Aufgaben:Exercise 4.5: On the Extrinsic L-values again - Revision history

Guenter at 15:06, 4 December 2022

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Guenter at 14:45, 4 December 2022

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Guenter at 14:25, 4 December 2022

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Guenter at 15:04, 3 December 2022

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Hwang at 19:15, 1 November 2022

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Noah at 23:01, 30 October 2022

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Noah at 21:39, 30 October 2022

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Noah at 21:39, 30 October 2022

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Noah at 00:03, 30 October 2022

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Guenter: Guenter moved page Aufgaben:Aufgabe 4.5: Nochmals zu den extrinsischen L–Werten to Aufgaben:Exercise 4.5: On the Extrinsic L-values again

2022-10-27T13:29:01Z

Guenter moved page Aufgabe 4.5: Nochmals zu den extrinsischen L–Werten to Exercise 4.5: On the Extrinsic L-values again

@@ Line 145: / Line 145: @@
 {{ML-Kopf}}
 [[File:P_ID3027__KC_A_4_5a_v2.png|right|frame|Results for&nbsp; $\underline{L}=(+1.0, +0.4, –1.0)$]]
-'''(1)'''&nbsp; According to the second $L_{\rm E}(i)$ approach holds:
+'''(1)'''&nbsp; According to the second&nbsp; $L_{\rm E}(i)$&nbsp; approach holds:
 :$${\rm sign} [L_{\rm E}(1)] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm sign} [L_{\rm E}(2)] \cdot {\rm sign} [L_{\rm E}(3)] = -1 \hspace{0.05cm},$$
-:$$|L_{\rm E}(1)| \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Min} \left ( |L_{\rm E}(2)|\hspace{0.05cm}, \hspace{0.05cm}|L_{\rm E}(3)| \right )  = {\rm Min} \left ( 0.4\hspace{0.05cm}, \hspace{0.05cm}1.0 \right ) = 0.4\hspace{0.3cm}
+:$$|L_{\rm E}(1)| \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Min} \left ( |L_{\rm E}(2)|\hspace{0.05cm}, \hspace{0.05cm}|L_{\rm E}(3)| \right )  = {\rm Min} \left ( 0.4\hspace{0.05cm}, \hspace{0.05cm}1.0 \right ) = 0.4$$
-\Rightarrow \hspace{0.3cm}L_{\rm E}(1) \hspace{0.15cm} \underline{-0.4}\hspace{0.05cm}.$$
+:$$\Rightarrow \hspace{0.3cm}L_{\rm E}(1) \hspace{0.15cm} \underline{-0.4}\hspace{0.05cm}.$$
 *In the same way you get:
-:$$L_{\rm E}(2) \hspace{0.15cm} \underline{-1.0}\hspace{0.05cm}, \hspace{0.3cm}
+:$$L_{\rm E}(2) \hspace{0.15cm} \underline{-1.0}\hspace{0.05cm}, $$
-L_{\rm E}(3) \hspace{0.15cm} \underline{+0.4}\hspace{0.05cm}.$$
+:$$L_{\rm E}(3) \hspace{0.15cm} \underline{+0.4}\hspace{0.05cm}.$$
-'''(2)'''&nbsp; The a posteriori $L$ values at the beginning of the first iteration $(I = 1)$ are the sum of the previous $L$&ndash;values (for $I = 0$) and the extrinsic values calculated in (1):
