Quantizers for the gamma distribution and other symmetrical distributions

This paper discusses minimum mean-square error quantization for symmetric distributions. If the distribution satisfies a log-concavity condition, the optimal quantizer is itself symmetric. For the gamma distribution often used to model speech signals, the log-concavity condition is not satisfied. It is shown that for this distribution both the uniformly spaced and the nonuniformly spaced optimal quantizers are not symmetrical for even numbers of quantizer levels. New quantization tables giving the optimal levels for quantizers for the gamma distribution are presented. A simple family of symmetric distributions is also examined. This family shows that as the distribution gets concentrated near the point of symmetry, nonsymmetric solutions become optimal.