line 44: @@
 *In contrast, the Viterbi receiver will correctly decide this sequence of length &nbsp;$10$,&nbsp; as shown in the next sections.}}<br>
-== Error quantities and total error quantities==
+== Metric and accumulated metric==
 <br>
 As in the chapter&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies|"Optimal Receiver Strategies"]],&nbsp; &nbsp;$Q \in \{Q_i\}$&nbsp; specifies the time-limited source symbol sequence consisting of &nbsp;$N$&nbsp; binary symbols.
@@ Line 54: / Line 54: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-*The &nbsp;'''error quantity'''&nbsp; $\varepsilon_{\nu}(i)$&nbsp; denotes the squared deviation between the actual noisy sample &nbsp;$d_\nu$&nbsp; and the noiseless sample &nbsp;$d_{ {\rm S}\nu}$ belonging to the sequence &nbsp;$Q_i$:&nbsp;
+*The &nbsp;'''metric'''&nbsp; $\varepsilon_{\nu}(i)$&nbsp; denotes the squared deviation between the actual noisy sample &nbsp;$d_\nu$&nbsp; and the noiseless sample &nbsp;$d_{ {\rm S}\nu}$ belonging to the sequence &nbsp;$Q_i$:&nbsp;
 :$$\varepsilon_{\nu}(i) = \vert d_{\nu} - d_{ {\rm S}\nu}(i) \vert^2
   \hspace{0.05cm}.$$
-:In the literature, this is also referred to as "metric".
+:In the literature, this is sometimes referred to as&nbsp; '''error quantity'''.
-*The &nbsp;'''total error quantity'''&nbsp; $\gamma_{\nu}(i)$ characterizes the sum of all error quantities up to time &nbsp;$\nu$:<br>
+*The &nbsp;'''accumulated metric'''&nbsp; $\gamma_{\nu}(i)$ characterizes the sum of all metrics up to time &nbsp;$\nu$:<br>
 :$$\gamma_{\nu}(i)= \sum_{k=0}^{\nu}\varepsilon_{k}(i)
   \hspace{0.05cm}.$$
-:In this context,&nbsp; one also speaks of the&nbsp; "accumulated metric".}}
+:In this context,&nbsp; one also speaks of the&nbsp; '''total error quantity'''.}}
-The graph illustrates the quantities defined above in a tree structure,&nbsp; from which it can be seen that the total error quantities can be calculated iteratively:
+The graph illustrates the quantities defined above in a tree structure,&nbsp; from which it can be seen that the accumulated metric can be calculated iteratively:
-[[File:P ID1469 Dig T 3 8 S2b version1.png|right|frame|Representation of the error quantities&nbsp; $\varepsilon_{\nu}(i)$&nbsp; and accumulated error quantities&nbsp; $\gamma_{\nu}(i)= \gamma_{\nu-1}(i\hspace{0.05cm}')+\varepsilon_{\nu}(i\hspace{0.05cm}'')
+[[File:P ID1469 Dig T 3 8 S2b version1.png|right|frame|Representation of the metrics&nbsp; $\varepsilon_{\nu}(i)$&nbsp; and accumulated metrics&nbsp; $\gamma_{\nu}(i)= \gamma_{\nu-1}(i\hspace{0.05cm}')+\varepsilon_{\nu}(i\hspace{0.05cm}'')
   \hspace{0.05cm}$&nbsp; in the tree diagram. &nbsp; <u>Notes:</u><br>&nbsp; $\bullet$ &nbsp; $i$, &nbsp;$i\hspace{0.05cm}'$, &nbsp;$i\hspace{0.05cm}''$ are different index variables.<br>&nbsp; $\bullet$ &nbsp; Definition is valid for a basic pulse with main value &nbsp;$g_{0}$&nbsp; and precursor &nbsp;$g_{-1}$.<br>&nbsp; $\bullet$ &nbsp;  With &nbsp;$v$&nbsp; precursors the above sum would have to start at &nbsp;$k = 1 -v$.<br>&nbsp; $\bullet$ &nbsp; The parameter &nbsp;$i \in  \{0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, 2^{N+1}-1\}$&nbsp;  is usually represented in binary.<br>&nbsp; $\bullet$ &nbsp; It describes the amplitude coefficients &nbsp;$a_1$, ... , &nbsp;$a_{\nu +1}$ $($each &nbsp;$0$&nbsp; or &nbsp;$1)$.|class=fit]]