+'''(2)'''&nbsp; The a-posteriori&nbsp; $L$&ndash;values at the beginning of the first iteration&nbsp; $(I = 1)$&nbsp; are the sum
 :$$L_1 = L_{\rm APP}(1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}1.0 + (-0.4)\hspace{0.15cm} \underline{=+0.6}\hspace{0.05cm},$$
 :$$L_2 = L_{\rm APP}(2) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 0.4 + (-1.0)\hspace{0.15cm} \underline{=-0.6}\hspace{0.05cm},$$
@@ Line 161: / Line 163: @@
-'''(3)'''&nbsp; As can be seen from the above table, <u>solutions 1 and 2</u> are correct in contrast to answer 3:
+'''(3)'''&nbsp; As can be seen from the above table,&nbsp; the&nbsp; <u>solutions 1 and 2</u>&nbsp; are correct in contrast to answer 3:
-*With each new iteration, the amounts of $L(1), \ L(2)$ and $L(3)$ become significantly larger.
+*With each new iteration,&nbsp; the magnitudes of&nbsp; $L(1), \ L(2)$ and $L(3)$&nbsp; become significantly larger.
- [[File:P_ID3030__KC_A_4_5d_v2.png|right|frame|Results for&nbsp; $\underline{L}=(+0.6, +1.0, –0.4)$]]
+[[File:P_ID3030__KC_A_4_5d_v2.png|right|frame|Results for&nbsp; $\underline{L}=(+0.6, +1.0, –0.4)$]]
-'''(4)'''&nbsp; As can be seen from the adjacent table, <u>answers 1 and 3</u> are correct:
+<br><br>
-*So the decision is made for the code word $\underline{x}_0 = (+1, +1, +1)$.
+'''(4)'''&nbsp; As can be seen from the adjacent table,
-*From $I = 1$ this would also be the decision of <i>hard decision</i>.
+&nbsp; the&nbsp; <u>answers 1 and 3</u>&nbsp; are correct:
 <br clear=all>
 [[File:P_ID3028__KC_A_4_5e_v2.png|right|frame|Results for&nbsp; $\underline{L}=(+0.6, +1.0, –0.8)$]]
-'''(5)'''&nbsp; Correct are <u>answers 2 and 3</u>:
+'''(5)'''&nbsp; Correct are the&nbsp; <u>answers 2 and 3</u>:
-*Because $|L(3)| > |L(1)|$ the following is valid from $I = 1$: &nbsp; $L_1 < 0 \hspace{0.05cm},\hspace{0.2cm}
+*Because of&nbsp; $|L(3)| > |L(1)|$&nbsp; the following is valid for $I /ge 1$: &nbsp; $L_1 < 0 \hspace{0.05cm},\hspace{0.2cm}
   L_2 > 0 \hspace{0.05cm},\hspace{0.2cm}
 L_3 < 0 \hspace{0.05cm}.$
-*From this iteration loop, <i>hard decision</i> returns the code word $\underline{x}_2 = (-1, +1, -1)$.
 <br clear=all>
   [[File: P_ID3029__KC_A_4_5f_v1.png|right|frame|Results for&nbsp; $\underline{L}=(+0.6, +1.0, –0.6)$]]
-'''(6)'''&nbsp; Correct is the <u>proposed solution 3</u>:
+'''(6)'''&nbsp; Correct is the&nbsp; <u>proposed solution 3</u>:
-*The adjacent table shows that under the condition $|L(1)| = |L(3)|$, starting from the iteration loop $I = 1$, all extrinsic $L$&ndash;values are zero.
+*The adjacent table shows that under the condition&nbsp; $|L(1)| = |L(3)|$,&nbsp; starting from the iteration loop&nbsp; $I = 1$,&nbsp; all extrinsic&nbsp; $L$&ndash;values are zero.
 *Thus, the a posteriori $L$ values remain constantly equal to $\underline{L} = (0., +0.4, 0.)$ even for $I > 1$, which cannot be assigned to any code word.
 {{ML-Fuß}}

