# 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 ...