Recursive GF(2 N) encoders using left-circulate function for optimum PSK–TCM schemes

Abstract

In this paper, phase shift keying–trellis coded modulation (PSK–TCM) schemes are designed using recursive convolutional encoders over Galois field GF(2 N ). These encoders are designed using the nonlinear left-circulate (LCIRC) function. The LCIRC function performs a bit left circulation over the representation word. Different encoding rates are obtained for these encoders when using different representation wordlengths at the input and the output, denoted as N in and N, respectively. A generalized 1-delay GF(2 N ) encoder scheme using LCIRC is proposed for performance analysis and optimization, for any possible encoding rate, N in / N. The minimum Euclidian distance is estimated for these PSK–TCM schemes and a general expression is found as a function of the wordlengths N in and N. The symbol error rate (SER) is estimated by simulation for PSK–TCM transmissions over an additive white Gaussian noise (AWGN) channel.