@@ Line 2: / Line 2: @@
 [[File:P_ID3026__KC_A_4_5_v2.png|right|frame|Table for first&nbsp; $L_{\rm E}(i)$&nbsp; approach]]
-We assume as in the&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"theory section"]]&nbsp; the&nbsp; "single parity&ndash;check code" &nbsp; $\rm SPC \, (3, \, 2, \, 2)$.&nbsp;
+We assume as in the&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_log_likelihood_ratios|"theory section"]]&nbsp; the&nbsp; "single parity&ndash;check code" &nbsp; $\rm SPC \, (3, \, 2, \, 2)$.&nbsp;
 The possible code words are&nbsp; $\underline{x} \hspace{-0.01cm}\in \hspace{-0.01cm}
@@ Line 94: / Line 94: @@
 *The exercise belongs to the chapter&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder|"Soft&ndash;in Soft&ndash;out Decoder"]].
-*Referred to in particular&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"Calculation of extrinsic LLRs"]].
+*Referred to in particular&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_log_likelihood_ratios|"Calculation of extrinsic log likelihood ratios"]].
 * Only the&nbsp; '''second solution approach'''&nbsp; is treated here.
@@ Line 105: / Line 105: @@
 ===Questions===
 <quiz display=simple>
-{It holds&nbsp; $\underline{L} = (+1.0, +0.4, -1.0)$. Determine the extrinsic&nbsp; $L$&ndash;values according to the '''second&nbsp; $L_{\rm E}(i)$&ndash;approach''' without previous iteration&nbsp; $\underline{(I = 0)}$.
+{It holds&nbsp; $\underline{L} = (+1.0, +0.4, -1.0)$.&nbsp; Determine the extrinsic&nbsp; $L$&ndash;values according to the&nbsp; '''second&nbsp; $L_{\rm E}(i)$&ndash;approach'''&nbsp; without previous iteration&nbsp; $\underline{(I = 0)}$.
 |type="{}"}
 $L_{\rm E}(1) \ = \ ${ -0.412--0.388 }
@@ Line 111: / Line 111: @@
 $L_{\rm E}(3) \ = \ ${ 0.4 3% }
-{What are the a-posteriori $L$&ndash;values for the first iteration&nbsp; $\underline{(I = 1)}$?
+{What are the a-posteriori $L$&ndash;values&nbsp; $L_i = L_{\rm APP} (i)$&nbsp; for the first iteration&nbsp; $\underline{(I = 1)}$?
 |type="{}"}
-$L_(1) \ = \ ${ 0.6 3% }
+$L_1 \ = \ ${ 0.6 3% }
-$L_(2) \ = \ ${ -0.618--0.582 }
+$L_2 \ = \ ${ -0.618--0.582 }
-$L_(3) \ = \ ${ -0.618--0.582 }
+$L_3 \ = \ ${ -0.618--0.582 }
 {Which of the following statements are true for&nbsp; $\underline{L} = (+1.0, +0.4, -1.0)$?
 |type="[]"}
-+ <i>Hard decision</i>&nbsp; after&nbsp; $I = 1$&nbsp; leads to the code word&nbsp; $\underline{x}_1 = (+1, -1, -1)$.
++ Hard decision&nbsp; after&nbsp; $I = 1$&nbsp; leads to the code word&nbsp; $\underline{x}_1 = (+1, -1, -1)$.
 + This does not change after further iterations.
 - Further iterations do not increase the reliability for&nbsp; $\underline{x}_1$&nbsp;.
@@ Line 127: / Line 127: @@
 + The iterative decoding leads to the result&nbsp; $\underline{x}_0 = (+1, +1, +1)$.
 - The iterative decoding leads to the result&nbsp; $\underline{x}_2 = (-1, +1, -1)$.
-+ This result also returns <i>hard decision</i>&nbsp; $I = 1$.
++ Hard decision also returns this result for&nbsp; $I \ge 1$.
 {Which of the following statements are true for&nbsp; $\underline{L} = (+0.6, +1.0, -0.8)$?
 |type="[]"}
-- The iterative decoding produces the result&nbsp; $\underline{x}_0 = (+1, +1, +1)$.
+- The iterative decoding leads to the result&nbsp; $\underline{x}_0 = (+1, +1, +1)$.
 + The iterative decoding leads to the result&nbsp; $\underline{x}_2 = (-1, +1, -1)$.
-+ This result also returns <i>hard decision</i>&nbsp; $I = 1$.
++ Hard decision also returns this result for&nbsp; $I \ge 1$.
 {Which of the following statements are true for&nbsp; $\underline{L} = (+0.6, +1.0, -0.6)$?

