Let and be jointly GACS processes, that is (Definition 2.2.10)
where , and let and be LTV filtered versions of and , respectively, obtained by systems with impulse response functions and .
Under assumption (2.47), substituting (2.109) into ((2.46)) leads to
Let and be LAPTV systems, that is, according to (1.107), with impulse-response functions
Substituting (2.111) into (2.110) leads to (Section 3.3)
where is defined in (1.115)
Under assumptions
(where conditions (2.115) and (2.116) on h1(t, u) and h2(t, u) can be possibly interchanged) one obtains (Section 3.3)
where .
Due to the presence of the Kronecker delta in the integrand function in (2.117), Ry(β, τ) can be nonzero only if for some n, σ1, and σ2 the function is nonzero in a set of values of s with positive Lebesgue measure. That is, only if x1(t) and x2(t) exhibit a joint ACS component in the cross-correlation function. In particular, from (2.117) with x1 ≡ x2 and h1 ≡ h2 LTI filters, accordingly with the results of Section 2.2.3, it follows that low-pass or band-pass filtering of a purely GACS signal (Section 2.2.2) leads to a zero-power signal (see Section 2.7.7 for a numerical example) (Izzo and Napolitano 2002a,b, 2005).
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