Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

1.3 Almost-Cyclostationary Processes

Many processes encountered in telecommunications, radar, mechanics, radio astronomy, biology, atmospheric science, and econometrics, are generated by underlying periodic phenomena. These processes, even if not periodic, give rise to random data whose statistical functions vary periodically with time and are called cyclostationary processes. In this section, the properties of cyclostationary, or more generally, of almost-cyclostationary processes are briefly reviewed. For extensive treatments, see (Gardner 1985, Chapter 12, 1987d), (Gardner et al. 2006), (Giannakis 1998), (Hurd and Miamee 2007).

1.3.1 Second-Order Wide-Sense Statistical Characterization

The (finite-power) process x(t) is said to be second-order cyclostationary in the wide sense or periodically correlated with period T₀ if its first-and second-order moments are periodic functions of time with period T₀. More generally, first-and second-order moments can be almost-periodic functions of time (in one of the senses considered in Section 1.2) and the process is said to be second-order almost-cyclostationary (ACS) in the wide sense or almost-periodically correlated (Gardner 1985, Chapter 12, 1987d). In such a case, its (conjugate) autocorrelation function (1.6) under mild regularity conditions can be expressed by the (generalized) Fourier series expansion

(1.87)

where subscript , A