Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

4.8 Discrete-Time SC Processes

In this section, discrete-time spectrally correlated processes are defined and characterized (Napolitano 2011). For the sake of generality, a joint characterization of two processes x₁(n) and x₂(n) in terms of cross-statistics is provided. The characterization of a single process can be obtained as a special case by taking x₁ ≡ x₂.

The discrete-time processes x₁(n) and x₂(n) are said to be second-order jointly harmonizable if (Loève 1963)

(4.182)

with spectral covariance of bounded variation:

(4.183)

Under the harmonizability assumption, we have

(4.184)

where χ_i(ν) is the integrated spectrum of x_i(n). We can formally write dχ_i(ν) = X_i(ν) dν (Gardner 1985, Chapter 10.1.2), (Papoulis 1991, Chapter 12-4), where

(4.185)

is the Fourier transform of x_i(n) to be considered in the sense of distributions (Gelfand and Vilenkin 1964, Chapter 3), (Henniger 1970), provided that χ_i(ν) does not contain singular component (Hurd and Miamee 2007).

Definition 4.8.1 Let ...