In this section, for jointly SC processes, the time-smoothed bifrequency cross-periodogram is considered as an estimator of the bifrequency spectral cross-correlation density function when the location of the support curves is unknown. For such an estimator, bias (Lemma 4.5.5) and covariance (Lemma 4.5.6) are determined. Moreover, its asymptotic biasedness and consistency are discussed (Theorems 4.5.7 and 4.5.9 and Corollary 4.5.8) (Napolitano 2001, 2003).
Definition 4.5.1 Given two stochastic processes and , their time-smoothed bifrequency cross-periodogram is defined as
where Y1/Δf(t, f1) and X1/Δf(t, f2) are the STFTs of y(t) and x(t), respectively, defined according to (4.94a), and aT(t) is a T-duration time-smoothing window.
Accordingly with the definition of STFT (4.94a), Y1/Δf(t, f1) represents the output of a low-pass filter with impulse-response function b1/Δf(t) (assumed to be an even function) with bandwidth ...