These theorems, relationships, and properties are generalizations of those
given in Chapter (Reference) and (Reference) with some outline proofs or
derivations given in the Appendix. For the multiplicity-MM problem, if the
support of the scaling function and wavelets and their respective coefficients
is finite and the system is
orthogonal or a tight frame, the length of the scaling function vector or
filter h(n)h(n) is a multiple of the multiplier MM. This is N=MGN=MG, where
Resnikoff and Wells (Reference) call MM the rank of the system and GG the
genus.
The results of (Reference), (Reference), (Reference), and (Reference) become
Theorem 28 If φ(t)φ(t) is an L1L1 solution to Equation 4 and
∫φ(t)dt≠0∫φ(t)dt≠0, then
∑
n
h
(
n
)
=
M
.
∑
n
h
(
n
)
=
M
.
(7)This is a generalization of the basic multiplicity-2 result in (Reference) and
does not depend on any particular normalization or orthogonality of φ(t)φ(t).
Theorem 29 If integer translates of the solution to Equation 4 are orthogonal, then
∑
n
h
(
n
+
M
m
)
h
(
n
)
=
δ
(
m
)
.
∑
n
h
(
n
+
M
m
)
h
(
n
)
=
δ
(
m
)
.
(8)This is a generalization of (Reference) and also does not depend on any normalization.
An interesting corollary of this theorem is
Corollary 3 If integer translates of the solution to Equation 4 are orthogonal, then
∑
n
|
h
(
n
)
|
2
=
1
.
∑
n
|
h
(
n
)
|
2
=
1
.
(9)A second corollary to this theorem is
Corollary 4 If integer translates of the solution to Equation 4 are orthogonal, then
∑
n
h
(
M
n
+
m
)
=
1
/
M
.
m
∈
Z
∑
n
h
(
M
n
+
m
)
=
1
/
M
.
m
∈
Z
(10)This is also true under weaker conditions than orthogonality as was
discussed for the M=2M=2 case.
Using the Fourier transform, the following relations can be derived:
Theorem 30 If φ(t)φ(t) is an L1L1 solution to Equation 4 and
∫φ(t)dt≠0∫φ(t)dt≠0, then
H
(
0
)
=
M
H
(
0
)
=
M
(11)which is a frequency domain existence condition.
Theorem 31 The integer translates of the solution to Equation 4 are orthogonal if
and only if
∑
ℓ
|
Φ
(
ω
+
2
π
ℓ
)
|
2
=
1
∑
ℓ
|
Φ
(
ω
+
2
π
ℓ
)
|
2
=
1
(12)Theorem 32 If φ(t)φ(t) is an L1L1 solution to Equation 4 and
∫φ(t)dt≠0∫φ(t)dt≠0, then
∑
n
h
(
n
+
M
m
)
h
(
n
)
=
δ
(
m
)
∑
n
h
(
n
+
M
m
)
h
(
n
)
=
δ
(
m
)
(13)if and only if
|
H
(
ω
)
|
2
+
|
H
(
ω
+
2
π
/
M
)
|
2
+
|
H
(
ω
+
4
π
/
M
)
|
2
+
⋯
+
|
H
(
ω
+
2
π
(
M
-
1
)
/
M
)
|
2
=
M
.
|
H
(
ω
)
|
2
+
|
H
(
ω
+
2
π
/
M
)
|
2
+
|
H
(
ω
+
4
π
/
M
)
|
2
+
⋯
+
|
H
(
ω
+
2
π
(
M
-
1
)
/
M
)
|
2
=
M
.
(14)This is a frequency domain orthogonality condition on h(n)h(n).
Corollary 5
H
(
2
π
ℓ
/
M
)
=
0
,
for
ℓ
=
1
,
2
,
⋯
,
M
-
1
H
(
2
π
ℓ
/
M
)
=
0
,
for
ℓ
=
1
,
2
,
⋯
,
M
-
1
(15)which is a generalization of (Reference) stating where the zeros of H(ω)H(ω),
the frequency response of the scaling filter, are located. This is an interesting
constraint on just where certain zeros of H(z)H(z) must be located.
Theorem 33 If ∑nh(n)=M∑nh(n)=M, and h(n)h(n) has finite support or decays fast
enough, then a φ(t)∈L2φ(t)∈L2 that
satisfies Equation 4 exists and is unique.
Theorem 34 If ∑nh(n)=M∑nh(n)=M and if ∑nh(n)h(n-Mk)=δ(k)∑nh(n)h(n-Mk)=δ(k), then φ(t)φ(t) exists, is integrable, and generates
a wavelet system that is a tight frame in L2L2.
