Channel Coding/Bounds for Block Error Probability - Revision history

Guenter at 15:34, 22 November 2022

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Guenter at 15:33, 22 November 2022

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Guenter at 09:58, 22 November 2022

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Guenter at 09:55, 22 November 2022

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Guenter at 09:53, 22 November 2022

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Hwang at 14:27, 18 November 2022

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Hwang at 14:24, 18 November 2022

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Hwang at 14:18, 18 November 2022

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Hwang at 14:08, 18 November 2022

2022-11-18T14:08:03Z

Hwang at 14:04, 18 November 2022

2022-11-18T14:04:21Z

@@ Line 192: / Line 192: @@
 *$W(X)$&nbsp; is the above defined&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{weight enumerator function}$]] characterizing the channel code used.<br>
-*The&nbsp; $\text{Bhattacharyya parameter}$&nbsp; $\beta$&nbsp; identifies the digital channel.&nbsp; For example:
+*The&nbsp; &raquo;$\text{Bhattacharyya parameter}$&laquo;&nbsp; $\beta$&nbsp; identifies the digital channel.&nbsp; For example:
 ::<math>\beta = \left\{ \begin{array}{c} \lambda \\  \sqrt{4 \cdot \varepsilon \cdot (1- \varepsilon)}\\

@@ Line 192: / Line 192: @@
 *$W(X)$&nbsp; is the above defined&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{weight enumerator function}$]] characterizing the channel code used.<br>
-*The&nbsp; "Bhattacharyya parameter"&nbsp; $\beta$&nbsp; identifies the digital channel.&nbsp; For example:
+*The&nbsp; $\text{Bhattacharyya parameter}$&nbsp; $\beta$&nbsp; identifies the digital channel.&nbsp; For example:
 ::<math>\beta = \left\{ \begin{array}{c} \lambda \\  \sqrt{4 \cdot \varepsilon \cdot (1- \varepsilon)}\\

@@ Line 265: / Line 265: @@
 [[File:EN_KC_T_1_6_S5.png|right|frame|Block error probability and bounds of the&nbsp; $\text{HC (7, 4, 3)}$|class=fit]]
-*In contrast to the graph in the section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|$\text{Coding gain &ndash; bit error rate with AWGN}$]],&nbsp; the&nbsp; &raquo;'''block error rate'''&laquo;&nbsp; is given here and not the bit error rate.
+*In contrast to the graph in the section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|"Coding gain &ndash; bit error rate with AWGN"]],&nbsp; the&nbsp; &raquo;'''block error rate'''&laquo;&nbsp; is given here and not the bit error rate.
 *Approximately,&nbsp; the latter is smaller by the factor&nbsp; $d_{\rm min}/k$&nbsp; if,&nbsp; as here&nbsp; $d_{\rm min}< k$.&nbsp; In the present example:&nbsp; $d_{\rm min}/k = 0.75$.<br>

@@ Line 147: / Line 147: @@
 Then holds:
-#The code words&nbsp; $\underline{x}_0 = (0, 0, 0, 0, 0)$&nbsp; and&nbsp; $\underline{x}_1 = (0, 1, 0, 1, 1)$&nbsp; differ in&nbsp; $d = 3$&nbsp; bits;&nbsp; $d$&nbsp; is the [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{Hamming distance}$]]&nbsp; between&nbsp; $\underline{x}_0$&nbsp; and&nbsp; $\underline{x}_1$.
+#The code words&nbsp; $\underline{x}_0 = (0, 0, 0, 0, 0)$&nbsp; and&nbsp; $\underline{x}_1 = (0, 1, 0, 1, 1)$&nbsp; differ in&nbsp; $d = 3$&nbsp; bits;&nbsp; $d$&nbsp; is the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{Hamming distance}$]] &nbsp; between&nbsp; $\underline{x}_0$&nbsp; and&nbsp; $\underline{x}_1$.
 #An incorrect decoding result &nbsp; $\big[\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}1}\big] $&nbsp; is obtained whenever at least two of the three bits at bit positions 2, 4, and 5 are falsified.
 #In contrast, the bit positions 1 and 3 do not matter here, since they are the same for&nbsp; $\underline{x}_0$&nbsp; and &nbsp;$\underline{x}_1$.
@@ Line 161: / Line 161: @@
 *Wrong decoding of the block therefore certainly occurs if four or three bits are falsified.
-*A corruption '''KORREKTUR: falsification''' of two bits results in a block error with &nbsp;$50\%$&nbsp; probability,&nbsp; if one assumes a random decision for this&nbsp; (last term).
+*A falsification of two bits results in a block error with &nbsp;$50\%$&nbsp; probability,&nbsp; if one assumes a random decision for this&nbsp; (last term).
 ::<math>{\rm Pr}\big[\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}3}\big]  = \varepsilon^4 + 4 \cdot \varepsilon^3 \cdot (1 - \varepsilon)    +  {1}/{2} \cdot 6 \cdot \varepsilon^2 \cdot (1 - \varepsilon)^2 \hspace{0.05cm}.</math>

