Wavelets trong Electromagnetics và mô hình thiết bị P6 potx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (536.9 KB, 59 trang )
CHAPTER SIX
Canonical Multiwavelets
As discussed in the previous chapters, wavelets have provided many beneficial
features, including orthogonality, vanishing moments, regularity (continuity and
smoothness), multiresolution analysis, among these features. Some wavelets are
compactly supported in the time domain (Coifman, Daubechies) or in the fre-
quency domain (Meyer), and some are symmetrical (Haar, Battle–Lemarie). On
many occasions it would be very useful if the basis functions were symmetrical.
For instance, it would be better to expand a symmetric object such as the human
face using symmetric basis functions rather than asymmetric ones. In regard to
boundary conditions, magnetic wall and electric wall are symmetric and antisym-
metric boundaries, respectively. It might be ideal to create a wavelet basis that is
symmetric, smooth, orthogonal, and compactly supported. Unfortunately, the previ-
ous four properties cannot be simultaneously possessed by any wavelets, as proved
in [1].
To overcome the limitations of the regular (i.e., scalar) wavelets, mathematicians
have proposed multiwavelets. There are two categories of multiwavelets, and both
of them are defined on finite intervals. The first class is that of the canonical multi-
wavelets that are based upon the vector-matrix dilation equation [2–4]; this class will
be studied in this chapter. The second class is based on the Lagrange or Legendre in-
terpolating polynomials [5], which is similar in some respects to the pseudospectral
domain method and as such facilitates MRA.
6.1 VECTOR-MATRIX DILATION EQUATION
Multiwavelets offer more flexibility than traditional wavelets by extending the scalar
dilation equation
ϕ(t) =
h
k
ϕ(2t − k)
240
Wavelets in Electromagnetics and Device Modeling. George W. Pan
Copyright
¶ 2003 John Wiley & Sons, Inc.
ISBN: 0-471-41901-X
VECTOR-MATRIX DILATION EQUATION 241
into the matrix-vector version
|φ(t)=
k
C
k
|φ(2t − k),
where C
k
=[C
k
]
r×r
is a matrix of r ×r, |φ(t)=(φ
0
(t) ···φ
r−1
(t))
T
is a column
vector of r × 1, and r is the multiplicity of the multiwavelets. By taking the jth
derivative, we have
|φ
( j)
(t)=
k
C
k
2
j
|φ
( j)
(2t − k).
Let us denote a matrix
(t) =
φ
0
(t)φ
0
(t) ··· φ
(r−1)
0
(t)
φ
1
(t)φ
1
(t)
.
.
.
.
.
.
φ
r−1
(t)
.
.
.φ
(r−1)
r−1
(t)
. (6.1.1)
Then
(t) =
|φ(t)|φ
(t) ··· |φ
(r−1)
(t)
=
C
k
|φ(2t − k) 2
C
k
|φ
(2t − k) ··· 2
r−1
C
k
|φ
(r−1)
(2t − k)
=
k
C
k
|φ(2t − k)
1
2
.
.
.
2
r−1
r×r
,
or
(t) =
C
k
(2t − k)
−1
, (6.1.2)
where
−1
= diag{1, 2, ,2
r−1
}. (6.1.3)
Equation (6.1.2) can be verified as follows:
Show.
LHS = (t)
=
φ
0
(t)φ
(r−1)
0
(t)
··· ··· ···
φ
r−1
(t)φ
(r−1)
r−1
(t)
242 CANONICAL MULTIWAVELETS
=
|φ(t)|φ
(1)
(t) ··· |φ
(r−1)
(t)
=
C
k
|φ(2t −k) 2
C
k
|φ
(1)
2t−k
···2
(r−1)
C
(r−1)
k
|φ(2t −k)
.
RHS =
k
C
k
(2t − k)
−1
=
(C
k
)
r×r
|φ(2t − 2)|φ
(1)
(2t − 2)···|φ
(r−1)
(2t − k)
·
1 ··· ··· ···0
020 ···0
002
2
···0
··· ··· ··· ···
000···2
r−1
r×r
=
k
(C
k
)
r×r
φ
0
(2t − k)φ
0
(2t − k) ··· φ
(r−1)
0
(2t − k)
φ
1
(2t − k)φ
1
(2t − k) ··· φ
(r−1)
1
(2t − k)
··· ··· ··· ···
φ
r−1
(2t − k)φ
r−1
(2t − k) ··· φ
(r−1)
r−1
(2t − k)
·
1
2
···
2
r−1
=
k
(C
k
)
r×r
φ
0
(2t − k) 2φ
0
(2t − k) ··· 2
(r−1)
φ
(r−1)
0
(2t − k)
φ
1
(2t − k) 2φ
1
(2t − k) ··· ···
··· ···
φ
r−1
(2t − k) 2φ
r−1
(2t − k) ··· 2
(r−1)
φ
(r−1)
r−1
(2t − k)
.
Hence
(t) =
C
k
(2t − k)
−1
. (6.1.4)
In the construction of the multiwavelets, we may use either the frequency domain
approach or the time domain approach. The frequency approach is more elegant but
requires more extensive mathematical background. We select the latter approach,
which seems to be easier to follow despite being more cumbersome.
6.2 TIME DOMAIN APPROACH
We begin with the vector dilation equation
|φ(t)=
k
C
k
|φ(2t − k), (6.2.1)
TIME DOMAIN APPROACH 243
which has an explicit form of
φ
0
(t)
φ
1
(t)
·
·
·
φ
r−1
(t)
=
n−1
k=0
[C
k
]
r×r
φ
0
(2t − k)
φ
1
(2t − k)
·
·
·
φ
r−1
(2t − k)
,
where n is the order of approximation (see Eq. (6.2.6)).
