Image Processing Reference
In-Depth Information
To determine the coef
cients of the AR model, the mean-squared error (MSE)
between the predicted output and the measured output is minimized. From Equations
7.63 and 7.64, the error between the measured output and the predicted output is
given by
X
P
e ( n ) ¼ y ( n ) þ
a i y ( n i )
(
7
:
65
)
i ¼ 1
The MSE can be expressed as the experted value of the square of the error
2
3
(
) 2
X
¼ E ( n ) þ
P
4
5
Ee 2
( n )
a i y ( n i )
(
7
:
66
)
i ¼ 1
Equation 7.66 can be expanded as
"
(
)
(
)
#
X
X
¼ E ( n ) þ
P
P
Ee 2
( n )
a i y ( n i )
y ( n ) þ
a i y ( n i )
(
:
)
7
67
i ¼ 1
i ¼ 1
Expanding the above equation results in
"
#
"
#
þ E X
X
¼ Ey 2
P
P
Ee 2
( n )
( n )
a i y ( n i ) y ( n )
þ Ey ( n )
a i y ( n i )
i ¼ 1
i ¼ 1
"
#
þ E X
X
P
P
a i y ( n i )
a j y ( n j )
(
7
:
68
)
i ¼ 1
j ¼ 1
Interchanging the expectation and summation, we have
X
X
p
p
¼ Ey 2
þ
Ee 2
( n )
( n )
a i Ey ( n i ) y ( n )
½
þ
a i Ey ( n ) y ( n i )
½
i ¼ 1
i ¼ 1
X
X
p
p
þ
a i a j Ey ( n i )
½
y ( n j )
(
7
:
69
)
i ¼
1
j ¼
1
Performing the expectation, we get
X
X
X
X
P
P
P
P
¼ r yy (
Ee 2
( n )
0
) þ
a i r yy ( i ) þ
a i r yy ( i ) þ
a i a j r yy ( j i )(
7
:
70
)
i ¼
1
i ¼
1
i ¼
1
j ¼
1
Therefore,
2 X
X
X
¼ r yy (
P
P
P
Ee 2
( n )
0
) þ
a i r yy ( i ) þ
a i a j r yy ( j i )
(
7
:
71
)
i ¼ 1
i ¼ 1
j ¼ 1
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