These results are a significant generalization of the basic M=2M=2 wavelet system
that we discussed in the earlier chapters. The definitions, properties, and
generation of these more general scaling functions have the same form as for
M=2M=2, but there is no longer a single wavelet associated with the scaling
function. There are M-1M-1 wavelets. In addition to Equation 4 we now have M-1M-1
wavelet equations, which we denote as
ψ
ℓ
(
t
)
=
∑
n
M
h
ℓ
(
n
)
φ
(
M
t
-
n
)
ψ
ℓ
(
t
)
=
∑
n
M
h
ℓ
(
n
)
φ
(
M
t
-
n
)
(16)for
ℓ
=
1
,
2
,
⋯
,
M
-
1
.
ℓ
=
1
,
2
,
⋯
,
M
-
1
.
(17)Some authors use a notation h0(n)h0(n) for h(n)h(n) and φ0(t)φ0(t) for ψ(t)ψ(t),
so that hℓ(n)hℓ(n) represents the coefficients for the scaling function and
all the wavelets and φℓ(t)φℓ(t) represents the scaling function and all
the wavelets.
Just as for the M=2M=2 case, the multiplicity-M scaling function and scaling
coefficients are unique and are simply the solution of the basic recursive
or refinement equation Equation 4. However, the wavelets and wavelet coefficients
are no longer unique or easy to design in general.
We now have the possibility of a more general and more flexible multiresolution expansion
system with the M-band scaling function and wavelets. There are now M-1M-1 signal
spaces spanned by the M-1M-1 wavelets at each scale jj. They are denoted
W
ℓ
,
j
=
Span
k
{
ψ
ℓ
(
M
j
t
+
k
)
W
ℓ
,
j
=
Span
k
{
ψ
ℓ
(
M
j
t
+
k
)
(18)for ℓ=1,2,⋯,M-1ℓ=1,2,⋯,M-1. For example with M=4M=4,
V
1
=
V
0
⊕
W
1
,
0
⊕
W
2
,
0
⊕
W
3
,
0
V
1
=
V
0
⊕
W
1
,
0
⊕
W
2
,
0
⊕
W
3
,
0
(19)and
V
2
=
V
1
⊕
W
1
,
1
⊕
W
2
,
1
⊕
W
3
,
1
V
2
=
V
1
⊕
W
1
,
1
⊕
W
2
,
1
⊕
W
3
,
1
(20)or
V
2
=
V
0
⊕
W
1
,
0
⊕
W
2
,
0
⊕
W
3
,
0
⊕
W
1
,
1
⊕
W
2
,
1
⊕
W
3
,
1
.
V
2
=
V
0
⊕
W
1
,
0
⊕
W
2
,
0
⊕
W
3
,
0
⊕
W
1
,
1
⊕
W
2
,
1
⊕
W
3
,
1
.
(21)In the limit as j→∞j→∞, we have
L
2
=
V
0
⊕
W
1
,
0
⊕
W
2
,
0
⊕
W
3
,
0
⊕
W
1
,
1
⊕
W
2
,
1
⊕
W
3
,
1
⊕
⋯
⊕
W
3
,
∞
.
L
2
=
V
0
⊕
W
1
,
0
⊕
W
2
,
0
⊕
W
3
,
0
⊕
W
1
,
1
⊕
W
2
,
1
⊕
W
3
,
1
⊕
⋯
⊕
W
3
,
∞
.
(22)Our notation for M=2M=2 in Chapter (Reference) is W1,j=WjW1,j=Wj
This is illustrated pictorially in (Reference) where we see the nested scaling
function spaces VjVj but each annular ring is now divided into M-1M-1
subspaces, each spanned by the M-1M-1 wavelets at that scale. Compare (Reference)
with (Reference) for the classical M=2M=2 case.
The expansion of a signal or function in terms of the M-band wavelets now involves
a triple sum over ℓ,jℓ,j, and kk.
f
(
t
)
=
∑
k
c
(
k
)
φ
k
(
t
)
+
∑
k
=
-
∞
∞
∑
j
=
0
∞
∑
ℓ
=
1
M
-
1
M
j
/
2
d
ℓ
,
j
(
k
)
ψ
ℓ
(
M
j
t
-
k
)
f
(
t
)
=
∑
k
c
(
k
)
φ
k
(
t
)
+
∑
k
=
-
∞
∞
∑
j
=
0
∞
∑
ℓ
=
1
M
-
1
M
j
/
2
d
ℓ
,
j
(
k
)
ψ
ℓ
(
M
j
t
-
k
)
(23)where the expansion coefficients (DWT) are found by
c
(
k
)
=
∫
f
(
t
)
φ
(
t
-
k
)
d
t
c
(
k
)
=
∫
f
(
t
)
φ
(
t
-
k
)
d
t
(24)and
d
ℓ
,
j
(
k
)
=
∫
f
(
t
)
M
j
/
2
ψ
ℓ
(
M
j
t
-
k
)
d
t
.
d
ℓ
,
j
(
k
)
=
∫
f
(
t
)
M
j
/
2
ψ
ℓ
(
M
j
t
-
k
)
d
t
.
(25)We now have an M-band discrete wavelet transform.