 :$$\gamma_{\nu}(i)= \gamma_{\nu-1}(i\hspace{0.05cm}')+\varepsilon_{\nu}(i\hspace{0.05cm}'')
@@ Line 71: / Line 71: @@
 It should be noted about this graph:
-*The nodes of the tree diagram represent the&nbsp; "total error quantities"&nbsp; $\gamma_{\nu}(i)$.&nbsp; Their number is doubled with each iteration step.&nbsp; At time &nbsp;$\nu$,&nbsp; there are &nbsp;$2^{\nu+1}$&nbsp; such nodes.&nbsp; For example,&nbsp; for &nbsp;$\nu = 3$,&nbsp; exactly &nbsp;$2^4 = 16$&nbsp; nodes can be seen.<br>
+*The nodes of the tree diagram represent the&nbsp; "accumulated metric"&nbsp; $\gamma_{\nu}(i)$.&nbsp; Their number is doubled with each iteration step.&nbsp; At time &nbsp;$\nu$,&nbsp; there are &nbsp;$2^{\nu+1}$&nbsp; such nodes.&nbsp; For example,&nbsp; for &nbsp;$\nu = 3$,&nbsp; exactly &nbsp;$2^4 = 16$&nbsp; nodes can be seen.<br>
-*The&nbsp; "amplitude coefficients"&nbsp; associated with the total error quantities &nbsp;$\gamma_{\nu}(i)$&nbsp; are obtained by following the path from the initial node to the node under consideration.&nbsp; It is agreed that an upward branch is assigned the coefficient &nbsp;$a_\nu=1$&nbsp; and a downward branch is assigned the coefficient &nbsp;$a_\nu=0$. <br>
+*The&nbsp; "amplitude coefficients"&nbsp; associated with the accumulated metric &nbsp;$\gamma_{\nu}(i)$&nbsp; are obtained by following the path from the initial node to the node under consideration.&nbsp; It is agreed that an upward branch is assigned the coefficient &nbsp;$a_\nu=1$&nbsp; and a downward branch is assigned the coefficient &nbsp;$a_\nu=0$. <br>
-*The green highlighted node &nbsp;$\gamma_{3}(\rm 1100)$&nbsp; denotes the total error quantity under the hypothetical assumption that the symbols &nbsp;$a_1=1$, &nbsp;$a_2=1$, &nbsp;$a_3=0$, &nbsp;$a_4=0$ were sent.&nbsp; This assignment can also be read from the directions of the arrows in the tree diagram: &nbsp; First twice upwards,&nbsp; then twice downwards.<br>
+*The green highlighted node &nbsp;$\gamma_{3}(\rm 1100)$&nbsp; denotes the accumulated metric under the hypothetical assumption that the symbols &nbsp;$a_1=1$, &nbsp;$a_2=1$, &nbsp;$a_3=0$, &nbsp;$a_4=0$ were sent.&nbsp; This assignment can also be read from the directions of the arrows in the tree diagram: &nbsp; First twice upwards,&nbsp; then twice downwards.<br>
 *Due to the precursor,&nbsp; the coefficient &nbsp;$a_4$&nbsp; must already be taken into account at time &nbsp;$\nu = 3$.&nbsp; All nodes &nbsp;$\gamma_{\nu}(i)$ computed under the assumption &nbsp;$a_{\nu +1}=1$&nbsp; are represented by rectangles,&nbsp; while the hypothesis &nbsp;$a_{\nu +1}=0$&nbsp; is symbolized by a rounded rectangle in each case,&nbsp; for example &nbsp;$\gamma_{2}(\rm 110)$&nbsp; or &nbsp;$\gamma_{3}(\rm 1100)$.<br>
-*The branches in the tree diagram are assigned to&nbsp; "error quantities"&nbsp; $\varepsilon_{\nu}(i)$.&nbsp; With the assumed basic pulse&nbsp; $($only &nbsp;$g_{0}$&nbsp; and &nbsp;$g_{-1})$,&nbsp; there are exactly four different quantities at each time point except for the initial state &nbsp;$(\nu = 0)$:&nbsp;
