From the previous approximation theorems, we see that a combination of
zero wavelet and zero scaling function moments used with samples of the
signal may give superior results to wavelets with only zero wavelet
moments. Not only does forcing zero scaling function moments give a
better approximation of the expansion coefficients by samples, it often
causes
the scaling function to be more symmetric. Indeed, that characteristic
may be more important than the sample approximation in certain
applications.
Daubechies considered the design of these wavelets which were suggested by
Coifman (Reference), (Reference), (Reference). Gopinath (Reference), (Reference) and Wells
(Reference), (Reference) show how zero scaling function moments give a better
approximation of highresolution scaling coefficients by samples. Tian
and Wells (Reference), (Reference) have also designed biorthogonal systems with
mixed zero moments with very interesting properties.
The Coifman wavelet system (Daubechies named the basis functions
“coiflets") is an orthonormal multiresolution wavelet system with
∫
t
k
φ
(
t
)
d
t
=
m
(
k
)
=
0
,
for
k
=
1
,
2
,
⋯
,
L

1
∫
t
k
φ
(
t
)
d
t
=
m
(
k
)
=
0
,
for
k
=
1
,
2
,
⋯
,
L

1
(40)
∫
t
k
ψ
(
t
)
d
t
=
m
1
(
k
)
=
0
,
for
k
=
1
,
2
,
⋯
,
L

1
.
∫
t
k
ψ
(
t
)
d
t
=
m
1
(
k
)
=
0
,
for
k
=
1
,
2
,
⋯
,
L

1
.
(41)This definition imposes the requirement that there be at least L1L1 zero
scaling function moments and at least L1L1 wavelet moments in addition to
the one zero moment of m1(0)m1(0) required by orthogonality. This system is
said to be of order or degree LL and sometime has the additional
requirement that the length of the scaling function filter h(n)h(n), which
is denoted NN, is minimum (Reference), (Reference). In the
design of these coiflets, one obtains more total zero moments than
N/21N/21. This was first noted by Beylkin, et al (Reference).
The length4 wavelet system
has only one degree of freedom, so it cannot have both a scaling function
moment and wavelet moment of zero (see Table 6). Tian
(Reference), (Reference) has derived formulas for four length6 coiflets. These are:
h
=

