4 - Convolutional Codes

Abstract

This chapter focuses on convolutional codes. Convolutional code generates a codeword of n symbols from some consecutive message blocks of k symbols. A convolutional encoder consists of k m-stage shift registers containing message symbols and circuits that perform some linear function to generate the codeword. The message symbols are fed into the shift registers k symbols at a time. The contents of the shift registers are shifted each time a k-symbol block is fed in. The linear function generator generates an n-symbol block from both the contents of the shift registers and the k input symbols each time an input block is fed in. The number of message symbols from which an output block is generated is called the constraint length of the code. A convolutional code can be defined using a parity check matrix as well. The encoder of a convolutional code is a finite state machine and the contents of the shift registers represent its state. The error-correction capability of a convolutional code is measured by the free distance in most cases. The free distance of a convolutional code is the minimum Hamming distance between the semi-infinite code sequences generated by the encoder, which is equal to the minimum Hamming weight of the nonzero code sequences generated by the encoder because a convolutional code is linear.