Channel Coding/The Basics of Turbo Codes - Revision history

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Guenter at 11:54, 21 December 2022

2022-12-21T11:54:21Z

@@ Line 221: / Line 221: @@
 *The inner decoder&nbsp; $($code&nbsp; $\mathcal{C}_2)$&nbsp; receives the intrinsic information&nbsp; $\underline{L}_{\rm K}(\underline{x})$&nbsp; from the channel and  a-priori information&nbsp; $\underline{L}_{\rm A}(\underline{w})$&nbsp; with&nbsp; $\underline{w} = \pi(\underline{c})$&nbsp; from the outer decoder&nbsp; $($after interleaving$)$&nbsp; and delivers the extrinsic information&nbsp; $\underline{L}_{\rm E}(\underline{w})$&nbsp; to the outer decoder.<br>
-*The outer decoder&nbsp; $(\mathcal{C}_1)$&nbsp; processes the apriori information&nbsp; $\underline{L}_{\rm A}(\underline{c})$, i.e. the extrinsic information&nbsp; $\underline{L}_{\rm E}(\underline{w})$&nbsp; after de&ndash;interleaving.&nbsp; It provides the extrinsic information&nbsp; $\underline{L}_{\rm E}(\underline{c})$.<br>
+*The outer decoder&nbsp; $(\mathcal{C}_1)$&nbsp; processes the a-priori information&nbsp; $\underline{L}_{\rm A}(\underline{c})$, i.e. the extrinsic information&nbsp; $\underline{L}_{\rm E}(\underline{w})$&nbsp; after de&ndash;interleaving.&nbsp; It provides the extrinsic information&nbsp; $\underline{L}_{\rm E}(\underline{c})$.<br>
 *After a sufficient number of iterations,&nbsp;  the desired decoding result is obtained in the form of the a-posteriori&nbsp; $L$&ndash;values&nbsp; $\underline{L}_{\rm APP}(\underline{u})$&nbsp; of the information sequence&nbsp; $\underline{u}$.<br><br>

@@ Line 169: / Line 169: @@
 * the information sequence&nbsp; $\underline{u}$&nbsp; into blocks of&nbsp; $K$&nbsp; information bits each,&nbsp; and<br>
-*the code sequence&nbsp; $\underline{x}$&nbsp; to blocks with each&nbsp; $N = 3 \cdot K$&nbsp; encoded bits.<br><br>
+*the encoded sequence&nbsp; $\underline{x}$&nbsp; to blocks with each&nbsp; $N = 3 \cdot K$&nbsp; encoded bits.<br><br>
 All results apply to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN channel}$]]. Data are taken from the lecture notes&nbsp; [Liv15]<ref>Liva, G.:&nbsp; Channels Codes for Iterative Decoding.&nbsp; Lecture notes, Department of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2015.</ref>.
@@ Line 214: / Line 214: @@
 *The information sequence&nbsp; $\underline{u}$&nbsp; is located at the outer convolutional encoder&nbsp; $\mathcal{C}_1$.&nbsp; Let its output sequence be&nbsp; $\underline{c}$. <br>
-*After the interleaver&nbsp; $(\Pi)$&nbsp; follows the inner convolutional encoder&nbsp; $\mathcal{C}_2$.&nbsp; The code sequence is called&nbsp; $\underline{x}$&nbsp;.<br>
+*After the interleaver&nbsp; $(\Pi)$&nbsp; follows the inner convolutional encoder&nbsp; $\mathcal{C}_2$.&nbsp; The encoded sequence is called&nbsp; $\underline{x}$&nbsp;.<br>
 *The resulting code rate is&nbsp; $R = R_1 \cdot R_2$.&nbsp; For rate&nbsp; $1/2$&nbsp; component codes: &nbsp; $R = 1/4$.<br><br>

@@ Line 46: / Line 46: @@
 {{GraueBox|TEXT=
-[[File:EN_KC_T_4_3_S1d_v1.png|right|frame|Example sequences at the rate&nbsp; $1/3$&nbsp; turbo encoder '''KORREKTUR: verwuerfelte Informationssequenz'''|class=fit]]
+[[File:EN_KC_T_4_3_S1d_v2.png|right|frame|Example sequences at the rate&nbsp; $1/3$&nbsp; turbo encoder|class=fit]]