3
+
7
16
2
,
1

7
16
2
,
7

7
8
2
,
7
+
7
8
2
,
5
+
7
16
2
,
1

7
16
2
,
h
=

3
+
7
16
2
,
1

7
16
2
,
7

7
8
2
,
7
+
7
8
2
,
5
+
7
16
2
,
1

7
16
2
,
(42)or
h
=

3

7
16
2
,
1
+
7
16
2
,
7
+
7
8
2
,
7

7
8
2
,
5

7
16
2
,
1
+
7
16
2
,
h
=

3

7
16
2
,
1
+
7
16
2
,
7
+
7
8
2
,
7

7
8
2
,
5

7
16
2
,
1
+
7
16
2
,
(43)or
h
=

3
+
15
16
2
,
1

15
16
2
,
3

15
8
2
,
3
+
15
8
2
,
13
+
15
16
2
,
9

15
16
2
,
h
=

3
+
15
16
2
,
1

15
16
2
,
3

15
8
2
,
3
+
15
8
2
,
13
+
15
16
2
,
9

15
16
2
,
(44)or
h
=

3

15
16
2
,
1
+
15
16
2
,
3
+
15
8
2
,
3

15
8
2
,
13

15
16
2
,
9
+
15
16
2
,
h
=

3

15
16
2
,
1
+
15
16
2
,
3
+
15
8
2
,
3

15
8
2
,
13

15
16
2
,
9
+
15
16
2
,
(45)with the first formula Equation 42 giving the same result as Daubechies
(Reference), (Reference) (corrected) and that of Odegard (Reference) and the third
giving the same result as Wickerhauser (Reference). The results from
Equation 42 are included in Table 4 along with the discrete
moments of the scaling function and wavelet, μ(k)μ(k) and μ1(k)μ1(k) for
k=0,1,2,3k=0,1,2,3. The design of a length6 Coifman system specifies one zero
scaling function moment and one zero wavelet moment (in addition to
μ1(0)=0μ1(0)=0), but we, in fact, obtain one extra zero scaling function
moment. That is the result of m(2)=m(1)2m(2)=m(1)2 from (Reference). In other
words, we get one more zero scaling function moment than the two degrees
of freedom would seem to indicate. This is true for all lengths N=6ℓN=6ℓ
for ℓ=1,2,3,⋯ℓ=1,2,3,⋯ and is a result of the interaction between the
scaling function moments and the wavelet moments described later.
The property of zero wavelet moments is shift invariant, but the zero
scaling function moments are shift dependent (Reference). Therefore,
a particular shift for the scaling function must be used. This shift is
two for the length6 example in Table 4, but is different for the
solutions in Equation 44 and Equation 45. Compare this table to the
corresponding one for Daubechies length6 scaling functions and wavelets
given in Table 2 where there are two zero discrete wavelet
moments – just as many as the degrees of freedom in that design.
The scaling function from Equation 42 is fairly symmetric, but not around
its center and the other three designs in Equation 43, Equation 44, and
Equation 45 are not symmetric at all. The scaling function from
Equation 42 is also fairly smooth, and from Equation 44 only slightly less
so but the scaling function from Equation 43 is very rough and from
Equation 45 seems to be fractal. Examination of the frequency response
H(ω)H(ω) and the zero location of the FIR filters h(n)h(n) shows very
similar frequency responses for Equation 42 and Equation 44 with
Equation 43 having a somewhat irregular but monotonic frequency response
and Equation 45 having a zero on the unit circle at ω=π/3ω=π/3,
i.e., not satisfying Cohen's condition (Reference) for an orthognal basis.
It is also worth noticing that the design in Equation 42 has the largest
Hölder smoothness. These four designs, all satisfying the same
necessary conditions, have very different characteristics. This tells us
to be very careful in using zero moment methods to design wavelet systems.
The designs are not unique and some are much better than others.
Table 4 contains the scaling function and wavelet coefficients
for the length6 and 12 designed by Daubechies and length8 designed by
Tian together with their discrete moments. We see the extra zero scaling
function moments for lengths 6 and 12 and also the extra zero for
lengths 8 and 12 that occurs after a nonzero one.
The continuous moments can be calculated from the discrete moments and
lower order continuous moments (Reference), (Reference), (Reference) using Equation 9
and Equation 10. An important relationship of the discrete moments for a
system with K1K1 zero wavelet moments is found by calculating the
derivatives of the magnitude squared of the discrete time Fourier
transform of h(n)h(n) which is H(ω)=∑nh(n)eiωnH(ω)=∑nh(n)eiωn and
has 2K12K1 zero derivatives of the magnitude squared at ω=0ω=0. This
gives (Reference) the kthkth derivative for kk even and 1<k<2K11<k<2K1
∑
ℓ
=
0
k
k
ℓ
(

1
)
ℓ
μ
(
ℓ
)
μ
(
k

ℓ
)
=
0
.
∑
ℓ
=
0
k
k
ℓ
(

1
)
ℓ
μ
(
ℓ
)
μ
(
k

ℓ
)
=
0
.
(46)Solving for μ(k)μ(k) in terms of lower order discrete moments and using
μ(0)=2μ(0)=2 gives for kk even
μ
(
k
)
=
1
2
2
∑
ℓ
=
1
k

1
k
ℓ
(

1
)
ℓ
μ
(
ℓ
)
μ
(
k

ℓ
)
μ
(
k
)
=
1
2
2
∑
ℓ
=
1
k

1
k
ℓ
(

1
)
ℓ
μ
(
ℓ
)
μ
(
k

ℓ
)
(47)which allows calculating the evenorder discrete scaling function moments
in terms of the lower oddorder discrete scaling function moments for
k=2,4,⋯,2K2k=2,4,⋯,2K2. For example:
μ
(
2
)
=
1
2
μ
2
(
1
)
μ
(
4
)
=

1
2
2
[
8
μ
(
1
)
μ
(
3
)

3
μ
4
(
1
)
]
⋯
⋯
μ
(
2
)
=
1
2
μ
2
(
1
)
μ
(
4
)
=

1
2
2
[
8
μ
(
1
)
μ
(
3
)

3
μ
4
(
1
)
]
⋯
⋯
(48)which can be seen from values in Table 2.
Johnson (Reference) noted from Beylkin (Reference) and Unser
(Reference) that by using the moments of the autocorrelation function
of the scaling function, a relationship of the continuous scaling function
moments can be derived in the form
∑
ℓ
=
0
k
k
ℓ
(

