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

In this paper, phase shift keying–trellis coded modulation (PSK–TCM) schemes are designed using recursive convolutional encoders over Galois field GF(2N). 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 Nin and N, respectively. A generalized 1-delay GF(2N) encoder scheme using LCIRC is proposed for performance analysis and optimization, for any possible encoding rate, Nin/N. The minimum Euclidian distance is estimated for these PSK–TCM schemes and a general expression is found as a function of the wordlengths Nin and N. The symbol error rate (SER) is estimated by simulation for PSK–TCM transmissions over an additive white Gaussian noise (AWGN) channel.