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Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications by Antonio Napolitano

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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) equation

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 WA(f) the Fourier transform of img, one obtains that the Fourier transform img of the time-smoothing window aT(t) can be expressed as

(5.26) equation

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

(5.27) equation

If is continuous in t = 0, then .

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

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