Aufgaben:Exercise 3.8: Rate Compatible Punctured Convolutional Codes - Revision history

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 {{quiz-Header|Buchseite=Channel_Coding/Code_Description_with_State_and_Trellis_Diagram}}
-[[File:P_ID2708__KC_A_3_8.png|right|frame|RCPC Puncturing Matrices]]
+[[File:P_ID2708__KC_A_3_8.png|right|frame|RCPC puncturing matrices]]
 An important application for&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Punctured_convolutional_codes|$\text{punctured convolutional codes}$]]&nbsp; are the&nbsp; "Rate Compatible Punctured Convolutional Codes"&nbsp; $($for short:&nbsp; '''RCPC&ndash;Codes'''$)$&nbsp; proposed by Joachim Hagenauer in&nbsp; '''[Hag88]'''.

← Older revision		Revision as of 17:01, 14 November 2022
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	*The reference  '''[Hag88]''' refers to the paper  "Hagenauer, J.:  Rate Compatible Punctured Convolutional Codes (RCPC codes) and their Applications. In:  IEEE Transactions on Communications,  vol COM-36, pp 389 - 400, 1988."		*The reference  '''[Hag88]''' refers to the paper  "Hagenauer, J.:  Rate Compatible Punctured Convolutional Codes (RCPC codes) and their Applications. In:  IEEE Transactions on Communications,  vol COM-36, pp 389 - 400, 1988."

−	*Professor  [https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/\|$\text{Joachim Hagenauer}$]  was head of the Institute of Communications Engineering  $\rm(LNT)$  at the Technical University of Munich from 1993 to 2006.  The initiators of the learning tutorial  $($Günter Söder and Klaus Eichin$)$  thank their long-time boss for supporting and promoting our  $\rm LNTwww$  project during the first six years.	+	*Professor  [https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/ $\text{Joachim Hagenauer}$]  was head of the Institute of Communications Engineering  $\rm(LNT)$  at the Technical University of Munich from 1993 to 2006.  The initiators of the learning tutorial  $($Günter Söder and Klaus Eichin$)$  thank their long-time boss for supporting and promoting our  $\rm LNTwww$  project during the first six years.

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 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>solutions 1 and 2</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>solutions 1 and 2</u>:
-*The number of rows of the puncturing matrices gives the parameter $n$ of the $(n, \ k = 1)$&ndash;RCPC mother code.
+*The number of rows of the puncturing matrices gives the parameter&nbsp; $n$&nbsp; of the&nbsp; $(n, \ k = 1)$&nbsp; RCPC mother code.
-*From this, its rate is $R_0 = 1/3$. The column number is equal to the puncturing period $p$. For the class of codes under consideration, $p = 8$.
-* In contrast, the puncturing matrices do not provide any information about the memory of the code &nbsp;&#8658;&nbsp;
+*From this,&nbsp; its rate is&nbsp; $R_0 = 1/3$.&nbsp; The column number is equal to the puncturing period&nbsp; $p$.&nbsp; For the class of codes under consideration:&nbsp; $p = 8$.
-'''(2)'''&nbsp; For the rate of the code $\mathcal{C}_l = p/N_l$, where $N_l$ denotes the number of all ones in the puncturing matrix $\mathbf{P}_l$ and $p$ denotes the puncturing period. Starting from the rate $R_0 = 1/3$ of the mother code $\mathcal{C}_0$, we obtain:
 * $R_1 = 8/20 = 2/5 = \underline{0.400}$,
 * $R_2 = 8/16 = 1/2 = \underline{0.500}$,
@@ Line 81: / Line 85: @@
-'''(3)'''&nbsp; Correct are the <u>solutions 1 and 4</u>:
+'''(3)'''&nbsp; Correct are the&nbsp; <u>solutions 1 and&nbsp; 4</u>:
-*All ones in the matrix $\mathbf{P}_4$ are also in the matrices $\mathbf{P}_3, \hspace{0.05cm}\text{ ...} above it. \hspace{0.05cm}, \ \mathbf{P}_0$.
+*All ones in the matrix&nbsp; $\mathbf{P}_4$&nbsp; are also in the matrices&nbsp; $\mathbf{P}_3, \hspace{0.05cm}\text{ ...}  \hspace{0.05cm}, \ \mathbf{P}_0$.
 *In the matrix $\mathbf{P}_3$ three ones are added compared to $\mathbf{P}_4$, in the matrix $\mathbf{P}_2$ compared to $\mathbf{P}_3$ again four, etc.
 {{ML-Fuß}}
 [[Category:Channel Coding: Exercises|^3.3 State and Trellis Diagram^]]