@@ Line 25: / Line 25: @@
 #However,&nbsp; the code is well suited for demonstration purposes because of the manageable effort involved.
 #With&nbsp; "iterative symbol-wise decoding"&nbsp; one can also correct one error.
-#In the present code,&nbsp; the extrinsic&nbsp; L&ndash;values&nbsp; $\underline{L}_{\rm E} = \big (L_{\rm E}(1), \ L_{\rm E}(2), \ L_{\rm E}(3)\big )$&nbsp; must be calculated according to the following equation:
+#In the present code,&nbsp; the extrinsic&nbsp; $L$&ndash;values&nbsp; $\underline{L}_{\rm E} = \big (L_{\rm E}(1), \ L_{\rm E}(2), \ L_{\rm E}(3)\big )$&nbsp; must be calculated according to the following equation:
 :$$L_{\rm E}(i) = {\rm ln} \hspace{0.15cm}\frac{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \hspace{0.05cm} \right ]}{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \hspace{0.05cm}  \hspace{0.05cm}\right ]}.$$
@@ Line 36: / Line 36: @@
 :$$L_{\rm E}(3) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm tanh}^{-1} \left [{\rm tanh}(L_1/2) \cdot {\rm tanh}(L_2/2) \right ] \hspace{0.05cm}.$$
-'''(1)'''&nbsp; This&nbsp; $L_{\rm E}(i)$&nbsp; approach underlies the results table above&nbsp; $($red entries$)$,&nbsp; assuming the following a-posteriori L&ndash;values:
+'''(1)'''&nbsp; This&nbsp; $L_{\rm E}(i)$&nbsp; approach underlies the results table above&nbsp; $($red entries$)$,&nbsp; assuming the following a-posteriori $L$&ndash;values:
 :$$\underline {L}_{\rm APP} = (+1.0\hspace{0.05cm},\hspace{0.05cm}+0.4\hspace{0.05cm},\hspace{0.05cm}-1.0)  \hspace{0.5cm}\Rightarrow \hspace{0.5cm}
 L_1 = +1.0\hspace{0.05cm},\hspace{0.15cm}
@@ Line 42: / Line 42: @@
 L_3 = -1.0\hspace{0.05cm}.$$
-'''(2)'''&nbsp; The extrinsic&nbsp; $L$&ndash;values for the zeroth iteration result in&nbsp; (derivation in&nbsp; [[Aufgaben:Exercise_4.5Z:_Tangent_Hyperbolic_and_Inverse|"Exercise 4.5Z"]]):
+'''(2)'''&nbsp; The extrinsic&nbsp; $L$&ndash;values for the zeroth iteration result in&nbsp; $($derivation in&nbsp; [[Aufgaben:Exercise_4.5Z:_Tangent_Hyperbolic_and_Inverse|$\text{Exercise 4.5Z})$]]:
 :$$L_{\rm E}(1) = -0.1829, \ L_{\rm E}(2) = -0.4337, \  L_{\rm E}(3) = +0.1829.$$
-'''(3)'''&nbsp; The a posteriori values at the beginning of the first iteration are thus
+'''(3)'''&nbsp; The a-posteriori&nbsp; $L$&ndash;values at the beginning of the first iteration are thus
-:$$\underline{L}^{(I=1)} = \underline{L}^{(I=0)}  + \underline{L}_{\hspace{0.02cm}\rm E}^{(I=0)}  =
+:$$\underline{L_{\rm APP} }^{(I=1)} = \underline{L_{\rm APP} }^{(I=0)}  + \underline{L}_{\hspace{0.02cm}\rm E}^{(I=0)}  =
 (+0.8171\hspace{0.05cm},\hspace{0.05cm}-0.0337\hspace{0.05cm},\hspace{0.05cm}-0.8171)
 \hspace{0.05cm} .  $$
-'''(4)'''&nbsp; From this, the new extrinsic values for the iteration loop&nbsp; $I = 1$&nbsp; are as follows:
+'''(4)'''&nbsp; From this,&nbsp; the new extrinsic&nbsp; $L$&ndash;values for the iteration loop&nbsp; $I = 1$&nbsp; are as follows:
 :$$L_{\rm E}(1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm tanh}^{-1} \big [{\rm tanh}(-0.0337/2) \cdot {\rm tanh}(-0.8171/2) \big ] = 0.0130 = -L_{\rm E}(3)\hspace{0.05cm},$$
 :$$L_{\rm E}(2) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm tanh}^{-1} \big [{\rm tanh}(+0.8171/2) \cdot {\rm tanh}(-0.8171/2) \big ]  = - 0.3023\hspace{0.05cm}.$$
-Further, one can see from the above table:
+Further,&nbsp; one can see from the above table:
-* A hard decision according to the signs before the first iteration&nbsp; $(I = 0)$ fails, since&nbsp; $(+1, +1, -1)$&nbsp; is not a valid&nbsp; $\rm SPC \, (3, \, 2, \, 2)$&ndash;code word.
+* A hard decision according to the signs before the first iteration&nbsp; $(I = 0)$ fails,&nbsp; since&nbsp; $(+1, +1, -1)$&nbsp; is not a valid&nbsp; $\rm SPC \, (3, \, 2, \, 2)$&nbsp; code word.
-* But already after&nbsp; $I = 1$&nbsp; iterations, a hard decision yields a valid code word, namely&nbsp; $\underline{x}_2 = (+1, -1, -1)$. Also in later graphs, the rows with correct HD decisions for the first time are highlighted in blue.
 * Hard decisions after further iterations&nbsp; $(I &#8805; 2)$&nbsp; each lead to the same code word&nbsp; $\underline{x}_2$. This statement is not only valid for this example, but in general.
