Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

2.8 Summary

In this chapter, GACS processes are characterized in the time domain in the strict and wide senses, at second-and higher-orders. Moreover, a heuristic characterization in the frequency domain is provided. The class of such nonstationary processes includes, as a special case, the ACS processes. Moreover, ACS processes filtered by Doppler channels and communications signals with time-varying parameters are further examples. The problem of estimating second-order statistical functions of (continuous-time) GACS processes is addressed. The cyclic cross-correlogram is proposed as an estimator of the cyclic cross-correlation function of jointly GACS processes and its expected value and covariance are determined for finite data-record length (Theorems 2.4.6 and 2.4.7). It is shown that, for GACS processes satisfying some mixing assumptions expressed in terms of summability of cumulants, the cyclic cross-correlogram is a mean-square consistent (Theorems 2.4.11 and 2.4.13) and asymptotically complex Normal (Theorem 2.4.18) estimator of the cyclic cross-correlation function. Specifically, the covariance and conjugate covariance of the cyclic-cross correlogram are shown to approach zero as the reciprocal of the data-record length, when the data-record length approaches infinity. Moreover, the rate of convergence to zero of the bias is shown to depend on the rate of decay to zero of the Fourier transform of the data-tapering window (Theorem 2.4.12). An asymptotic bound for the covariance ...