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Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications by Antonio Napolitano

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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 x1x2.

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

(4.182) equation

with img spectral covariance of bounded variation:

(4.183) equation

Under the harmonizability assumption, we have

(4.184) equation

where χi(ν) is the integrated spectrum of xi(n). We can formally write dχi(ν) = Xi(ν) (Gardner 1985, Chapter 10.1.2), (Papoulis 1991, Chapter 12-4), where

(4.185) equation

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 ...

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