Theorem 35 If the scaling function φ(t)φ(t) satisfies the conditions for existence
and orthogonality and the wavelets are defined by
Equation 16 and if the integer translates of these wavelets span Wℓ,0Wℓ,0
the orthogonal compliments of V0V0, all being in V1V1,
i.e., the wavelets are orthogonal to the scaling function at the same
scale; that is, if
∫
φ
(
t
-
n
)
ψ
ℓ
(
t
-
m
)
d
t
=
0
∫
φ
(
t
-
n
)
ψ
ℓ
(
t
-
m
)
d
t
=
0
(26)for ℓ=1,2,⋯,M-1ℓ=1,2,⋯,M-1, then
∑
n
h
(
n
)
h
ℓ
(
n
-
M
k
)
=
0
∑
n
h
(
n
)
h
ℓ
(
n
-
M
k
)
=
0
(27)for all integers kk and for ℓ=1,2,⋯,M-1ℓ=1,2,⋯,M-1.
Combining Equation 8 and Equation 27 and calling h0(n)=h(n)h0(n)=h(n) gives
∑
n
h
m
(
n
)
h
ℓ
(
n
-
M
k
)
=
δ
(
k
)
δ
(
m
-
ℓ
)
∑
n
h
m
(
n
)
h
ℓ
(
n
-
M
k
)
=
δ
(
k
)
δ
(
m
-
ℓ
)
(28)as necessary conditions on hℓ(n)hℓ(n) for an orthogonal system.
Unlike the M=2M=2 case, for M>2M>2 there is no formula for hℓ(n)hℓ(n) and
there are many possible wavelets for a given scaling function.
Mallat's algorithm takes on a more complex
form as shown in (Reference). The advantage is a more flexible
system that allows a mixture of linear and logarithmic tiling of the
time–scale plane. A powerful tool that removes the ambiguity is choosing
the wavelets by “modulated cosine" design.
(Reference) shows the frequency response of
the filter band, much as (Reference) did for M=2M=2. Examples of scaling
functions and wavelets are illustrated in (Reference), and the tiling
of the time-scale plane is shown in (Reference).
(Reference) shows the time-frequency resolution
characteristics of a four-band DWT basis. Notice how it is different from the
Standard, Fourier, DSTFT and two-band DWT bases shown in earlier chapters.
It gives a mixture of a logarithmic and linear frequency resolution.
We next define the kthkth moments of ψℓ(t)ψℓ(t) as
m
ℓ
(
k
)
=
∫
t
k
ψ
ℓ
(
t
)
d
t
m
ℓ
(
k
)
=
∫
t
k
ψ
ℓ
(
t
)
d
t
(29)and the kthkth discrete moments of hℓ(n)hℓ(n) as
μ
ℓ
(
k
)
=
∑
n
n
k
h
ℓ
(
n
)
.
μ
ℓ
(
k
)
=
∑
n
n
k
h
ℓ
(
n
)
.
(30)Theorem 36 (Equivalent Characterizations of K-Regular M-Band Filters) A unitary
scaling filter is K-regular if and only if the following equivalent
statements are true:
- All moments of the wavelet filters are zero, μℓ(k)=0μℓ(k)=0, for k=0,1,⋯,(K-1)k=0,1,⋯,(K-1) and for ℓ=1,2,⋯,(M-1)ℓ=1,2,⋯,(M-1)
- All moments of the wavelets are zero, mℓ(k)=0mℓ(k)=0, for k=0,1,⋯,(K-1)k=0,1,⋯,(K-1) and for ℓ=1,2,⋯,(M-1)ℓ=1,2,⋯,(M-1)
- The partial moments of the scaling filter are equal for
k=0,1,⋯,(K-1)k=0,1,⋯,(K-1)
- The frequency response of the scaling filter has zeros of order KK at the
MthMth roots of unity, ω=2πℓ/Mω=2πℓ/M for ℓ=1,2,⋯,M-1ℓ=1,2,⋯,M-1.
- The magnitude-squared frequency response of the scaling filter is flat to
order 2K at ω=0ω=0. This follows from (Reference).
- All polynomial sequences up to degree (K-1)(K-1) can be expressed as a linear
combination of integer-shifted scaling filters.
- All polynomials of degree up to (K-1)(K-1) can be expressed as a linear combination
of integer-shifted scaling functions for all jj.
This powerful result [16], (Reference) is similar to the M=2M=2 case
presented in Chapter (Reference). It not only ties the number of zero
moments to the regularity but also to the degree of polynomials that can
be exactly represented by a sum of weighted and shifted scaling functions.
Note the location of the zeros of H(z)H(z) are equally spaced around the
unit circle, resulting in a narrower frequency response than for the
half-band filters if M=2M=2. This is consistent with the requirements
given in Equation 14 and illustrated in (Reference).
Sketches of some of the derivations in this section are given in the
Appendix or are simple extensions of the M=2M=2 case. More details are
given in [3], [16], (Reference).
