Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

4.5 Measurement of Spectral Correlation –Unknown Support Curves

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

(4.123)

where Y_1/Δf(t, f₁) and X_1/Δf(t, f₂) are the STFTs of y(t) and x(t), respectively, defined according to (4.94a), and a_T(t) is a T-duration time-smoothing window.

Accordingly with the definition of STFT (4.94a), Y_1/Δf(t, f₁) represents the output of a low-pass filter with impulse-response function b_1/Δf(t) (assumed to be an even function) with bandwidth ...