Digital Signal Processing Reference
In-Depth Information
where
p
X(t
1
)
,
X(t
2
)
(
is the second-order pdf of the process. We can also
define the autocovariance function
x
1
,
x
2
)
E
[
X
]
∗
C
X
(
t
1
,
t
2
)
=
(
t
1
)
−
(
X
,
t
1
)
][
X
(
t
2
)
−
(
X
,
t
2
)
κ
1
κ
1
κ
1
(
=
R
X
(
t
1
,
t
2
)
−
κ
1
(
X
,
t
1
)
·
X
,
t
2
)
(2.83)
In order to evaluate the second-order moments in different time instants,
let us create a vector
x
]
T
. Then, if we compute
=
[
X
(
t
1
)
X
(
t
2
)
···
X
(
t
n
)
E
xx
H
for a zero-mean process, we obtain the autocorrelation matrix
E
xx
H
R
xx
=
⎡
⎣
⎤
⎦
X
∗
(
X
∗
(
X
(
t
1
)
t
1
)
···
X
(
t
1
)
t
n
)
.
.
.
.
.
.
.
.
.
=
X
∗
(
X
∗
(
X
(
t
n
)
t
1
)
···
X
(
t
n
)
t
n
)
⎡
⎤
R
X
(
t
1
,
t
1
)
···
R
X
(
t
1
,
t
n
)
⎣
.
.
.
.
.
.
⎦
.
.
.
=
(2.84)
R
X
(
t
n
,
t
1
)
···
R
X
(
t
n
,
t
n
)
H
stands for Hermitian transposi-
tion. The autocovariance matrix is obtained if the autocorrelation function is
replaced by the autocovariance function in (2.84).
Another important measure is the cross-correlation function, which
expresses the correlation between different processes. Given two different
stochastic processes
X
In the above definition, the superscript
(
·
)
(
t
)
and
Y
(
t
)
, the two cross-correlation functions can be
defined as [135]
E
X
t
2
)
Y
∗
(
R
XY
(
t
1
,
t
2
)
=
(
t
1
)
(2.85)
and
E
Y
t
2
)
X
∗
(
R
YX
(
t
1
,
t
2
)
=
(
t
1
)
(2.86)
We can also define a cross-correlation matrix, given by
R
X
(
t
1
,
t
2
)
R
XY
(
t
1
,
t
2
)
R
XY
(
t
1
,
t
2
)
=
(2.87)
R
YX
(
t
1
,
t
2
)
R
Y
(
t
1
,
t
2
)
So far, there has been a strong dependence of the definitions with respect
to multiple time indices. However, some random signals show regularities
that can be extremely useful, as we shall now see.