@@ Line 32: / Line 32: @@
 Hints:
-* This exercise refers to the section&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Punctured_convolutional_codes|"punctured convolutional codes"]]&nbsp; in the chapter "Code description with state and trellis diagram".
+* This exercise refers to the section&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Punctured_convolutional_codes|"Punctured convolutional codes"]]&nbsp; in the chapter&nbsp; "Code description with state and trellis diagram".
 *The reference&nbsp; '''[Hag88]''' refers to the paper&nbsp; "Hagenauer, J.:&nbsp; Rate Compatible Punctured Convolutional Codes (RCPC codes) and their Applications. In:&nbsp; IEEE Transactions on Communications,&nbsp; vol COM-36, pp 389 - 400, 1988."

@@ Line 36: / Line 36: @@
 *The reference&nbsp; '''[Hag88]''' refers to the paper&nbsp; "Hagenauer, J.:&nbsp; Rate Compatible Punctured Convolutional Codes (RCPC codes) and their Applications. In:&nbsp; IEEE Transactions on Communications,&nbsp; vol COM-36, pp 389 - 400, 1988."
-*Professor&nbsp; [https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/|$\text{Joachim Hagenauer}$]&nbsp; was head of the Institute of Communications Engineering&nbsp; $\rm(LNT)$&nbsp; at the Technical University of Munich from 1993 to 2006.
+*Professor&nbsp; [https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/|$\text{Joachim Hagenauer}$]&nbsp; was head of the Institute of Communications Engineering&nbsp; $\rm(LNT)$&nbsp; at the Technical University of Munich from 1993 to 2006.&nbsp; The initiators of the learning tutorial&nbsp; $($Günter Söder and Klaus Eichin$)$&nbsp; thank their long-time boss for supporting and promoting our&nbsp; $\rm LNTwww$&nbsp; project during the first six years.
@@ Line 53: / Line 51: @@
 - The memory of the RCPC code class is&nbsp; $M = 4$.
-{Which code rates do the codes&nbsp; $\mathcal{C}_1$, ... , $\mathcal{C}_4$&nbsp; have?
+{Which code rates do the codes &nbsp; $\mathcal{C}_1$, ... , $\mathcal{C}_4$ &nbsp; have?
 |type="{}"}
-${\rm Matrix \ P_1} \Rightarrow {\rm Code \ \mathcal{C}_1} \text{:} \hspace{0.4cm} R_1 \ = \ ${ 0.4 3% }
+${\rm matrix \ P_1} \ \Rightarrow \  {\rm code \ \mathcal{C}_1} \text{:} \hspace{0.4cm} R_1 \ = \ ${ 0.4 3% }
-${\rm Matrix \ P_2} \Rightarrow {\rm Code \ \mathcal{C}_2} \text{:} \hspace{0.4cm}R_2 \ = \ ${ 0.5 3% }
+${\rm matrix \ P_2} \ \Rightarrow       ß {\rm code \ \mathcal{C}_2} \text{:} \hspace{0.4cm}R_2 \ = \ ${ 0.5 3% }
-${\rm Matrix \ P_3} \Rightarrow {\rm Code \ \mathcal{C}_3} \text{:} \hspace{0.4cm} R_3 \ = \ ${ 0.667 3% }
+${\rm matrix \ P_3} \ \Rightarrow \ {\rm code \ \mathcal{C}_3} \text{:} \hspace{0.4cm} R_3 \ = \ ${ 0.667 3% }
-${\rm Matrix \ P_4} \Rightarrow {\rm Code \ \mathcal{C}_4} \text{:} \hspace{0.4cm} R_4 \ = \ ${ 0.889 3% }
+${\rm matrix \ P_4} \ \Rightarrow \ {\rm code \ \mathcal{C}_4} \text{:} \hspace{0.4cm} R_4 \ = \ ${ 0.889 3% }
-{Which statements are valid for the matrix elements $P_{ij}^{(l)}$?
+{Which statements are valid for the matrix elements&nbsp; $P_{ij}^{(l)}$?
 |type="[]"}
 + From&nbsp; $P_{ij}^{(l)} = 1$&nbsp; follows&nbsp; $P_{ij}^{(\lambda)} = 1$&nbsp; for all&nbsp; $\lambda < l$.

