Wavelets trong Electromagnetics và mô hình thiết bị P3 docx

The MRA allows us to expand a function f t in terms of basis functions, consisting of the scalets and wavelets.. 32 BASIC ORTHOGONAL WAVELET THEORY3.2 CONSTRUCTION OF SCALETS ϕτ Haar wav

ϕ (k) (t) ≤ C pk (1 + | t |) −p , k = 0, 1, 2, , r, (3.1.1)

p ∈ Z, t ∈ R, C pk − constants.

Thus we have defined a set, S r, which will be used in the text;

ϕ ∈ S r = {ϕ : ϕ (k) (t) exist with rapid decay as in (3.1.1)}.

30

Wavelets in Electromagnetics and Device Modeling George W Pan

Copyright ¶ 2003 John Wiley & Sons, Inc.

ISBN: 0-471-41901-X

MULTIRESOLUTION ANALYSIS 31

A multiresolution analysis of L2(R) is defined as a nested sequence of closed

sub-spaces{V j}j ∈Z of L2(R), with the following properties [1]:

(5) There existsϕ(t) ∈ V0such that set{ϕ(t − n)} forms a Riesz basis of V0.

A Riesz basis of a separable Hilbert space H is a basis { f n} that is close to beingorthogonal That is, there exists a bounded invertible operator which maps{ f n} onto

an orthonormal basis

Let us explain these mathematical properties intuitively:

• In property (1) we form a nested sequence of closed subspaces This sequencerepresents a causality relationship such that information at a given level is suf-ficient to compute the contents of the next coarser level

• Property (2) implies that V j is a dilation invariant subspace As will be seen inlater sections, this property allows us to build multigrid basis functions accord-ing to the nature of the solution In the rapidly varying regions the resolutionwill be very fine, while in the slowly fluctuating regions the bases will be coarse

• Property (3) suggests that V j is invariant under translation (i.e., shifting).

• Property (4) relates residues or errors to the uniform Lipschitz regularity of the

function, f , to be approximated by expansion in the wavelet bases.

• In property (5) the Riesz basis condition will be used to derive and prove gence The last two properties are more suitable for mathematicians; interestedreaders are referred to [1–3]

conver-Clearly,√

2ϕ(2t − n) is an orthonormal basis for V1, since the map f √2 f (2·) is

isometric from V0 onto V1 Since ϕ ∈ V1, we have

Equation (3.1.2) is called the dilation equation, and is one of the most useful

equa-tions in the field of wavelets The MRA allows us to expand a function f (t) in

terms of basis functions, consisting of the scalets and wavelets Any function f ∈

L2(R) can be projected onto V m by means of a projection operator P V m, defined

as P V m f = f m := n f m ,n ϕ m ,n , where f m ,n is the coefficient of expansion of f

on the basisϕ m ,n From the previously listed MRA properties, it can be proved thatlimm→∞|| f − f m|| = 0, that is to say, that by increasing the resolution in MRA, afunction can be approximated with any precision

32 BASIC ORTHOGONAL WAVELET THEORY

3.2 CONSTRUCTION OF SCALETS ϕ(τ)

Haar wavelets are the simplest wavelet system, but their discontinuities hinder theireffectiveness Naturally people have found it useful to switch from a piecewise con-stant “box” to a piecewise linear “triangle.” Unfortunately, the triangles are no longerorthogonal Thus an orthogonalization procedure must be conducted, which leads tothe Franklin wavelets

3.2.1 Franklin Scalet

Consider a triangle function depicted in Fig 3.1

θ(t) = (1 − | t − 1 |)χ [0,2] (t).

This function is the convolution of two pulse functions ofχ [0,1] (t), where χ [0,1] (t) is

the characteristic function that is 1 in[0, 1] and 0 outside this interval The Fourier

transform of the pulse function can be obtained using the following relationships:

{1(t) − 1(t − 1)} ↔ 1

s (1 − e −s ) = 1

i ω (1 − e −iω ),

where 1(t) is the Heaviside step function By the convolution theorem, the triangle

has as its Fourier transform

2

= ˆθ(ω).

Notice thatθ(t) is centered at t = 1 Let us define θ c (t) := θ(t + 1), a triangle

centered at t= 0 with a real spectrum of

ˆθ c (ω) =

sinω/2 ω/2

2

.

Occasionally we will use T (t) := θ c (t) to denote the triangle centered at the origin.

