Channel Coding/Distance Characteristics and Error Probability Bounds - Revision history

Guenter at 17:24, 20 December 2022

2022-12-20T17:24:57Z

2022-12-07T16:12:32Z

2022-12-02T18:00:44Z

2022-12-01T11:33:19Z

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 {{GraueBox|TEXT=
 $\text{Example 5:}$&nbsp; The graph illustrates the good correction capability &nbsp; &rArr; &nbsp; small bit error rate&nbsp; $\rm (BER)$&nbsp; of convolutional codes at the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{$\rm AWGN$&nbsp; channel}$]].
-[[File:EN_KC_T_3_5_S6.png|right|frame|AWGN bit error probability of convolutional codes '''Korrektur''']]
+[[File:EN_KC_T_3_5_S6.png|right|frame|AWGN bit error probability of convolutional codes]]
 *Red circles denote the&nbsp; $\rm BER$&nbsp; for our&nbsp; "rate&nbsp; $1/2$&nbsp; standard  encoder"&nbsp; with memory&nbsp; $m = 2$.<br>
@@ Line 458: / Line 458: @@
-* Codes with large memory&nbsp; $m$&nbsp; lead to large gains over uncoded transmission&nbsp;  $($dashed curve$)$.
+* Codes with large memory&nbsp; $m$&nbsp; lead to large gains over uncoded transmission.

@@ Line 180: / Line 180: @@
 First,&nbsp; the state transition diagram must be redrawn for this purpose.&nbsp; The graphic shows this on the left in the previous form as diagram&nbsp; $\rm (A)$,&nbsp; while the new diagram&nbsp; $\rm (B)$&nbsp; is given on the right.&nbsp; It can be seen:
-[[File:KC T 3 5 Diagram.png|right|frame|State transition diagram in two different variants '''KORREKTUR: Diagram'''|class=fit]]
+[[File:KC_T_3_5_Diagram_v3.png|right|frame|State transition diagram in two different variants|class=fit]]
 # &nbsp; The state&nbsp; $S_0$&nbsp; is split into the start state&nbsp; $S_0$&nbsp; and the end state&nbsp; $S_0\hspace{0.01cm}'$.&nbsp; Thus,&nbsp; all paths of the trellis that start in state&nbsp; $S_0$&nbsp; and return to it at some point can also be traced in the right graph&nbsp; $\rm (B)$.&nbsp; Excluded are direct transitions from&nbsp; $S_0$&nbsp; to&nbsp; $S_0\hspace{0.01cm}'$&nbsp; and thus also the zero path.<br>

@@ Line 156: / Line 156: @@
 {{GraueBox|TEXT=
-$\text{Example 2:}$&nbsp; Thus, the enhanced path weighting enumerator function of our standard coder '''KORREKTUR: encoder''' is:
+$\text{Example 2:}$&nbsp; Thus, the enhanced path weighting enumerator function of our standard encoder is:
 ::<math>T_{\rm enh}(X,\ U) =  U \cdot X^5 + 2  \cdot U^2 \cdot X^6  + 4  \cdot  U^3 \cdot X^7+ \text{ ...} \hspace{0.1cm}

@@ Line 472: / Line 472: @@
 == Exercises for the chapter ==
 <br>
-[[Aufgaben:Exercise_3.12:_Path_Weighting_Function|"Exercise 3.12: Path Weighting Function"]]
+[[Aufgaben:Exercise_3.12:_Path_Weighting_Function|Exercise 3.12: Path Weighting Function]]
-[[Aufgaben:Exercise_3.12Z:_Ring_and_Feedback|"Exercise 3.12Z: Ring and Feedback"]]
+[[Aufgaben:Exercise_3.12Z:_Ring_and_Feedback|Exercise 3.12Z: Ring and Feedback]]
-[[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again|"Exercise 3.13: Path Weighting Function again"]]
+[[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again|Exercise 3.13: Path Weighting Function again]]
-[[Aufgaben:Exercise_3.14:_Error_Probability_Bounds|"Exercise 3.14: Error Probability Bounds"]]
+[[Aufgaben:Exercise_3.14:_Error_Probability_Bounds|Exercise 3.14: Error Probability Bounds]]
 ==References==

