2

Generalized Almost-Cyclostationary Processes

In this chapter, the statistical characterization of generalized almost-cyclostationary (GACS) processes is presented. Then the problem of estimating second-order cross-moments of GACS processes is addressed. GACS processes have statistical functions that are almost-periodic functions of time whose (generalized) Fourier series expansions have both frequencies and coefficients that depend on the lag shifts of the processes. The class of such nonstationary processes includes the almost-cyclostationary (ACS) processes which 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 second-order cross-moment of two jointly GACS processes is shown to be completely characterized by the cyclic cross-correlation function. Moreover, the cyclic cross-correlogram is proved to be a mean-square consistent, asymptotically Normal estimator, of the cyclic crosscorrelation function. It is shown that continuous-time GACS processes do not have a discretetime counterpart. The discrete-time cyclic cross-correlogram of the discrete-time ACS process obtained by uniformly sampling aGACS process is considered as an 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. The asymptotic properties of the hybrid cyclic cross-correlogram are shown to be 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. Well-known consistency and asymptotic Normality results for ACS processes are obtained by specializing the results for GACS processes.