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 ε. The upper bound asymptotically equals 2^-rRg(ε,k,r). where k/(k +r) [1 - log₂(l +2√(ε(1-ε))] ⩽g(ε,k,r)⩽1) for all ε⩾0, lim_ε→0 g(ε,k,r)=1, and lim_k→∞g(ε,k,r)=1. Numerical computations of g(ε,k,r) are also given. This result is analogous to Zador´s (1982) asymptotic distortion rate of 2^-rR for quantization on noiseless channels. Next, using a random coding argument on nonredundant index assignments, a useful upper bound is derived in terms of point density functions, on the minimum mean squared error of high resolution, regular, vector quantizers in the presence of channel noise. The formula provides an accurate approximation to the distortion of a noisy channel quantizer whose codebook is arbitrarily ordered. Finally, it is shown that the minimum mean squared distortion of a regular, noisy channel VQ with a randomized nonredundant index assignment, is, in probability, asymptotically bounded away from zero