Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

5.5 Proofs for Section 4.7.1 “Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram”

In this section, proofs of lemmas and theorems presented in Section 4.7.1 on the mean-square consistency of the frequency-smoothed cross-periodogram are reported.

Lemma 5.5.1 Let W(f) be a.e. continuous and regular as |f|→ ∞, and (that is, W can be either W_A satisfying Assumption 4.4.5 or W_B satisfying Assumption 4.6.2). We have the following results.

a. Let {Ψ⁽ⁿ⁾(λ)} be a set of a.e. derivable functions such that, for n ≠ m, Ψ⁽ⁿ⁾(λ) = Ψ^(m)(λ) at most in a set of zero Lebesgue measure in . It results that

(5.70)

for almost all λ.

Proof: For n = m, the left-hand side of (5.70) can be written as

(5.71)

for almost all λ, provided that ν ≠ 0, where the a.e. continuity of W(f) is accounted for.

For n ≠ m, the left-hand side of (5.70) can be written as

(5.72)

for ν ≠ 0 and almost all