Let us denote an infinite-dimensional matrix
L =
··· ···
··· C
3
C
2
C
1
C
0
··· ··· C
3
C
2
C
1
C
0
C
3
C
2
C
1
C
0
··· ···
.
Then (6.2.1) becomes
|(t)=L |(2t), (6.2.2)
where |(t)=[···φ(t − 1) |φ(t)|φ(t +1) |···]
T
. The explicit form of (6.2.2)
is
···
|φ(t − 1)
|φ(t)
|φ(t + 1)
···
=
···
C
3
C
2
C
1
C
0
··· C
3
C
2
C
1
C
0
··· C
3
C
2
C
1
C
0
···
···
|φ(2t − 1)
|φ(2t)
|φ(2t + 1)
···
,
or
···
C
3
C
2
C
1
C
0
··· C
3
C
2
C
1
C
0
··· C
3
C
2
C
1
C
0
···
···
|φ(2t − 1)
|φ(2t)
|φ(2t + 1)
···
=
···
|φ(t − 1)
|φ(t)
|φ(t + 1)
···
.
(6.2.3)
Let us pick out the row that represents |φ(2t) and |φ(t) in (6.2.3), namely
···+C
2
|φ(2t − 2)+C
1
|φ(2t − 1)+C
0
|φ(2t)=|φ(t).
If we replace t by t − 1, then (6.2.1) becomes
k
C
k
|φ(2t − k − 2)=|φ(t − 1).
244 CANONICAL MULTIWAVELETS
Explicitly, the equation above is
···+C
1
|φ(2t − 3)+C
0
|φ(2t − 2)=|φ(t − 1),
which is one row above in (6.2.3). Notice the two-unit shift in the row of the matrix
L that corresponds to the equation above.
Now consider the monomials t
j
, j = 0, 1, ,r − 1, which span the scaling
subspace. The φ(·) are the basis function in V
r
. Therefore
t
j
:= G
j
(t) =
∞
k=−∞
y
[j]
k
|φ(t − k)=y
[j]
|(t), (6.2.4)
where
y
[j]
|=
···y
[j]
0
|y
[j]
1
|y
[j]
2
|···
and each piece y
[j]
k
| is a row vector with r components that matches the vectors
|φ(t − k). Substituting (6.2.2) into (6.2.4), we obtain
G
j
(t) =y
[j]
|(t)=y
[j]
| L |(2t).
On the other hand, we may rewrite this as
G
j
(t) = t
j
= 2
−j
(2t)
j
= 2
−j
y
[j]
|(2t) .
Hence
y
[j]
| L |(2t)=2
−j
y
[j]
|(2t),
and therefore
y
[j]
| L = 2
−j
y
[j]
|. (6.2.5)
The previous equation implies that L has eigenvalue 2
−j
for the left eigenvector
y
[j]
|.Thatistosay,ifL has eigenvalues 1, 2
−1
, 2
−2
, ,2
−( p−1)
with left eigen-
vectors y
[j]
|, then
G
j
(t) =
∞
k=−∞
y
[j]
k
|φ(t − k).
A special and important case is j = 0, in which case
k
y
[0]
k
|φ(t − k)=1 = t
0
.
In the remainder of this section, we will list definitions, lemmas and theorems that
will form a solid foundation of multiwavelets in the time domain.
Definition. A multiscalet |φ(t) has approximation order n if each monomial
t
j
, j = 0, ,n − 1 is a linear summation of integer translations |φ(t − k)
CONSTRUCTION OF MULTISCALETS 245
such that
t
j
=
∞
k=−∞
y
[j]
k
|φ(t − k), j = 0, 1, ,n − 1, (6.2.6)
almost everywhere.
Lemma 1. Suppose that φ
j
(t) ∈ L
1
for j = 0, ,r − 1 and the translates
φ
j
(t − k), k ∈ Z, are linearly independent. Then |φ(t) provides an approximation of
order n if and only if L has eigenvalues 2
−j
corresponding to the left eigenvectors
y
[j]
|=
···y
[j]
0
|y
[j]
1
y
[j]
2
|···
with a component
y
[j]
k
|=
j
=0
j
l
(−k)
j−
u
[]
|, j = 0, 1, ,n − 1, (6.2.7)
where u
[]
| are constant vectors that will be given in (6.12.12).
Lemma 2. Suppose that y
[j]
| is given by (6.2.7) and that L corresponds to a multiscalet
with an approximation order n.Then
y
[j]
|L = 2
−j
y
[j]
|, j = 0, ,n − 1,
if and only if the following finite equations are held:
k
y
[j]
k
|C
2k+1
= 2
−j
u
[j]
| (6.2.8)
k
y
[j]
k
|C
2k
= 2
−j
y
[j]
1
|=2
−j
j
=0
(−1)
j−
j
u
[]
| for j = 0, 1, ,n − 1.
(6.2.9)
Equations (6.2.8) and (6.2.9) are referred to as the approximation conditions. The proofs of
Lemma 1 and Lemma 2 are provided in the Appendix to this chapter.
6.3 CONSTRUCTION OF MULTISCALETS
We begin with the approximation conditions (6.2.8) and (6.2.9):
k
y
[j]
k
|C
2k+1
= 2
−j
u
[]
|, (6.3.1)
k
y
[j]
k
|C
2k
= 2
−j
y
[j]
1
|
= 2
−j
j
=0
(−1)
j−
j
u
[]
|, (6.3.2)
246 CANONICAL MULTIWAVELETS
which are a system of nonlinear equations in terms of matrix components and
the starting vectors u
[j]
|. These equations can be solved effectively only for low
approximation orders with a small number of dilation coefficients. Fortunately, in
electromagnetics, the order is usually ≤ 4. An intervallic function of order r is a
multiscalet
|φ(t)=(φ
0
(t) φ
r−1
(t))
T
(6.3.3)
consisting of intervallic φ
j
, which are piecewise polynomials of degree 2r − 1 with
r − 1 continuous derivatives. For all r, φ
j
(t) = 0 only on two intervals [0, 1] and
[1, 2]. The function value and its r −1 derivatives are specified at each integer node.
