University of Queensland School of Information Technology and Electrical Engineering

Adaptive Decomposition in Electromagnetics

Submitted for the Degree of Bachelor of Engineering (Electrical) Honours Thesis By: Wayne Wai Keong Wong Supervisor: Associate Professor Dr. Nick Shuley October 2003

Wayne Wong

29 October 2003

Head of School

Department of Information Technology and Electrical Engineering

Professor Simon Kaplan

Dear Professor Kaplan,

In accordance with the requirement of the Degree of Bachelor of Engineering (Honours) in the Division of

Electrical Engineering, I hereby present the following thesis entitled “Adaptive Decomposition in

Electromagnetics”. This work was performed under the supervision of Dr. Nick Shuley.

I declared that the work submitted in this thesis is of my own, except as acknowledged or as referenced, and

has not been previously submitted for a degree at the University of Queensland or any other institutions.

Faithfully Yours,

Wayne Wong

ACKNOWLEDGEMENT

Special thanks and gratitude goes to Dr. Nick Shuley for his patience and kind assistance.

Thanks also goes to Mr. Eric Phua for his previous work and research.

I

CONTENT PAGE

ACKNOWLEDGEMENTS……………………………………………………………………….I

CONTENT PAGE………………………………………………………………………...…II - IV

ABSTRACT………………………………………………………………………………………V

CHAPTER 1:

1.1) Introduction………………………………………………………………………..1

1.2) Purposes and Aims………………………………………………………………...2

1.3) Report Structure………………………………………………………………...…3

CHAPTER 2:

2.1) Background………………………………………………………………………..5

2.2) Types of Dictionaries……………………………………………………………...8

2.2.1) Frequency Dictionaries……………………………………………9

2.2.2) Time-frequency Dictionaries…………………………………….10

2.2.3) Trivial Dictionaries………………………………………………11

2.2.4) Time-scale Dictionaries………………………………………….12

2.3) Methods of Decomposition………………………………………………………14

2.3.1) Methods of Frame (MOF)………………………………………..15

2.3.2) Best Orthogonal Basis (BOB)……………………………………15

2.3.3) Basic Pursuits (BP)………………………………………………16

2.3.4) Matching Pursuits (MP)………………………………………….17

II

CONTENT PAGE

CHAPTER 3:

3.1) Wave-Based Dictionaries………………………………………………………...19

3.1.1) Resonance Dictionary (WD)……………………………………..20

3.1.2) Wave front Dictionary (RD)……………………………………..22

CHAPTER 4:

4.1) Software Implementation……………………………………………………...…25

4.2) Modules Overview……………………………………………………………….26

4.3) Structure flow…………………………………………………………………….27

4.3.1) Resonance Dictionary (RD)……………………………………...28

4.3.2) Wave front Dictionary (WD)………………………………….…32

4.4) Matching Pursuit Algorithm………………………………………………….….36

CHAPTER 5:

5.1) Discussion on Simulation Results………………………………………………..39

5.2) Resonance Dictionary (RD)……………………………………………………...40

5.2.1) Natural Frequencies……………………………………………...40

i) Dictionary size = 10; Error tolerance = 0………………..40

ii) Dictionary size = 10; Error tolerance = 0.2……………...42

5.2.2) Simulated Frequencies…………………………………………...43

i) Dictionary size = 20; Error tolerance = 0…………….….43

ii) Dictionary size = 20; Error tolerance = 0.1……………...44

III

CONTENT PAGE

5.3) Wave front Dictionary (WD)…………………………………………………….45

5.3.1) Results……………………………………………………………45

i) Dictionary size = 30; Error tolerance = 0………………..45

ii) Dictionary size = 30; Error tolerance = 0.2……………...46

iii) Dictionary size = 40; Error tolerance = 0………………..47

iv) Dictionary size = 40; Error tolerance = 0.3……………...48

CHAPTER 6:

Conclusions………………………………………………………………………………50

References……………………………………………………………………………………..i - iii

Appendices:

Software Source Codes………………………………………………………….…Pg 1 - 6

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

ABSTRACT

Signal is usually represented by a linear sum of functions that can be selected either by physical

descriptive or mathematical analysis. There has been a large number of over complete waveform

dictionaries developed. Stationary wavelets, wavelet packets and the chirplets are such examples.

