A NOISE-SHAPING THEOREM

Noise shaping is a process which aims to remove as much quantization n oise as possible from the spectrum of a given bandlimited signal when quantizing it (recall that the spectrum of a signal is the support of its Fourier transform). We provide a mathematical analysis of such a p rocess, using methods of harmonic analysis on the unit disc. We are es pecially interested in conditions under which an analytic (but not nec essary polynomial) function may be used as a transfer function of the process. Stability conditions will be given under terms of metric char acteristics of the transfer function (stability means that the quantiz er is never overloaded). Several kinds of quantizer transfer functions will be considered, especially midtread and midriser ones. We shall r estrict our analysis to the case of deterministic (i.e., not random) s ignals. Some knowledge of Fourier analysis and signal processing will be presupposed. (C) 1996 Academic PRess, Inc.