Optimum GF(2N) encoders using left-circulate function for PSK-TCM schemes

Phase shift keying - trellis coded modulation (PSK-TCM) schemes are designed using recursive systematic convolutional (RSC) 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) RSC 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.