To find the orthogonal functionϕ(t), we employ the isometric property of the

Fourier transform First, we may show that

CONSTRUCTION OF SCALETSϕ(τ) 33

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−∞ϕ(t − n)ϕ(t) dt,

34 BASIC ORTHOGONAL WAVELET THEORY

whereδ0,nis the Kronecker delta By employing (3.2.1), we arrive at

Comparing (3.2.2) with (3.2.3), we conclude that c0 = 1, and c n = 0 for n = 1.

This conclusion can also be drawn from the uniqueness of the Fourier transform asfollows We know that

1

2π

2π0

| ˆϕ†(ω) |2:=

k

| ˆϕ(ω + 2kπ) |2= 1. (3.2.4)

In the following paragraphs we will construct the scaletϕ(t) using translated

trian-glesθ(t + 1 − n) as building blocks.

Since ϕ ∈ V0, we have ϕ(t) = n a n θ(t + 1 − n) for a sequence {a n } ∈ l2,meaning that

n | a n|2< +∞ Taking the Fourier transform, we immediately have

36 BASIC ORTHOGONAL WAVELET THEORY

It will be seen in the next paragraph that



= 6π21−2

3sin2πx

sin4πx

38 BASIC ORTHOGONAL WAVELET THEORY

TABLE 3.1 First Ten Coefficients of a n = a −n for the Franklin Scalet

2

φ (t) ψ(t)

cos k ω



1−2

3sin2(ω/2) d ω.

This equation provides a numerical expression for the evaluation of a n, which can

be accomplished by imposing Gaussian–Legendre quadrature The values of a nare

displayed in Table 3.1 Using these values of a nand the translated triangle functions

θ(t +1−n), the Franklin scalet is constructed according to (3.2.12) From the integral

expression of a k , we observe that a −k = a k Alsoθ c (t) is symmetric Therefore the

Franklin scalet is an even function The Franklin wavelet is symmetric about t = 1

2,and will be studied in the next section The Franklin scalet and wavelets are depicted

in Fig 3.2

3.2.2 Battle–Lemarie Scalets

The Franklin wavelets employ the triangle functions as building blocks in the struction of an orthogonal system These triangles are continuous functions but notsmooth; their derivatives are discontinuous at certain points If we convolve the tri-angle with the box one more time, the resulting function will be smooth The trans-lations of this smooth function can then be used as building blocks to build smoothorthogonal wavelet systems The greater the number of convolutions conducted, thesmoother the building block functions become This smoothness is achieved at theexpense of larger support widths of the resulting scalets In general, the B-spline of

con-degree N is obtained by convolving the “box” N times Hence

ˆθ N (ω) = e −iκ(ω/2)

sin(ω/2) ω/2

Detailed construction of higher-order Battle–Lemarie wavelets is left to readers

as an exercise problem in this chapter

40 BASIC ORTHOGONAL WAVELET THEORY

−5 −4 −3 −2 −1 0 1 2 3 4 5

−1.5

−1

−0.5 0 0.5 1

1.5

φ (t) ψ(t)

−20 −15 −100 −5 0 5 10 15 20 0.2

0.4 0.6 0.8

ψ(ω)

FIGURE 3.3 Battle–Lemarie scaletϕ and wavelet ψ.

3.2.3 Preliminary Properties of Scalets

In the previous discussions we used the triangle functions as building blocks to erate the Franklin wavelets, according toϕ(t) =k a k θ c (t − k) If the triangles are

gen-replaced by smoother building blocks, higher-order Battle–Lemarie wavelets may be

obtained in the same manner Unfortunately, the number of nonzero coefficients a k are infinite, although a k decays very rapidly, meaning that a k = O(e −a| k | ) A chal-

lenging question arises: Is it possible to have a finite number of nonzero coefficientsthat generate orthogonal wavelets? This query leads us to the Daubechies wavelet

We seek h k in the dilation equation

is given in the latter, while ˆϕ (ω/2) remains

unknown here Equation (3.2.15) translates the orthonormal condition

| ˆϕ†(ω) |2= 1into



 ˆh ω

2

 2+ ˆh ω



k (−1) nk h k e −ik(ω/2)

42 BASIC ORTHOGONAL WAVELET THEORY

we can refer to the orthogonal condition (3.2.6) and obtain

where we have used| ˆϕ†(·) |2= 1

Note that ˆh is a periodic function with period 2π By setting ω = 0 in (3.2.15),

After the scalets are obtained, we can create the corresponding waveletsψ(t) In this

process we may take advantage of the MRA structure by choosing{ψ(t − n)} as an orthonormal basis of W0, which is the orthogonal complement of V0 in V1, namely

