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Sec. 6–2 Power Spectral Density 425
which indicates that X T is itself a random process, since x T is a random process. We will now
simplify the notation and denote these functions simply by X T ( f ), x T (t), and x(t), because it is
clear that they are all random processes.
The normalized † energy in the time interval (T2, T2) is
q
q
E T = x 2 T (t) dt = ƒ X T (f) ƒ 2 df
(6–38)
L L
Here, Parseval’s theorem was used to obtain the second integral. E T is a random variable
because x(t) is a random process. Furthermore, the mean normalized energy is obtained by
taking the ensemble average of Eq. (6–38):
(6–39)
The normalized average power is the energy expended per unit time, so the normalized average
power is
or
(6–40)
In the evaluation of the limit in Eq. (6–40), it is important that the ensemble average be evaluated
before the limit operation is carried out, because we want to ensure that X T ( f ) is finite.
[Since x(t) is a power signal, X( f ) = lim T : q X T ( f ) may not exist.] Note that Eq. (6–40) indicates
that, for a random process, the average normalized power is given by the time average of
the second moment. Of course, if x(t) is wide-sense stationary, 8x 2 (t)9 = x 2 (t) because x 2 (t)
is a constant.
From the definition of the PSD in Chapter 2, we know that
(6–41)
Thus, we see that the following definition of the PSD is consistent with that given by
Eq. (2–66) in Chapter 2.
DEFINITION.
P =
-T/2
lim
T: q
P =
L
q
-q
-q
T/2
q
q
E T = x 2 (t) dt = x 2 2
T
(t) dt = ƒ X T
(f) ƒ df
L L L
1
T L
T/2
-T/2
c lim
T: q
- q
x 2 (t) dt =
1
T |X T(f)| 2 d df = x 2 (t)
q
P = (f) df
L
-q
-q
lim
T: q
1
T L
The power spectral density (PSD) for a random process x(t) is given by
- q
q
-q
x T 2 (t) dt
where
x (f) = lim a [|X T(f)| 2 ]
T: q
b
T
T/2
X T (f) = x(t)e -j2pft dt
L
-T/2
(6–42)
(6–43)
† If x(t) is a voltage or current waveform, E T is the energy on a per-ohm (i.e., R = 1) normalized basis.