Page 715 - The Mechatronics Handbook

0066_Frame_C23 Page 23 Wednesday, January 9, 2002 1:53 PM

FIGURE 23.15 (a) Reconstruction of analog signal, (b) effect of anti-image postfiltering.

If the spectrum of a signal exists, then we can find the signal from its spectrum through the inverse
DTFT. The inverse DTFT of X(f ) is given as

1
s
xn() = ---- ∫ F /2 Xf()e j2πfn f d
F s – F /2
s
The properties of the DTFT of a DT signal are shown in Table 23.6.
For DT signals, the convolution of two sequences x(n) and y(n) is expressed as
∞
xn() ∗yn() = ∑ xm()yn m) (23.40)
(
–
m=∞
–
while the cross-correlation function of x(n) and y(n) is given by
∞
R xy n() = xn() ⊕ yn() = ∑ xm()y mn)
(
∗
–
m=∞
–
When x(n) = y(n), then the autocorrelation of x(n) is
∞
R xx n() = ∑ xm()x mn)
∗
(
–
m=∞
–
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