Digital Signal Processing Reference
In-Depth Information
energy of the signal must increase at the same rate as
T
, the duration of the signal.
Under this condition, it is permissible to interchange the order of the limiting action
on
T
and the integration over
v
so that
q
1
2p
L
1
T
ƒ
F
T
(v)
ƒ
2
dv.
P =
lim
T:
q
-
q
In this form of the average power equation, the integrand is called the
power spec-
tral density
and is denoted by the symbol
1
T
ƒ
F
T
(v)
ƒ
2
.
f
(v) K
lim
T:
q
(5.53)
In terms of the power spectral density function, the equation for normalized aver-
age signal power is
q
-
q
f
(v)dv =
q
0
f
(v)dv,
1
2p
L
1
p
L
P =
(5.54)
f
(v)
because is an even function.
For periodic signals, the normalized average power can be determined from
the Fourier series as
q
q
k= 1
ƒ
C
k
ƒ
2
= C
2
2
.
ƒ
C
k
ƒ
P =
a
+ 2
a
(5.55)
k=-
q
Using the relationship shown in (5.35),
ƒ
F(kv
0
)
ƒ
= 2p
ƒ
C
k
ƒ
,
We write the normalized average power of a signal
f(t)
in terms of the Fourier
transform as
q
q
k= 1
ƒ
F(kv
0
)
ƒ
2
1
2p
1
4p
2
ƒ
F(0)
ƒ
1
2p
2
a
2
2
2
.
¢
≤
ƒ
F(kv
0
)
ƒ
P =
=
+
(5.56)
a
k=-
q
It is seen that for a periodic signal, the power distribution over any band of fre-
quencies can be determined from the Fourier transform of the signal.
Power spectral density of a periodic signal
EXAMPLE 5.20
The magnitude frequency spectrum of a periodic signal is shown in Figure 5.35(a). According
to (5.56), the power spectral density can be displayed by squaring the magnitude of each dis-
crete frequency component and dividing by
4p
2
.
This result is shown in Figure 5.35(b). It
