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Random Processes and Spectral Analysis Chap. 6
Wiener–Khintchine Theorem
Often, the PSD is evaluated from the autocorrelation function for the random process by using
the following theorem.
When x(t) is a wide-sense stationary process, the
PSD can be obtained from the Fourier transform of the autocorrelation function:
WIENER–KHINTCHINE THEOREM †
Conversely,
provided that R(t) becomes sufficiently small for large values of t, so that
(6–44)
(6–45)
ƒ tR(t) ƒ dt 6q
(6–46)
L-q
This theorem is also valid for a nonstationary process, provided that we replace R x (t) by
R x (t, t + t).
where
Proof.
q
x(f) = [R x (t)] = R x (t)e -j2pft dt
L
q
R x (t) = -1 [ x (f)] = x (f)e j2pft df
L
From the definition of PSD,
x (f) = lim
T : q a ƒ X 2
T(f) ƒ
b
T
|X T (f)| 2 = `
L
q
T/2
-T/2
=
L
T/2
T/2
-T/2 L-T/2
-q
-q
x(t)e -jvt dt `
2
x(t 1 )x(t 2 )e -jvt 1
e jvt 2
dt 1 dt 2
and x(t) is assumed to be real. But x(t 1 )x(t 2 ) = R x (t 1 , t 2 ). Furthermore, let t = t 2 - t 1 , and
make a change in variable from t 2 to t t 1 . Then
t 1 = T/2 t = T/2- t 1
ƒ X T (f) ƒ 2 = B R x (t 1, t 1 + t)e -jvt dtR dt 1 (6–47)
L t 1 = -T/2 L t =-T/2- t 1
1
The area of this two-dimensional integration is shown in Fig. 6–4. In the figure, 1 denotes
the area covered by the product of the inner integral and the differential width dt 1 . To evaluate
this two-dimensional integral easily, the order of integration will be exchanged.
⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
† Named after the American mathematician Norbert Wiener (1894–1964) and the German mathematician
A. I. Khintchine (1894–1959). Other spellings of the German name are Khinchine and Khinchin.