 $\text{Example 1:}$&nbsp; The graph shows the different sequences in matched colors. To note:
 #For&nbsp; $\underline{u}_{\Pi}$&nbsp; a&nbsp; $3×4$&nbsp; interleaver matrix is considered according to&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving|"Exercise 4.8Z"]].<br><br>
@@ Line 130: / Line 130: @@
 Assumed is a rate&nbsp; $1/3$&nbsp; turbo code according to the description in the&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Basic_structure_of_a_turbo_code| "first section of this page"]].&nbsp; Also,&nbsp; the color scheme for the information sequence&nbsp; $\underline{u}$&nbsp; and the two parity sequences&nbsp; $\underline{p}_1$&nbsp; and&nbsp; $\underline{p}_2$&nbsp; are adapted from the earlier graphics.&nbsp; Further,&nbsp; it should be noted:
-[[File:EN_KC_T_4_3_S4a_v2_neu_v2.png|right|frame|Iterative turbo decoder for rate&nbsp; $R = 1/3$ '''KORREKTUR: a-priori'''|class=fit]]
+[[File:EN_KC_T_4_3_S4a_v2.png|right|frame|Iterative turbo decoder for rate&nbsp; $R = 1/3$ |class=fit]]
 *The received vectors&nbsp; $\underline{y}_0,\hspace{0.15cm} \underline{y}_1,\hspace{0.15cm} \underline{y}_2$&nbsp; are real-valued and provide the respective soft information with respect to the information sequence&nbsp; $\underline{u}$&nbsp; and the sequences&nbsp; $\underline{p}_1$&nbsp; $($parity for encoder 1$)$&nbsp; and&nbsp; $\underline{p}_2$&nbsp; $($parity for encoder 2$)$.<br>

@@ Line 190: / Line 190: @@
 &rArr; &nbsp; We now try to explain the flatter drop in bit error probability with larger&nbsp; $E_{\rm B}/N_0$.&nbsp; This is called an&nbsp; &raquo;$\text{error floor}$&laquo;:
-#The reason for this asymptotically worse behavior with better channel&nbsp; $($in the example: &nbsp; ab&nbsp; $10 \cdot {\rm lg} \, E_{\rm B}/N_0 \ge 2 \ \rm dB)$&nbsp; is the period&nbsp; $P$&nbsp; of the encoder impulse response&nbsp; $\underline{g}$,&nbsp; as demonstrated in the section&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_requirement_for_turbo_codes: _Interleaving|$\rm Interleaving$]],&nbsp;  and explained  in the&nbsp; [[Aufgaben:Exercise_4.10:_Turbo_Encoder_for_UMTS_and_LTE|"Exercise 4.10"]]&nbsp; with examples. <br>
+#The reason for this asymptotically worse behavior with better channel&nbsp; $($in the example: &nbsp; from&nbsp; $10 \cdot {\rm lg} \, E_{\rm B}/N_0 \ge 2 \ \rm dB)$&nbsp; is the period&nbsp; $P$&nbsp; of the encoder impulse response&nbsp; $\underline{g}$,&nbsp; as demonstrated in the section&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_requirement_for_turbo_codes: _Interleaving|$\rm Interleaving$]],&nbsp;  and explained  in the&nbsp; [[Aufgaben:Exercise_4.10:_Turbo_Encoder_for_UMTS_and_LTE|"Exercise 4.10"]]&nbsp; with examples. <br>
 #For&nbsp; $m = 2$,&nbsp; the period is&nbsp; $P = 2^m -1 = 3$.&nbsp; Thus,&nbsp; for&nbsp; $\underline{u} = (1, 1, 1) &nbsp; &#8658; &nbsp; w_{\rm H}(\underline{u}) = 3$&nbsp;  the parity sequence is bounded: &nbsp; $\underline{p} = (1, 0, 1)$ &nbsp; &#8658; &nbsp; $w_{\rm H}(\underline{p}) = 2$ &nbsp; despite unbounded impulse response&nbsp; $\underline{g}$.<br>
 #The sequence &nbsp; $\underline{u} = (0, \ \text{...}\hspace{0.05cm} , \ 0, \ 1, \ 0, \ 0, \ 1, \ 0, \ \text{...}\hspace{0.05cm})$ &nbsp; &#8658; &nbsp; $U(D) = D^x \cdot (1 + D^P)$&nbsp; also leads to a small Hamming weight&nbsp; $w_{\rm H}(\underline{p})$&nbsp; at the output,&nbsp; which complicates the iterative decoding process.<br>