-Besides, in this exercise we consider a '''second $L_{\rm E}(i)$ approach''', which is given here for the example of the first symbol $(i = 1)$:
+Besides,&nbsp; in this exercise we consider a&nbsp; &raquo;'''second $L_{\rm E}(i)$ approach'''&laquo;,&nbsp; which is given here for the example of the first symbol&nbsp; $(i = 1)$:
-:$${\rm sign} \big[L_{\rm E}(1)\big] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm sign} \big[L_{\rm E}(2)\big] \cdot {\rm sign} \big[L_{\rm E}(3)\big]\hspace{0.05cm},\hspace{0.8cm}
+:$${\rm sign} \big[L_{\rm E}(1)\big] \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm sign} \big[L_{\rm E}(2)\big] \cdot {\rm sign} \big[L_{\rm E}(3)\big]\hspace{0.05cm},$$
-|L_{\rm E}(1)| \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Min} \left ( |L_{\rm E}(2)|\hspace{0.05cm}, \hspace{0.05cm}|L_{\rm E}(3)| \right )  \hspace{0.05cm}.$$
+:$$|L_{\rm E}(1)| \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Min} \left ( |L_{\rm E}(2)|\hspace{0.05cm}, \hspace{0.05cm}|L_{\rm E}(3)| \right )  \hspace{0.05cm}.$$
-This second approach is based on the assumption that the reliability of&nbsp; $L_{\rm E}(i)$&nbsp; is essentially determined by the most unreliable neighbor symbol. The better (larger) input&ndash;LLR is completely disregarded. &ndash; Let us consider two examples for this:
+This second approach is based on the assumption that the reliability of&nbsp; $L_{\rm E}(i)$&nbsp; is essentially determined by the most unreliable neighbor symbol.&nbsp; The better&nbsp; $($larger$)$&nbsp; the input log likelihood ratio is completely disregarded.
-'''(1)'''&nbsp; For&nbsp; $L_2 = 1.0$&nbsp; and&nbsp; $L_3 = 5.0$&nbsp; we get for example
 * after the first approach: &nbsp; $L_{\rm E}(1) =2 \cdot {\rm tanh}^{-1} \big [{\rm tanh}(0.5) \cdot {\rm tanh}(2.5) \big ] =2 \cdot {\rm tanh}^{-1}(0.4559) = 0.984\hspace{0.05cm},$
 * according to the second approach: &nbsp; $|L_{\rm E}(1)| \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Min} \big ( 1.0\hspace{0.05cm}, \hspace{0.05cm}5.0 \big )  = 1.000 \hspace{0.05cm}.$
@@ Line 74: / Line 81: @@
 '''(2)'''&nbsp; On the other hand one obtains for&nbsp; $L_2 = L_3 = 1.0$
 * according to the first approach: &nbsp; $L_{\rm E}(1) =2 \cdot {\rm tanh}^{-1} \big [{\rm tanh}(0.5) \cdot {\rm tanh}(0.5) \big ] =2 \cdot {\rm tanh}^{-1}(0.2135) = 0.433\hspace{0.05cm},$
 * according to the second approach: &nbsp; $|L_{\rm E}(1)| \hspace{-0.15cm} \ = \ \hspace{-0.15cm} {\rm Min} \big ( 1.0\hspace{0.05cm}, \hspace{0.05cm}1.0 \big )  = 1.000 \hspace{0.05cm}.$
-One can see the clear discrepancy between the two approaches. The second approach (approximation) is clearly more positive than the first (correct) approach. However, it is actually only important that the iterations lead to the desired decoding result.
+One can see the clear discrepancy between the two approaches.&nbsp; The second approach&nbsp; $($approximation$)$&nbsp; is clearly more positive than the first&nbsp; $($correct$)$&nbsp; approach. However,&nbsp; it is actually only important that the iterations lead to the desired decoding result.
@@ Line 83: / Line 91: @@
-Hints:
+<u>Hints:</u>
 *The exercise belongs to the chapter&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder|"Soft&ndash;in Soft&ndash;out Decoder"]].
 *Referred to in particular&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"Calculation of extrinsic LLRs"]].
-* Only the&nbsp; '''second solution approach''' is treated here.
+*Referred to in particular&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"Calculation of extrinsic LLRs"]].
-* For the first solution approach we refer to the&nbsp; [[Aufgaben:Exercise_4.5Z:_Tangent_Hyperbolic_and_Inverse|"Exercise 4.5Z"]] .
@@ Line 100: / Line 111: @@
 $L_{\rm E}(3) \ = \ ${ 0.4 3% }
-{What are the a posteriori&ndash;$L$&ndash;values for the first iteration&nbsp; $\underline{(I = 1)}$?
+{What are the a-posteriori $L$&ndash;values for the first iteration&nbsp; $\underline{(I = 1)}$?
 |type="{}"}
 $L_(1) \ = \ ${ 0.6 3% }

