Biomedical Engineering Reference
In-Depth Information
In addition, the Fourier transform of the mother wavelet
ψ
(
t
) vanishes in a
neighborhood of the origin. We denote by
W
m
the closed linear span of
{
ψ
(2
m
t
−
n
)
}
. This set of functions form an orthogonal basis of
L
2
(
R
)
.
That is,
V
m
=
V
m
−
1
⊕
W
m
−
1
L
2
(
R
)
=⊕
m
=−∞
W
m
For
f
∈
L
2
(
R
), we have the projections onto the subspace
V
m
and
W
m
re-
spectively given by
∞
a
m
,
n
2
m
/
2
φ
(2
m
t
−
n
)
,
f
m
(
t
)
=
P
m
f
(
t
)
=
(5.64)
n
=−∞
∞
f
m
(
t
)
=
P
m
f
(
t
)
=
b
m
,
n
2
m
/
2
ψ
(2
m
t
−
n
)
,
(5.65)
n
=−∞
where
a
m
,
n
=
2
−
m
/
2
∞
−∞
f
(
x
)
φ
(2
m
t
−
n
)
dx
,
b
m
,
n
=
2
−
m
/
2
∞
−∞
f
(
x
)
ψ
(2
m
t
−
n
)
dx
.
The coefficients
a
j
,
n
and
b
j
,
n
at resolution
j
=
m
and
j
=
m
+
1 are related
by a tree algorithm. To see this, we space
V
1
, we have two distinct orthonormal
bases:
√
2
φ
(2
x
−
n
)
∞
n
=−∞
and
{
φ
(
x
−
n
)
,ψ
(
x
−
k
)
}
n
,
k
=−∞
.
Hence each
f
∈
V
1
has an expansion
∞
a
1
,
n
√
2
φ
(2
x
−
n
)
f
(
x
)
=
n
=−∞
b
0
,
n
ψ
(
x
−
n
)
.
∞
a
0
,
n
φ
(
x
−
n
)
+
=
n
=−∞
By (5.63) we have
∞
∞
(
−
1)
n
−
1
c
1
−
n
+
2
k
b
0
,
k
,
a
1
,
n
=
c
n
−
2
k
a
0
,
k
+
(5.66)
k
=−∞
k
=−∞
