2.3 Linear Time-Variant Filtering of GACS Processes

Let and be jointly GACS processes, that is (Definition 2.2.10)

(2.109)

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

(2.110)

Let and be LAPTV systems, that is, according to (1.107), with impulse-response functions

(2.111)

Substituting (2.111) into (2.110) leads to (Section 3.3)

(2.112)

where is defined in (1.115)

(2.113)

Under assumptions

(2.114)

(2.115)

(2.116)

(where conditions (2.115) and (2.116) on h₁(t, u) and h₂(t, u) can be possibly interchanged) one obtains (Section 3.3)

(2.117)

where .

Due to the presence of the Kronecker delta in the integrand function in (2.117), R_y(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).