@@ Line 130: / Line 130: @@
 Assumed is a rate&nbsp; $1/3$&nbsp; turbo code according to the description in the&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Basic_structure_of_a_turbo_code| "first section of this page"]].&nbsp; Also,&nbsp; the color scheme for the information sequence&nbsp; $\underline{u}$&nbsp; and the two parity sequences&nbsp; $\underline{p}_1$&nbsp; and&nbsp; $\underline{p}_2$&nbsp; are adapted from the earlier graphics.&nbsp; Further,&nbsp; it should be noted:
-[[File:EN_KC_T_4_3_S4a_v2_neu_v2.png|right|frame|Iterative turbo decoder for rate&nbsp; $R = 1/3$|class=fit]]
+[[File:EN_KC_T_4_3_S4a_v2_neu_v2.png|right|frame|Iterative turbo decoder for rate&nbsp; $R = 1/3$ '''KORREKTUR: a-priori'''|class=fit]]
 *The received vectors&nbsp; $\underline{y}_0,\hspace{0.15cm} \underline{y}_1,\hspace{0.15cm} \underline{y}_2$&nbsp; are real-valued and provide the respective soft information with respect to the information sequence&nbsp; $\underline{u}$&nbsp; and the sequences&nbsp; $\underline{p}_1$&nbsp; $($parity for encoder 1$)$&nbsp; and&nbsp; $\underline{p}_2$&nbsp; $($parity for encoder 2$)$.<br>

@@ Line 42: / Line 42: @@
 #The code word sequence&nbsp; $\underline{x} = (\underline{X}_1, \underline{X}_2, \ \text{...}\hspace{0.05cm}  , \ \underline{X}_i, \ \text{...}\hspace{0.05cm} )$&nbsp; is generated.&nbsp;  To avoid confusion,&nbsp; the code words&nbsp; $\underline{X}_i = (u, \ p_1, \ p_2)$&nbsp; with capital letters were introduced in the last section.<br>
 #The encoders&nbsp; $\mathcal{C}_1$&nbsp; and&nbsp; $\mathcal{C}_2$&nbsp; are conceived&nbsp; $($at least in thought$)$&nbsp; as&nbsp; [[Theory_of_Stochastic_Signals/Digital_Filters| $\text{digital filters}$]]&nbsp; and are thus characterized by the&nbsp; [[Channel_Coding/Algebraic_and_Polynomial_Description#Application_of_the_D.E2.80.93transform_to_rate_.7F.27.22.60UNIQ-MathJax158-QINU.60.22.27.7F_convolution_encoders| $\text{transfer functions}$]]&nbsp; $G_1(D)$&nbsp; and&nbsp; $G_2(D)$.<br>
-#For various reasons &nbsp; &#8658; &nbsp; see&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_Requirement_f.C3. BCr_Turbocodes:_Interleaving| "two sections ahead"]]&nbsp; the input sequence of the second encoder &nbsp; &#8658; &nbsp; $\underline{u}_{\pi}$&nbsp; should be scrambled with respect to the sequence&nbsp; $\underline{u}$&nbsp; by an interleaver&nbsp; $(\Pi)$.<br>
+#For various reasons &nbsp; &#8658; &nbsp; see&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_requirement_for_turbo_codes:_Interleaving| "two sections ahead"]]&nbsp; the input sequence of the second encoder &nbsp; &#8658; &nbsp; $\underline{u}_{\pi}$&nbsp; should be scrambled with respect to the sequence&nbsp; $\underline{u}$&nbsp; by an interleaver&nbsp; $(\Pi)$.<br>
 #Thereby there is nothing against choosing both encoders the same:&nbsp; $G_1(D) = G_2(D) = G(D)$.&nbsp; Without interleaver the correction capability would be extremely limited.<br><br>

@@ Line 9: / Line 9: @@
 <br>
 All communications systems current today&nbsp; $($2017$)$,&nbsp; such as&nbsp;
