Performance Analysis and Optimal Filter Design for Sigma-Delta Modulation via Duality with DPCM

TL;DR

This paper characterizes the optimal trade-off between quantization rate and distortion in sigma-delta modulation by establishing a duality with differential pulse-code modulation, enabling full performance analysis and filter design.

Contribution

It introduces a duality relation between sigma-delta modulation and DPCM, allowing complete characterization of optimal filters and performance trade-offs.

Findings

Derived the optimal feedback filters for sigma-delta modulation.

Established a duality linking sigma-delta and DPCM performance.

Provided a framework for optimal filter design under various constraints.

Abstract

Sampling above the Nyquist rate is at the heart of sigma-delta modulation, where the increase in sampling rate is translated to a reduction in the overall (mean-squared-error) reconstruction distortion. This is attained by using a feedback filter at the encoder, in conjunction with a low-pass filter at the decoder. The goal of this work is to characterize the optimal trade-off between the per-sample quantization rate and the resulting mean-squared-error distortion, under various restrictions on the feedback filter. To this end, we establish a duality relation between the performance of sigma-delta modulation, and that of differential pulse-code modulation when applied to (discrete-time) band-limited inputs. As the optimal trade-off for the latter scheme is fully understood, the full characterization for sigma-delta modulation, as well as the optimal feedback filters, immediately follow.