If the intervallic functions are defined on [0, 2], then they are alternatively symmetric
and antisymmetric about t = 1. The translations of these functions span V
0
.
The dilation equation may be written as
|φ(t)=
k
C
k
|φ(2t − k)
= C
0
|φ(2t)+C
1
|φ(2t − 1) + C
2
|φ(2t − 2). (6.3.4)
Since the support is [0, 2], the only nonzero coefficients are C
0
, C
1
,andC
2
.There
are r basis functions at each node, and C
i
are matrices of r × r (i = 0, 1, 2).The
polynomials of degree 2r − 1on[0, 1] and [1, 2] can be determined by
d
dt
k
φ
j
(1) = δ
k, j
, k, j = 0, ,r − 1, (6.3.5)
d
dt
k
φ
j
(0) = 0 =
d
dt
k
φ
j
(2), k, j = 0, ,r − 1, (6.3.6)
where δ
k, j
is the Kronecker delta.
The symmetry and antisymmetry about t = 1aregivenby
φ
j
(2 − t) = (−1)
j
φ
j
(t), j = 0, ,r − 1. (6.3.7)
Notice that C
0
|φ(2)=0 = C
2
|φ(0) by (6.3.6). Equations (6.3.5) and (6.3.6) may
be expressed compactly as
(n) = δ
1,n
I,
where δ
1,n
is the Kronecker delta, I is the identity matrix of r × r,and(t) was
defined in (6.1.4) as
(t) = (|φ(t)|φ
(t)···|φ
(r−1)
(t))
=
φ
0
(t)φ
(r−1)
0
(t)
··· ··· ···
φ
r−1
(t)φ
(r−1)
r−1
(t)
with φ
( j)
i
(t) :=(d/dt)
j
φ
i
(t), i, j = 0, ,r − 1.
CONSTRUCTION OF MULTISCALETS 247
Example 1
The multiscalets for multiplicity r = 2are
φ
0
(t) = (3t
2
− 2t
3
), φ
1
(t) = t
3
− t
2
for t ∈[0, 1], (6.3.8)
φ
0
(t) = φ
0
(2 − t), φ
1
(t) =−φ
1
(2 −t) for t ∈[1, 2]. (6.3.9)
We can verify that
(t)|
t=1
=
φ
0
(t)φ
0
(t)
φ
1
(t)φ
1
(t)
|
t=1
=
10
01
.
It is easy to find that
φ
0
(1) = 1,
φ
1
(1) = 0,
φ
0
(t)|
t=1
=[6t − 6t
2
]
t=1
= 0,
φ
1
(t)|
t=1
=[3t
2
− 2t]|
t=1
= 1.
The curves of φ
0
(t) and φ
1
(t) with explicit expressions are plotted in Fig. 6.1.
Recall from (6.1.2) and (6.1.3) that
(t) =
C
k
(2t − k)
−1
−1
= diag{1, 2, ,2
r−1
}.
Let us evaluate the dilation coefficients by taking t = m/2, m ∈ Z in (6.1.4),
m
2
=
k
C
k
(m − k)
−1
=
k
C
k
δ
1,m−k
−1
= C
m−1
−1
. (6.3.10)
Since has a support of [0, 2],allC
k
= 0fork ≥ 3. For the three nonzero coeffi-
cients, we have from (6.3.10) that
C
0
=
1
2
,
C
1
= (1) = I = = diag
1,
1
2
, ,
1
2
r−1
, (6.3.11)
C
2
=
3
2
.
While C
1
was given in (6.3.11) for any multiplicity r, C
0
and C
2
can be obtained for
the case of r = 2 in the next example. For arbitrary r, the general expressions of C
0
and C
2
will be derived later in this section.
248 CANONICAL MULTIWAVELETS
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
φ
0
(t),
φ
1
(t)
φ
0
(t)
φ
1
(t)
FIGURE 6.1 Multiscalets of r = 2 from analytic expression.
Example 2
Evaluate C
0
and C
2
for r = 2.
Solution
(t) =
φ
0
(t)φ
0
(t)
φ
1
(t)φ
1
(t)
=
(3t
2
− 2t
3
) 6(t − t
2
)
t
3
− t
2
(3t
2
− 2t)
for t ≤ 1. (6.3.12)
Hence
1
2
=
1
2
3
2
−
1
8
−
1
4
,
C
0
=
1
2
=
1
2
3
2
−
1
8
−
1
4
10
0
1
2
=
1
2
3
4
−
1
8
−
1
8
.
. (6.3.13)
To evaluate C
2
, we need (
3
2
). However, we cannot set t =
3
2
in (6.3.12). In-
stead, (3/2) may be found from (
1
2
) by symmetry/antisymmetry about t = 1
(see Fig. 6.2), yielding
3
2
=
1
2
−
3
2
1
8
−
1
4
.
CONSTRUCTION OF MULTISCALETS 249
Therefore
C
2
=
3
2
=
1
2
−
3
2
1
8
−
1
4
10
0
1
2
=
1
2
−
3
4
1
8
−
1
8
. (6.3.14)
Next let us derive C
0
and C
2
for arbitrary r . The property (6.3.7) may be written in
amatrixformas
|φ((2 − t)=S|φ((t), (6.3.15)
where
S =
1
−1
···
(−1)
r−1
= S
−1
.
