Wavelets trong Electromagnetics và mô hình thiết bị P6 potx

Unfortunately, the previ-ous four properties cannot be simultaneously possessed by any wavelets, as proved in-6.1 VECTOR-MATRIX DILATION EQUATION Multiwavelets offer more flexibility tha

CHAPTER SIX

Canonical Multiwavelets

As discussed in the previous chapters, wavelets have provided many beneficialfeatures, including orthogonality, vanishing moments, regularity (continuity andsmoothness), multiresolution analysis, among these features Some wavelets arecompactly supported in the time domain (Coifman, Daubechies) or in the fre-quency domain (Meyer), and some are symmetrical (Haar, Battle–Lemarie) Onmany 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 humanface using symmetric basis functions rather than asymmetric ones In regard toboundary conditions, magnetic wall and electric wall are symmetric and antisym-metric boundaries, respectively It might be ideal to create a wavelet basis that issymmetric, smooth, orthogonal, and compactly supported Unfortunately, the previ-ous four properties cannot be simultaneously possessed by any wavelets, as proved

in-6.1 VECTOR-MATRIX DILATION EQUATION

Multiwavelets offer more flexibility than traditional wavelets by extending the scalardilation equation

ϕ(t) =
h k ϕ(2t − k)

240

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) = (φ0(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

6.2 TIME DOMAIN APPROACH

We begin with the vector dilation equation

| φ(t) =
C k | φ(2t − k), (6.2.1)

TIME DOMAIN APPROACH 243

which has an explicit form of

where n is the order of approximation (see Eq (6.2.6)).

Let us denote an infinite-dimensional matrix

· · · + C2| φ(2t − 2) + C1| φ(2t − 1) + C0| φ(2t) = | φ(t).

If we replace t by t− 1, then (6.2.1) becomes

C k | φ(2t − k − 2) = | φ(t − 1).

Explicitly, the equation above is

· · · + C1| φ(2t − 3) + C0| 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 Theφ(·) are the basis function in V r Therefore

and each piece y k [ j] | is a row vector with r components that matches the vectors

| φ(t − k) Substituting (6.2.2) into (6.2.4), we obtain

The previous equation implies that L has eigenvalue 2 − j for the left eigenvector

y [ j] | That is to say, if L has eigenvalues 1, 2−1, 2−2, , 2 −(p−1)with left

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)

Lemma 1. Suppose that φ j (t) ∈ L1 for j = 0, , r − 1 and the translates

order n if and only if L has eigenvalues 2 − jcorresponding to the left eigenvectors

y [ j]| =· · · y0[ j] | y1[ j] y2[ j]| · · · with a component

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

which are a system of nonlinear equations in terms of matrix components andthe starting vectors u [ j]| These equations can be solved effectively only for lowapproximation orders with a small number of dilation coefficients Fortunately, inelectromagnetics, the order is usually≤ 4 An intervallic function of order r is a

multiscalet

| φ(t) = (φ0(t) φ r−1(t)) T

(6.3.3)consisting of intervallicφ 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 V0.The dilation equation may be written as

| φ(t) =

k

C k | φ(2t − k)

= C0| φ(2t) + C1| φ(2t − 1) + C2| φ(2t − 2). (6.3.4)Since the support is[0, 2], the only nonzero coefficients are C0, C1, and C2 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 − 1 on [0, 1] and [1, 2] can be determined by

k

φ j (2), k, j = 0, , r − 1, (6.3.6)

whereδ k , jis the Kronecker delta

The symmetry and antisymmetry about t = 1 are given by

The curves ofφ0(t) and φ1(t) with explicit expressions are plotted in Fig 6.1.

Recall from (6.1.2) and (6.1.3) that

Since has a support of [0, 2], all C k = 0 for k ≥ 3 For the three nonzero

coeffi-cients, we have from (6.3.10) that

While C1was given in (6.3.11) for any multiplicity r , C0and C2can be obtained for

the case of r = 2 in the next example For arbitrary r, the general expressions of C0

and C will be derived later in this section

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

FIGURE 6.1 Multiscalets of r= 2 from analytic expression

Example 2 Evaluate C0and C2for r= 2

By linear independence of translationsφ(2t − k), we claim that

As a result of (6.3.16), C2remains to be determined The coefficient C2of arbitrary

r can be obtained from the following theorem:

Theorem 1 The eigenvalues of C2 are (1

2) r , (1

2) r+1, , (1

2) 2r−1, and C

2can befound from the similarity transformation of a diagonal matrix

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

CONSTRUCTION OF MULTISCALETS 251

The resultant matrices C0, C1, and C2in this example agree exactly with (6.3.13) and(6.3.14) in Example 2 This implies that the multiscalets constructed by the analyticexpressions and by the numerical (iterative or cascade) methods are identical.Using the vector dilation equations, we obtain

it-tained from analytic expressions For multiplicity r = 3, the corresponding lowpass

matrices C0, C1, and C2can be calculated in the same manner outlined in Example 3and are given below:

−5

32 3 8 1

64 1

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.

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

(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 = δ0,n

In order to simplify notation, we have denotedφ, without subscript, as a vector The

simplified notation ofφ and ψ will be carried out throughout the chapter The result

above may be written in vector form as

Unfortunately, we do not have orthogonality of{φ(t − m)} with regard to these inner

ORTHOGONAL MULTIWAVELETS ˘ψ(t) 255

Hence

... multiwavelets ˘ψ0(t) and ˘ψ1(t) of r = 2.

6.5 INTERVALLIC MULTIWAVELETS ψ(t )

The orthogonal multiwavelets... orthogonal finite element multiwavelets [4], which we referred to as theintervallic multiwavelets to avoid confusion with the finite element method (FEM)

intro-in electromagnetics This multiwavelet... 6.7.

Example 4 Derive explicit expressions of the multiwavelets and dual multiwavelets

−3 8

Hence, the intervallic multiwavelets are

t+29)