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
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 , one obtains that the Fourier transform of the time-smoothing window aT(t) can be expressed as
with , continuous, and infinitesimal for |f|→ ∞. Moreover, in the sense of distributions, it results that
If is continuous in t = 0, then .