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In-Depth Information
correlation. In the case of
no
sample-to-sample correlation, such as white noise, the autocorre-
lation function is zero for lags greater than zero, as seen in Equation (
88
). The autocorrelation
function for the AR(
N
) process can be obtained as follows:
E
x
n
x
n
−
k
R
xx
(
k
)
=
(91)
E
N
=
a
i
x
n
−
i
+
n
(
x
n
−
k
)
(92)
i
=
1
E
N
E
n
x
n
−
k
=
a
i
x
n
−
i
x
n
−
k
+
(93)
i
=
1
i
=
1
a
i
R
xx
(
k
−
i
)
for
k
>
0
=
(94)
i
=
1
a
i
R
xx
(
2
i
)
+
σ
for
k
=
0
where
σ
E
n
x
n
−
k
=
2
for
k
=
0
(95)
0for
k
>
0
Example8.6.1:
Suppose we have an AR(3) process. Let us write out the equations for the autocorrelation
coefficient for lags 1, 2, and 3:
R
xx
(
1
)
=
a
1
R
xx
(
0
)
+
a
2
R
xx
(
1
)
+
a
3
R
xx
(
2
)
R
xx
(
2
)
=
a
1
R
xx
(
1
)
+
a
2
R
xx
(
0
)
+
a
3
R
xx
(
1
)
R
xx
(
3
)
=
a
1
R
xx
(
2
)
+
a
2
R
xx
(
1
)
+
a
3
R
xx
(
0
)
If we know the values of the autocorrelation function
R
xx
(
k
)
,for
k
=
0
,
1
,
2
,
3, we can use
this set of equations to find the AR(3) coefficients
{
a
1
,
a
2
,
a
3
}
. On the other hand, if we know
2
the model coefficients and
σ
, we can use the above equations along with the equation for
to find the first four autocorrelation coefficients. All of the other autocorrelation values
can be obtained by using Equation (
94
).
R
xx
(
0
)
To see how the autocorrelation function is related to the temporal behavior of the sequence,
let us look at the behavior of a simple AR(1) source.
Example8.6.2:
An AR(1) source is defined by the equation
x
n
=
a
1
x
n
−
1
+
n
(96)
The autocorrelation function for this source (see Problem 8 at the end of this chapter) is given
by
1
a
1
σ
2
R
xx
(
k
)
=
(97)
a
1
−
1