@@ Line 449: / Line 449: @@
 {{GraueBox|TEXT=
-$\text{Example 5:}$&nbsp; The graph illustrates the good correction capability &nbsp; &rArr; &nbsp; small bit error rate&nbsp; $\rm (BER)$&nbsp; of convolutional codes at the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|$\text{$\rm AWGN$&nbsp; channel}$]].
+$\text{Example 5:}$&nbsp; The graph illustrates the good correction capability &nbsp; &rArr; &nbsp; small bit error rate&nbsp; $\rm (BER)$&nbsp; of convolutional codes at the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{$\rm AWGN$&nbsp; channel}$]].
 [[File:EN_KC_T_3_5_S6.png|right|frame|AWGN bit error probability of convolutional codes '''Korrektur''']]

@@ Line 386: / Line 386: @@
 Finally,&nbsp; an upper bound is given for the bit error probability.&nbsp; According to the graph,&nbsp; we proceed as in&nbsp; [Liv10]<ref name='Liv10'>Liva, G.:&nbsp; Channel Coding.&nbsp; Lecture manuscript, Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010.</ref>&nbsp; assume the following conditions:<br>
-*Sent the zero sequence&nbsp; $\underline{x} = \underline{0}$ &nbsp; &#8658; &nbsp; Path&nbsp; $\varphi_0$.<br>
+*The zero sequence&nbsp; $\underline{x} = \underline{0}$ was sent &nbsp; &#8658; &nbsp; Path&nbsp; $\varphi_0$.<br>
 *The duration of a path deviation&nbsp; $($"error burst duration"$)$&nbsp; is denoted by&nbsp; $L$&nbsp;.<br>

@@ Line 349: / Line 349: @@
 {{BlaueBox|TEXT=
-By&nbsp; &raquo;'''summing over all paths'''&laquo;&nbsp; and comparing with the (simple)&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Path_weighting_enumerator_function|$\text{path weighting function}$]]&nbsp; $T(X)$&nbsp; we get the result:
+By&nbsp; &raquo;'''summing over all paths'''&laquo;&nbsp; and comparing with the (simple)&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds#Path_weighting_enumerator_function|$\text{path weighting function}$]]&nbsp; $T(X)$&nbsp; we get the result:
 ::<math>{\rm Pr(burst\:error)} \le T(X = \beta),\hspace{0.5cm}{\rm with}\hspace{0.5cm}

@@ Line 325: / Line 325: @@
 :$$\Rightarrow \hspace{0.3cm}{\rm Pr(burst\:error)}= {\rm Pr} \left ( \cup_{\varphi_{\hspace{0.02cm}i} \in {\it \Phi}} \hspace{0.15cm} \big[\varphi_{\hspace{0.02cm}0} \mapsto \varphi_{\hspace{0.02cm}i}\big] \right )\hspace{0.05cm}.$$
-*An upper bound for this is provided by the so-called&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|"$\text{Union Bound}$"]]:
+*An upper bound for this is provided by the so-called&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|$\text{Union Bound}$]]:
 ::<math>{\rm Pr(Burst\:error)} \le \sum_{\varphi_{\hspace{0.02cm}i} \in {\it \Phi}}\hspace{0.15cm}

@@ Line 305: / Line 305: @@
 *All other paths&nbsp; $\varphi_1, \ \varphi_2, \ \varphi_3, \ \text{...} $&nbsp; $($and many more$)$&nbsp; leave&nbsp; $\varphi_0$&nbsp; at time&nbsp; $t$.
-*They all belong to the path set&nbsp; ${\it \Phi}$&nbsp; which is defined as &nbsp;"Viterbi decoder leaves the correct path at time&nbsp; $t$".&nbsp; This probability is calculated in the next section}}.<br>
+*They all belong to the path set&nbsp; ${\it \Phi}$&nbsp; which is defined as &nbsp;"Viterbi decoder leaves the correct path at time&nbsp; $t$".&nbsp; This probability is calculated in the next section.}}<br>
 == Burst error probability and Bhattacharyya bound ==

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 * the block error probability would always result in&nbsp; ${\rm Pr(block\hspace{0.1cm} error)}\equiv 1$,
-*even if the bit error probability is extremely small \hspace{0.05cm} $($but non-zero$)$.
+*even if the bit error probability is extremely small $\hspace{0.05cm}$ $($but non-zero$)$.
 <br clear=all>
 {{BlaueBox|TEXT=