Several different methods of decomposition have also been proposed. These methods include the

Method of Frames (MOF), Best Orthogonal Basis (BOB), Basic Pursuits (BP) and the Matching

Pursuits (MP). These methods will be subsequently reviewed in the later part of the report. The

concept of adaptive algorithm will also be touched on.

A fuller discussion on the different explicit definition of functions used for defining the Wave-

Based Dictionaries will be reviewed. Resonance and Wave front Dictionaries are examples of

such Wave-Based Dictionaries and they will be mainly covered in this thesis.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

CHAPTER ONE

(1.1) Introduction Signal decompositions for the detection and the identification of targets have often been a

popular option in radar signal image processing. Wave patterns such as wave front, resonance

and chirps are a few examples that could be obtained from this signal decomposition of targets.

There is an approach known as Adaptive Decomposition [1]. In this approach, the signals are

decomposed into a linear expansion of waveforms. Upon decomposition, a general dictionary

comprising of all the basic functions is used to closely represent the anticipated signal. There are

a number of decomposition techniques available. There are the Method of Frames (MOF) [2],

Best Orthogonal Basis (BOB) [3], Basic Pursuits (BP) [4] and the Matching Pursuits (MP) [5].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

In this thesis, the focus will be dealing with the Matching Pursuits (MP) technique, developed by

Mallet and Zhang [5]. In this approach, an algorithm slowly builds up from a sequence of sparse

approximations starting from an empty decomposition and then gradually into a greedy fashion

[1].

(1.2) Purposes and Aims The main purpose for doing this thesis topic is to carry out a series of research and analysis on

the performance of approximate comparison of wave signals as they hit on different surfaces and

objects. As this is a relatively difficult process to be monitored in real-life environment, a

software program is thus developed to perform the necessary calculations and comparison

between the different wave signals in a simulated environment.

Listed are some of the aims of the thesis project:

Using Mat lab software program to implement the matching pursuits algorithm.

Construction of the Waved-Based dictionaries [6], namely the Resonance and the

Wave front Dictionaries for the algorithm.

Illustrations of the resultant approximate signals from both the Resonance and the

Wave front input signals.

Carrying out logical explanations and discussions for the results obtained from the

resultant approximate signals.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(1.3) Report Structures The thesis report is broken down into different chapters and sections. Below listed are the

different breakdown structure of the whole report.

In Chapter 1, the report looks at some of the purposes and aims of doing this thesis project. This

chapter also shows the breakdown structure of the whole report.

In Chapter 2, the report will be looking at the background of the Adaptive Decomposition

technique. Relevant functions and equations will be used to support the theory behind this

technique. Different types of dictionaries will also be discussed over here. This Chapter also

addresses on the different methods of decomposition used. An overview of the different

methods will be given. Relevant equations will also be given in these methods, to provide a

better understanding for the reader.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

In Chapter 3, Wave-Based Dictionaries are discussed. They are namely the Resonance

Dictionaries (RD) and the Wave front Dictionaries (WD). The algorithms for both the

Dictionaries will also be presented.

In Chapter 4, the report will look into the actual Software Implementation for the whole thesis

topic. Under this chapter, the Modules Overview will be presented to give a clearer

understanding on the different modules used for the software program. The structure flow and

the algorithms for both the Resonance and the Wave front Dictionaries will also be looked into.

In Chapter 5, some of the results obtained from both the Resonance and the Wave front

Dictionaries will be presented. They will be presented in the forms of illustrations and

appropriate discussions will also be given.