@@ Line 57: / Line 57: @@
 *Often one describes the set&nbsp; $\hspace{0.05cm}\{W_i \}\hspace{0.05cm}$&nbsp; also as a polynomial with a pseudo-variable&nbsp; $X$:
 ::<math>\left \{ \hspace{0.05cm} W_i \hspace{0.05cm} \right \} \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm}
@@ Line 100: / Line 101: @@
 :$${\rm Pr(block\:error)} = {\rm Pr}\left (\big [\underline{x}_{\hspace{0.02cm}0} \hspace{-0.07cm}\mapsto \hspace{-0.07cm}\underline{x}_{\hspace{0.02cm}1}\big ] \hspace{-0.07cm}\cup\hspace{-0.07cm}\big [\underline{x}_{\hspace{0.02cm}0} \hspace{-0.07cm}\mapsto \hspace{-0.07cm} \underline{x}_{\hspace{0.02cm}2}\big ] \hspace{-0.07cm}\cup\hspace{-0.07cm}\big [\underline{x}_{\hspace{0.02cm}0} \hspace{-0.07cm}\mapsto \hspace{-0.07cm} \underline{x}_{\hspace{0.02cm}3}\big ] \right ).$$
-&rArr; &nbsp;The event&nbsp; "falsification from &nbsp; $\underline{x}_0$ &nbsp; to&nbsp; $\underline{x}_1$"&nbsp; occurs for a given received word &nbsp; $\underline{y}$ &nbsp; according to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|$\text{maximum likelihood block-wise decision rule}$]]&nbsp; exactly when for the conditional probability density function  holds:
+&rArr; &nbsp;The event&nbsp; "falsification from &nbsp; $\underline{x}_0$ &nbsp; to&nbsp; $\underline{x}_1$"&nbsp; occurs for a given received word &nbsp; $\underline{y}$ &nbsp; according to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|"maximum likelihood block-wise decision rule"]]&nbsp; exactly when for the conditional probability density function  holds:
 ::<math>\big [\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}1}\big ] \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm}
@@ Line 111: / Line 112: @@
 #One calls this upper bound for the block error probability the&nbsp; &raquo;'''Union Bound'''&laquo;.
-#This was already used in the chapter&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Union_Bound_-_Upper_bound_for_the_error_probability|$\text{Approximation of the error probability}$]]&nbsp; of the book&nbsp; "Digital Signal Transmission".<br>
+#This was already used in the chapter&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Union_Bound_-_Upper_bound_for_the_error_probability|"Approximation of the error probability"]]&nbsp; of the book&nbsp; "Digital Signal Transmission".<br>
@@ Line 146: / Line 147: @@
 Then holds:
-#The code words&nbsp; $\underline{x}_0 = (0, 0, 0, 0, 0)$&nbsp; and&nbsp; $\underline{x}_1 = (0, 1, 0, 1, 1)$&nbsp; differ in&nbsp; $d = 3$&nbsp; bits;&nbsp; $d$&nbsp; indicates the [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{Hamming distance}$]]&nbsp; between&nbsp; $\underline{x}_0$&nbsp; and&nbsp; $\underline{x}_1$.
+#The code words&nbsp; $\underline{x}_0 = (0, 0, 0, 0, 0)$&nbsp; and&nbsp; $\underline{x}_1 = (0, 1, 0, 1, 1)$&nbsp; differ in&nbsp; $d = 3$&nbsp; bits;&nbsp; $d$&nbsp; is the [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|$\text{Hamming distance}$]]&nbsp; between&nbsp; $\underline{x}_0$&nbsp; and&nbsp; $\underline{x}_1$.
-#An incorrect decoding result &nbsp; $\big[\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}1}\big] $&nbsp; is obtained whenever at least two of the three bits at bit positions 2, 4, and 5 are corrupted '''KORREKTUR: falsified'''.
+#An incorrect decoding result &nbsp; $\big[\underline{x}_{\hspace{0.02cm}0} \mapsto \underline{x}_{\hspace{0.02cm}1}\big] $&nbsp; is obtained whenever at least two of the three bits at bit positions 2, 4, and 5 are falsified.
 #In contrast, the bit positions 1 and 3 do not matter here, since they are the same for&nbsp; $\underline{x}_0$&nbsp; and &nbsp;$\underline{x}_1$.
@@ Line 158: / Line 159: @@
 On the other hand,&nbsp; the code words &nbsp; $\underline{x}_0 = (0, 0, 0, 0, 0)$ &nbsp; and &nbsp; $\underline{x}_3 = (1, 1, 1, 0, 1)$ &nbsp; differ in four bit positions:
-*Wrong decoding of the block therefore certainly occurs if four or three bits are corrupted '''KORREKTUR: falsified'''.
+*Wrong decoding of the block therefore certainly occurs if four or three bits are falsified.
 *A corruption '''KORREKTUR: falsification''' of two bits results in a block error with &nbsp;$50\%$&nbsp; probability,&nbsp; if one assumes a random decision for this&nbsp; (last term).
@@ Line 230: / Line 231: @@
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;  The table summarizes the Bhattacharyya results for different values of the BSC parameter&nbsp; $\varepsilon$&nbsp; valid for the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Distance_spectrum_of_a_linear_code|$\text{example $\text{(5, 2)}$ code}$]].
-[[File:P ID2370 KC T 1 6 S4.png|right|frame|"Union Bound"&nbsp; vs.&nbsp; "Bhattacharyya Barrier '''KORREKTUR: Bound'''"&nbsp; valid for BSC model|class=fit]]
+[[File:P ID2370 KC T 1 6 S4.png|right|frame|"Union Bound"&nbsp; vs.&nbsp; "Bhattacharyya Bound"&nbsp; valid for BSC model|class=fit]]
 *For this one applies:
@@ Line 284: / Line 285: @@
 ::<math>{\rm Pr(Union\hspace{0.15cm} Bound)} \approx W_{d_{\rm min} } \cdot {\rm Q} \left ( \sqrt{d_{\rm min} \cdot {2R \cdot E_{\rm B} }/{N_0} } \right ) = 1.16 \cdot 10^{-5}
   \hspace{0.05cm}.</math>
-*However,&nbsp; this so-called&nbsp; "Truncated Union Bound" &nbsp;is no longer a true boundary '''KORREKTUR: bound''' for the block error probability in all applications,&nbsp; but is to be understood as an approximation.<br>
+*However,&nbsp; this so-called&nbsp; "Truncated Union Bound" &nbsp;is no longer a true bound for the block error probability in all applications,&nbsp; but is to be understood as an approximation.<br>
 *The&nbsp; "Bhattacharyya Bound"&nbsp; is marked by red dots in the graph.&nbsp; This is well above the Union Bound due to the highly simplified&nbsp; [https://en.wikipedia.org/wiki/Chernoff_bound $\text{Chernoff&ndash;Rubin Bound}$]&nbsp;

