New Results From a Mathematical Study of an Adaptive Quantizer

In a recent paper 1 we obtained a number of fundamental properties of a class of two-bit (four-level) adaptive quantizers useful for coding speech and other continuous signals with a large dynamic range. We also developed formulas for the quantitative analysis of the device. In the present paper, we consider a general, multibit adaptive quantizer and obtain extensions to all the results previously reported. A feature new with this work is a unified treatment and a common body of results for quantizers with both bounded and unbounded range, the former being the case of practical interest. In the final section of the paper, Section IV, we present results from a computational investigation on adaptive quantizers up to four bits. 335 Readers familiar with quantizers and whose primary interest is in the performance of the device may skip the earlier sections that contain the development of the mathematical results. Section IV includes a comparison of the performances of uniform and nonuniform quantizers for normally distributed input sequences. A quantizer with 2N levels is shown in Fig. 1. In the figure, input refers to the nth sample of the continuous signal, x(n), where n = 0, 1, · · ·; output refers to the level that is coded before transmission at that time. We let = 1 and call A the step size.* In uniform quantizers, = i and the vertical axis is also subdivided into equal intervals in the range (771 A, tj^A). In adaptive quantizers which are of interest here, the step size, and hence the entire quantizer function, is timevariable, and the step size at the nth sampling instant is denoted by A(n).