-* [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_.C3.9CBERBLICK_ZUM_VIERTEN_HAUPTKAPITEL_.23|$\rm UMTS$]]&nbsp; $($"Universal Mobile Telecommunications System" &nbsp; &rArr; &nbsp; 3rd generation mobile communications$)$ and&nbsp;
+* [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm UMTS$]]&nbsp; $($"Universal Mobile Telecommunications System" &nbsp; &rArr; &nbsp; 3rd generation mobile communications$)$ and&nbsp;
 *[[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm LTE$]]&nbsp; $($"Long Term Evolution"&nbsp; &rArr; &nbsp; 4th generation mobile communications$)$
-use the concept of&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Symbol-wise_soft-in_soft-out_decoding|$\text{symbol-wise iterative decoding}$]].&nbsp; This  is directly related to the invention of&nbsp; &raquo;'''turbo codes'''&laquo;&nbsp; in 1993 by&nbsp; [https://en.wikipedia.org/wiki/Claude_Berrou $\text{Claude Berrou}$],&nbsp; [https://en.wikipedia.org/wiki/Alain_Glavieux $\text{Alain Glavieux}$]&nbsp; and&nbsp; [https://scholar.google.com/citations?user=-UZolIAAAAAJ $\text{Punya Thitimajshima}$]&nbsp; because it was only with these codes that the Shannon bound could be approached with reasonable decoding effort.<br>
+use the concept of&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Bit-wise_soft-in_soft-out_decoding|$\text{symbol-wise iterative decoding}$]].&nbsp; This  is directly related to the invention of&nbsp; &raquo;'''turbo codes'''&laquo;&nbsp; in 1993 by&nbsp; [https://en.wikipedia.org/wiki/Claude_Berrou $\text{Claude Berrou}$],&nbsp; [https://en.wikipedia.org/wiki/Alain_Glavieux $\text{Alain Glavieux}$]&nbsp; and&nbsp; [https://scholar.google.com/citations?user=-UZolIAAAAAJ $\text{Punya Thitimajshima}$]&nbsp; because it was only with these codes that the Shannon bound could be approached with reasonable decoding effort.<br>
 Turbo codes result from the parallel or serial concatenation of convolutional codes.&nbsp;

@@ Line 13: / Line 13: @@
-use the concept of&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Symbol-wise_soft-in_soft-out_decoding| "symbol-wise iterative decoding"]].&nbsp; This  is directly related to the invention of&nbsp; '''turbo codes'''&nbsp; in 1993 by&nbsp; [https://en.wikipedia.org/wiki/Claude_Berrou $\text{Claude Berrou}$],&nbsp; [https://en.wikipedia.org/wiki/Alain_Glavieux $\text{Alain Glavieux}$]&nbsp; and&nbsp; [https://scholar.google.com/citations?user=-UZolIAAAAAJ $\text{Punya Thitimajshima}$]&nbsp; because it was only with these codes that the Shannon bound could be approached with reasonable decoding effort.<br>
+use the concept of&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Symbol-wise_soft-in_soft-out_decoding|$\text{symbol-wise iterative decoding}$]].&nbsp; This  is directly related to the invention of&nbsp; &raquo;'''turbo codes'''&laquo;&nbsp; in 1993 by&nbsp; [https://en.wikipedia.org/wiki/Claude_Berrou $\text{Claude Berrou}$],&nbsp; [https://en.wikipedia.org/wiki/Alain_Glavieux $\text{Alain Glavieux}$]&nbsp; and&nbsp; [https://scholar.google.com/citations?user=-UZolIAAAAAJ $\text{Punya Thitimajshima}$]&nbsp; because it was only with these codes that the Shannon bound could be approached with reasonable decoding effort.<br>
 Turbo codes result from the parallel or serial concatenation of convolutional codes.&nbsp;
@@ Line 48: / Line 48: @@
