ASYMPTOTIC BOUNDS ON OPTIMAL NOISY CHANNEL QUANTIZATION VIA RANDOM CODING

Asymptotically optimal zero-delay vector quantization in the presence of channel noise is studied using random coding techniques. First, an upper bound is derived for the average rth-power distortion of channel optimized k-dimensional vector quantization at transmission rate R on a binary symmetric channel with bit error probability epsilon. The up per bound asymptotically equals 2-rRg(epsilon,k,r), where k/(k + r) [1 - log2(1 + 2square-rootepsilon(1 - epsilon))] less-than-or-equal-to g (epsilon,k,r) less-than-or-equal-to 1 for all epsilon greater-than-or- equal-to 0, lim(epsilon --> 0) g(epsilon,k,r) = 1, and lim(k --> infin ity) g(epsilon,k,r) = 1. Numerical computations of g(epsilon,k,r) are also given. This result is analogous to Zador's asymptotic distortion rate of 2-rR for quantization on noiseless channels. Next, using a ran dom coding argument on nonredundant index assignments, a useful upper bound is derived in terms of point density functions, on the minimum m ean squared error of high resolution, regular, vector quantizers in th e presence of channel noise. The formula provides an accurate approxim ation to the distortion of a noisy channel quantizer whose codebook is arbitrarily ordered. Finally, it is shown that the minimum mean squar ed distortion of a regular, noisy channel VQ with a randomized nonredu ndant index assignment, is, in probability, asymptotically bounded awa y from zero.