Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

TL;DR

This paper demonstrates that decimation combined with Sigma-Delta quantization on finite frames leads to exponential decay in reconstruction error relative to bit-rate, improving data efficiency in A/D conversion.

Contribution

It introduces an alternative decimation process compatible with Sigma-Delta quantization for finite frames, extending exponential error decay results.

Findings

Decimation with Sigma-Delta quantization achieves exponential error decay.

The proposed alternative decimation process is compatible with second-order Sigma-Delta schemes.

Results improve data storage efficiency while maintaining high accuracy.

Abstract

In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the $\Sigma\Delta$ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results have been proven for finite unitarily generated frames. We have devised a process called alternative decimation on finite frames that is compatible with $\Sigma\Delta$ quantization up to the second order. In both cases, decimation results in exponential error decay with respect to the bit usage.