# 2.6 Discrete-Time Estimator of the Cyclic Cross-Correlation Function

In this section, the discrete-time cyclic cross-correlogram of the discrete-time process obtained by uniformly sampling a continuous-time GACS process is shown to be a mean-square consistent and asymptotically complex Normal estimator of the continuous-time cyclic cross-correlation function as the data-record length approaches infinity and the sampling period approaches zero (Napolitano 2009).

Note that the results of Section 2.6 for discrete-time processes cannot be obtained straightforwardly by substituting integrals with sums into the results of Sections 2.4.1 –2.4.3 which have been obtained for continuous-time processes as is made, in the stationary case, in (Brillinger and Rosenblatt 1967).

## 2.6.1 Discrete-Time Cyclic Cross-Correlogram

In this section, the discrete-time cyclic cross-correlogram is defined and its mean and covariance are evaluated for finite number of samples and sampling period.

**Definition 2.6.1** *Let* y_{d}(n) *and* x_{d}(n) *be the discrete-time processes defined in (2.172). Their* discrete-time cyclic cross-correlogram (DT-CCC) *at cycle frequency is defined as*

(2.181)

where

(2.182)

*is a data-tapering ...*