Biomedical Engineering Reference
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VVWj
−
=⊕
∈
Z
(3.34)
j
1
j
j
where
⊕
refers to concatenation. Thus, we have
VW V
VWW V
VWWWV
=⊕
=⊕⊕
=⊕⊕⊕
0
1
1
(3.35)
0
1
2
2
0
1
2
3
3
Thus, the closed subspaces
V
j
at level
j
are the sum of the whole function space
L
2
(
R
):
VW
=
⊕⊕⊕⊕∈
W
W
j
Z
(3.36)
j
j
+
1
j
+
2
j
+
3
Figure 3.17 depicts the MRWA described by (3.32).
Consequently,
φ
(
t
)
∈
V
1
⊂
V
0
and
ψ
(
t
/2)
∈
V
1
⊂
V
0
can be expressed as linear
combinations of the basis function of
V
0
, {
φ
(
t
)
= φ
(
t
−
k
):
k
∈
Z
}, that is:
()
()(
)
∑
φ
t
=
2
h k
φ
2
t
−
k
(3.37)
k
and
()
∑
()(
)
ψ
t
=
2
g k
φ
2
t
−
k
(3.38)
k
where
t
he
coefficients
h
(
k
)
and
g
(
k
)
are
defined
as
the
inner
products
(
)
φ
(),
t
22
φ
t
−
k
and
ψ
(),
t
22
φ
(
t
−
k
)
, respectively. The sequences {
h
(
k
),
k
∈
Z
}
and {
g
(
k
),
k
),
respectively. They form a pair of quadrature mirror filters that is used in the MRWA
[52]. There are many scaling functions in the literature including the Haar,
Daubechies, biorthogonal, Coiflets, Symlets, Morlet, Mexican hat and Meyer func-
∈
Z
} are coefficients of a lowpass filter
H
(
ω
) and a highpass filter
G
(
ω
V
3
W
3
V
2
W
2
V
1
WW
1
V
0
Figure 3.17
Multiresolution wavelet analysis.
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