Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

4.1 Introduction

It is well known that wide-sense stationary (WSS) stochastic processes do not exhibit correlation between spectral components at distinct frequencies. That is, by passing a WSS process throughout two bandpass filters with nonoverlapping passbands, and then frequency shifting the two output processes to a common band, one obtains two uncorrelated stochastic processes (Gardner 1987d). Equivalently, for WSS processes, the Loève bifrequency spectrum (Loève 1963) (also called dual-frequency spectrum (Øigård et al. 2006) or cointensity spectrum (Middleton 1967)) has support contained in the main diagonal of the bifrequency plane. The presence of spectral correlation between spectral components at distinct frequencies is an indicator of the nonstationarity of the process (Loève 1963), (Gardner 1987d), (Hurd and Gerr 1991), (Genossar 1992), (Dehay and Hurd 1994), (Napolitano 2003), (Dmochowski et al. 2009). When correlation exists only between spectral components that are separated by quantities belonging to a countable set of values, the process is said almost-cyclostationary (ACS) or almost-periodically correlated (Gardner 1985, 1987d), (Gardner et al. 2006), (Hurd and Miamee 2007). The values of the separation between correlated spectral components are called cycle frequencies and are the frequencies of the (generalized) Fourier series expansion of the almost-periodically time-variant statistical autocorrelation function (Gardner 1987d), (Gardner 1991b), (Dehay and ...