Mean-square consistency and asymptotic normality of a hybrid estimator of the cyclic autocorrelation function of generalized almost-cyclostationary processes

In this paper, the problem of estimating the cyclic autocorrelation function of a continuous-time generalized almost-cyclostationary (GACS) process 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. The discrete-time cyclic correlogram of the discrete-time process obtained by uniformly sampling a GACS process is considered as estimator of samples of the continuous-time cyclic autocorrelation function. The asymptotic performance analysis is carried out by resorting to the hybrid cyclic correlogram which is partially continuous-time and partially discrete-time. It is shown that its asymptotic properties are coincident with those of the continuous-time cyclic correlogram. Hence, discrete-time estimation does not give rise to any loose in asymptotic performance with respect to continuous-time estimation.