Aufgaben:Exercise 3.8: Rate Compatible Punctured Convolutional Codes

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RCPC Puncturing Matrices

An important application for  $\text{punctured convolutional codes}$  are the  "Rate Compatible Punctured Convolutional Codes"  $($for short:  RCPC–Codes$)$  proposed by Joachim Hagenauer in  [Hag88].

Starting from a mother code   $\mathcal{C}_0$   with rate  $R_0 = 1/n$,  other codes   $\mathcal{C}_l$   with higher code rate  $(R_l > R_0)$  are determined by different puncturing matrices  $\mathbf{P}_l$.

The puncturing matrices   $\mathbf{P}_0, \hspace{0.05cm}\text{ ...} \hspace{0.05cm} , \ \mathbf{P}_4$  to be analyzed are shown on the right.

• If for the matrix  $\mathbf{P}_l$  the matrix element  $P_{ij} = 1$,  the corresponding code bit is transmitted,  while  $P_{ij} = 0$  indicates puncturing.

• In the questionnaire,  we also use the shorter notation  $P_{ij}^{(l)}$  for the element  $P_{ij}$  of the matrix  $\mathbf{P}_l$.

• In the graph:  All the zeros in the matrix  $\mathbf{P}_l$  that were still ones in the matrix  $\mathbf{P}_{l–1}$  are marked in red.  This measure increases the code rate  $R_{l}$  compared to  $R_{l-1}$.

The RCPC–codes are well suited for the realization of

• "unequal error protection"  for hybrid ARQ procedures,

• systems with  "incremental redundancy".  This means that after conventional convolutional coding,  bits corresponding to the puncturing matrix  $\mathbf{P}_l$  are omitted from the code word  $\underline{x}^{(0)}$  and the shortened code word  $\underline{x}^{(l)}$  is transmitted:

• 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.

Joachim Hagenauer

Hints:

• This exercise refers to the section  "punctured convolutional codes"  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."

• Professor  $\text{Joachim Hagenauer}$  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.

Questions

1

What statements do the given puncturing matrices provide?

	The rate of the RCPC mother code is  $R_0 = 1/3$.
	The puncturing period is  $p = 8$.
	The memory of the RCPC code class is  $M = 4$.

2

Which code rates do the codes  $\mathcal{C}_1$, ... , $\mathcal{C}_4$  have?

${\rm Matrix \ P_1} \Rightarrow {\rm Code \ \mathcal{C}_1} \text{:} \hspace{0.4cm} R_1 \ = \ $

${\rm Matrix \ P_2} \Rightarrow {\rm Code \ \mathcal{C}_2} \text{:} \hspace{0.4cm}R_2 \ = \ $

${\rm Matrix \ P_3} \Rightarrow {\rm Code \ \mathcal{C}_3} \text{:} \hspace{0.4cm} R_3 \ = \ $

${\rm Matrix \ P_4} \Rightarrow {\rm Code \ \mathcal{C}_4} \text{:} \hspace{0.4cm} R_4 \ = \ $

3

Which statements are valid for the matrix elements $P_{ij}^{(l)}$?

	From  $P_{ij}^{(l)} = 1$  follows  $P_{ij}^{(\lambda)} = 1$  for all  $\lambda < l$.
	From  $P_{ij}^{(l)} = 1$  follows  $P_{ij}^{(\lambda)} = 1$  for all  $\lambda > l$.
	From  $P_{ij}^{(l)} = 0$  follows  $P_{ij}^{(\lambda)} = 0$  for all  $\lambda < l$.
	From  $P_{ij}^{(l)} = 0$  follows  $P_{ij}^{(\lambda)} = 0$  for all  $\lambda > l$.

Solution

Solution

(1)  Correct are solutions 1 and 2:

• The number of rows of the puncturing matrices gives the parameter $n$ of the $(n, \ k = 1)$–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  ⇒ 

(2)  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:

• $R_1 = 8/20 = 2/5 = \underline{0.400}$,
• $R_2 = 8/16 = 1/2 = \underline{0.500}$,
• $R_3 = 8/12 = 2/3 = \underline{0.667}$,
• $R_4 = 8/9 = \underline{0.889}$.

(3)  Correct are the solutions 1 and 4:

• All ones in the matrix $\mathbf{P}_4$ are also in the matrices $\mathbf{P}_3, \hspace{0.05cm}\text{ ...} above it. \hspace{0.05cm}, \ \mathbf{P}_0$.
• 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.