+*The branches in the tree diagram are assigned to the&nbsp; "metrics"&nbsp; $\varepsilon_{\nu}(i)$.&nbsp; With the assumed basic pulse&nbsp; $($only &nbsp;$g_{0}$&nbsp; and &nbsp;$g_{-1})$,&nbsp; there are exactly four different metrics at each time point except for the initial state &nbsp;$(\nu = 0)$:&nbsp;
 :$$\varepsilon_{\nu}(00) = |d_{\nu}|^2\hspace{0.05cm},\hspace{0.5cm}\varepsilon_{\nu}(01) = |d_{\nu}-g_{-1}|^2\hspace{0.05cm},$$
 :$$
 \varepsilon_{\nu}(10) = |d_{\nu}-g_{0}|^2\hspace{0.05cm},\hspace{0.2cm}\varepsilon_{\nu}(11) = |d_{\nu}-g_{0}-g_{-1}|^2\hspace{0.05cm}.$$
-*The total error quantity &nbsp;$\gamma_{\nu}(i)$&nbsp; is the sum of the preceding node &nbsp;$\gamma_{\nu-1}(i\hspace{0.05cm}')$&nbsp; and the intervening branch &nbsp;$\varepsilon_{\nu}(i\hspace{0.05cm}'')$.&nbsp; For example,&nbsp; for the highlighted nodes:
+*The accumulated metric &nbsp;$\gamma_{\nu}(i)$&nbsp; is the sum of the preceding node &nbsp;$\gamma_{\nu-1}(i\hspace{0.05cm}')$&nbsp; and the intervening branch &nbsp;$\varepsilon_{\nu}(i\hspace{0.05cm}'')$.&nbsp; For example,&nbsp; for the highlighted nodes:
 :$$\gamma_{1}(11)=\gamma_{0}(1)+\varepsilon_{1}(11) ,$$
 :$$ \gamma_{2}(110)=\gamma_{1}(11)+\varepsilon_{2}(10) ,$$
@@ Line 106: / Line 106: @@
   \hspace{0.05cm}.$$
-Then, for the error quantities &nbsp;$\varepsilon_{\nu}(i)$&nbsp; at time points &nbsp;$\nu = 0$&nbsp; to &nbsp;$\nu = 3$:
+Then,&nbsp; for the metrics &nbsp;$\varepsilon_{\nu}(i)$&nbsp; at time points &nbsp;$\nu = 0$&nbsp; to &nbsp;$\nu = 3$:
 :$$\nu = 0\text{:} \hspace{0.2cm}\varepsilon_{0}(00)  =  \big [0.2- (0 \cdot 0.7 + 0 \cdot
 .3) \big ]^2=0.04 \hspace{0.05cm},\hspace{0.4cm} \varepsilon_{0}(01)  =  \big [0.2- (0 \cdot 0.7 + 1 \cdot
@@ Line 134: / Line 134: @@
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; With the error quantities &nbsp;$\varepsilon_{\nu}(i)$&nbsp; determined in &nbsp;$\text{Example 2}$,&nbsp; the total error quantities &nbsp;$\gamma_{\nu}(i)$&nbsp; can now also be calculated. Below is the tree diagram with the total error quantities &nbsp;$\gamma_{\nu}(i)$&nbsp; as nodes for the time points &nbsp;$\nu = 0$&nbsp; to &nbsp;$\nu = 3$.&nbsp;
+$\text{Example 3:}$&nbsp; With the metrics &nbsp;$\varepsilon_{\nu}(i)$&nbsp; determined in &nbsp;$\text{Example 2}$,&nbsp; the accumulated metrics &nbsp;$\gamma_{\nu}(i)$&nbsp; can now also be calculated. Below is the tree diagram with the accumulated metrics &nbsp;$\gamma_{\nu}(i)$&nbsp; as nodes for the time points &nbsp;$\nu = 0$&nbsp; to &nbsp;$\nu = 3$.&nbsp;
-[[File:P ID1470 Dig T 3 8 S2c version1.png|center|frame|Iterative calculation of the total error quantities (example)|class=fit]]
+[[File:P ID1470 Dig T 3 8 S2c version1.png|frame|right|Iterative calculation of the accumulated metrics (example)|class=fit]]
-The minimum total error quantity at time &nbsp;$\nu = 3$&nbsp; is &nbsp;$\gamma_{3}(\rm 1100) = 0.14$. From this, the coefficients of the most likely transmitted (unipolar) sequence according to the present signal values &nbsp;$d_0 = 0.2$, &nbsp;$d_1 = 0.7$, &nbsp;$d_2 = 0.5$&nbsp; and &nbsp;$d_3 = 0$&nbsp; are &nbsp;$a_1 = 1$, &nbsp;$a_2 = 1$&nbsp; and &nbsp;$a_3 = 0$ (green path).
