Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

5.3 Proofs for Section 4.5 “Measurement of Spectral Correlation –Unknown Support Curves”

In this section, proofs of lemmas and theorems presented in Section 4.5 on bias and covariance of the time-smoothed bifrequency cross-periodogram are reported.

Accounting for the properties of the Fourier transform of a summable function (Champeney 1990) and observing that from Assumption 4.5.2, one obtains

(5.25)

the following result can easily be proved.

Lemma 5.3.1 (Napolitano 2003, Lemma B.1). Under Assumption 4.5.2 (time-smoothing window regularity) and denoting by W_A(f) the Fourier transform of , one obtains that the Fourier transform of the time-smoothing window a_T(t) can be expressed as

(5.26)

with , continuous, and infinitesimal for |f|→ ∞. Moreover, in the sense of distributions, it results that

(5.27)

If is continuous in t = 0, then .

5.3.1 Proof of Lemma 4.5.5 Expected Value of the Time-Smoothed ...