In Chapter 6, conclusions will be drawn from this thesis topic. Future recommendations and

improvements will also be included in this chapter, to give further suggestions as to how this

topic could be further expands and develops.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

CHAPTER 2

(2.1) Background

In this section, a short review on the adaptive representation of the signals is discussed. Firstly,

let H be a Hibert space. Secondly, let D be a dictionary of functions in an N-dimensional Hilbert

space, H. The smallest possible dictionary is a basis of H. Given that a dictionary is filled with

basic functions of

D = {φγ : γ∈ Γ} in H,

with || φγ || = 1 and the span (D ) = H . [2.1.a]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

It is desirable to decompose a complex-valued signal, S onto a set of basis functions, φγ, as

S = ∑αγ φγ , [2.1.b] γ∈ Γ

where αγ is the complex-valued weight and φγ is a parameterized function.

The approximation decomposition can also be set by

M

S = ∑αγi φγi +R (M) , i = 1

where R (M) is the residual. [2.1.c]

Usually, the dimension of the Dictionary D is larger than N (i.e. the size of the index set). Thus,

there will be a redundant or over complete set of basis functions in H for signal decomposition

[12].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Adaptive decomposition allows the use of composite bases, which are closely modeled into the

physical characteristics of the signals. The set of the basis function is over complete and will

comprise of a set of disparate functions that describe the different particular signal

characteristics. By allowing an over complete representation, we are more interested in the

reconstruction of the signal using elementary waveforms, that will be super-positioned. These

waveforms are energies corresponding to the different specific signal components that are

consecutively mapped, in order to present a single dictionary element [1].

There are two types of dictionaries used, which define the specific analytical models or classes of

functions. This models or classes of functions closely approximate the anticipated signal

characteristics. There are namely, the “Physics-based dictionaries” and the “Data-based

dictionaries” [1]. “Physics-based dictionaries” consists of functions, which define the principles

analysis of the underlying physical processes, while the “Data-based dictionaries” consists of

functions that describe the characteristics of the component signals in the data.

In conclusion, the ability to decompose signals into general, over-determined dictionary allows a

better representation of the dictionary elements.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(2.2) Types of Dictionaries In the past, a single orthonormal basis for expansion has always been used. Signals are

not always compactly represented using these expansions. Hence, an alternative has been

developed. Using this alternative approach, a collection of waveforms (i.e. the basic functions of

a signal) can be used to describe the different signal characteristics. Thus, these waveforms can

be used for the construction of the actual signal and this set of waveforms is often referred to as a

“Dictionary”.

According to Mallet and Zhang [5], a dictionary is a collection of parameterized waveforms,

being represented by

D = {φγ : γ∈ Γ }. [2.2.a]

Each function φγ can be regarded as an “atom” in the decomposition of the complex-valued

signal, S as shown in [2.1]. Some of the examples of the dictionary elements are pure tones (i.e.

Fourier Dictionary) and bumps (ie. Wavelet Dictionary).

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Dictionaries can either contains the exact numbers of atoms and they are known as “Complete

Dictionary”. Dictionaries containing more numbers of atoms are known as “Over complete

Dictionary”. Incidentally, Dictionaries containing fewer atoms are known as “Under complete

Dictionary” [5].

In the subsequent sections in this chapter, a brief description for the different dictionaries which

have been proposed over the years will be accounted for.

(2.2.1) Frequency Dictionaries Frequency Dictionary is an example of a Fourier Dictionary. The Fourier Dictionary

comprises of a collection of sinusoidal waveforms,

φγ , whereby γ = ( w, v ). [2.2.1.a]

In this case, w ∈ [0, 2π] is a frequency variable and V ∈ {0, 1} is the phase type (i.e. sine or

cosine). The equations below illustrate this point.

φ (w , 0) = cos (wt). [2.2.1.b]

φ (w , 1) = sin (wt). [2.2.1.c]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

For a standard Fourier Dictionary, γ will spread through the set of the Cosine Fourier

frequencies with the expression,

wk = 2πk /n, where k = 0,1,…n/2. [2.2.1.d]

Incidentally, for all Sine Fourier frequencies with the expression,

wk , where k = 1,…n/2-1. [2.2.1.e]

In the dictionary, there will be an “n” number of waveforms available. The atoms in this

dictionary will be mutually orthogonal [12].