1
)
k

ℓ
m
(
ℓ
)
m
(
k

ℓ
)
=
0
∑
ℓ
=
0
k
k
ℓ
(

1
)
k

ℓ
m
(
ℓ
)
m
(
k

ℓ
)
=
0
(49)where 0<k<2K0<k<2K if K1K1 wavelet moments are zero. Solving for m(k)m(k)
in terms of lower order moments gives for kk even
m
(
k
)
=

1
2
∑
ℓ
=
1
k

1
k
ℓ
(

1
)
ℓ
m
(
ℓ
)
m
(
k

ℓ
)
m
(
k
)
=

1
2
∑
ℓ
=
1
k

1
k
ℓ
(

1
)
ℓ
m
(
ℓ
)
m
(
k

ℓ
)
(50)which allows calculating the evenorder scaling function moments
in terms of the lower oddorder scaling function moments for
k=2,4,⋯,2K2k=2,4,⋯,2K2. For example (Reference):
m
(
2
)
=
m
2
(
1
)
m
(
4
)
=
4
m
(
3
)
m
(
1
)

3
m
4
(
1
)
m
(
6
)
=
6
m
(
5
)
m
(
1
)
+
10
m
2
(
3
)
+
60
m
(
3
)
m
3
(
1
)
+
45
m
6
(
1
)
m
(
8
)
=
8
m
(
7
)
m
(
1
)
+
56
m
(
5
)
m
(
3
)

168
m
(
5
)
m
3
(
1
)
+
2520
m
(
3
)
m
5
(
1
)

840
m
(
3
)
m
2
(
1
)

1575
m
8
(
1
)
⋯
⋯
m
(
2
)
=
m
2
(
1
)
m
(
4
)
=
4
m
(
3
)
m
(
1
)

3
m
4
(
1
)
m
(
6
)
=
6
m
(
5
)
m
(
1
)
+
10
m
2
(
3
)
+
60
m
(
3
)
m
3
(
1
)
+
45
m
6
(
1
)
m
(
8
)
=
8
m
(
7
)
m
(
1
)
+
56
m
(
5
)
m
(
3
)

168
m
(
5
)
m
3
(
1
)
+
2520
m
(
3
)
m
5
(
1
)

840
m
(
3
)
m
2
(
1
)

1575
m
8
(
1
)
⋯
⋯
(51)Table 4: Coiflet Scaling Function and Wavelet Coefficients plus their Discrete Moments

LengthN=6N=6, 
Degree L=2L=2 



n
n

h
(
n
)
h
(
n
)

h
1
(
n
)
h
1
(
n
)

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

k
k

2 
0.07273261951285 
0.01565572813546 
1.414213 
0 
0 
1 
0.33789766245781 
0.07273261951285 
0 
0 
1 
0 
0.85257202021226 
0.38486484686420 
0 
1.163722 
2 
1 
0.38486484686420 
0.85257202021226 
0.375737 
3.866903 
3 
2 
0.07273261951285 
0.33789766245781 
2.872795 
10.267374 
4 
3 
0.01565572813546 
0.07273261951285 




LengthN=8N=8, 
Degree L=3L=3 



n
n

h
(
n
)
h
(
n
)

h
1
(
n
)
h
1
(
n
)

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

k
k

4 
0.04687500000000 
0.01565572813546 
1.414213 
0 
0 
3 
0.02116013576461 
0.07273261951285 
0 
0 
1 
2 
0.14062500000000 
0.38486484686420 
0 
0 
2 
1 
0.43848040729385 
1.38486484686420 
2.994111 
0.187868 
3 
0 
1.38486484686420 
0.43848040729385 
0 
11.976447 
4 
1 
0.38486484686420 
0.14062500000000 
45.851020 
43.972332 
5 
2 
0.07273261951285 
0.02116013576461 
63.639610 
271.348747 
6 
3 
0.01565572813546 
0.04687500000000 




LengthN=12N=12, 
Degree L=4L=4 



n
n

h
(
n
)
h
(
n
)

h
1
(
n
)
h
1
(
n
)

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

k
k

4 
0.016387336463 
0.000720549446 
1.414213 
0 
0 
3 
0.041464936781 
0.001823208870 
0 
0 
1 
2 
0.067372554722 
0.005611434819 
0 
0 
2 
1 
0.386110066823 
0.023680171946 
0 
0 
3 
0 
0.812723635449 
0.059434418646 
0 
11.18525 
4 
1 
0.417005184423 
0.076488599078 
5.911352 
175.86964 
5 
2 
0.076488599078 
0.417005184423 
0 
1795.33634 
6 
3 
0.059434418646 
0.812723635449 
586.341304 
15230.54650 
7 
4 
0.023680171946 
0.386110066823 
3096.310009 
117752.68833 
8 
5 
0.005611434819 
0.067372554722 