Applying the dilation equation (6.3.4) to (6.3.15), we obtain
LHS =|φ(2 −t)
= C
0
|φ(4 − 2t) +C
1
|φ(3 − 2t) +C
2
|φ(2 − 2t)
= C
0
|φ[2 −(2t − 2)] + C
1
|φ[2 −(2t − 1)] + C
2
|φ(2 − 2t)
= C
0
S|φ(2t − 2)+C
1
S|φ(2t − 1)+C
2
S|φ(2t),
where the last equality was arrived at by using the symmetry–antisymmetry property
of (6.3.15).
Applying the dilation equation (6.3.4) to the right-hand side of (6.3.15), we have
RHS = S[C
0
|φ(2t)+C
1
|φ(2t − 1)+C
2
|φ(2t − 2)].
Equating both sides, we have
C
0
|φ(2t)+C
1
|φ(2t − 1)|C
2
|φ(2t − 2)=S
−1
C
2
S|φ(2t)
+ S
−1
C
1
S |φ(2t − 1)
+ S
−1
C
0
S |φ(2t − 2).
By linear independence of translations φ(2t − k), we claim that
C
0
= S
−1
C
2
S
= SC
2
S
−1
. (6.3.16)
The component expression of (6.3.16) is
[C
0
]
ij
= (−1)
i+j
[C
2
]
ij
.
250 CANONICAL MULTIWAVELETS
As a result of (6.3.16), C
2
remains to be determined. The coefficient C
2
of arbitrary
r can be obtained from the following theorem:
Theorem 1. The eigenvalues of C
2
are (
1
2
)
r
,(
1
2
)
r+1
, ,(
1
2
)
2r−1
,andC
2
can be
found from the similarity transformation of a diagonal matrix by
C
2
= U
−1
U,
where the transformation matrix U is given by
[U]
mn
= (−1)
r+m−n
(r + m − 1)!
[r + m − n]!
. (6.3.17)
Note that a similarity transform does not change eigenvalues. Therefore
= diag
1
2
r
,
1
2
r+1
, ,
1
2
2r−1
.
The proof of this theorem is provided in the Appendix to this chapter.
Example 3
The piecewise cubic case r = 2.
=
1
4
0
0
1
8
U =
2!
2!
−
2!
1!
−
3!
3!
3!
2!
=
1 −2
−13
,
U
−1
=
32
11
.
Thus
C
2
= U
−1
U =
1
2
−
3
4
1
8
−
1
8
,
C
0
= SC
2
S
−1
=
1
2
3
4
−
1
8
−
1
8
.
Recall that C
1
was given in Eq. (6.3.11) as
10
0
1
2
.
CONSTRUCTION OF MULTISCALETS 251
The resultant matrices C
0
, C
1
,andC
2
in this example agree exactly with (6.3.13) and
(6.3.14) in Example 2. This implies that the multiscalets constructed by the analytic
expressions and by the numerical (iterative or cascade) methods are identical.
Using the vector dilation equations, we obtain
|φ(t)=
k
C
k
|φ(2t − k)
with given matrices C
0
, C
1
,andC
2
, we may construct the scalets either by the it-
erative method or the cascade method, as in Chapter 3 for the Daubechies scalet.
Figure 6.2 depicts the two multiscalets, φ
0
and φ
1
; they are identical to those ob-
tained from analytic expressions. For multiplicity r = 3, the corresponding lowpass
matrices C
0
, C
1
,andC
2
can be calculated in the same manner outlined in Example 3
andaregivenbelow:
C
0
=
1
2
15
16
0
−
5
32
−
7
32
3
8
1
64
1
64
−
1
16
,
C
1
=
100
0
1
2
0
00
1
4
, (6.3.18)
C
2
=
1
2
−
15
16
0
5
32
−
7
32
−
3
8
1
64
−
1
64
−
1
16
.
Iterative φ
0
(x)
Iterative φ
1
(x)
Explicit
φ
0
(x)
Explicit φ
1
(x)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
FIGURE 6.2 Multiscalets of r = 2, analytical and iterated.
252 CANONICAL MULTIWAVELETS
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
φ
0
(x)
φ
1
(x)
φ
2
(x
)
FIGURE 6.3 Multiscalets of r = 3.
The corresponding multiscalets are plotted in Fig. 6.3. The explicit polynomials of
φ
0
(t), φ
1
(t) and φ
2
(t) are
φ
0
(t) = 6t
5
− 15t
4
+ 10t
3
φ
1
(t) =−3t
5
+ 7t
4
− 4t
3
φ
2
(t) =
1
2
t
5
− t
4
+
1
2
t
3
on the interval [0, 1]. Using the symmetry/antisymmetry, we can obtain the closed
form expressions on the interval [1, 2].
In general multiscalets with arbitrary r have the form
φ
0
(t) = p
1,1
t
2r−1
+ p
1,2
t
2r−2
+···+p
1,r
t
r
···=···
φ
r−1
(t) = p
r,1
t
2r−1
+ p
r,2
t
2r−2
+···+p
r,r
t
r
where the coefficients p
i, j
are obtained by inverting the matrix whose entries are
a(k,)=
(2r − k)!
(2r − k − + 1)!
, k,= 1, 2, ,r. (6.3.19)
Thus far we have constructed the multiscalets that are compactly supported on
[0, 2]. These multiscalets do not satisfy orthogonality in the usual sense
φ
i
(t)φ
i
(t − n) dt = δ
0,n
.
