Mathematical Analysis of an Adaptive Quantizer

Vol. 53, No. 5, M a y - J u n e 1974 Printed in U.S.A. Mathematical Analysis of an Adaptive Quantizer By DEBASIS MITRA ( M a n u s c r i p t r e c e i v e d D e c e m b e r 4, 1973) This paper presents a mathematical analysis of an adaptive quantizer, a pulse code modulator, which is used for coding speech and other continuous signals with a large dynamic range into digital form. The device is a two-bit quantizer in which the step size is modified at every sampling instant with the object of adapting the range of the device to the intensity level of the signal. In the adaptation algorithm analyzed in the paper, the encoded information of the previous sampling instant is used either to increase or to decrease the step size by fixed, but not necessarily equal, proportions. Initially, the stochastic stability of the device is established by constructing a stochastic Liapunov f unction. Various basic identities and bounds on aspects of the behavior of the device are obtained. The qualitative results obtained indicate the nature of the trade-offs betiveen the quality of the steady state and the transient performance of the device. Also, formidas are developed for the purpose of evaluating the mean time required for the step size to adapt from arbitrary initial conditions to certain optimal values.