Hence {ψ mn}n ,m∈Z forms an orthogonal basis of L2(R) In the Fourier transform

domain the two orthogonal equations (3.3.2) and (3.3.3) become, respectively,

it follows thatψ(t) ∈ V1 Since ψ(t) ∈ V1, ψ(t) can be represented in terms of basis

functionsϕ(2t − k) in V1, yielding the dilation equation

44 BASIC ORTHOGONAL WAVELET THEORY

Show. Owing to this analogy, we will only show the second equation

Following the steps in the derivation from (3.2.1) through (3.2.4), we have

46 BASIC ORTHOGONAL WAVELET THEORY

= √12

√2

How-ϕ(t) and ψ(t) constructed in this way may have noncompact supports.

We conclude this section by quoting several theorems [3] and [4] The proofs arequite abstract and are printed in a smaller font Readers who are not interested inmathematical rigor may always skip material printed in smaller fonts without jeop-ardizing their understanding of the course

Theorem 1 Assume thatψ(t) ∈ S r, meaning thatψ(t) has rth continuous

deriva-tives and they are fast decaying according to (3.1.1);ψ m ,n (t) := 2 m /2 ψ(2 m t − n) form an orthonormal system in L2(R) Then ψ(t) has rth zero moments, namely

∞

−∞t k ψ(t) dt = 0, k = 0, 1, , r.

The significance of this theorem is its generality For instance, the Battle–Lemarie

wavelets of N = 2 are built from convolving the box function consequently twice

No zero moment requirement was forced explicitly However, from the theorem, it isguaranteed that ψ(t)t  dt = 0,  = 0, 1.

Proof We prove the theorem by induction on k.

(1) k= 0, we wish to show that −∞∞ ψ(t) dt = 0 ∃N = 2 j0k0thatψ(N) = 0.

Let j ∈ Z, 2 j N is an integer (all j ≥ j0) By orthogonality, we may write

j→∞ψ(2 − j y + N)ψ(y) dy.

The change of the limit with integral is permitted because| ψ(2 − j y + N)ψ(y) | ≤

c | ψ(y) |, and Lebesgue dominated convergence allows the commutation Thus

48 BASIC ORTHOGONAL WAVELET THEORY

Theorem 2 (Vanishing Moments) Assume thatϕ and ψ form an orthogonal basis,

The following four statements are equivalent:

• The waveletψ has p vanishing moments.

• ˆψ(ω) and its first p − 1 derivatives are zero at ω = 0.

• ˆh(ω) and its first p − 1 derivatives are zero at ω = π.

In the previous section we derived the relationship between the bandpass filter g k

of the wavelets and the lowpass filter h k of the scalets Now we may construct theFranklin wavelet from Franklin scalets by applying the results from the previoussection We begin with

where T (t) = θ c (t) is the triangle centered at the origin Using the dilation equation

and orthogonality, we have

50 BASIC ORTHOGONAL WAVELET THEORY

Since a −k = a kfor Franklin, we use absolute signs in the subscripts of (3.4.5)

In general, the wavelet is given by the dilation equation

PROPERTIES OF SCALETS ˆϕ(ω) 51 TABLE 3.2 Coefficients for the Franklin Scalet and

52 BASIC ORTHOGONAL WAVELET THEORY

This property can be seen from the Franklin scalet

ˆϕ(ω) =

sin(ω/2) ω/2

In this section we will prove the basic and most useful property of ˆϕ(ω) for scalets

in general, that is, ˆϕ(0) = 1 The proof is printed in the following paragraph, and it

lasts several pages

Proof Consider the characteristic function

The projection of f (t) onto V mis

 f (t), p(t) = 1

2π F(ω), P(ω).

Equation (3.5.3) can be considered in two different ways

(1) A function defined on[0, 1] can always be extended into a periodic function with

[−2m π, 2 m π] as one period, and be analyzed as a periodic function We have a finite

power signal (periodic) and a discrete spectrum As a result the Fourier coefficient of

54 BASIC ORTHOGONAL WAVELET THEORY

2 ∞

−∞

sin(t/2)

2 + π 2

= 1

ˆh(0) = 1 ˆh(π) = 0.

56 BASIC ORTHOGONAL WAVELET THEORY

(5) ∞

n=−∞ϕ(t − n) = 1.