(2.2.2) Time-frequency Dictionaries One of the examples for a Time-frequency Dictionary will be a Gabor Dictionary, developed by

Gabor [7]. The parameters of the Gabor functions (i.e. time-frequency atoms) will be

γ = {w, T, θ, δt },

where w ∈ [0, π] is a frequency.

T = Location.

θ

δ

= Phase.

t = Duration. [2.2.2.a]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

The time-frequency atoms can be expressed as:

φγ (t) = exp {- (t –T) ^2 / (δt) ^2}. cos (w (t -T) + θ). [2.2.2.b]

Usually these atoms will consist of frequencies close to w and will gradually disappear far away

from T [12].

(2.2.3) Trivial Dictionaries

In the case of the Dirac dictionary, it has a collection of waveforms that are zeroed, except in one

point:

γ = {0, 1, n –1} and φγ (t) = 1{t = γ}. [2.2.3.a]

This dictionary is also the orthogonal basis of R^n. However, in the case of the Heavside

dictionary, it has a collection of waveforms that jumps at one particular point:

γ = {0, 1, n –1} and φγ (t) = 1{t ≥ γ}. [2.2.3.b]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

The atoms in this dictionary are not orthogonal, but every signal has a representation of:

n-1

S = So φ + ∑ (Sγ - Sγ - 1) φγ [2.2.3.c] o γ = 1

(2.2.4) Time-scale Dictionaries The time-scale dictionary is a collection of translations and dilations of both “the basic father and

the mother wavelets”. One of such examples will be the Haar dictionary, whereby the “father

wavelet” is

ϕ = 1[0,1]. [2.2.4.a]

and the “mother wavelet” is

ψ = 1[1/2,1] – 1[0,1/2]. [2.2.4.b]

These wavelets will be indexed by

γ = (a, b, v ),

where a ∈ (0, ∞) is a scale variable.

b ∈ [0, n] is the location.

v ∈ {0, 1} is the gender. [2.2.4.c]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

In order for the dictionary to work properly, there are several requirements that the dictionary has

to meet. The two main basic requirements that the dictionary needs to meet are “Orthogonality”

and “Orthonormal”.

ORTHOGONALITY

Interval between a ≤ x ≤ b are called “Orthogonal” if b

⟨ f (x)| g(x) ⟩ ≡ ∫ f (x)g(x) dx = 0. [2.2.4.d] a

ORTHONORMAL

Both functions φi (x) and φj (x) are othonormal if they are orthogonal and each normalized. These

two conditions can be written as

b

∫ φi (x)φj (x)w(x) dx = δi j, [2.2.4.e] a where δi j = Kronecker function [24].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(2.3) Methods of Decomposition In the following sections, an overview of the different methods of decomposition will be

accounted for. These methods include the “Method of Frame (MOF)”, “Best Orthogonal Basis

(BOB)”, “Basic Pursuits (BP)” and lastly the “Matching Pursuits (MP)”. A brief algorithm for

each of the different methods will be presented.

The first technique discussed here will be the MOF. It does not generally result in a sparse

decomposition. However, it was one of the first few approaches to represent a signal in terms of

an over determined dictionary [1].

The second technique, BOB does not decompose a signal onto a general dictionary. However,

due to the compactly represented signals nature, it can produce a near-optimally sparse

representation [1].

The last two techniques, namely BP and MP all achieve the desired result of generating sparse

signal decompositions onto general, over determined dictionaries. When a dictionary is over

complete, there will be more than one solution and this solution will be known as a non-unique

solution [1].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(2.3.1) Methods of Frame (MOF) The Methods of Frame (MOF) is a technique for creating an over determined dictionary

described by Daubechies [2]. MOF has coefficients that have the minimum l 2 norm:

Min || α ||2 subject to φα = s. [2.3.1.a]

The solution is unique. Sometimes it is known as minimum-length solution or least square

solution. However, there are two problems with the MOF. Firstly, it has very poor sparsity and

secondly, it has a limited intrinsic resolution [1].

