Image Processing Reference
In-Depth Information
where r
yy
(i) is the autocorrelation of the output process, y(n), at ith lag. As the signal
y(n) is real, autocorrelation function r
yy
is symmetric (r
yy
(i)
¼
r
yy
(
i)). We minimize
the above expression to obtain the optimum values for a
i
, i
¼
1, 2, . . . , P. Applying
the orthogonality principle, we obtain the well-known Yule
-
Walker equations
X
P
r
yy
(
j
i
)
a
j
¼
r
yy
(
i
)
i
¼
1, 2,
...
, P
(
7
:
72
)
j
¼
1
The above equations can be written in matrix form as
2
4
3
5
2
4
3
5
2
4
3
5
r
yy
(
0
)
r
yy
(
1
)
r
yy
(
2
)
r
yy
(
P
1
)
r
yy
(
1
)
a
1
a
2
a
3
.
a
P
r
yy
(
1
)
r
yy
(
0
)
r
yy
(
1
)
r
yy
(
P
2
)
r
yy
(
2
)
.
.
r
yy
(
3
)
.
.
¼
(
7
:
73
)
.
r
yy
(
P
)
.
.
.
r
yy
(
2
P
)
r
yy
(
1
P
)
r
yy
(
0
)
Since the output of the system y(n) is real, its autocorrelation is symmetric, that is,
r
yy
(
i
) ¼
r
yy
(
i
)
, so Equation 7.73 can be rewritten as
2
4
3
5
2
4
3
5
2
4
3
5
r
yy
(
0
)
r
yy
(
1
)
r
yy
(
2
)
r
yy
(
P
1
)
r
yy
(
1
)
a
1
a
2
a
3
.
a
P
r
yy
(
1
)
r
yy
(
0
)
r
yy
(
1
)
r
yy
(
P
2
)
r
yy
(
2
)
.
.
r
yy
(
3
)
.
.
¼
(
7
:
74
)
.
r
yy
(
P
)
.
.
.
r
yy
(
P
2
)
r
yy
(
P
)
r
yy
(
)
1
0
And the variance of the error function e(n), MSE is given by
2
s
e
¼
r
yy
(
0
) þ
r
yy
(
1
)
a
1
þ
r
yy
(
2
)
a
2
þþ
r
yy
(
P
)
a
P
(
7
:
75
)
Combining the above two equations we obtain
2
3
2
3
2
3
r
yy
(
0
)
r
yy
(
1
)
r
yy
(
2
)
r
yy
(
P
1
)
r
yy
(
P
)
1
a
1
a
2
.
a
P
1
a
P
2
0
0
.
0
0
s
r
yy
(
1
)
r
yy
(
0
)
r
yy
(
1
)
r
yy
(
P
2
)
r
yy
(
P
1
)
4
5
4
5
4
5
r
yy
(
2
)
r
yy
(
1
)
r
yy
(
0
)
r
yy
(
P
3
)
r
yy
(
P
2
)
.
.
¼
(
7
:
76
)
.
.
.
.
.
r
yy
(
P
1
)
r
yy
(
1
)
r
yy
(
P
)
...
r
yy
(
0
)
The AR coef
cients and the variance of error are determined by solving the above
system of linear equations using least-squares regression equations for a set of N
number of measured data samples. The autocorrelation and variance of the data can
be found using MATLAB functions.
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