A Deterministic Analysis of Decimation for Sigma-Delta Quantization of Bandlimited Functions

TL;DR

This paper provides a deterministic analysis of decimation in Sigma-Delta quantization, showing it can efficiently reduce bit-rate while maintaining accurate reconstruction of bandlimited functions, with exponential error decay.

Contribution

It offers a novel deterministic proof that decimation effectively compresses Sigma-Delta bit-streams without sacrificing reconstruction quality.

Findings

Decimation reduces bit-rate significantly while preserving reconstruction accuracy.

Reconstruction error decays exponentially with bit-rate in stable $r$th order Sigma-Delta schemes.

Applicable to 1-bit, greedy, first-order Sigma-Delta schemes.

Abstract

We study Sigma-Delta ($\Sigma\Delta$) quantization of oversampled bandlimited functions. We prove that digitally integrating blocks of bits and then down-sampling, a process known as decimation, can efficiently encode the associated $\Sigma\Delta$ bit-stream. It allows a large reduction in the bit-rate while still permitting good approximation of the underlying bandlimited function via an appropriate reconstruction kernel. Specifically, in the case of stable $r$th order $\Sigma\Delta$ schemes we show that the reconstruction error decays exponentially in the bit-rate. For example, this result applies to the 1-bit, greedy, first-order $\Sigma\Delta$ scheme.