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Soft-in_Soft-Out_Decoder}}
-[[File:P_ID3026__KC_A_4_5_v2.png|right|frame|Table according to the first&nbsp; $L_{\rm E}(i)$ approach]]
+[[File:P_ID3026__KC_A_4_5_v2.png|right|frame|Table for first&nbsp; $L_{\rm E}(i)$&nbsp; approach]]
-We assume as in&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"theory section"]]&nbsp; from&nbsp; <i>single parity&ndash;check code</i> &nbsp; $\rm SPC \, (3, \, 2, \, 2)$&nbsp;. The possible code words are&nbsp; $\underline{x} \hspace{-0.01cm}\in \hspace{-0.01cm}
+We assume as in the&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"theory section"]]&nbsp; the&nbsp; "single parity&ndash;check code" &nbsp; $\rm SPC \, (3, \, 2, \, 2)$.&nbsp;
 \{ \underline{x}_0,\hspace{0.05cm}
 \underline{x}_1,\hspace{0.05cm}
 \underline{x}_2,\hspace{0.05cm}
 \underline{x}_3\}$&nbsp; with
-:$$\underline{x}_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (0\hspace{-0.03cm},\hspace{0.05cm}0\hspace{-0.03cm},\hspace{0.05cm}0)\hspace{0.35cm}{\rm bzw. } \hspace{0.35cm}
+:$$\underline{x}_0 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (0\hspace{-0.03cm},\hspace{0.05cm}0\hspace{-0.03cm},\hspace{0.05cm}0)\hspace{0.35cm}{\rm resp. } \hspace{0.35cm}
 \underline{x}_0 \hspace{-0.05cm}=\hspace{-0.05cm} (+1\hspace{-0.03cm},\hspace{-0.05cm}+1\hspace{-0.03cm},\hspace{-0.05cm}+1)\hspace{0.05cm},$$
-:$$\underline{x}_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (0\hspace{-0.03cm},\hspace{0.05cm}1\hspace{-0.03cm},\hspace{0.05cm}1)\hspace{0.35cm}{\rm bzw. } \hspace{0.35cm}
+:$$\underline{x}_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (0\hspace{-0.03cm},\hspace{0.05cm}1\hspace{-0.03cm},\hspace{0.05cm}1)\hspace{0.35cm}{\rm resp. } \hspace{0.35cm}
 \underline{x}_1 \hspace{-0.05cm}=\hspace{-0.05cm} (+1\hspace{-0.03cm},\hspace{-0.05cm}-1\hspace{-0.03cm},\hspace{-0.05cm}-1)\hspace{0.05cm},$$
-:$$\underline{x}_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (1\hspace{-0.03cm},\hspace{0.05cm}0\hspace{-0.03cm},\hspace{0.05cm}1)\hspace{0.35cm}{\rm bzw. } \hspace{0.35cm}
+:$$\underline{x}_2 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (1\hspace{-0.03cm},\hspace{0.05cm}0\hspace{-0.03cm},\hspace{0.05cm}1)\hspace{0.35cm}{\rm resp. } \hspace{0.35cm}
 \underline{x}_2 \hspace{-0.05cm}=\hspace{-0.05cm} (-1\hspace{-0.03cm},\hspace{-0.05cm}+1\hspace{-0.03cm},\hspace{-0.05cm}-1)\hspace{0.05cm},$$
-:$$\underline{x}_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (1\hspace{-0.03cm},\hspace{0.05cm}1\hspace{-0.03cm},\hspace{0.05cm}0)\hspace{0.35cm}{\rm bzw. } \hspace{0.35cm}
+:$$\underline{x}_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (1\hspace{-0.03cm},\hspace{0.05cm}1\hspace{-0.03cm},\hspace{0.05cm}0)\hspace{0.35cm}{\rm resp. } \hspace{0.35cm}
 \underline{x}_3 \hspace{-0.05cm}=\hspace{-0.05cm} (-1\hspace{-0.03cm},\hspace{-0.05cm}-1\hspace{-0.03cm},\hspace{-0.05cm}+1)\hspace{0.05cm}.$$
