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Random Processes and Spectral Analysis Chap. 6
The output PSD is
y (f) = ƒ H(f) ƒ 2 x (f)
(6–83)
where H( f ) = [h(t)].
Equation (6–83) shows that the power transfer function of the network is
G h (f) = P y(f)
P x (f)
= ƒ H(f) ƒ
2
(6–84)
as cited in Eq. (2–143).
But
Proof. From Eq. (6–80),
R y (t) ! y(t)y(t + t)
q
q
= c h(l 1 )x(t - l 1 )dl 1 d c h(l 2 )x(t + t - l 2 ) dl 2 d
L L
- q
=
L
q
q
-q L-q
-q
h(l 1 )h(l 2 )x(t - l 1 )x(t + t - l 2 ) dl 1 dl 2
(6–85)
x(t - l 1 )x(t + t - l 2 ) = R x (t + t - l 2 - t + l 1 ) = R x (t - l 2 + l 1 )
so Eq. (6–85) is equivalent to Eq. (6–82a). Furthermore, Eq. (6–82a) may be written in terms
of convolution operations as
q
q
R y (t) = h(l 1 ) e h(l 2 )R x [(t + l 1 ) - l 2 ] dl 2 f dl 1
L L
-q
q
= h(l 1 ){h(t + l 1 ) * R x (t + l 1 )} dl 1
L
-q
q
= h(l 1 ){h[-((-t) - l 1 )] * R x [-((-t) - l 1 )]} dl 1
L
-q
-q
= h(-t) * h[-(-t)] * R x [-(-t)]
which is equivalent to the convolution notation of Eq. (6–82b).
The PSD of the output is obtained by taking the Fourier transform of Eq. (6–82b).
We obtain
or
[R y (t)] = [h(-t)][h(t) * R x (t)]
y (f) = H * (f)H(f)P x (f)
where h(t) is assumed to be real. This equation is equivalent to Eq. (6–83).