Optimum PAM-TCM schemes using left-circulate function over GF(2<sup>N</sup>)

Abstract

In this paper, optimum recursive systematic convolutional (RSC) encoders over Galois field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> ) 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> and N, respectively. A generalized 1-delay GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> ) RSC encoder scheme using LCIRC is proposed for performance analysis and optimization, for any possible encoding rate, N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> /N. The minimum Euclidian distance is estimated for these optimum encoders and a general expression is found as a function of the wordlengths N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> 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.