6 
0.001823208870 
0.041464936781 



7 
0.000720549446 
0.016387336463 



if the wavelet moments are zero up to k=K1k=K1. Notice that setting m(1)=m(3)=0m(1)=m(3)=0 causes m(2)=m(4)=m(6)=m(8)=0m(2)=m(4)=m(6)=m(8)=0 if sufficient wavelet
moments are zero. This explains the extra zero moments in
Table 4. It also shows that the traditional specification of
zero scaling function moments is redundant. In Table 4 m(8)m(8)
would be zero if more wavelet moments were zero.
Table 5: Discrete and Continuous Moments for the Coiflet Systems

N=6N=6, 
L
=
2
L
=
2



k
k

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

m
(
k
)
m
(
k
)

m
1
(
k
)
m
1
(
k
)

0 
1.4142135623 
0 
1.0000000000 
0 
1 
0 
0 
0 
0 
2 
0 
1.1637219122 
0 
0.2057189138 
3 
0.3757374752 
3.8669032118 
0.0379552166 
0.3417891854 
4 
2.8727952940 
10.2673737288 
0.1354248688 
0.4537580992 
5 
3.7573747525 
28.0624304008 
0.0857053279 
0.6103378310 

N=8N=8, 
L
=
3
L
=
3



k
k

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

m
(
k
)
m
(
k
)

m
1
(
k
)
m
1
(
k
)

0 
1.4142135623 
0 
1.0000000000 
0 
1 
0 
0 
0 
0 
2 
0 
0 
0 
0 
3 
2.9941117777 
0.1878687376 
0.3024509630 
0.0166054072 
4 
0 
11.9764471108 
0 
0.5292891854 
5 
45.8510203537 
43.9723329775 
1.0458570134 
0.9716604635 
To see the continuous scaling function and wavelet moments for these systems,
Table 5 shows both the continuous and discrete
moments for the length6 and 8 coiflet systems. Notice
the zero moment
m(4)=μ(4)=0m(4)=μ(4)=0 for length8. The length14, 20, and 26 systems also have
the “extra" zero scaling moment just after the first nonzero moment.
This always occurs for lengthN=6ℓ+2N=6ℓ+2 coiflets.
(Reference) shows the length6, 8, 10, and 12
coiflet scaling functions φ(t)φ(t) and wavelets ψ(t)ψ(t). Notice their approximate symmetry
and compare this to Daubechies' classical wavelet systems and her more
symmetric ones achieved by using the different factorization mentioned in
"Daubechies' Method for Zero Wavelet Moment Design" and shown in (Reference). The difference between these
systems and truly symmetric ones (which requires giving up orthogonality,
realness, or finite support) is probably negligible in many applications.
3: Length6, 8, 10, and 12 Coiflet Scaling Functions and Wavelets








The preceding section shows that Coifman systems do not necessarily have
an equal number of scaling function and wavelet moments equal to zero.
Lengths N=6ℓ+2N=6ℓ+2 have equal numbers of zero scaling function and wavelet
moments, but always have evenorder “extra" zero scaling function moments
located after the first nonzero one. Lengths N=6ℓN=6ℓ always have an
“extra" zero scaling function moment. Indeed, both will have several
evenorder “extra" zero moments for longer NN as a result of the
relationships illustrated in Equation 51 through Equation 51. Lengths
N=6ℓ2N=6ℓ2 do not occur for the original definition of a Coifman system
if one looks only at the degree with minimum length. If we specify the
length of the coefficient vector, all even lengths become possible, some
with the same coiflet degree.
We examine the general Coifman wavelet system defined in Equation 40 and
Equation 41 and allow the number of specified zero scaling function and
wavelet moments to differ by at most one. That will include all the
reported coiflets plus length10, 16, 22, and N=6ℓ2N=6ℓ2.
The length10 was designed by Odegard (Reference) by setting
the number of zero scaling functions to 3 and the number of zero wavelet
moment to 2 rather than 2 and 2 for the length8 or 3 and 3 for the
length12 coiflets. The result in Table 6 shows that the
length10 design again gives one extra zero scaling function moment which
is two more than the number of zero wavelet moments. This is an
evenorder moment predicted by Equation 51 and results in a total number
of zero moments between that for length8 and length12, as one would
expect. A similar approach was used to design length16, 22, and 28.
Table 6: Coiflet Scaling Function and Wavelet Coefficients
plus their Discrete Moments

