Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

3.4 Proofs for Section 2.4.1 “The Cyclic Cross-Correlogram”

In this section, proofs of lemmas and theorems presented in Section 2.4.1 on the bias and covariance of the cyclic cross-correlogram are reported.

In the following, all the functions are assumed to be Lebesgue measurable. Consequently, without recalling the measurability assumption, we use the fact that if the functions ϕ₁ and ϕ₂ are such that |ϕ₁| ≤ |ϕ₂|, ϕ₁ is measurable and ϕ₂ is integrable (i.e., ϕ₂ is measurable and |ϕ₂| is integrable), then ϕ₁ is integrable (Prohorov and Rozanov 1989, p. 82). Furthermore, if and , then |ϕ₁ϕ₂| ≤ |ϕ₁|||ϕ₂||_∞ almost everywhere and, hence, .

3.4.1 Proof of Theorem 2.4.6 Expected Value of the Cyclic Cross-Correlogram

By using (2.31c) and (2.118) one has

(3.38a)

(3.38b)

(3.38c)

(3.38d)

from which ...