-In the exercise we mostly use the second (bipolar) representation of the code symbols: &nbsp; $x_i &#8712; \{+1, -1\}$.
+In the exercise we mostly use the second (bipolar) representation of the code symbols: &nbsp;
-*It is not that the&nbsp; $\rm SPC \, (3, \, 2, \, 2)$&nbsp; would be of much practical interest, since, for example, in&nbsp; <i>hard decision</i>&nbsp; because of&nbsp; $d_{\rm min} = 2$&nbsp; only one error can be detected and none can be corrected. However, the code is well suited for practice and demonstration purposes because of the manageable effort involved.
+Note:
-*With&nbsp; ''iterative symbol-wise decoding''&nbsp; one can also correct one error. In the present code, the extrinsic&nbsp; $L$&ndash;values&nbsp; $\underline{L}_{\rm E} = \big (L_{\rm E}(1), \ L_{\rm E}(2), \ L_{\rm E}(3)\big )$&nbsp; must be calculated according to the following equation.
+#It is not that the&nbsp; $\rm SPC \, (3, \, 2, \, 2)$&nbsp; would be of much practical interest,&nbsp; since,&nbsp; for example,&nbsp; in&nbsp; "hard decision"&nbsp; because of&nbsp; $d_{\rm min} = 2$&nbsp; only one error can be detected and none can be corrected.
 :$$L_{\rm E}(i) = {\rm ln} \hspace{0.15cm}\frac{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \hspace{0.05cm} \right ]}{{\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \hspace{0.05cm}  \hspace{0.05cm}\right ]}.$$
@@ Line 25: / Line 31: @@
-As the&nbsp; '''first $L_{\rm E}(i)$ approach'''&nbsp; we refer to the approach corresponding to the equations
+&rArr; &nbsp; As the&nbsp; &raquo;'''first $L_{\rm E}(i)$ approach'''&laquo;&nbsp; we refer to the approach corresponding to the equations
 :$$L_{\rm E}(1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm tanh}^{-1} \left [{\rm tanh}(L_2/2) \cdot {\rm tanh}(L_3/2) \right ] \hspace{0.05cm},$$
 :$$L_{\rm E}(2) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm tanh}^{-1} \left [{\rm tanh}(L_1/2) \cdot {\rm tanh}(L_3/2) \right ] \hspace{0.05cm},$$
 :$$L_{\rm E}(3) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm tanh}^{-1} \left [{\rm tanh}(L_1/2) \cdot {\rm tanh}(L_2/2) \right ] \hspace{0.05cm}.$$
-'''(1)'''&nbsp; This&nbsp; $L_{\rm E}(i)$ approach underlies the results table above (red entries), assuming the following a posteriori $L$ values:
+'''(1)'''&nbsp; This&nbsp; $L_{\rm E}(i)$&nbsp; approach underlies the results table above&nbsp; $($red entries$)$,&nbsp; assuming the following a-posteriori L&ndash;values:
-:$$\underline {L}_{\rm APP} = (+1.0\hspace{0.05cm},\hspace{0.05cm}+0.4\hspace{0.05cm},\hspace{0.05cm}-1.0)  \hspace{0.5cm}{\rm kurz\hspace{-0.1cm}:}\hspace{0.25cm}
+:$$\underline {L}_{\rm APP} = (+1.0\hspace{0.05cm},\hspace{0.05cm}+0.4\hspace{0.05cm},\hspace{0.05cm}-1.0)  \hspace{0.5cm}\Rightarrow \hspace{0.5cm}
-L_1 = +1.0\hspace{0.05cm},\hspace{0.05cm}
+L_1 = +1.0\hspace{0.05cm},\hspace{0.15cm}
-L_2 = +0.4\hspace{0.05cm},\hspace{0.05cm}
+L_2 = +0.4\hspace{0.05cm},\hspace{0.15cm}
 L_3 = -1.0\hspace{0.05cm}.$$