LengthN=4N=4, 
Degree L=1L=1 



n
n

h
(
n
)
h
(
n
)

h
1
(
n
)
h
1
(
n
)

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

k
k

1 
0.224143868042 
0.129409522551 
1.414213 
0 
0 
0 
0.836516303737 
0.482962913144 
0 
0.517638 
1 
1 
0.482962913144 
0.836516303737 
0.189468 
0.189468 
2 
2 
0.129409522551 
0.224143868042 
0.776457 
0.827225 
3 

LengthN=10N=10, 
Degree L=3L=3 



n
n

h
(
n
)
h
(
n
)

h
1
(
n
)
h
1
(
n
)

μ
(
k
)
μ
(
k
)

μ
1
(
k
)
μ
1
(
k
)

k
k

2 
0.032128481856 
0.000233764788 
1.414213 
0 
0 
1 
0.075539271956 
0.000549618934 
0 
0 
1 
0 
0.096935064502 
0.013550370057 
0 
0 
2 
1 
0.491549094027 
0.033777338659 
0 
3.031570 
3 
2 
0.805141083557 
0.304413564385 
0 
24.674674 
4 
3 
0.304413564385 
0.805141083557 
14.709025 
138.980052 
5 
4 
0.033777338659 
0.491549094027 
64.986095 
710.373341 
6 
5 
0.013550370057 
0.096935064502 



6 
0.000549618934 
0.075539271956 



7 
0.000233764788 
0.032128481856 



We have designed these “new" coiflet systems (e.g., N=10,16,22,28N=10,16,22,28)
by using the Matlab optimization toolbox constrained optimization
function. Wells and Tian (Reference) used Newton's method to design
lengths N=6ℓ+2N=6ℓ+2 and N=6ℓN=6ℓ coiflets up to length 30 (Reference).
Selesnick (Reference) has used a filter design approach. Still
another approach is given by Wei and Bovik (Reference).
Table 6 also shows the result of designing a length4 system,
using the one degree of freedom to ask for one zero scaling function
moment rather than one zero wavelet moment as we did for the Daubechies
system. For length4, we do not get any “extra" zero moments because
there are not enough zero wavelet moments. Here we see a direct tradeoff
between zero scaling function moments and wavelet moments. Adding these
new lengths to our traditional coiflets gives Table 7.
Table 7: Moments for Various LengthNN and DegreeLL Coiflets, where (*) is the number
of zero wavelet moments, excluding the m1(0)=0m1(0)=0
N
N

