In this section, proofs of results presented in Section 2.4.2 on the mean-square consistency of the cyclic cross-correlogram are reported.

**Lemma 3.5.1** *Let a(t) be such that Assumption 2.4.5 is satisfied. We have the following.*

a. *The Fourier transform*

(3.60)

*Proof:* Since , item (a) is a consequence of the properties of the Fourier transforms (Champeney 1990). In Assumption 2.4.5 it is also assumed that there exists γ > 0 such that

In reference to item (b), observe that from (2.125) we have

where, in the third equality, the variable change s = t/T is made. Thus, for f = 0,

(see (2.126)). For f ≠ 0, since , we have ...

Start Free Trial

No credit card required