Punctured 8-PSK Turbo-TCM transmissions using recursive systematic convolutional GF(2N) encoders

A new Galois field GF(2
^N
) structure for designing recursive systematic convolutional (RSC) encoders is presented in this article. The proposed RSC encoders are used as components of a parallel-concatenated Turbo trellis coded modulation (TTCM) scheme. The GF(2
^N
) RSC encoders include a nonlinear function named left-circulate (LCIRC), which is performing a left circulation over the N bits representation word. A closed-form expression was found for the minimum Euclidian distance of rate (N − 1) =N LCIRC-RSC encoders for (2
^N
)-ary PSK-TCM, and its values show that these RSC-LCIRC encoders are optimum, having the same performances as the corresponding conventional binary encoders. However, the RSC-LCIRC encoders are less complex than the corresponding binary encoders for a given encoding rate. TTCM transmission schemes with optimum RSC-LCIRC component encoders are investigated using the iterative symbol-by-symbol log-MAP decoding algorithm and symbol puncturing. The bit error rate (BER) is estimated by simulation for the proposed TTCM scheme with 8-PSK modulation when transmitting over additive white Gaussian noise (AWGN) channel.