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Table 8.1.
Conjugate mirror filter h[k] for linear and cubic splines [116].
n
k
h[k]
n
k
h[k]
0
0.817645956
7,-7
-0.017982291
1,-1
0.397294430
8,-8
0.008685294
2,-2
-0.069101020
9,9
0.008201477
2
3,-3
-0.051945337 4 10,-10
-0.004353840
4,-4
0.016974805
11,-11
-0.003882426
5,-5
0.009990599
12,-12
0.002186714
6,-6
-0.003883261
13,-13
0.001882120
7,-7
-0.002201945
14,-14
-0.001103748
8,-8
0.000923371
15,-15
-0.000927187
9,-9
0.000511636
16,-16
0.000559952
10,-10
-0.000224296
17,-17
0.000462093
11,-11
-0.000122686
18,-18
-0.000285414
0
0.766130398 4 19,-19 -0.000232304
1,-1
0.433923147
20,-10
0.000146098
2,-2
-0.050201753
4
3,-3
-0.110036987
4,-4
0.032080869
5,-5
0.042068328
6,-6
-0.017176331
In a biorthogonal system, two pairs of filters are normally used. One pair is
called the analysis filter and the other pair is called the synthesis filter. Hence,
if
h, g
are the analysis filters and
h
and
g
are the synthesis filters, then they
should be connected to each other suitably. According to Cohen, Daubechies,
and Feauveau [40], they are connected by
1)
k
h
(1
g
[
k
]=(
−
−
k
)
,
1)
k
h
[1
g
[
k
]=(
−
−
k
]
,
i.e., they are cross related by time reversal and flipping signs of every other
member. When
h
[
k
]=
h
[
k
],
g
[
k
] reduces to
g
[
k
]=(
1)
k
h
[1
k
]. This tells
us about scaling and wavelet coecients for orthogonal wavelets, wherein
g
[
k
]=(
−
−
1)
k
h
[1
−
−
k
]. From the perfect reconstruction condition, we can write
[
k
]
h
[
k
+2
r
]=
δ
(
r
)
.
(8.42)
k
Thus,
h
is orthogonal to
h
. Hence, if we assume
h
[
k
] is not zero for
N
1
≤
r
≤
N
2
and
h
[
k
] is not zero for
N
1
≤
r
≤
N
2
, then
N
1
=2
l
+1
,
N
2
−
N
2
−
N
1
=2
l
+1
,
l, l
∈
Z.
h
[
k
]and
h
[
k
] are called the coe
cients of the scaling and dual scaling func-
tions. Similarly,
g
[
k
] and
g
[
k
] are called the coe
cients of the wavelet and
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