CONSTRUCTION OF MULTISCALETS 253
Instead, condition (6.1.2) leads to another type of orthogonality of a Sobolev-type
inner product. We define
f, g
0
:=
r−1
j=0
k
f
( j)
(k)g
( j)
(k), (6.3.20)
where the subscript 0 indicates scaling level 0, the overbar denotes the complex
conjugate, and f and g are in C
r−1
0
, which is r − 1 times differentiable and satisfies
zero boundary conditions. Then we have
φ
i
,φ
k
(·−m)
0
=
r−1
j=0
p∈Z
φ
( j)
i
( p)φ
( j)
k
( p − m)
=
r−1
j=0
φ
( j)
i
(1)φ
( j)
k
(1 − m)
=
r−1
j=0
δ
i, j
δ
k, j
δ
0,m
= δ
i,k
δ
0,m
,
where we have used the property of (6.3.5) that
φ
( j)
i
(1) = δ
i, j
for j = 0, 1, ,r − 1,
and
φ(0) = 0,
φ(2) = 0.
In order to simplify notation, we have denoted φ, without subscript, as a vector. The
simplified notation of φ and ψ will be carried out throughout the chapter. The result
above may be written in vector form as
φ, φ
T
(·−m)
0
= δ
0,m
I,
or in matrix form as
k
(k)
T
(k − m) = δ
0,m
I. (6.3.21)
In a similar manner we define at level p,
f (t), g(t)
p
:=
r−1
j=0
k
f
( j)
(t) g
( j)
(t)|
t=2
−p
k
, p ∈ Z. (6.3.22)
254 CANONICAL MULTIWAVELETS
Unfortunately, we do not have orthogonality of {φ(t −m)} with regard to these inner
products.
For p = 1, we obtain
φ, φ
T
(·−m)
1
=
k
k
2
T
k
2
− m
=
k
C
k−1
−1
(C
k−2m−1
−1
)
T
=
k
C
k−1
−2
C
k−2m−1
.
Note that for m = 1,
φ, φ
T
(t − 1)
1
=
k
C
k−1
−2
C
T
k−3
= C
2
−2
C
T
0
.
For m = 0,
φ, φ
T
1
= C
0
−2
C
T
0
+ C
1
−2
C
1
+ C
2
−2
C
2
= C
0
−2
C
T
0
+ I
2
+ C
2
−2
C
2
.
For m =−1,
φ, φ
T
(t + 1)=
k
C
k−1
−2
C
T
k+1
.
Following the same derivation, we may show that
φ(2t), φ(2t − m)
1
= δ
0,m
−2
.
In fact
φ
p
(2t), φ
q
(2t − m)
1
=
r−1
j=0
k∈Z
2
j
φ
( j)
p
1
2
· 2k
2
j
φ
( j)
q
1
2
· 2k − m
=
r−1
j=0
k∈Z
2
2 j
φ
( j)
p
(k)φ
( j)
q
(k − m)
=
r−1
j=0
k∈Z
2
2 j
δ
j, p
δ
1,k
δ
j,q
δ
k,m+1
= 2
2 p
k∈Z
δ
p,q
δ
1,k
δ
k,m+1
.
ORTHOGONAL MULTIWAVELETS
˘
ψ(
t
) 255
Hence
φ(2t), φ(2t − m)
1
= δ
0,m
−2
. (6.3.23)
Equation (6.3.23) will be used in Section 6.5.
6.4 ORTHOGONAL MULTIWAVELETS
˘
ψ(
t
)
In the previous section the orthogonal multiscalets were constructed. Naturally, one
expects to build the corresponding multiwavelets. Multiwavelets
˘
ψ
0
(t), ,
˘
ψ
r−1
(t)
are orthogonal to multiscalets φ
0
(t), ,φ
r−1
(t), and they satisfy the dilation equa-
tion
˘
ψ
0
(t)
.
.
.
˘
ψ
r−1
(t)
=
4
k=0
[G
k
]
φ
0
(2t − k)
.
.
.
φ
r−1
(2t − k)
. (6.4.1)
Note that there are five nonzero matrices G. The support of (2t − k) is [k/2,(k +
2)/2]. With coefficients G
0
, ,G
4
, the support of
˘
ψ(t) will be [0, 3]. In fact the
left endpoint is for k = 0, and the right endpoint is for k = 4.
The orthogonality against (t) and its translations provide equations for the G,
and they are
˘
(t)
T
(t + 1) = 0,
˘
(t)
T
(t) = 0,
˘
(t)
T
(t − 1) = 0,
˘
(t)
T
(t − 2) = 0. (6.4.2)
Note that
supp{(t)}=[0, 2],
supp{
˘
(t)}=[0, 3].
The translations of k > 2ork < −1 in (6.4.2) shift the functions so that there is no
overlap between and
˘
. Therefore they are ruled out from (6.4.2).
Equation (6.4.2) involves integrals of
˘
ψ
i
(t)φ
j
(t − k). By using the dilation equa-
tions of both scalets and wavelets, we integrate φ
i
(2t − k)φ
j
(2t − m). A change of
variable converts these inner product integrals into
φ
i
(2t − k)φ
j
(2t − m) dt =
1
2
φ
i
(t)φ
j
(t − m + k) dt, (6.4.3)
where φ
i
and φ
j
are supported on [0, 2]. Hence the only inner products needed are
the two matrices
256 CANONICAL MULTIWAVELETS
X = (t)
T
(t),
Y = (t)
T
(t − 1) = (t + 1)
T
(t). (6.4.4)
To avoid an explicit evaluation of polynomials φ
i
(t) and their inner products, we sub-
stitute the dilation equations into (6.4.4) and impose (6.4.3) to convert all arguments
involving 2t into t. The resultant two matrix equations are
2X = C
0
XC
T
0
+ C
1
Y
T
C
T
0
+ C
0
YC
T
1
+ C
1
XC
T
1
+ C
2
Y
T
C
T
1
+ C
1
YC
T
2
+ C
2
XC
T
2
,
2Y = C
1
YC
T
0
+ C
2
XC
T
0
+ C
2
YC
T
1
.
