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Random Processes and Spectral Analysis Chap. 6
This equation can be written compactly as
T
ƒ X T (f) ƒ 2 = (T - |t|)R x (t)e -jvt dt
L
-T
(6–49)
By substituting Eq. (6–49) into Eq. (6–42), we obtain
x (f) =
lim
T
T: q
L-T
a T - |t| bR x (t)e -jvt dt
T
(6–50a)
or
q
x (f) = R x (t)e jvt dt -
L
-q
lim
T
T: q L-T
|t|
T R x(t)e -jvt dt
(6–50b)
Using the assumption of Eq. (6–46), we observe that the right-hand integral is zero and
Eq. (6–50b) reduces to Eq. (6–44). Thus, the theorem is proved. The converse relationship
follows directly from the properties of Fourier transforms. Furthermore, if x(t) is not
stationary, Eq. (6–44) is still obtained if we replace R x (t 1 , t 1 + t) into Eq. (6–48) by
R x 1t 1 , t 1 + t2 = R x 1t2.
Comparing the definition of the PSD with results of the Wiener–Khintchine theorem,
we see that there are two different methods that may be used to evaluate the PSD of a random
process:
1. The PSD is obtained using the direct method by evaluating the definition as given by
Eq. (6–42).
2. The PSD is obtained using the indirect method by evaluating the Fourier transform of
R x (t), where R x (t) has to be obtained first.
Both methods will be demonstrated in Example 6–4.
Properties of the PSD
Some properties of the PSD are:
1. x (f) is always real.
(6–51)
2. x (f) Ú 0.
(6–52)
3. When x(t) is real, x (-f) = x (f).
(6–53)
4. 1-q q x (f) df = P = total normalized power.
(6–54a)
When x(t) is wide-sense stationary,
q
x (f) df = P = x 2 = R x (0)
(6–54b)
L-q
5. x (0) = 1-q q R x (t) dt.
(6–55)
These properties follow directly from the definition of a PSD and the use of the Wiener–
Khintchine theorem.