+*The minimum accumulated metric at time &nbsp;$\nu = 3$&nbsp; is &nbsp;$\gamma_{3}(\rm 1100) = 0.14$.&nbsp; From this,&nbsp; the coefficients of the most likely transmitted&nbsp; (unipolar)&nbsp; sequence according to the present signal values &nbsp;$d_0 = 0.2$, &nbsp;$d_1 = 0.7$, &nbsp;$d_2 = 0.5$&nbsp; and &nbsp;$d_3 = 0$&nbsp; are &nbsp;$a_1 = 1$, &nbsp;$a_2 = 1$&nbsp; and &nbsp;$a_3 = 0$&nbsp; (green path).
-Further, it should be noted:
+*If the sequence length is &nbsp;$N = 3$&nbsp; $($that is: &nbsp; only three symbols are jointly decided by the Viterbi receiver$)$,&nbsp; the decision &nbsp;$a_4 = 0$&nbsp; is also certainly the correct one,&nbsp; since all coefficients &nbsp;$a_{\nu>3}$&nbsp; were assumed to be zero.<br>
-*However, for longer sequences &nbsp;$(N > 3)$,&nbsp; it cannot necessarily be concluded from the minimum value &nbsp;$\gamma_{3}(\rm 1100)$&nbsp; that &nbsp;$a_1 = 1$, &nbsp;$a_2 = 1$,&nbsp; $a_3 = 0$&nbsp; is actually part of the most probable sequence. Considering further sample values &nbsp;$(d_4$, &nbsp;$d_5$, ...$)$ this preliminary result could well change.}}<br><br>
+*However,&nbsp; for longer sequences &nbsp;$(N > 3)$,&nbsp; it cannot necessarily be concluded from the minimum value &nbsp;$\gamma_{3}(\rm 1100)$&nbsp; that &nbsp;$a_1 = 1$, &nbsp;$a_2 = 1$,&nbsp; $a_3 = 0$&nbsp; is actually part of the most probable sequence.

@@ Line 33: / Line 33: @@
 <u>Note:</u> &nbsp; The algorithm is not more complex with bipolar signaling. &nbsp; $\bullet$ &nbsp; In contrast, the computational cost increases when the basic detection pulse becomes broader and has more than one precursor &nbsp;$g_{-1}$. &nbsp; $\bullet$ &nbsp; The neglect of precursors in the description is not a fundamental limitation because any pulse can satisfy this condition by appropriate choice of the detection time &nbsp;$T_{\rm D}$. &nbsp; $\bullet$ &nbsp; In the following all signal values are normalized to &nbsp;$1$.&nbsp; <br>
 [[File:P ID1468 Dig T 3 8 S1b version1.png|right|frame|Sampling values to illustrate the Viterbi algorithm|class=fit]]
-The basic pulse values in this example are assumed to be &nbsp;$g_0 = 0.7$&nbsp; and &nbsp;$g_{-1} = 0.3$.&nbsp; It can be further seen that &nbsp;$d_{ {\rm S}\nu}$&nbsp; can take only four different values, namely &nbsp;$0$, &nbsp;$g_{-1}=0.3$, &nbsp;$g_0= 0.7$&nbsp; and &nbsp;$g_0 +g_{-1}= 1$.
+* The basic pulse values are assumed to be &nbsp;$g_0 = 0.7$&nbsp; and &nbsp;$g_{-1} = 0.3$.&nbsp; It can be further seen that &nbsp;$d_{ {\rm S}\nu}$&nbsp; can take only four different values, namely &nbsp;$0$, &nbsp;$g_{-1}=0.3$, &nbsp;$g_0= 0.7$,&nbsp; $g_0 +g_{-1}= 1$.
