Asymptotic Analysis of Synchronous Signal Processing

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.