Note that∞

n=−∞ϕ(t −n) is a periodic function of period 1 Therefore it can

be expanded into the Fourier series

DAUBECHIES WAVELETS 57

and the properties of ˆϕ(ω),

ˆh(0) = 1, ˆh(π) = 0,

2 Furthermore

Let us use a trial-and-error method to find a set of nonzero lowpass filter coefficients

h kunder conditions of (i) through (iii):

(1) Set h k = 0 for k = 0, 1, and find h0 , h1 From properties (i) through (iii) wehave

58 BASIC ORTHOGONAL WAVELET THEORY

(2) Set h k = 0 for k = 0, 1, 2; find h0 , h1, h2:

√

2− h0

2+

1

2+



h1−

√24

2

=

12

2

.

This equation represents a circle in the h0 − h1 plane that is centered at

(√2/4,√2/4) with radius 1/2, and passes through the origin The last

equa-tion in (3.6.1) gives

h0

1

2+



h1− 1

2√2

2

=

12

2

,

We can verify that for anyν, the four equations in (3.6.1) are all satisfied We

need one more equation to specifyν, which will be obtained as follows The

wavelets have the frequency domain expression

60 BASIC ORTHOGONAL WAVELET THEORY



k

h k i

2(k − 1)e i (ω/2)(k−1) (−1) k ,

ˆg(0) = i

2√2

2 ,

h3= 1−

√3

c1=3+

√3

c2=3−

√3

c3=1−

√3

The corresponding scalet and wavelet are illustrated in Fig 3.4 It is expected thathigher order wavelets are smoother, but their supports are wider Higher-order Dau-bechies wavelets can be derived in the same way presented here The coefficients aretabulated in Table 3.3 and will be employed to construct the Daubechies scalets and

wavelets of different orders In general, for Daubechies scalets h n = 0 for n < 0 and

n > 2N + 1; the support ϕ = [0, 2N − 1], and the support ψ = [1 − N, N], where

N is the order, or the number of vanishing moments, meaning that, ∞

−∞t k ψ(t) dt =

0, k = 0, 1, , N − 1 A detailed discussion may be found in [1].

62 BASIC ORTHOGONAL WAVELET THEORY

1 5

2

φ (t) ψ(t)

−20 −15 −10 −5 0 5 10 15 20 0

0.2 0.4 0.6 0.8

ψ(ω)

FIGURE 3.4 Daubechies scaletϕ and wavelet ψ (N = 2).

TABLE 3.3 Coefficients for Compactly Supported Daubechies Wavelets

64 BASIC ORTHOGONAL WAVELET THEORY

3.7 COIFMAN WAVELETS (COIFLETS)

An orthonormal wavelet system with compact support is called the Coifman wavelet

system of order L if ϕ(t) and ψ(t) have L−1 and L vanishing moments, respectively,

dt t l ψ(t) = 0, l = 0, 1, , L − 1, (3.7.1)

dt t l ϕ(t) = 0, l = 1, 2, , L − 1. (3.7.2)

As usual, the scalet still has a normalized d.c component

The nonzero support of the Coiflets of order L = 2K is 3L − 1.

Consider the case L = 2 Notice that (3.7.1) states the vanishing moments forthe wavelets, and (3.7.3) is the normalization of the scalet, with respect to the d.c.component Both of these two equations are shared by other wavelets The uniqueproperty of the Coiflets is contained in (3.7.2), namely the vanishing moments of thescalets This property can be shown to yield

COIFMAN WAVELETS (COIFLETS) 65

solve the h k for the Coiflets of order 2 They form a set of nonlinear equations thatare shown explicitly below:

The six-variable nonlinear equations for Coiflets of L = 2 were solved numerically

using the software package Maple, yielding two sets of solutions:

66 BASIC ORTHOGONAL WAVELET THEORY

TABLE 3.4 Comparison between the Daubechies and Coifman Wavelets

Daubechies Basic Equations Coifman

COIFMAN WAVELETS (COIFLETS) 67

TABLE 3.5 Filter Bank of Coiflets

68 BASIC ORTHOGONAL WAVELET THEORY

The second set of the solutions can be derived in irrational numbers [5] as

h√ −2

2 =1 −√7 32

h√ −1

2 =5 +√7 32

h0

√

2 =7 +√7 16

h1

√

2 =7 −√7 16

h2

√

2 =1 −√7 32

2

φ (t) ψ(t)

−20 −15 −100 −5 0 5 10 15 20 0.2

0.4 0.6 0.8

ψ(ω)

FIGURE 3.5 Coifman scaletϕ and wavelet ψ(L = 4).