-The samples present at the Viterbi decision (red dots) are &nbsp;$d_0 = 0.2$, &nbsp;$d_1 = 0.7$, &nbsp;$d_2 = 0.5$, &nbsp;$d_3 = 0$, ... , where the differences &nbsp;$d_{ {\rm N}\nu} = d_\nu - d_{ {\rm S}\nu}$&nbsp; originate from an AWGN noise source.<br>
+* The noisy samples present at the Viterbi decision&nbsp; (red dots)&nbsp; are &nbsp;$d_0 = 0.2$, &nbsp;$d_1 = 0.7$, &nbsp;$d_2 = 0.5$, &nbsp;$d_3 = 0$, ... ,&nbsp; where the differences &nbsp;$d_{ {\rm N}\nu} = d_\nu - d_{ {\rm S}\nu}$&nbsp; originate from an AWGN noise source.<br>
-*A threshold decision $($with the threshold at &nbsp;$E = 0.5)$&nbsp; would make at least one wrong decision at these ten bits shown $($at time &nbsp;$t  = 4T)$, and possibly another at &nbsp;$t  = 2T$ if the sample &nbsp;$d_2 = 0.5$&nbsp; is nevertheless slightly smaller than the threshold &nbsp;$E = 0.5$.
+*A threshold decision&nbsp; $($with the threshold at &nbsp;$E = 0.5)$&nbsp; would make at least one wrong decision at these ten bits shown&nbsp; $($at time &nbsp;$t  = 4T)$,&nbsp; and possibly another at &nbsp;$t  = 2T$&nbsp; if the sample &nbsp;$d_2 = 0.5$&nbsp; is nevertheless slightly smaller than the threshold &nbsp;$E = 0.5$.
 *In contrast, the Viterbi receiver will correctly decide this sequence of length &nbsp;$10$,&nbsp; as shown in the next sections.}}<br>
 == Error quantities and total error quantities==
 <br>
-As in the chapter&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies|"Optimal Receiver Strategies"]],&nbsp; &nbsp;$Q \in \{Q_i\}$&nbsp; specifies the bounded source symbol sequence consisting of &nbsp;$N$&nbsp; binary symbols.
+As in the chapter&nbsp; [[Digital_Signal_Transmission/Optimal_Receiver_Strategies|"Optimal Receiver Strategies"]],&nbsp; &nbsp;$Q \in \{Q_i\}$&nbsp; specifies the time-limited source symbol sequence consisting of &nbsp;$N$&nbsp; binary symbols.
-*Thus, the number of possible symbol sequences &nbsp;$Q_i$&nbsp; is &nbsp;$2^N$.
+*Thus,&nbsp; the number of possible symbol sequences &nbsp;$Q_i$&nbsp; is &nbsp;$2^N$.
 *$V$&nbsp; denotes the sink symbol sequence of length $N$ set equal to the most probable sequence &nbsp;$Q_j$&nbsp; by the Viterbi decision.<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The &nbsp;'''error quantity'''&nbsp; $\varepsilon_{\nu}(i)$&nbsp; denotes the squared deviation between the actual, noisy sample &nbsp;$d_\nu$&nbsp; and the useful sample &nbsp;$d_{ {\rm S}\nu}$ belonging to the sequence &nbsp;$Q_i$:&nbsp;
+$\text{Definitions:}$&nbsp;
 :$$\varepsilon_{\nu}(i) = \vert d_{\nu} - d_{ {\rm S}\nu}(i) \vert^2
   \hspace{0.05cm}.$$
-In the literature, this is also referred to as "metric".}}
+:In the literature, this is also referred to as "metric".
