4.11 Discrete-Time Estimators of the Spectral Cross-Correlation Density

Let

(4.298)

be the discrete-time sequences obtained by uniformly sampling with period T_s = 1/f_s the continuous-time (jointly) SC processes x(t) and y(t).

The discrete-time frequency-smoothed cross-periodogram along the support curve is defined as

(4.299)

where denotes periodic convolution (with period 1) with respect to ν₁, Y_N(n, ν₁) and X_N(n, ν₂) are discrete-time STFTs defined according to

(4.300)

and

(4.301)

is a frequency-smoothing window with summable in [− 1/2, 1/2] and such that

(4.302)

In the absence of aliasing, statistical functions of discrete-time sampled processes in the principal frequency domain are scaled versions of statistical functions of the continuous-time processes (Sections 4.9.2 and 7.7.5). In such a case, consistency results for the discrete-time frequency-smoothed cross-periodogram can be proved with obvious changes analogously to the case of continuous-time processes. Moreover, even in the case of non-strictly band-limited continuous-time processes, consistency results can be proved provided that the amount of aliasing is controlled by taking the sampling period sufficiently small. In such a case, under suitable conditions, the result is that

(4.303)

where the order of the three limits cannot be interchanged.