Distortion bounds for vector quantizers with finite codebook size

Upper and lower bounds are presented for the distortion of the optimal N-point vector quantizer applied to k-dimensional signals. Under certain smoothness conditions on the source distribution, the bounds are shown to hold for each and every value of N, the codebook size. These results extend bounds derived in the high-resolution limit, which assume that the number of code vectors is arbitrarily large. Two approaches to the upper bound are presented. The first, constructive construction, achieves the correct asymptotic rate of convergence as well as the correct dependence on the source density, although leading to an inferior value for the constant. The second construction, based on a random coding argument, is shown to additionally achieve a value of the constant which is much closer to the best known result derived within the asymptotic theory. Lower bound results derived in the correspondence are again shown to possess the correct asymptotic form and yield a constant which is almost indistinguishable from the best value achieved in the asymptotic regime. Finally, application of the results to the problem of source coding yields upper bounds on the distortion rate function for a wide class of processes.