Signal Processing for Cognitive Radios

6.5.1 Temporally Windowed Observations

With the notation in (6.59), we see that TVFT-DTFT (6.62) is equivalent to the DTFT of the observation signal :

(6.67a)

(6.67b)

Hence, X(F, n;N) is the convolution of the spectral component x[n]e^–j2πFn in x[n] with the rectangular function centered at time n = 0 with a finite width N. The final result is to smoothen, or broaden, the spectral component by an order of .

Then it is also possible to replace the rectangular window in (6.59) by a general data tapering function w_N[n] that is nonzero only for . Let us denote by x_w;n₀ [n] the tapered data obtained by using a data taper W_N[n]:

(6.68)

The resulting DTFT of the tapered data x[n]w_N[n – n₀] is

(6.69)

where last step follows by noting that w_N[n] is an N-length taper centered at n = 0. ...