A Noise-Shaping Theorem

Abstract

Noise shaping is a process which aims to remove as much quantization noise 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 process, using methods of harmonic analysis on the unit disc. We are especially interested in conditions under which an analytic (but not necessary polynomial) function may be used as a transfer function of the process. Stability conditions will be given under terms of metric characteristics of the transfer function (stability means that the quantizer is never overloaded). Several kinds of quantizer transfer functions will be considered, especially midtread and midriser ones. We shall restrict our analysis to the case of deterministic (i.e., not random) signals. Some knowledge of Fourier analysis and signal processing will be presupposed.

On the choice of an auxiliary function in the estimation

“…Some other numerical stability issues related to Fourier analysis can be found e.g. in Deraedt (1996) and Korn (2005). In the next section, we propose a simple solution by reformulating the problem.…”