Optimum quantization of signals†

Abstract

Criteria for optimum quantization with minimum mean square error of speech and random signals obeying negative exponential and normal amplitude distributions are derived and discussed. The treatment takes into account the error probabilities when the signal amplitude exceeds the finite range of quantization, with emphasis on obtaining certain general conclusions from analytical approaches. It was found that the quantization error has a sharp minimum at an optimum ratio of range to signal r.m.s. amplitude for a fixed number of uniform quantizing levels. The value of this optimum range varies between two and seven times the r.m.s. amplitude, depending on the number of quantizing levels and signal amplitude statistics. By introducing instantaneous companding according to certain variational solutions pertaining to the particular amplitude statistics, it is possible to achieve further lower error-to-signal power ratio over wide variations of signal strength. It is demonstrated quantitatively that these desired quantizer performance characteristics are only partially realized by the commonly used logarithmic companding techniques.