Asymptotic Analysis of Synchronous Signal Processing

TL;DR

This paper develops a unified theoretical framework for analyzing the asymptotic behavior of smoothing, filtering, and prediction of cyclostationary processes, linking Fourier and KL expansions and providing practical performance bounds.

Contribution

It introduces a novel asymptotic eigenbasis linking Fourier and KL expansions for cyclostationary processes, enabling performance analysis without explicit implementations.

Findings

Derived asymptotic eigenbasis linking Fourier and KL expansions

Obtained MMSE expressions depending on cyclic spectrum

Quantified performance gains in practical scenarios

Abstract

This paper extends various theoretical results from stationary data processing to cyclostationary (CS) processes under a unified framework. We first derive their asymptotic eigenbasis, which provides a link between their Fourier and Karhunen-Lo\`eve (KL) expansions, through a unitary transformation dictated by the cyclic spectrum. By exploiting this connection and the optimalities offered by the KL representation, we study the asymptotic performance of smoothing, filtering and prediction of CS processes, without the need for deriving explicit implementations. We obtain minimum mean squared error expressions that depend on the cyclic spectrum and include classical limits based on the power spectral density as particular cases. We conclude this work by applying the results to a practical scenario, in order to quantify the achievable gains of synchronous signal processing.