# Adapted Decimation on Finite Frames for Arbitrary Orders of Sigma-Delta Quantization

## TL;DR

This paper introduces adapted decimation for finite frames in Sigma-Delta quantization, achieving polynomial error decay of any order with oversampling and exponential decay with bit-rate, enhancing A/D conversion efficiency.

## Contribution

It extends decimation to arbitrary order in finite frames, providing a new scheme that improves error decay rates in Sigma-Delta quantization.

## Key findings

- Polynomial reconstruction error decay of arbitrary order.
- Exponential decay of error with respect to bit-rate.
- Enhanced efficiency in A/D conversion processes.

## 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. We build on our previous result, which extends signal decimation to finite frames, albeit only up to the second order. In this study, we introduce a new scheme called adapted decimation, which yields polynomial reconstruction error decay rate of arbitrary order with respect to the oversampling rate, and exponential with respect to the bit-rate.