+*The &nbsp;'''total error quantity'''&nbsp; $\gamma_{\nu}(i)$ characterizes the sum of all error quantities up to time &nbsp;$\nu$:<br>
 :$$\gamma_{\nu}(i)= \sum_{k=0}^{\nu}\varepsilon_{k}(i)
   \hspace{0.05cm}.$$
-In this context, one also speaks of the "accumulated metric".}}
+:In this context,&nbsp; one also speaks of the&nbsp; "accumulated metric".}}
-[[File:P ID1469 Dig T 3 8 S2b version1.png|right|frame|Representation of the error quantities and accumulated error quantities in the tree diagram|class=fit]]
+The graph illustrates the quantities defined above in a tree structure,&nbsp; from which it can be seen that the total error quantities can be calculated iteratively:
-The graph illustrates the quantities defined above in a tree structure, from which it can be seen that the total error quantities can be calculated iteratively:
+[[File:P ID1469 Dig T 3 8 S2b version1.png|right|frame|Representation of the error quantities&nbsp; $\varepsilon_{\nu}(i)$&nbsp; and accumulated error quantities&nbsp; $\gamma_{\nu}(i)= \gamma_{\nu-1}(i\hspace{0.05cm}')+\varepsilon_{\nu}(i\hspace{0.05cm}'')
 :$$\gamma_{\nu}(i)= \gamma_{\nu-1}(i\hspace{0.05cm}')+\varepsilon_{\nu}(i\hspace{0.05cm}'')
   \hspace{0.05cm}.$$
-''Notes:''
+It should be noted about this graph:
-*$i$, &nbsp;$i\hspace{0.05cm}'$, &nbsp;$i\hspace{0.05cm}''$ are different index variables.
+*The nodes of the tree diagram represent the&nbsp; "total error quantities"&nbsp; $\gamma_{\nu}(i)$.&nbsp; Their number is doubled with each iteration step.&nbsp; At time &nbsp;$\nu$,&nbsp; there are &nbsp;$2^{\nu+1}$&nbsp; such nodes.&nbsp; For example,&nbsp; for &nbsp;$\nu = 3$,&nbsp; exactly &nbsp;$2^4 = 16$&nbsp; nodes can be seen.<br>
-*The <i>amplitude coefficients</i>&nbsp; associated with the total error quantities &nbsp;$\gamma_{\nu}(i)$&nbsp; are obtained by following the path from the initial node to the node under consideration. It is agreed that an upward branch is assigned the coefficient &nbsp;$a_\nu=1$&nbsp; and a downward branch is assigned the coefficient &nbsp;$a_\nu=0$.&nbsp; <br>
+*The&nbsp; "amplitude coefficients"&nbsp; associated with the total error quantities &nbsp;$\gamma_{\nu}(i)$&nbsp; are obtained by following the path from the initial node to the node under consideration.&nbsp; It is agreed that an upward branch is assigned the coefficient &nbsp;$a_\nu=1$&nbsp; and a downward branch is assigned the coefficient &nbsp;$a_\nu=0$. <br>
-*For example, the green highlighted node &nbsp;$\gamma_{3}(\rm 1100)$&nbsp; denotes the total error quantity under the hypothetical assumption that the symbols &nbsp;$a_1=1$, &nbsp;$a_2=1$, &nbsp;$a_3=0$, &nbsp;$a_4=0$ were sent. This assignment can also be read from the directions of the arrows in the tree diagram: &nbsp; First twice upwards, then twice downwards.<br>
+*The green highlighted node &nbsp;$\gamma_{3}(\rm 1100)$&nbsp; denotes the total error quantity under the hypothetical assumption that the symbols &nbsp;$a_1=1$, &nbsp;$a_2=1$, &nbsp;$a_3=0$, &nbsp;$a_4=0$ were sent.&nbsp; This assignment can also be read from the directions of the arrows in the tree diagram: &nbsp; First twice upwards,&nbsp; then twice downwards.<br>
