Bounds on the a priori index crossover probabilities for trellis-based channel codes

This paper derives truncated union bounds on the a priori index crossover probabilities p(j|i) that result when an n-bit data index i is convolutionally encoded, transmitted over a noisy channel, and decoded with the Viterbi algorithm, giving received index j. The bounds are derived with a modified transfer function technique, using n-stage state transition matrices with symbolic labels. The technique is easily automated with commercial symbolic algebra packages. Bounds are obtained for convolutional and trellis-coded modulation (TCM) codes, over binary symmetric and additive white Gaussian noise (AWGN) channels. A joint source channel coding example demonstrates that the bounds on p(j|i) developed in this paper can give a 13-dB accuracy improvement in end-to-end signal-to-noise ratio (SNR) predictions, when compared to predictions based on bounds on the delivered bit error probability P_b