Bounds on Quantizer Performance in the Low Bit-Rate Region

This paper investigates the optimum entropy versus distortion performance of quantizers optimized for uniform, Gaussian, Laplacian, and gamma-distributed memoryless sources which are useful models of the quantizer input signals in speech or picture coding schemes. We list the maximally obtainable signal-to-quantization noise ratios for one-dimensional optimum (i.e. entropy-coded) quantizers in the important low bit-rate region. These results have been obtained by an iterative solution of a set of nonlinear equations. Additionally we have also computed the corresponding rate-distortion functions by employing the Blahut-algorithm. These latter results upperbound the performances of multi-dimensional quantization schemes, and a comparison with the former results indicates the penalty to be paid for restricting a coder to perform a one-dimensionai quantization. It will be shown that the differences can be significant in the low bit-rate region.