L
L

m
=
0
m
=
0

m
1
=
0
m
1
=
0

m
=
0
m
=
0

m
1
=
0
m
1
=
0

Total zero 
Hölder 


set 
set* 
actual 
actual* 
moments 
exponent 
4 
1 
1 
0 
1 
0 
1 
0.2075 
6 
2 
1 
1 
2 
1 
3 
1.0137 
8 
3 
2 
2 
2 
2 
4 
1.3887 
10 
3 
3 
2 
4 
2 
6 
1.0909 
12 
4 
3 
3 
4 
3 
7 
1.9294 
14 
5 
4 
4 
4 
4 
8 
1.7353 
16 
5 
5 
4 
6 
4 
10 
1.5558 
18 
6 
5 
5 
6 
5 
11 
2.1859 
20 
7 
6 
6 
6 
6 
12 
2.8531 
22 
7 
7 
6 
8 
6 
14 
2.5190 
24 
8 
7 
7 
8 
7 
15 
2.8300 
26 
9 
8 
8 
8 
8 
16 
3.4404 
28 
9 
9 
8 
10 
8 
18 
2.9734 
30 
10 
9 
9 
10 
9 
19 
3.4083 
The fourth and sixth columns in Table 7 contain the number of zero wavelet
moments, excluding the m1(0)=0m1(0)=0 which is zero because of orthogonality in
all of these systems. The extra zero scaling function moments that occur
after a nonzero moment for N=6ℓ+2N=6ℓ+2 are also excluded from the count.
This table shows coiflets for all even lengths. It shows the extra zero scaling
function moments that are sometime achieved and how the total number
of zero moments monotonically increases and how the “smoothness" as measured
by the Hölder exponent (Reference), (Reference), (Reference) increases with NN and LL.
When both scaling function and wavelet moments are set to zero, a larger
number can be obtained than is expected from considering the degrees of
freedom available. As stated earlier, of the NN degrees of freedom
available from the NN coefficients, h(n)h(n), one is used to insure
existence of φ(t)φ(t) through the linear constraint Equation 1, and
N/2N/2 are used to insure orthonormality through the quadratic constraints
Equation 1. This leaves N/21N/21 degrees of freedom to achieve other
characteristics. Daubechies used these to set the first N/21N/21 wavelet
moments to zero. If setting scaling function moments were independent of
setting wavelet moments zero, one would think that the coiflet
system would allow (N/21)/2(N/21)/2 wavelet moments to be set zero and the same
number of scaling function moments. For the coiflets
described in Table 7, one always obtains more than this.
The structure of this problem allows more zero moments to be both set and
achieved than the simple degrees of freedom would predict. In fact, the
coiflets achieve approximately 2N/32N/3 total zero moments as compared with
the number of degrees of freedom which is approximately N/2N/2, and which
is achieved by the Daubechies wavelet system.
As noted earlier and illustrated in Table 8, these
coiflets fall into three classes. Those with scaling filter lengths of
N=6ℓ+2N=6ℓ+2 (due to Tian) have equal number of zero scaling function and
wavelet moments, but always has “extra" zero scaling function moments
located after the first nonzero one. Lengths N=6ℓN=6ℓ (due to
Daubechies) always have one more zero scaling function moment than zero
wavelet moment and lengths N=6ℓ2N=6ℓ2 (new) always have two more zero
scaling function moments than zero wavelet moments. These “extra" zero
moments are predicted by Equation 51 to Equation 51, and there will be
additional evenorder zero moments for longer lengths. We have observed
that within each of these classes, the Hölder exponent increases
monotonically.
Table 8: Number of Zero Moments for The Three Classes of Coiflets
(ℓ=1,2,⋯ℓ=1,2,⋯),
*excluding μ1(0)=0μ1(0)=0, †excluding NonContiguous zeros
N
N

m=0m=0† 
m1=0m1=0* 
Total zero 
Length 
achieved 
achieved 
moments 
N
=
6
ℓ
+
2
N
=
6
ℓ
+
2

(
N

2
)
/
3
(
N

2
)
/
3

(
N

2
)
/
3
(
N

2
)
/
3

(
2
/
3
)
(
N

2
)
(
2
/
3
)
(
N

2
)

N
=
6
ℓ
N
=
6
ℓ

N
/
3
N
/
3

(
N

3
)
/
3
(
N

3
)
/
3

(
2
/
3
)
(
N

3
/
2
)
(
2
/
3
)
(
N

3
/
2
)

N
=
6
ℓ

2
N
=
6
ℓ

2

(
N
+
2
)
/
3
(
N
+
2
)
/
3

(
N

4
)
/
3
(
N

4
)
/
3

(
2
/
3
)
(
N

1
)
(
2
/
3
)
(
N

1
)

The approach taken in some investigations of coiflets would specify the
coiflet degree and then find the shortest filter that would achieve that
degree. The lengths N=6ℓ2N=6ℓ2 were not found by this approach because
they have the same coiflet degree as the system just two shorter.
However, they achieve two more zero scaling function moments than the
shorter length with the same degree. By specifying the number of zero
moments and/or the filter length, it is easier to see the complete picture.
Table 7 is just part of a large collection of zero
moment wavelet system designs with a wide variety of tradeoffs that would
be tailored to a particular application. In addition to the variety
illustrated here, many (perhaps all) of these sets of specified zero
moments have multiple solutions. This is certainly true for length6 as
illustrated in Equation 42 through Equation 45 and for other lengths that
we have found experimentally. The variety of solutions for each length
can have different shifts, different Hölder exponents, and different
degrees of being approximately symmetric.
The results of this chapter and section show the importance of moments to
the characteristics of scaling functions and wavelets. It may not,
however, be necessary or important to use the exact criteria of Daubechies
or Coifman, but understanding the effects of zero moments is very
important. It may be that setting a few scaling function moments and a
few wavelets moments may be sufficient with the remaining degrees of
freedom used for some other optimization, either in the frequency domain
or in the time domain. As is noted in the next section, an alternative
might be to minimize a larger number of various moments rather than to
zero a few (Reference).
Examples of the Coiflet Systems are shown in (Reference).