Error estimates in Second-order Continuous-Time Sigma-Delta modulators

TL;DR

This paper develops a theoretical framework to estimate the error between the filtered output and input in second-order continuous-time Sigma-Delta modulators, showing it can decrease faster than 1/N^2 with oversampling ratio, validated by numerical experiments.

Contribution

It introduces a general error analysis framework for second-order CT-Sigma-Delta modulators, providing new error bounds under regularity assumptions.

Findings

Error estimate can be in o(1/N^2) for second-order CT-Sigma-Delta.

Theoretical results are validated through numerical experiments.

Provides a foundation for understanding accuracy in continuous-time Sigma-Delta converters.

Abstract

Continuous-time Sigma-Delta (CT-$\Sigma\Delta$) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete time $\Sigma\Delta$ modulators, theoretical analysis of the error between the filtered output and the input remain scarce. This paper presents a general framework to study this error: under regularity assumptions on the input and the filtering kernel, we prove for a second-order CT-$\Sigma\Delta$ that the error estimate may be in $o(1/N^2)$, where $N$ is the oversampling ratio. The whole theory is validated by numerical experiments.