(2.3.2) Best Orthogonal Basis (BOB)

Wavelet packet and cosine packet are dictionaries with special properties and these properties

allow certain special sub-collections of the elements in the dictionaries, so as to amount to the

orthogonal bases. Coifman and Wickerhauser [3] have proposed a method that is capable of

adaptively picking a single orthogonal basis, which will be the “best basis”, from among many

others. By denoting the vector coefficients of s in the orthogonal basis B as (s |B| I) I, the

“entropy” can be defined as

E (s |B|) = ∑ 1 e (s |B| I),

where e (s) is a scalar function of a scalar argument. [2.3.2.a]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Thus, the algorithm is

Min {E (s |B|); B ortho basis ⊂ D} [2.3.2.b]

The technique starts by constructing a large dictionary, in which all the elements are orthogonal.

They are then constructed into sub-collections of the basic functions. Subsequently, the inner

product of the data for each element in the dictionary and the entropy of collection for the

resulting coefficients will be computed. The minimized entropy is later retained and the whole

procedure repeats itself, until the minimum entropy basis has been obtained. However, there is a

disadvantage to this technique. It does not work well with dictionaries comprising of disparate,

non-orthogonal functions [1].

(2.3.3) Basic Pursuits (BP)

The Basic pursuits is a method proposed by Chen and Donoho [4]. By assuming that the

dictionary is over complete, the signal can be represented using,

S = ∑αγ φγ. [2.3.3.a]

BP works on the principles of solving φα = s. By picking α whose coefficients have the minimal

l 1 norm, the following representation is obtained,

min ||α||1 subject to φα = s [2.3.3.b]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

In addition, BP looks very much similar to MOF, except that the l 2 norm is being replaced by the

l 1 norm. Due to this change, it will lead to a totally different set of results. Usually, MOF has a

quadratic optimization problem with linear equality constraints. Thus, it will always results in a

system of linear equations solution. BP on the other hand, requires solution of a convex, non-

quadratic optimization problem. This would usually involve a considerably larger amount of

effort and complexity [1].

(2.3.4) Matching Pursuits (MP)

The Matching Pursuits is an algorithm that builds up starting from an empty decomposition into

a greedy fashion, by means of a sequence of sparse approximations. Mallet and Zhang [5] are the

ones who propose this algorithm. At any given stage of the construction, the element of the

dictionary that best correlates with the residual is added to the approximation. This procedure

will continue until it is terminated when a desired level of accuracy has been achieved.

Let γ = (s, u, v, w) and f = signal to be analyzed. γ0 is then chosen to maximize |⟨ f, gγ ⟩|.

f is then written as f = ⟨ f, gγ 0⟩ gγ 0 + Rf ,

where ⟨ f, gγ 0⟩ = Inner product of f.

gγ 0 = Sampled data.

Rf = Residual. [2.3.4.a]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

The process is repeated on the residual, Rf. After M iterations, the following equation is obtained

M -1

f = ∑ ⟨ R k f, g k⟩ gγk + R M f [2.3.4.b] k =0

where R k f is the residual after k steps. The whole process will be terminated when the residual

is being regarded as too small or typically when the ||R M f || 2 is less than the prescribed tolerance

[13].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

CHAPTER 3

(3.1) Wave-Based Dictionaries Over the past years, there has been a lot of focus on the development of algorithms for the

purpose of target identification. These algorithms focus on the extraction of the wave objects,

from the scattering data. These can be classified as either the resonance or the wave front,

whereby the two are being connected via the wave-front-resonance algorithm [8].

In the context of a wide-band or time scattering domain, late-time resonance [9] – [12] has

always been a mean of interest for target identification. This is due to the fact that the research

has been greatly motivated by Baum’s [9] singularity expansion method (SEM). In SEM, it was

shown that the late-time portion of time-domain scattered fields could be represented as a sum of

damped oscillations. Incidentally, this is the characteristic of the aspect-independent resonance

[15].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Alongside with the development of the resonance-based processing, several researchers have

recently been considering the development of the techniques, for extracting the wave fronts from

the scattering data as well. Altes [16] considers a matched-filter approach that is designed to

operate on the time-domain data directly. In the case of Hurst and Mittra [17] and Walton [18],

they demonstrated that the model-based processing of frequency-domain data could actually

produce a temporal resolution, which is far better and attainable via Fourier-based techniques.

