# Exercise 3.8: Rate Compatible Punctured Convolutional Codes

An important application for  "punctured convolutional codes"  are the  Rate Compatible Punctured Convolutional Codes  (or RCPC–Codes for short)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  $\mathbf{P}_l$ .

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

• If for the matrix  $\mathbf{P}_l$  the matrix element  $P_{ij} = 1$, the corresponding codebit 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.

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

• If the punctured codeword 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.

Hints:

• 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  Joachim Hagenauer  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 – 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.