Digital Signal Processing Reference
In-Depth Information
70
67.88
60
53.31
50
40
30
22.63
20
10
1.94
0
1.08
4.53
10
4000
3000
2000
1000 0
Frequency (rad/s)
(a)
1000
2000
3000
4000
120
116.7
100
80
72.0
60
40
20
13.0
0
4000
3000
2000
1000 0
Frequency (rad/s)
(b)
1000
2000
3000
4000
Figure 5.35
Power spectral density of a
periodic signal.
should be noted that the values of the spectral components at frequencies above 1000 rad/s
are small in magnitude, but not zero as may be implied by Figure 5.35(b).
The normalized average power in the frequency band
ƒ
v
ƒ
F 1000 rad/s
is found by
summing the power of the discrete frequency components in that range:
116.7 + 2(72.0 + 13.0) = 286.7 W.
■
The input-output relationship of a system
G(v) = H(v)F(v)
(5.57)
can also be expressed in terms of the energy or power spectral densities of the input
and output signals. First, we conjugate both sides of (5.57):
G
*
(v) = [H(v)F(v)]
*
= H
*
(v)F
*
(v).