 [[File:EN_KC_T_4_3_S1d_v1.png|right|frame|Example sequences at the rate&nbsp; $1/3$&nbsp; turbo encoder|class=fit]]
 $\text{Example 1:}$&nbsp; The graph shows the different sequences in matched colors. To note:
-#For&nbsp; $\underline{u}_{\Pi}$&nbsp; a&nbsp; $3×4$&nbsp; interleaver matrix is considered according to&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving|$\text{Exercise 4.8Z}$]].<br><br>
+#For&nbsp; $\underline{u}_{\Pi}$&nbsp; a&nbsp; $3×4$&nbsp; interleaver matrix is considered according to&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving|"Exercise 4.8Z"]].<br><br>
-#The parity sequences are obtained according to&nbsp; $G_1(D) = G_2(D) = 1 + D^2$ &nbsp; &#8658; &nbsp; see&nbsp; [[Aufgaben:Exercise_4.08:_Repetition_to_the_Convolutional_Codes|$\text{Exercise 4.8}$]].}}<br>
+#The parity sequences are obtained according to&nbsp; $G_1(D) = G_2(D) = 1 + D^2$ &nbsp; &#8658; &nbsp; see&nbsp; [[Aufgaben:Exercise_4.08:_Repetition_to_the_Convolutional_Codes|"Exercise 4.8"]].}}<br>
 == First requirement for turbo codes: Recursive component codes ==
@@ Line 71: / Line 71: @@
-More details on the examples in this section can be found in the&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving|$\text{Exercise 4.8}$]]&nbsp; and the&nbsp; [[Aufgaben:Exercise_4.09:_Recursive_Systematic_Convolutional_Codes|$\text{Exercise 4.9}$]].<br>
+More details on the examples in this section can be found in the&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving|"Exercise 4.8"]]&nbsp; and the&nbsp; [[Aufgaben:Exercise_4.09:_Recursive_Systematic_Convolutional_Codes|"Exercise 4.9"]].<br>
 == Second requirement for turbo codes: Interleaving ==
@@ Line 109: / Line 109: @@
 :$$1, 9, 17, 25, 33, 41, 49, 57, 2, 10, 18, \ \text{...}\hspace{0.05cm} , 48, 56, 64.$$
-More information on block interleaving is available in the&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving| $\text{Exercise 4.8Z}$]].<br>
+More information on block interleaving is available in the&nbsp; [[Aufgaben:Exercise_4.08Z:_Basics_about_Interleaving|"Exercise 4.8Z"]].<br>
 [[File:P ID3050 KC T 4 3 S3b v5.png|left|frame|Clarification of $S$&ndash;random interleaving]]
@@ Line 138: / Line 138: @@
 *In the second decoder,&nbsp; the scrambling of the information sequence&nbsp; $\underline{u}$&nbsp; must be taken into account.&nbsp; Thus,&nbsp; the&nbsp; $L$ values to be processed are&nbsp; $\pi(L_{\rm K, \hspace{0.03cm}0})$&nbsp; and&nbsp; $L_{\rm K, \hspace{0.03cm}2}$.<br>
-*In the general&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Basic_structure_of_concatenated_coding_systems|"SISO decoder"]]&nbsp; the information exchange between the two component decoders was controlled with&nbsp; $\underline{L}_{\rm A, \hspace{0.03cm}2} = \underline{L}_{\rm E, \hspace{0.03cm}1}$&nbsp; and&nbsp; $\underline{L}_{\rm A, \hspace{0.03cm}1} = \underline{L}_{\rm E, \hspace{0.03cm}2}$.
+*In the general&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Basic_structure_of_concatenated_coding_systems|$\text{SISO decoder}$]]&nbsp; the information exchange between the two component decoders was controlled with&nbsp; $\underline{L}_{\rm A, \hspace{0.03cm}2} = \underline{L}_{\rm E, \hspace{0.03cm}1}$&nbsp; and&nbsp; $\underline{L}_{\rm A, \hspace{0.03cm}1} = \underline{L}_{\rm E, \hspace{0.03cm}2}$.
