Near Optimal Decoding for TCM Using the BIVA and Trellis Shaping
AAECC-13 Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
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We derive an upper bound on the bit-error probability (BEP) in limited-search detection over a finite interference channel. A unified channel model is presented; this includes finite-length intersymbol interference channels and multiuser CDMA channels as two special cases. We show that the BEP of the M-algorithm (MA) is bounded from above by the sum of three terms: an upper bound on the error probability of the Viterbi (1967) algorithm (VA) detection given by Forney Jr. (1972), and upper bounds on the error probabilities of two types of erroneous decision caused by the correct path loss event. We prove that error propagation (in terms of the mean recovery step number) is finite for all finite interference channels. The convergence and asymptotic behavior of the upper bounds are studied. The results show that, if a channel satisfies certain mild conditions, all series in the bounds are convergent. One of the key results is that, for any finite interference channel satisfying certain mild conditions. the asymptotic BEP of the MA is bounded by the same upper and lower bounds (which have the same asymptotical behavior) as those for the VA if the correct path loss probability is smaller than that of the VA. Furthermore, we extend the above results to near optimally decode long convolutional codes in a short packet format (about 200-300 bits). We present a nonsorting combined M/T algorithm and showed that the M/T algorithm with M>2( dfree/n) and T>(dfreeEb)/n can near-optimally decode the code. We also propose a hierarchical decoding algorithm (HDA) to further cut down the average decoding complexity. Numerical results show that the bounds are reasonably tight. The HDA can achieve a performance within about 0.8 dB of the sphere-packing lower bound for a packet error rate of 10-4 and a packet length below 200 bits, which is the best reported decoding performance so far for block sizes from 100 to 200 bits