These equations determine X and Y up to a scalar factor, and the X and Y enter
the orthogonality equation (6.4.2). Substituting in (6.4.2) for
˘
(t) introduces the
unknown G, and substituting for (t) links the known C. After some algebra we
obtain
G
0
(Y
T
C
T
1
+ XC
T
2
) + G
1
Y
T
C
T
2
= 0,
G
0
(XC
T
0
+ YC
T
1
) + G
1
(Y
T
C
T
0
+ XC
T
1
+ YC
T
2
)
+ G
2
(Y
T
C
T
1
+ XC
T
2
) + G
3
Y
T
C
T
2
= 0,
G
1
YC
T
0
+ G
2
(XC
T
0
+ YC
T
1
) + G
3
(Y
T
C
T
0
+ XC
T
1
+ YC
T
2
)
+ G
4
(Y
T
C
T
1
+ XC
T
2
) = 0,
G
3
YC
T
0
+ G
4
(XC
T
0
+ YC
T
1
) = 0. (6.4.5)
The equations above form a system of 4r
2
homogeneous equations that consist of
5r
2
entries in G
0
, ,G
4
. We pick out the solution with G
2
= I . In this case,
symmetry–antisymmetry is also held for the wavelets. The property C
0
= SC
2
S
may be extended to the G as
G
0
= SG
4
S
G
1
= SG
3
S,
where S = diag{1, −1, ,(−1)
r−1
}. As a result the first two equations in (6.4.5)
become identical to the remaining two. Employing this pattern of the G and also
X = SXS
Y = SY
T
S,
we obtain
[X]
ij
=[X ]
ji
=
φ
i
(t)φ
j
(t) dt
=
φ
i
(2 − t)φ
j
(2 − t) dt
ORTHOGONAL MULTIWAVELETS
˘
ψ(
t
) 257
= (−1)
i+j
φ
i
(t)φ
j
(t) dt
= (−1)
i+j
[X]
ij
and
[Y ]
ij
=
φ
i
(t)φ
j
(t − 1) dt
=
φ
i
(3 − t)φ (2 − t) dt
= (−1)
i+j
φ
i
(t − 1)φ
j
(t) dt
= (−1)
i+j
[Y ]
ji
.
Finally, Eq. (6.4.5) reduces to two matrix equations for two unknowns G
3
and G
4
.
Using Matlab, we have solved for the unknowns and listed them below
G
0
=
−
17
98
−
89
98
79
6438
137
2146
,
G
1
=
−
16
49
−
286
49
152
3219
550
1073
,
G
2
=
10
01
, (6.4.6)
G
3
=
−
16
49
286
49
−
152
3219
550
1073
,
G
4
=
−
17
98
89
98
−
79
6438
137
2146
.
The orthogonal wavelets are constructed according to Eq. (6.4.1). They are plotted in
Fig. 6.4. By construction, the scalets are orthogonal to the wavelets and their integer
translations. Hence
V
1
= V
0
⊕ W
0
,
V
2
= V
0
⊕ W
0
⊕ W
1
,
W
i
⊥ W
j
for i = j.
Unfortunately, a wavelet is not orthogonal to its translations, nor a scalet to its trans-
lations.
258 CANONICAL MULTIWAVELETS
0 0.5 1 1.5 2 2.5 3
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
FIGURE 6.4 Intervallic multiwavelets
˘
ψ
0
(t) and
˘
ψ
1
(t) of r = 2.
6.5 INTERVALLIC MULTIWAVELETS ψ(
t
)
The orthogonal multiwavelets constructed in the previous section are orthogonal to
the multiscalets in the standard L
2
sense. However, these multiwavelets are oscilla-
tory and have relatively wide supports. Most inconveniently, they are not orthogonal
to their translations. To improve the properties of the multiwavelet, Walter intro-
duced the orthogonal finite element multiwavelets [4], which we referred to as the
intervallic multiwavelets to avoid confusion with the finite element method (FEM)
in electromagnetics. This multiwavelet family is comprised of the intervallic multi-
wavelet and its dual, namely the intervallic dual multiwavelets.
As usual, we denote the closed linear span V
p
in L
2
(R) of
{φ
0
(2
p
t − k), ,φ
r−1
(2
p
t − k)},
with an inner product ,
p
. This is equivalent to the L
2
inner product in V
p
.Wenow
introduce a biorthogonal pair of wavelets (ψ,
˜
ψ), both of which belong to V
1
.The
first is in V
⊥
0
and is given by
ψ(t) =
k
D
k
φ(2t − k), (6.5.1)
where ψ(t) =[ψ
1
(t), ψ
2
(t), ,ψ
n
(t)]
T
.
Since ψ
i
∈ V
1
, we use its inner product to determine the D
k
so that ψ
1
(t) is
orthogonal in the sense of V
1
to V
0
. Namely we need
ψ, φ
T
(·−m)
1
= 0forallm ∈ Z .
The LHS can be evaluated by using (6.5.1), and the weighted orthogonality of
(6.3.23),
φ(2t), φ
T
(2t − m)
1
= δ
0,m
−2
.
Derivation
We had
φ(t), φ
T
(t − m)
0
= δ
0,m
I.
INTERVALLIC MULTIWAVELETS ψ(
t
) 259
Thus
ψ, φ
T
(·−m)
1
=
k
D
k
φ(2t − k), φ
T
(t − m)
1
.
