Optimum QAM - TCM Schemes Using Left - Circulate Function over GF ( 2 N )

In this paper, quadrature amplitude modulation – trellis coded modulation (QAM-TCM) schemes are designed using recursive convolutional (RC) encoders over Galois field GF(2 N ). These encoders are designed using the nonlinear leftcirculate (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 1delay GF(2 N ) RC 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 QAM-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 QAM-TCM transmissions over an additive white Gaussian noise (AWGN) channel. The proposed encoders outperform the corresponding binary encoders in terms of structural complexity and the availability of a general expression for the Euclidian distance. Keywords-Recursive convolutional GF(2 N ) encoders; Leftcirculate function; QAM-TCM.