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

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6.5 Proofs of FOT Counterparts of Some Results on ACS and GACS Signals

In this section, proofs of some results presented in Section 1.3 in the stochastic approach are carried out in the functional approach. Moreover, when possible, results are derived in the more general case of GACS signals.

6.5.1 Proof of FOT Counterpart of (1.114)

By substituting (1.112) into (1.113) we have

equation

(6.79) equation

where, in the last equality, the variable changes s1 = tu1 and s2 = tu2 are made. Thus by assuming that the order of sum, integral, and almost-periodic component extraction operations can be interchanged, and observing that almost-periodic functions can be led outside the almost-periodic component extraction operator, we get

(6.80) equation

where, in the last equality, the variable change s2s1 = u is made and, according to (1.95) (with the replacements img and img), we put

(6.81) equation

By substituting (1.115) into

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