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 [[File:P_ID2708__KC_A_3_8.png|right|frame|RCPC Puncturing Matrices]]
-An important application for&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Punctured_convolutional_codes|"punctured convolutional codes"]]&nbsp; are the&nbsp; <i>Rate Compatible Punctured Convolutional Codes</i>&nbsp; (or '''RCPC&ndash;Codes''' for short)proposed by&nbsp; [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|"Joachim Hagenauer"]]&nbsp; in [Hag88].  Starting from a mother code&nbsp; $\mathcal{C}_0$&nbsp; with rate&nbsp; $R_0 = 1/n$&nbsp; other codes&nbsp; $\mathcal{C}_l$&nbsp; with higher code rate&nbsp; $(R_l > R_0)$ are determined by different puncturing matrices&nbsp; $\mathbf{P}_l$&nbsp;.
+An important application for&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Punctured_convolutional_codes|$\text{punctured convolutional codes}$]]&nbsp; are the&nbsp; "Rate Compatible Punctured Convolutional Codes"&nbsp; $($for short:&nbsp; '''RCPC&ndash;Codes'''$)$&nbsp; proposed by Joachim Hagenauer in&nbsp; '''[Hag88]'''.
-On the right are the puncturing matrices to be analyzed&nbsp; $\mathbf{P}_0, \hspace{0.05cm}\text{ ...} \hspace{0.05cm} , \ \mathbf{P}_4$&nbsp; shown.
+Starting from a mother code &nbsp; $\mathcal{C}_0$ &nbsp; with rate&nbsp; $R_0 = 1/n$,&nbsp; other codes &nbsp; $\mathcal{C}_l$ &nbsp; with higher code rate&nbsp; $(R_l > R_0)$&nbsp; are determined by different puncturing matrices&nbsp; $\mathbf{P}_l$.
 The RCPC&ndash;codes are well suited for the realization of
-* <i>unequal error protection</i>&nbsp; for hybrid ARQ procedures,
+* "unequal error protection"&nbsp; for hybrid ARQ procedures,
+* systems with&nbsp; "incremental redundancy".&nbsp; This means that after conventional convolutional coding,&nbsp; bits corresponding to the puncturing matrix&nbsp; $\mathbf{P}_l$&nbsp; are omitted from the code word&nbsp; $\underline{x}^{(0)}$&nbsp; and the shortened code word&nbsp; $\underline{x}^{(l)}$&nbsp; is transmitted:
-Incremental redundancy systems" means that after conventional convolutional coding, bits corresponding to the puncturing matrix&nbsp; $\mathbf{P}_l$&nbsp; are omitted from the code word&nbsp; $\underline{x}^{(0)}$&nbsp; and the shortened code word&nbsp; $\underline{x}^{(l)}$&nbsp; is transmitted:
+:*If the punctured code word cannot be correctly decoded in the receiver,&nbsp; the receiver requests further redundancy from the transmitter in the form of the previously punctured bits.
-*If the punctured code word cannot be correctly decoded in the receiver, the receiver requests further redundancy from the transmitter in the form of the previously punctured bits.
+:*This prevents the transmission of redundancy that is not required and adapts the throughput to the channel conditions.
@@ Line 33: / Line 34: @@
 * This exercise refers to the section&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Punctured_convolutional_codes|"punctured convolutional codes"]]&nbsp; in the chapter "Code description with state and trellis diagram".
-*The reference [Hag88] refers to the paper "Hagenauer, J.: ''Rate Compatible Punctured Convolutional Codes (RCPC codes) and their Applications''. In: IEEE Transactions on Communications, vol COM-36, pp 389 - 400, 1988."
+*The reference&nbsp; '''[Hag88]''' refers to the paper&nbsp; "Hagenauer, J.:&nbsp; Rate Compatible Punctured Convolutional Codes (RCPC codes) and their Applications. In:&nbsp; IEEE Transactions on Communications,&nbsp; vol COM-36, pp 389 - 400, 1988."
-*Professor&nbsp; [[Biographies_and_Bibliographies/Chair_of_LNT#Prof._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29| Joachim Hagenauer]]&nbsp; was head of the Department of Communications Engineering (LNT) at the Technical University of Munich from 1993 to 2006.
-*The initiators of the learning tutorial you just used &ndash; Günter Söder and Klaus Eichin &ndash; thank their long-time boss for supporting and promoting our&nbsp; $\rm LNTwww$ project during the first six years.
+*Professor&nbsp; [[Biographies_and_Bibliographies/Chair_of_LNT#Prof._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|$\text{Joachim Hagenauer}$]]&nbsp; was head of the Institute of Communications Engineering&nbsp; $\rm(LNT)$&nbsp; at the Technical University of Munich from 1993 to 2006.