  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.
