Biomedical Engineering Reference
In-Depth Information
FIGURE 2.8:
Example of
AIC
function. a) Signal generated with AR model order
5(
a
1
=
0
.
2
,
a
2
=
−
0
.
5
,
a
3
=
−
0
.
3
,
a
4
=
0
.
1
,
a
5
=
0
.
2), b)
AIC
for models of order
p
estimated for that signal. Note a minimum at
p
=
5.
p
i
=
0
a
i
x
t
−
i
=
ε
t
(2.54)
Applying
Z
transform to both sides of the above equation and taking into account its
properties (Sect. 1.4.6) we get:
A
(
f
)
X
(
f
)=
E
(
f
)
(2.55)
multiplying equation (2.55) by
A
−
1
leads to:
A
−
1
X
(
f
)=
(
f
)
E
(
f
)=
H
(
f
)
E
(
f
)
(2.56)
A
−
1
where we defined
H
(
f
)=
(
f
)
. From the definition of power spectrum we obtain:
X
∗
(
VH
∗
(
(
)=
(
)
)=
(
)
)
S
f
X
f
f
H
f
f
(2.57)
Expressing H(s) in terms of model coefficients:
1
H
(
z
)=
(2.58)
∑
p
k
0
a
k
z
−
k
=
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