High Rate Quantization

Abstract

In this chapter we find bounds and approximations to the performance of memoryless vector quantizers or block source codes under the assumption that the rate of the code is high (and hence the distortion is low). These approximations are also referred to as asymptotic quantization approximations (asymptotic in the sense of large rate) and fine quantization. This approach contrasts with the usual Shannon theory in that we do not explicitly define an information theoretic optimization and then relate it to the performance of deterministic codes via a coding theorem. Instead we assume a certain asymptotic structure to a deterministic code and use basic calculus approximations to find the resulting distortion, rate, and entropy. The results so obtained do relate to the Shannon rate distortion functions and we close with some such comparisons. The theory will be presented in a less rigorous style than the Shannon theory because being precise would require an unpleasant amount of detailed integration theory.