Employing the dilation equation
φ
T
(t − m) =
j
C
j
φ(2t − 2m − j )
T
=
j
φ
T
(2t − 2m − j)
1×n
C
T
j
,
we have
ψ, φ
T
(·−m)
1
=
k
j
D
k
φ(2t − k), φ
T
(2t − 2m − j)
1
δ
k,2m+j
−2
C
T
j
=
j
D
2m+j
−2
C
T
j
= 0, m ∈ Z .
The previous inner products are identically zero for m < −1, or m > 4, because
there will be no overlap between ψ and φ.For−1 ≤ m ≤ 4, we end with the
following equations:
m =−1: D
0
−2
C
T
2
= 0
m = 0: D
0
−2
C
T
0
+ D
1
−2
C
T
1
+ D
2
−2
C
T
2
= 0
m = 1: D
2
−2
C
T
0
+ D
3
−2
C
T
1
+ D
T
4
−2
C
T
2
= 0,
m = 2: D
4
−2
C
T
0
= 0,
m = 3: D
6
−2
C
T
0
= 0.
Since both C
0
and C
2
are nonsingular, it follows that D
0
= 0 = D
4
= D
6
. Hence
only two equations are left
D
1
−2
C
T
1
=−D
2
−2
C
T
2
D
2
−2
C
T
0
=−D
3
−2
C
T
1
.
Also C
1
= from (6.3.11). One solution is to take
D
2
= C
1
= ,
which makes
D
1
−1
=−
−1
C
T
2
→ D
1
=−
−1
C
T
2
−1
C
T
0
=−D
3
−1
.
260 CANONICAL MULTIWAVELETS
In general,
D
m
= (−1)
m
−1
C
T
3−m
, m ∈ Z . (6.5.2)
Verification.
One can verify from (6.5.2) that
D
2
=
−1
C
T
1
,
D
0
=
−1
C
T
3
= 0 because C
3
= 0.
Since only C
0
, C
1
, C
2
= 0,(m = 3, 2, 1 in (6.5.2)), we obtain
ψ(t) = D
1
φ(2t − 1) + D
2
φ(2t − 2) + D
3
φ(2t − 3)
=−
−1
C
T
2
φ(2t −1) + φ(2t − 2) −
−1
C
T
0
φ(2t −3). (6.5.3)
Figure 6.5 illustrates the two multiwavelets, ψ
0
and ψ
1
, obtained by the iteration method and
by the explicit polynomial expressions of Example 2 in Section 6.3. Noticing that supp ψ(t) =
[
1
2
,
5
2
],andψ(t) =
D
k
φ(2t − k), we arrive at
ψ
( j)
(t) |
n/2
= ψ
( j)
n
2
=
k
D
k
φ
( j)
(n − k)2
j
= D
n−1
φ
( j)
(1)2
j
; j = 0, 1, ,r −1; n = 2, 3, 4. (6.5.4)
For integer values of t only φ
( j)
(1) may be nonzero, that is to have from (6.5.4) a sampling
property
ψ
( j)
q
p +
3
2
= δ
q, j
δ
p,0
. (6.5.5)
0.5 1 1.5 2 2.5
−1.5
−1
−0.5
0
0.5
1
1.5
t
ψ
0
(
t
)
, ψ
1
(
t
)
ψ
0
(t) by iteration
ψ
1
(t) by iteration
ψ
0
(t) by Hermitian
ψ
1
(t) by Hermitian
FIGURE 6.5 Intervallic multiwavelets of r = 2.
MULTIWAVELET EXPANSION 261
Show.
(1) For n = 3 in (6.5.4), namely p = 0 in (6.5.5), we have from (6.5.4),
ψ
( j)
3
2
= D
2
φ
( j)
(1)2
j
, D
2
=
that is,
ψ
( j)
0
(t)
···
ψ
( j)
r−1
(t)
t=
3
2
=
1
2
−1
···
2
−(r−1)
φ
( j)
0
(1)
···
φ
( j)
r−1
(1)
2
j
.
The (q +1)th element
ψ
( j)
q
(1) = 2
−q
φ
( j)
q
(1)2
j
= 2
−q
δ
q, j
2
j
= δ
q, j
, q = 0, 1, ,(r −1).
(2) For p = 0, say p = 1 in (6.5.5), we have
ψ
( j)
q
5
2
, n = 5.
D
n−1
= D
4
= 0 in (6.5.4) which makes ψ
( j)
q
(
5
2
) = 0. This is in agreement with
(6.5.5), that is,
ψ
( j)
q
p +
3
2
= δ
q, j
δ
p,0
= 0.
Regrettably, the wavelet ψ(t ) is not orthogonal to its translations ψ(t − ) in
the Sobolev sampling sense. Hence we introduce the dual multiwavelets. The dual
multiwavelets
˜
ψ are related to φ by
φ
0
(2t − 2)
···
···
···
φ
r−1
(2t − 2)
=
1
2
···
2
(r−1)
˜
ψ
0
(t)
···
···
˜
ψ
r−1
(t)
→ φ
j
(2t − 2) = 2
j
˜
ψ
j
(t),
j = 0, 1, ,r − 1.
Detailed study of the dual multiwavelets is deferred to Section 6.7.
6.6 MULTIWAVELET EXPANSION
Let us expand f (t) in terms of φ
p
(t − k):
f (t) =
r−1
p=0
k∈Z
a
k, p
φ
p
(t − k), (6.6.1)
262 CANONICAL MULTIWAVELETS
where the coefficients
a
k, p
= f
( p)
(k + 1). (6.6.2)
Show.