 *Written out at the bit level,&nbsp; these equations are denoted by&nbsp; $(1 &#8804; i &#8804; n)$:
@@ Line 190: / Line 190: @@
 &rArr; &nbsp; We now try to explain the flatter drop in bit error probability with larger&nbsp; $E_{\rm B}/N_0$.&nbsp; This is called an&nbsp; &raquo;$\text{error floor}$&laquo;:
-#The reason for this asymptotically worse behavior with better channel&nbsp; $($in the example: &nbsp; ab&nbsp; $10 \cdot {\rm lg} \, E_{\rm B}/N_0 \ge 2 \ \rm dB)$&nbsp; is the period&nbsp; $P$&nbsp; of the encoder impulse response&nbsp; $\underline{g}$,&nbsp; as demonstrated in the section&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_requirement_for_turbo_codes: _Interleaving|$\rm Interleaving$]],&nbsp;  and explained  in the&nbsp; [[Aufgaben:Exercise_4.10:_Turbo_Encoder_for_UMTS_and_LTE|$\text{Exercise 4.10}$]]&nbsp; with examples. <br>
+#The reason for this asymptotically worse behavior with better channel&nbsp; $($in the example: &nbsp; ab&nbsp; $10 \cdot {\rm lg} \, E_{\rm B}/N_0 \ge 2 \ \rm dB)$&nbsp; is the period&nbsp; $P$&nbsp; of the encoder impulse response&nbsp; $\underline{g}$,&nbsp; as demonstrated in the section&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_requirement_for_turbo_codes: _Interleaving|$\rm Interleaving$]],&nbsp;  and explained  in the&nbsp; [[Aufgaben:Exercise_4.10:_Turbo_Encoder_for_UMTS_and_LTE|"Exercise 4.10"]]&nbsp; with examples. <br>
 #For&nbsp; $m = 2$,&nbsp; the period is&nbsp; $P = 2^m -1 = 3$.&nbsp; Thus,&nbsp; for&nbsp; $\underline{u} = (1, 1, 1) &nbsp; &#8658; &nbsp; w_{\rm H}(\underline{u}) = 3$&nbsp;  the parity sequence is bounded: &nbsp; $\underline{p} = (1, 0, 1)$ &nbsp; &#8658; &nbsp; $w_{\rm H}(\underline{p}) = 2$ &nbsp; despite unbounded impulse response&nbsp; $\underline{g}$.<br>
 #The sequence &nbsp; $\underline{u} = (0, \ \text{...}\hspace{0.05cm} , \ 0, \ 1, \ 0, \ 0, \ 1, \ 0, \ \text{...}\hspace{0.05cm})$ &nbsp; &#8658; &nbsp; $U(D) = D^x \cdot (1 + D^P)$&nbsp; also leads to a small Hamming weight&nbsp; $w_{\rm H}(\underline{p})$&nbsp; at the output,&nbsp; which complicates the iterative decoding process.<br>
@@ Line 200: / Line 200: @@
 {{BlaueBox|TEXT=
-$\text{Conclusions:}$&nbsp; The exemplary shown bit error probability and block error probability curves also apply qualitatively for&nbsp; $m > 2$,&nbsp; e.g.&nbsp; for the  UMTS and LTE turbo codes&nbsp; $($each with&nbsp; $m = 3)$,&nbsp; which is analyzed in&nbsp; [[Aufgaben:Exercise_4.10:_Turbo_Encoder_for_UMTS_and_LTE|$\text{Exercise 4.10}$]].&nbsp; However,&nbsp; some quantitative differences emerge:
+$\text{Conclusions:}$&nbsp; The exemplary shown bit error probability and block error probability curves also apply qualitatively for&nbsp; $m > 2$,&nbsp; e.g.&nbsp; for the  UMTS and LTE turbo codes&nbsp; $($each with&nbsp; $m = 3)$,&nbsp; which is analyzed in&nbsp; [[Aufgaben:Exercise_4.10:_Turbo_Encoder_for_UMTS_and_LTE|"Exercise 4.10"]].&nbsp; However,&nbsp; some quantitative differences emerge:
 *The curve is steeper for small&nbsp; $E_{\rm B}/N_0$&nbsp; and the distance from the Shannon bound is slightly less than in the example shown here for&nbsp; $m = 2$.<br>
@@ Line 236: / Line 236: @@
 <br>
 [[File:EN_KC_T_4_3_S7b_v2.png|right|frame|Some standardized turbo codes compared to the Shannon bound]]
