Discrete-Time Estimation of Second-Order Statistics of Generalized Almost-Cyclostationary Processes

In this paper, the problem of estimating second-order statistics of continuous-time generalized almost-cyclostationary (GACS) processes is addressed. GACS processes in the wide sense have autocorrelation function almost-periodic in time whose generalized Fourier series expansion has both frequencies and coefficients that depend on the lag shifts. Almost-cyclostationary (ACS) processes are obtained as a special case when the frequencies do not depend on the lag shifts. ACS processes filtered by Doppler channels and communications signals with time-varying parameters are further examples. It is shown that continuous-time GACS processes do not have a discrete-time counterpart. The discrete-time cyclic cross-correlogram of the discrete-time ACS processes obtained by uniformly sampling GACS processes is considered as estimator of samples of the continuous-time cyclic cross-correlation function. The asymptotic performance analysis is carried out by resorting to the hybrid cyclic cross-correlogram which is partially continuous-time and partially discrete-time. Its mean-square consistency and asymptotic complex Normality as the number of data-samples approaches infinity and the sampling period approaches zero are proved under mild conditions on the regularity of the Fourier series coefficients and the finite or practically finite memory of the processes expressed in terms of summability of cumulants. It is shown that the asymptotic properties of the hybrid cyclic cross-correlogram are coincident with those of the continuous-time cyclic cross-correlogram. Hence, discrete-time estimation does not give rise to any loss in asymptotic performance with respect to continuous-time estimation.