Digital Signal Processing Reference
In-Depth Information
Multiplying both sides of the equation by (5.57) yields
G(v)G
*
(v) = H(v)H
*
(v)F(v)F
*
(v),
or
2
2
2
.
ƒG(v) ƒ
= ƒH(v) ƒ
ƒF(v) ƒ
(5.58)
If both sides on this expression are divided by and the equivalents from
(5.49) are substituted, we have an expression that describes the transmission of en-
ergy through a linear system:
p
2
g
(v) = ƒH(v) ƒ
f
(v).
(5.59)
For the case that the input to a system is a power signal, the time-averaging opera-
tion can be applied to both sides of (5.58):
1
T
ƒG
T
(v) ƒ
1
T
ƒF
T
(v) ƒ
2
2
2
.
lim
T:
q
= ƒH(v) ƒ
lim
T:
q
Then, from (5.53),
2
g
(v) = ƒH(v) ƒ
f
(v).
(5.60)
Usually, the exact content of an information signal in a communications
system cannot be predicted; however, its power spectral density can be deter-
mined statistically. Thus, (5.58) is often used in the analysis and design of these
systems.
Power spectral density of a system's output signal
EXAMPLE 5.21
A signal with power spectral density shown in Figure 5.36(a) is the input to a linear sys-
tem with the frequency response plotted in Figure 5.36(b). The power spectral density of the
output signal, is determined by application of Equation (5.60).
Because the power spectral density of the input signal is a discrete-frequency function,
x(t)
y(t),
2
y
(v) =
x
(v) ƒH(v) ƒ
can be determined by evaluating the equation at only those frequencies where the power
spectral density of
x(t)
is nonzero. For example, to determine the power density in the output
signal at
v = 60 (rad/s),
from the frequency response of the linear system, we find that
therefore,
2
ƒH(60) ƒ = 0.7071;
ƒH(60) ƒ
= 0.5.
From Figure 5.36(a),
x
(60) = 7.84.
We calcu-
late
y
(v) = 3.92.
These calculations are repeated for all frequencies of interest.
The power spectral density of the output signal is plotted in Figure 5.36(c).