In the following sections, both the resonance and the wave front characteristics and algorithms

will be looked much into details.

(3.1.1) Resonance Dictionary (RD) Resonance is actually synthesized by successive wave fronts. In other words, resonance is

the late-timer. They are represented compactly by sinusoids with characteristic oscillation

frequencies and damping constants [9]. Each set of targets can also be characterized by a set of

such modes. Implementation of the waved-based dictionary, including the sets of resonance

modes is as shown below:

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Assuming that the complex resonant frequencies of the “mth” mode for target “n” are expressed

as

Smn = jwmn - αmn [3.1.1.a]

For the “nth” target, the RD element [6] is

M

gr (t; T, n) = [Σ amn exp (Smnt) + c.c] U (t – T) [3.1.1.b] M = 1

where U (t) = Heavside function.

c.c = Complex Conjugate of the sum to the left.

T = Start times, depending on the continuously shifting of the element.

amn = Least-square projection of the data.

Upon determining the amn, the composite dictionary function gr (t; T, n), it is used to determine

the T- dependent correlation coefficient, as stated in [19].

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

The resonant correlations required for [19] are actually calculated, through the use of a

composite vector. This composite vector comprising of all the modes

m ∈ [1, M], [3.1.1.c]

for any given target n after a least-squares calculation of amn. The correlation of the resonant

data with any one of the modes is generally weaker. However, the correlation with the set of

resonant modes will generally be stronger, when T and n are consistent with the data.

(3.1.2) Wave front Dictionary (WD)

Wave fronts are characterized by a branch of narrow support in time and a branch of wide

support in frequency. The timing of the successive wave front can be used to represent the

geometrical structure of the target. Recently, researchers have discovered that by looking closer

into the frequency dependencies of the wave fronts, additional information for the target can be

obtained.

According to the 1st –order geometrical theory of diffraction (GTD) [20], [21], the wave fronts

can be scattered from an edge, endpoint and finite smooth surfaces. They all have frequency

dependencies proportional to w1/2, 1/w and w respectively, whereby w represents the angular

frequency. It is known that such an algorithm can be highly sensitive to improper model and

model-order selection.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

There has been a study done to examine the robustness of the wave front-based processing [22].

The results has been proven fail due to the following reasons:

1) It does not have sufficient resolution to resolve closely spaced multiple wave fronts.

2) The underlying assumption in the model is invalid.

In the case of point (2), the model is being assumed that the data can be represented; in terms of

wave fronts scattering from different isolated scattering centres. It also assumes that each wave

front can be described by a non-uniform GTD.

The WD elements [6] are defined as follows:

∞

gw (t; T, α) = ∫ [w exp (j π / 2)] ^ α h(w) exp [ jw ( t – T )] dw [3.1.2.a] -∞

∞ and h(w) = ∫ h(t) exp (-jwt) dt

-∞

where h(t) = Incident waveform.

α ∈ {0, ±1/2, ±1, ±3/2, ±2, ±5/2}.

T = Continuous shift parameter. [3.1.2.b]

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

In the case of incident field h(t), the positive integers α in the dictionary vector of [3.1.2.b]

correspond to the α differentiations. In the case of the negative integers α in the dictionary vector

of [3.1.2.b], they correspond to the α integrations. Take note that the integers α are used by Altes

[16] in the context of underwater scattering.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Chapter 4

(4.1) Software Implementation By using Mat lab 6.5, investigations can be carried out on the matching pursuits algorithm for

wave-based dictionaries, namely the Resonance Dictionary and the Wave front Dictionary. By

converting the algorithm into a source code format, investigations could thus be performed on

the signals behavior, in comparison to the theoretical aspects of the whole matching pursuits

phenomenon.

There are generally a few types of software programs, which allow the decoding of the source

code for the matching pursuits. Some of the following software is the Fortran and C++. Mat lab

is preferred in this case, as it has a greater capability for solving numerical problems. It is also

very much compatible to most operating systems.