-Turbo codes are used in almost all newer communication systems.&nbsp;  The graph shows their performance with the AWGN channel compared to&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_capacity_of_the_AWGN_model| "Shannon's channel capacity"]]&nbsp; $($blue curve$)$.<br>
+Turbo codes are used in almost all newer communication systems.&nbsp;  The graph shows their performance with the AWGN channel compared to&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_capacity_of_the_AWGN_model|$\text{Shannon's channel capacity}$]]&nbsp; $($blue curve$)$.<br>
-The green highlighted area&nbsp;  "BPSK"&nbsp;  indicates the Shannon bound for digital systems with binary input,&nbsp;  with which according to the&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_coding_theorem_and_channel_capacity| "channel coding theorem"]]&nbsp; an error-free transmission is just possible.<br>
+The green highlighted area&nbsp;  "BPSK"&nbsp;  indicates the Shannon bound for digital systems with binary input,&nbsp;  with which according to the&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#Channel_coding_theorem_and_channel_capacity|$\text{channel coding theorem}$]]&nbsp; an error-free transmission is just possible.<br>
 It should be noted that the bit error rate&nbsp; $\rm BER= 10^{-5}$&nbsp; is the basis here for the channel codes of standardized systems which are drawn in,&nbsp;  while the information-theoretical capacity curves&nbsp;  $($Shannon,&nbsp;  BPSK$)$&nbsp;  apply to the error probability&nbsp; $0$.
@@ Line 245: / Line 245: @@
 *The red crosses mark the turbo codes according to&nbsp;  $\rm CCSDS$&nbsp;  $($"Consultative Committee for Space Data Systems"$)$,&nbsp;  developed for use in space missions.&nbsp;  This class assumes the fixed interleaver size&nbsp; $K = 6920$&nbsp; and provides codes of rate&nbsp; $1/6$,&nbsp; $1/4$,&nbsp; $1/3$&nbsp; and&nbsp; $1/2$.&nbsp; The lowest code rates allow operation at&nbsp; $10 \cdot {\rm lg} \, (E_{\rm B}/N_0) \approx 0 \ \rm dB$.<br>
-*The green circle represents a very simple&nbsp; "Repeat Accumulate"&nbsp; $\rm (RA)$&nbsp;  code,&nbsp;  a serial concatenated turbo code.&nbsp;  The following is an outline of its structure: &nbsp; The outer decoder uses a&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes| "repetition code"]],&nbsp; in the drawn example with rate&nbsp; $R = 1/3$.&nbsp;  The interleaver is followed by an RSC code with&nbsp; $G(D) = 1/(1 + D)$ &nbsp; &#8658; &nbsp; memory&nbsp; $m = 1$.&nbsp;  When executed systematically,&nbsp;  the total code rate is&nbsp; $R = 1/4$&nbsp; $($three parity bits are added to each bit of information$)$.
+*The green circle represents a very simple&nbsp; "Repeat Accumulate"&nbsp; $\rm (RA)$&nbsp;  code,&nbsp;  a serial concatenated turbo code.&nbsp;  The following is an outline of its structure: &nbsp; The outer decoder uses a&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|$\text{repetition code}$]],&nbsp; in the drawn example with rate&nbsp; $R = 1/3$.&nbsp;  The interleaver is followed by an RSC code with&nbsp; $G(D) = 1/(1 + D)$ &nbsp; &#8658; &nbsp; memory&nbsp; $m = 1$.&nbsp;  When executed systematically,&nbsp;  the total code rate is&nbsp; $R = 1/4$&nbsp; $($three parity bits are added to each bit of information$)$.