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Sec. 6–4 Linear Systems 439
Example 6–6 AVERAGE POWER AND RMS VALUE FOR BIPOLAR LINE CODES
Using Eqs. (6–76) and (6–78), evaluate the RMS value and the average power for the bipolar line
codes that are described in Example 6–5. See Example6_06.m for the solution.
6–4 LINEAR SYSTEMS
Input–Output Relationships
As developed in Chapter 2, a linear time-invariant system may be described by its impulse
response h(t) or, equivalently, by its transfer function H(f). This is illustrated in Fig. 6–6, in
which x(t) is the input and y(t) is the output. The input–output relationship is
y(t) = h(t) * x(t)
(6–80)
The corresponding Fourier transform relationship is
Y(f) = H(f) X(f)
(6–81)
If x(t) and y(t) are random processes, these relationships are still valid (just as they were for
the case of deterministic functions). In communication systems, x(t) might be a random signal
plus (random) noise, or x(t) might be noise alone when the signal is absent. In the case of
random processes, autocorrelation functions and PSD functions may be used to describe the
frequencies involved. Consequently, we need to address the following question: What are the
autocorrelation function and the PSD for the output process y(t) when the autocorrelation and
PSD for the input x(t) are known?
THEOREM. If a wide-sense stationary random process x(t) is applied to the input of a
time-invariant linear network with impulse response h(t), the output autocorrelation is
or
R y (t) =
L
q
q
-q L-q
h(l 1 )h(l 2 )R x (t - l 2 + l 1 ) dl 1 dl 2
(6–82a)
R y (t) = h(-t) * h(t) * R x (t)
(6–82b)
Input
x (t)
Linear Network
h(t)
H(f)
Output
y (t)
X (f)
Y (f)
R x ()
R y ()
p x (f )
Figure 6–6
Linear system.
p y (f)