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h
1
(
ω
)
h
∗
(
ω
)+
h
1
(
ω
+
π
)
h
∗
(
ω
+
π
)=0
.
Thus, we have
ψ
(
ω
)=
√
2
h
1
(
ω/
2)
φ
(
ω/
2)
,
1
h
∗
(
ω/
2+
π
)
φ
(
ω/
2)
.
√
2
e
−iω/
2
1
=
Now from
h
(
ω
)-
φ
(
ω
) relation, we can write (from equation (8.32))
1
φ
(
ω
)=
√
2
h
(
ω/
2)
φ
(
ω/
2)
.
Therefore,
1
φ
(2
ω
)=
√
2
h
(
ω
)
φ
(
ω
)
or,
h
(
ω
)=
√
2
φ
(2
ω
)
φ
(
ω
)
,
(2
ω
)
n
√
S
2
n
(2
ω
)
.
ω
n
√
S
2
n
(
ω
)
=
√
2
e
−iβω
,
(8.40)
e
−iβω/
2
=
e
−iβω/
2
S
2
n
(
ω
)
2
2
n−
1
S
2
n
(2
ω
)
.
With this, we are in a position to compute Battle and Lemarie wavelets in a
straightforward way. The generalized form of the Fourier transform of these
wavelet functions from equation (8.39) can be written as
ψ
(
ω
)=
√
2
h
1
(
ω/
2)
φ
(
ω/
2)
,
1
h
∗
(
ω/
2+
π
)
φ
(
ω/
2)
,
√
2
e
−iω/
2
1
=
√
2
e
−iω/
2
(8.41)
S
2
n
(
ω/
2+
π
)
1
(
ω/
2)
n
√
S
2
n
(
ω/
2)
,
1
=
2
2
n−
1
S
2
n
(2(
ω/
2+
π
))
.
e
−
iω/
2
ω
n
S
2
n
(
ω/
2+
π
)
=
S
2
n
(
ω
)
S
2
n
(
ω/
2)
.
One can compute the spline of any degree following the procedure for com-
putation adopted for linear and cubic splines for
n
=2and
n
=4.The
conjugate mirror filters for
n
=2and
n
= 4 are given by respective
h
(
ω
), and
their impulse response
h
(
k
) is listed in Table 8.1.
8.7 Biorthogonal Spline Wavelets
We have already seen the underlying concept of splines in orthogonal wavelet
systems. Use of splines in biorthogonal systems is equally simple, straight-
forward, and ecient. The main advantages of biorthogonal systems over or-
thogonal systems are more flexibility and greater ease of design. As far as
the filter design is concerned, orthogonal wavelet and scaling filters must have
equal length. This restriction, however, is not present in biorthogonal systems.
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