Optimum PAM-TCM schemes using left-circulate function over GF(2N)

In this paper, optimum recursive systematic convolutional (RSC) encoders over Galois field GF(2^N) are designed using a nonlinear function, i.e., left-circulate function (LCIRC). The LCIRC function performs a bit left circulation over the representation word; it is used in microprocessors as an accumulator operation, and in chaotic sequence generators working in finite precision. 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 optimum encoders 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 a quaternary pulse amplitude modulation - trellis-coded modulation (PAM-TCM) transmission over an additive white Gaussian noise (AWGN) channel. The simulation results confirm the expected coding gains determined theoretically.