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)
with spectral covariance of bounded variation:
Under the harmonizability assumption, we have
where χi(ν) is the integrated spectrum of xi(n). We can formally write dχi(ν) = Xi(ν) dν (Gardner 1985, Chapter 10.1.2), (Papoulis 1991, Chapter 12-4), where
is the Fourier transform of xi(n) to be considered in the sense of distributions (Gelfand and Vilenkin 1964, Chapter 3), (Henniger 1970), provided that χi(ν) does not contain singular component (Hurd and Miamee 2007).
Definition 4.8.1 Let ...