Let
(4.298)
be the discrete-time sequences obtained by uniformly sampling with period Ts = 1/fs 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, YN(n, ν1) and XN(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.
18.117.232.239