Multiplying both sides of (6.6.1) by φ
q
(t) and taking the inner product ,
0
,we
arrive at
φ
q
(t), f (t)
0
=
r−1
p=0
k∈Z
a
k, p
φ
q
(t), φ
p
(t − k)
0
, (6.6.3)
where
φ
q
(t), φ
p
(t − k)
0
=
r−1
j=0
α∈Z
φ
( j)
q
(α)φ
( j)
p
(α −k).
Taking a close look of (6.6.3), we have the following:
RHS =
r−1
p=0
k∈Z
a
k, p
r−1
j=0
α∈Z
φ
( j)
q
(α)φ
( j)
(α −k)
=
k∈Z
r−1
p=0
a
k, p
r−1
j=0
φ
( j)
q
(1)φ
( j)
p
(1 − k)
=
k∈Z
r−1
p=0
a
k, p
δ
p,q
δ
0,k
=
k∈Z
a
k,q
δ
0,k
(6.6.4)
LHS =
r−1
j=0
∈Z
φ
( j)
q
() f
( j)
()
=
∈Z
r−1
j=0
δ
j,q
δ
1,
f
( j)
()
=
∈Z
f
(q)
() δ
1,
=
k∈Z
f
(q)
(k +1)δ
0,k
, (6.6.5)
wherewehaveused = k +1. Comparing (6.6.4) and (6.6.5), we obtain
a
k,q
= f
(q)
(k + 1).
Next, if we expand f (t) in terms of φ
p
(2t − k) as
f (t) =
r−1
p=0
k
a
1
k, p
φ
p
(2t − k), (6.6.6)
MULTIWAVELET EXPANSION 263
then the coefficients
a
1
k, p
= 2
−p
f
( p)
(2
−1
(k + 1)). (6.6.7)
Show.
Multiplying both sides of the expansion (6.6.6) by φ
q
(2t) and performing the inner
product operation ,
1
, we obtain
φ
q
(2t), f (t)
1
=
r−1
p=0
k∈Z
a
1
k, p
φ
q
(2t), φ
p
(2t − k)
1
(6.6.8)
whereby, from (6.3.22),
φ
q
(2t), φ
p
(2t − k)
1
=
r−1
j=0
α∈Z
2
2 j
φ
( j)
q
(α)φ
( j)
p
(α − k).
From (6.6.8) we have
RHS =
r−1
p=0
k∈Z
a
1
k, p
r−1
j=0
2
2 j
φ
( j)
q
(1)φ
( j)
p
(1 −k)
=
k∈Z
r−1
p=0
2
p+q
a
1
k, p
δ
p,q
δ
0,k
=
k∈Z
2
2q
a
1
k,q
δ
0,k
(6.6.9)
LHS =φ
q
(2t), f (t)
1
=
r−1
j=0
∈Z
2
j
φ
( j)
q
2
2
f
( j)
1
2
=
r−1
j=0
∈Z
2
j
δ
j,q
δ
1,
f
( j)
2
=
∈Z
2
q
f
(q)
2
δ
1,
=
k∈Z
2
q
f
(q)
k +1
2
δ
0,k
. (6.6.10)
Comparing (6.6.10) with (6.6.9), we find that
a
1
k,q
= 2
−q
f
(q)
k + 1
2
.
Let us denote as f
0
∈ V
0
the projection of f onto V
0
, namely
f
0
(t) =
r−1
p=0
k∈Z
f
( p)
(k + 1)φ
p
(t − k),
264 CANONICAL MULTIWAVELETS
and let f ∈ V
1
with expansion (6.6.6). Hence the difference
f (t) − f
0
(t) =
r−1
j=0
k
f
( j)
k +
1
2
2
−j
φ
j
(2t − 2k)
+
k
f
( j)
(k + 1)[2
−j
φ(2t − 2k − 1) − φ
j
(t − k)]
.
The first summation on the RHS of the previous equation is related to the intervallic
dual multiwavelet.
6.7 INTERVALLIC DUAL MULTIWAVELETS
˜
ψ(
t
)
The intervallic dual wavelet is defined as
˜
ψ(t) = φ(2t − 2). (6.7.1)
Lemma 3. Let ψ and
˜
ψ be defined by (6.5.3) and (6.7.1) respectively. Then:
(i) ψ
( j)
p
(k +
3
2
) = δ
p, j
δ
k,0
, j, p = 0, ,r − 1, k ∈ Z.
(ii)
˜
ψ
( j)
p
(k +
3
2
) = δ
p, j
δ
k,0
, j, p = 0, ,r − 1, k ∈ Z.
(iii)
˜
ψ
( j)
p
(k) = 0, j, p = 0, ,r − 1, k ∈ Z.
(iv) ψ
j
(·−k),
˜
ψ
j
(·−)
1
= δ
k,
, k,∈ Z .
We have proved (i) as (6.5.5). Property (ii) can be verified in the same manner. Let us verify
Property (iii).
Proof. From (6.7.1)
˜
ψ
( j)
(t) = (φ(2t −2))
( j)
= 2
j
1
2
−1
···
2
−(r−1)
φ
( j)
0
(2t − 2)
φ
( j)
1
(2t − 2)
···
φ
( j)
r−1
(2t − 2)
.
For 2t − 2 = k,the(p + 1)th element
2
j
· 2
−p
φ
( j)
p
(2t − 2)|
(2t−2)=k
= 2
j−p
φ
( j)
p
(k) = 2
j−p
δ
j, p
δ
1,k
.
For t ∈ Z , 2t − 2 = 1. Therefore
˜
ψ(k) = 0.
We now summarize the multiwavelet properties into a theorem.
Theorem 2. If W
0
= closure{ψ
j
(t − k)},
˜
W
0
= closure{
˜
ψ
j
(t − k)}, then W
0
⊥V
0
,
˜
W
0
V
0
={0},andV
1
=
˜
W
0
V
0
.
