Maximum-likelihood detection of uniform convolutional codes

Abstract

An algebraic decoding algorithm for the maximum-likelihood detection of uniform convolutional codes is developed. Coded signals are assumed transmitted over a coherent PSK channel which is corrupted by additive white Gaussian noise. This algorithm makes use of previously decoded information bits in order to reduce the size and complexity of the decoder. The probability of making a first decoding error is calculated for codes of one-half and one-fourth redundancy with the aid of an IBM 360 digital computer. This is accomplished, for the one-fourth redundancy code, by numerical integration of a four-dimensional Gaussian density function; for the one-half redundancy code, integration of a two-dimensional Gaussian density function is required. The use of decoded information bits by the decoder is shown to cause high probability for additional errors whenever a first decoding error occurs. The average number of decoding errors, which are triggered by a first decoding error, is found by applying the theory of finite Markov chains. This result is used, along with the first error probability, to determine the error rate of the code. For an information-bit-energy-to-noise-power-density ratio equal to 1.0, the error rate of the one-half redundancy code is found to be twice that of uncoded coherent PSK, while the error rate of the one-fourth redundancy code is about three and one-half times that of the uncoded case. However, at an information-bit-energy-to-noise-power-density ratio of 10.0, the one-fourth redundancy code is found to have an error rate of about one-third that of the uncoded case, while the one-half redundancy code has an error rate which remains at twice the value of the uncoded case. An experimental decoder for the one-fourth redundancy code is built and tested in the laboratory along with a decoder for uncoded coherent PSK. Experimental results of the coded and uncoded cases are compared and found to show relationships similar those calculated from theory.