Convolutional Codes

Abstract

Channel codes can be divided into two main categories. One is the block codes, and the other is the convolutional codes. In the previous chapters, we studied the block codes in details. In this chapter, we will explain different types of error-correcting codes which are convolutional codes. Convolutional codes as their names imply are types of codes based on the convolutional operation. There are fundamental differences between convolutional and block codes. For block codes, we have definite code-word lengths; however, for convolutional codes, the length of the code-words is not a fixed number. Convolutional encoder circuits are constructed using memory elements such as flip-flops. In this chapter, we provide information about convolutional encoding and decoding operations. The impulse responses of the convolutional encoders are inspected in details. We also considered the generator and parity check matrices of the convolutional codes. The Viterbi decoding of convolutional codes is explained in a clear manner.

Problems

• 1. Find the impulse responses of the convolutional encoder shown in Fig. P8.1, and find the output of the encoder for the input sequence d = [11010].

Fig. P8.1

Convolutional encoder for Problem 1

• 2. Find the impulse responses of the convolutional encoder shown in Fig. P8.2.

Fig. P8.2

Convolutional encoder for Problem 2

• 3. Find the transfer functions of the convolutional encoder shown in Fig. P8.3.

Fig. P8.3

Convolutional encoder for Problem 3

• 4. Find the generator matrix of the convolutional encoder shown in Fig. P8.4.

Fig. P8.4

Convolutional encoder for Problem 4

• 5. Find the transfer function of the convolutional encoder shown in Fig. P8.5.

Fig. P8.5

Convolutional encoder for Problem 5

• 6. Find the transfer function of the convolutional encoder shown in Fig. P8.6.

Fig. P8.6

Convolutional encoder for Problem 6

• 7. Find the generator matrix of the convolutional encoder shown in Fig. P8.7 in polynomial form.

Fig. P8.7

Convolutional encoder for Problem 7

• 8. Find the transfer function of the convolutional encoder shown in Fig. P8.8.

Fig. P8.8

Convolutional encoder for Problem 8

• 9. Find the transfer function of the convolutional encoder shown in Fig. P8.9.

Fig. P8.9

Convolutional encoder for Problem 9

• 10. Find the generator matrix of the convolutional encoder shown in Fig. P8.10 in polynomial form.

Fig. P8.10

Convolutional encoder for Problem 10

• 11. Draw the state diagram of the convolutional encoder shown in Fig. P8.11.

Fig. P8.11

Convolutional encoder for Problem 11

• 12. Draw the trellis diagram of the convolutional encoder shown in Fig. P8.12.

Fig. P8.12

Convolutional encoder for Problem 12

• 13. Considering the convolutional encoder given in Fig. P8.13:

  • (a) Obtain the state diagram.

  • (b) Draw the trellis diagram.

  • (c) Encode the data-word d = [101101] using the state diagram, and let c be the code-word obtained after encoding operation. Decode c using the Viterbi decoding algorithm.

Fig. P8.13

Convolutional encoder for Problem 13

• 14. Considering the convolutional encoder given in Fig. P8.14:

  • (a) Obtain the state diagram.

  • (b) Draw the trellis diagram.

  • (c) Encode the data-word d = [11011] using the state diagram, and let c be the code-word obtained after encoding operation. Decode c using the Viterbi decoding algorithm.

Fig. P8.14

Convolutional encoder for Problem 14