Rate-Distortion Analysis of Quantizers with Error Feedback

TL;DR

This paper analyzes the rate-distortion trade-off in quantizers with error feedback, including Delta-Sigma modulators, by deriving optimal filters and demonstrating their practical approximations.

Contribution

It introduces a parameterization of the optimal error feedback filter, enabling numerical determination and analysis of rate-distortion performance.

Findings

Optimal error feedback filter minimizes mean squared quantization error

Relationship between bit rate and mean squared error is clarified

Practical filter approximations effectively implement the optimal filters

Abstract

A Delta-Sigma modulator that is often utilized to convert analog signals into digital signals can be modeled as a static uniform quantizer with an error feedback filter. In this paper, we present a rate-distortion analysis of quantizers with error feedback including the Delta-Sigma modulators, assuming that the error owing to overloading in the static quantizer is negligible. We demonstrate that the amplitude response of the optimal error feedback filter that minimizes the mean squared quantization error can be parameterized by one parameter. This parameterization enables us to determine the optimal error feedback filter numerically. The relationship between the number of bits used for the quantization and the achievable mean squared error can be obtained using the optimal error feedback filter. This clarifies the rate-distortion property of quantizers with error feedback. Then, ideal optimal error feedback filters are approximated by practical filters using the Yule-Walker method and the linear matrix inequality-based method. Numerical examples are provided for demonstrating our analysis and synthesis.