Quantization Noise Spectra

Abstract

Uniform qunntizers play a fundamental role in digital communication systems and have been the subject of extensive study for many decades, The inherent nonlinearity of quantizers makes their analysis hoth difficult and Interesting, It usually has been accomplished either by assuming the quantizer noise to be a signal-independent, uniform white random process or by replacing the quantizer by a deterministic linear device, or by combining the two assumptions. Such linearizing approximations simplify the analysis and permit the use of linear systems techniques, but few results exist quantifying how good such approximations are for specific systems, These complications are magnified when the quantizer is inside a feedback loop, as for Sigma-Delta modulators. Exact descriptions of the moments and spectra of quantizer noise have been developed recently for the special case of single-loop, multistage and multiloop Signul-Delta modulators. These results demonstrate. that the white noise and linearization assumptions can be quite poor approximations in some systems and quite good in others. It turns out that many of the techniques used in the analysis were first applied to the analysis of quantizers by Clavier, Panter, and Grieg (1947) in pioneering (but often overlooked) work that preceded Bennett's (1948) classic study of quantization noise spectra. We take advantage of the benefit of hindsight to develop several results describing the behavior of quantization noise in a unified and simplified manner. Exact formulas for quantizer noise spectra are developed and applied to a variety of systems and inputs, including scalar quantization (PCM), dithered PCM, Sigma-Delta modulation, dithered Sigma-Delta modulation, two-stage Sigma-Delta modulation, and second-order Sigma-Delta modulation.