-*Due to the precursor, the coefficient &nbsp;$a_4$&nbsp; must already be taken into account at time &nbsp;$\nu = 3$.&nbsp; All nodes &nbsp;$\gamma_{\nu}(i)$ computed under the assumption &nbsp;$a_{\nu +1}=1$&nbsp; are represented by rectangles in the tree diagram, while the hypothesis &nbsp;$a_{\nu +1}=0$&nbsp; is symbolized by a rounded rectangle in each case, for example &nbsp;$\gamma_{2}(\rm 110)$&nbsp; or &nbsp;$\gamma_{3}(\rm 1100)$.<br>
+*Due to the precursor,&nbsp; the coefficient &nbsp;$a_4$&nbsp; must already be taken into account at time &nbsp;$\nu = 3$.&nbsp; All nodes &nbsp;$\gamma_{\nu}(i)$ computed under the assumption &nbsp;$a_{\nu +1}=1$&nbsp; are represented by rectangles,&nbsp; while the hypothesis &nbsp;$a_{\nu +1}=0$&nbsp; is symbolized by a rounded rectangle in each case,&nbsp; for example &nbsp;$\gamma_{2}(\rm 110)$&nbsp; or &nbsp;$\gamma_{3}(\rm 1100)$.<br>
-*The branches in the tree diagram are assigned to the <i>error quantities</i>&nbsp; $\varepsilon_{\nu}(i)$.&nbsp; With the assumed basic pulse (only &nbsp;$g_{0}$&nbsp; and &nbsp;$g_{-1}$&nbsp; are nonzero), there are exactly four different quantities at each time point except for the initial state &nbsp;$(\nu = 0)$:&nbsp;
+*The branches in the tree diagram are assigned to&nbsp; "error quantities"&nbsp; $\varepsilon_{\nu}(i)$.&nbsp; With the assumed basic pulse&nbsp; $($only &nbsp;$g_{0}$&nbsp; and &nbsp;$g_{-1})$,&nbsp; there are exactly four different quantities at each time point except for the initial state &nbsp;$(\nu = 0)$:&nbsp;
-:$$\varepsilon_{\nu}(00) = |d_{\nu}|^2\hspace{0.05cm},\hspace{0.5cm}\varepsilon_{\nu}(01) = |d_{\nu}-g_{-1}|^2\hspace{0.05cm},\hspace{0.2cm}
+:$$\varepsilon_{\nu}(00) = |d_{\nu}|^2\hspace{0.05cm},\hspace{0.5cm}\varepsilon_{\nu}(01) = |d_{\nu}-g_{-1}|^2\hspace{0.05cm},$$
 \varepsilon_{\nu}(10) = |d_{\nu}-g_{0}|^2\hspace{0.05cm},\hspace{0.2cm}\varepsilon_{\nu}(11) = |d_{\nu}-g_{0}-g_{-1}|^2\hspace{0.05cm}.$$
-*The total error quantity &nbsp;$\gamma_{\nu}(i)$&nbsp; is equal to the sum of the preceding node &nbsp;$\gamma_{\nu-1}(i\hspace{0.05cm}')$&nbsp; and the intervening branch &nbsp;$\varepsilon_{\nu}(i\hspace{0.05cm}'')$. For example, for the highlighted nodes:
+*The total error quantity &nbsp;$\gamma_{\nu}(i)$&nbsp; is the sum of the preceding node &nbsp;$\gamma_{\nu-1}(i\hspace{0.05cm}')$&nbsp; and the intervening branch &nbsp;$\varepsilon_{\nu}(i\hspace{0.05cm}'')$.&nbsp; For example,&nbsp; for the highlighted nodes:
-:$$\gamma_{1}(11)=\gamma_{0}(1)+\varepsilon_{1}(11) ,\hspace{0.25cm}
+:$$\gamma_{1}(11)=\gamma_{0}(1)+\varepsilon_{1}(11) ,$$
- \gamma_{2}(110)=\gamma_{1}(11)+\varepsilon_{2}(10) ,\hspace{0.25cm}
+:$$ \gamma_{2}(110)=\gamma_{1}(11)+\varepsilon_{2}(10) ,$$
- \gamma_{3}(1100)=\gamma_{2}(110)+\varepsilon_{3}(00).$$
+:$$ \gamma_{3}(1100)=\gamma_{2}(110)+\varepsilon_{3}(00).$$
-*For the first nodes &nbsp;$\gamma_{0}(0)$&nbsp; and &nbsp;$\gamma_{0}(1)$,&nbsp; it is considered that, by convention, the symbol &nbsp;$(a_1$, &nbsp;$a_2$, ...$)$ is always transmitted before the actual transmission &nbsp;$a_0 = 0$.&nbsp; It follows that:
+*For the first nodes &nbsp;$\gamma_{0}(0)$&nbsp; and &nbsp;$\gamma_{0}(1)$,&nbsp; it is considered that, by convention, the symbol &nbsp;$a_0=0$&nbsp; is always transmitted before the actual transmission &nbsp;&nbsp;$(a_1$, &nbsp;$a_2$, ...$)$.&nbsp; It follows that:
 :$$\gamma_{0}(0)=\varepsilon_{0}(00)= |d_{0}|^2 \hspace{0.05cm},\hspace{0.2cm}
   \gamma_{0}(1)=\varepsilon_{0}(01)=|d_{0}-g_{-1}|^2