@@ Line 246: / Line 246: @@
 {{BlaueBox|TEXT=
-$\text{Conclusions:}$&nbsp;  Based on&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_for_the_BSC_model|$\text{Example 3}$]]&nbsp; and&nbsp; $\text{Example 4}$&nbsp; (in this section)&nbsp; for the simple&nbsp; $\text{(5, 2)}$&nbsp;  block code,&nbsp; though of little practical relevance,&nbsp; and in anticipation of&nbsp; $\text{example 5}$&nbsp; (in the next section)&nbsp; for the&nbsp; $\text{(7,&nbsp;4,&nbsp;3)}$&nbsp; Hamming code,&nbsp; we summarize:
+$\text{Conclusions:}$&nbsp;  Based on&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_for_the_BSC_model|$\text{Example 3}$]]&nbsp; and&nbsp; $\text{Example 4}$&nbsp; (in this section)&nbsp; for the simple&nbsp; $\text{(5, 2)}$&nbsp;  block code,&nbsp; though of little practical relevance,&nbsp; and in anticipation of&nbsp; $\text{Example 5}$&nbsp; (in the next section)&nbsp; for the&nbsp; $\text{(7,&nbsp;4,&nbsp;3)}$&nbsp; Hamming code,&nbsp; we summarize:
 #The block error probability of a coding system is often not analytically specifiable and must be determined by simulation.&nbsp; The same applies to the bit error probability.<br>
 #The&nbsp; &raquo;'''Union Bound'''&laquo;&nbsp; provides an upper bound for the block error probability.&nbsp; For many applications&nbsp; $($especially short codes$)$&nbsp; the Union Bound is only slightly above the actual error probability.<br>