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Sec. 6–4 Linear Systems 441
x (t)
h 1 (t)
H 1 (f)
h 2 (t)
H 2 (f)
y (t)
Overall response: h(t)=h 1 (t) * h 2 (t)
H(f)=H 1 (f)H 2 (f)
Figure 6–7 Two linear cascaded networks.
x 1 (t)
h 1 (t)
H 1 (f)
y 1 (t)
R x1 x 2 ()
p x1 x 2
(f)
R y1 y 2 ()
p y1 y 2
(f)
x 2 (t)
h 2 (t)
H 2 (f)
y 2 (t)
Figure 6–8
Two linear systems.
This theorem may be applied to cascaded linear systems. For example, two cascaded
networks are shown in Fig. 6–7.
The theorem may also be generalized to obtain the cross-correlation or cross-spectrum
of two linear systems, as illustrated in Fig. 6–8. x 1 (t) and y 1 (t) are the input and output
processes of the first system, which has the impulse response h 1 (t). Similarly, x 2 (t) and y 2 (t)
are the input and output processes for the second system.
THEOREM. Let x 1 (t) and x 2 (t) be wide-sense stationary inputs for two time-invariant
linear systems, as shown in Fig. 6–8; then the output cross-correlation function is
R y1 y 2
(t) =
L
q
q
-q L-q
h 1 (l 1 )h 2 (l 2 )R x1 x 2
(t - l 2 + l 1 ) dl 1 dl 2
(6–86a)
or
R y1 y 2
(t) = h 1 (-t) * h 2 (t) * R x1 x 2
(t)
(6–86b)
Furthermore, by definition, the output cross power spectral density is the Fourier transform
of the cross-correlation function; thus,
y1 y 2
(f) = H 1
* (f)H2 (f) x1 x 2
(f)
(6–87)
where y1 y 2
(f) = [R y1 y 2
(t)], x1 x 2
(f) = [R x1 x 2
(t)], H 1 (f) = [h 1 (t)], and H 2 ( f ) =
[h 2 (t)].
The proof of this theorem is similar to that for the preceding theorem and will be left to
the reader as an exercise.