Digital Signal Processing Reference
In-Depth Information
In general, it is difficult to describe the probability distribution of the
output random process
Y
(
t
)
, even when the probability distribution of
the input random process
X
.
However, it is useful to perform an analysis in terms of the mean and
autocorrelation function of the output signal.
If we assume that the input signal
X
(
t
)
is completely specified for
−∞ ≤
t
≤+∞
is a stationary process, then we
can evaluate the mean of the output random process
Y
(
t
)
(
t
)
as follows:
E
∞
d
τ
κ
1
(
Y
,
t
)
=
h
(
τ
)
X
(
t
−
τ
)
−∞
∞
E
X
d
τ
=
h
(
τ
)
(
t
−
τ
)
−∞
∞
=
h
(
τ
)
κ
1
(
X
,
t
−
τ
)
d
τ
(2.110)
−∞
and, since we are dealing with a stationary process, we have κ
1
(
X
)
=
κ
1
(
X
,
t
)
,
hence
∞
κ
1
(
Y
)
=
κ
1
(
X
)
h
(
τ
)
d
τ
−∞
=
κ
1
(
X
,
t
)
H
(
0
)
(2.111)
where
H
is the zero frequency response of the system.
We can also evaluate the autocorrelation function of the output signal
(
0
)
Y
(
t
)
. Recalling that
E
Y
t
2
)
Y
∗
(
R
Y
(
t
1
,
t
2
)
=
(
t
1
)
(2.112)
E
∞
∞
h
∗
(
X
∗
(
R
Y
(
t
1
,
t
2
)
=
h
(
τ
1
)
X
(
t
1
−
τ
1
)
d
τ
1
τ
2
)
t
2
−
τ
2
)
d
τ
2
−∞
−∞
∞
∞
E
X
τ
2
)
d
τ
1
d
τ
2
h
∗
(
X
∗
(
=
h
(
τ
1
)
τ
2
)
(
t
1
−
τ
1
)
t
2
−
(2.113)
−∞
−∞
is stationary, then, as discussed in
Section 2.4.2
,
the autocorrelation function of
X
Now, since we assume that
X
(
t
)
(
t
)
is only a function of the
difference between the observation times
t
1
−
τ
1
and
t
2
−
τ
2
. Thus, letting