@@ Line 163: / Line 163: @@
 '''(6)'''&nbsp; Correct is the <u>proposed solution 3</u>:
 *The adjacent table shows that under the condition $|L(1)| = |L(3)|$, starting from the iteration loop $I = 1$, all extrinsic $L$&ndash;values are zero.
-*Thus, the a posteriori $L$ values remain constantly equal to $\underline{L} = (0., +0.4, 0.)$ even for $I > 1$, which cannot be assigned to any codeword.
+*Thus, the a posteriori $L$ values remain constantly equal to $\underline{L} = (0., +0.4, 0.)$ even for $I > 1$, which cannot be assigned to any code word.
 {{ML-Fuß}}

@@ Line 81: / Line 81: @@
 *Referred to in particular&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Calculation_of_extrinsic_LLRs|"For calculation of extrinsic L values"]].
 * Only the&nbsp; '''second solution approach''' is treated here.
-* For the first solution approach we refer to the&nbsp; [[Exercises:Exercise_4.5Z:_Tangent_Hyperbolic_and_Inverse|"Exercise 4.5Z"]] .
+* For the first solution approach we refer to the&nbsp; [[Aufgaben:Exercise_4.5Z:_Tangent_Hyperbolic_and_Inverse|"Exercise 4.5Z"]] .

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