In the following next few sections of this chapter, an overview of the modules used in the source

code is accounted for. The structure flow for both the Resonance Dictionary and the Wave front

Dictionary are also accounted for. In addition, the whole matching pursuit algorithm will be

discussed as well.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(4.2) Modules Overview Figure 4.1 shows a block diagram of the different modules involved in the source code.

1 3 2

INPUT SIGNAL

MATCHING PURSUITS ALGORITHM

DICTIONARY TO USE ? FOURIER

RESONANCE

WAVEFRONT

OUTPUT SIGNAL DONE

START

Figure 4.1

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

In the software program, there are mainly six modules. They are namely the “Input Signal”,

the “Matching Pursuit Algorithm”, the “Wave front Dictionary”, the “Resonance Dictionary”,

the “Fourier Dictionary” and the “Output Signal”. Depending on which dictionary and what type

of wave signal are being used, the matching pursuit algorithm could thus access and evaluate the

signal accordingly.

(4.3) Structure flow In this section, the structure flow of both the Resonance Dictionary (RD) and the Wave front

Dictionary (WD) will be accounted for. As the Fourier Dictionary has already been mentioned

and done by the previous guy working on the same thesis topic, it will not be included in this

report.

This whole thesis therefore, is simply an extension of what has been left off from the previous

guy. Instead, this thesis project will consist solely on the RD and the WD. Thus, the report will

concern mainly on the RD and the WD as well.

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ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(4.3.1) Resonance Dictionary (RD) The RD listed in [6] is as follows:

For the nth target, the RD element is

M

gr ( t; T, n ) = [ Σ amn exp (Smnt) + c.c ] U (t – T ), m = 1 where U( t ) = Heavside function.

c.c = Complex Conjugate of the sum to the left.

T = Start times, depending on the continuously shifting of the element.

amn = Least-square projection of the data. [4.3.1.a]

As stated that c.c denotes the complex conjugate of the sum to the left, which is imaginary and

that the real complex conjugate is required for the representation of a real signal. Thus, the above

RD element could be simplified as the following:

M

, n ) = m = 1

gr ( t; T Σ amn exp ( Smnt ). [4.3.1.b]

As seen from the simplified RD element, only the complex conjugate of the real signal

representation is taken into consideration. The imaginary value of the complex conjugate is thus

left out.

28

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

As shown from [4.3.1.b], the amplitude:

amn = αmn + jβmn,

where αmn = Attenuation constant (Real part).

βmn = Phase constant (Imaginary part). [4.3.1.c]

and that complex resonant frequencies:

Smn = jwmn - αmn

where the attenuation constant, αmn has to an negative value, [4.3.1.d]

due to the damping behavior of the resonance frequency.

In the case of the resonance frequency, the damping behavior lies basically on the complex

resonance frequencies, Smn and not on the amplitude, amn. It is noted that the data for the Smn will

be provided by another separate software program, known as “Proxy”. At this point of time, Mat

lab will only be interested in generating a bunch of random values for Smn, unless real data is

given.

29

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Both figure 4.2 and 4.3 show portions of how the Mat lab source codes are being presented out.

Figure 4.2 shows the “Pursuit” Mat lab coding. Figure 4.3 shows the “Resonance Dictionary”

Mat lab coding.

Figure 4.2 “Pursuit” Mat lab coding.

From the “Pursuit” Mat lab coding (figure 4.2), line 8 to 12 show how the random values for the

Smn are being simulated.

30

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Figure 4.3 “Resonance Dictionary” Mat lab coding

From the “Resonance Dictionary” Mat lab coding (figure 4.3), line 8 to 12 show how the actual

dictionary is created, assuming that only the complex conjugate for the real signal representation

is taken into consideration.

From the “Resonance Dictionary” Mat lab coding (figure 4.3), line 16 to 18 show how the

dictionary is being orthonormalized.

31

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(4.3.2) Wave front Dictionary (WD) The WD listed in [6] is as follows:

∞

gw ( t; T, α ) = ∫ [ w exp ( jπ / 2) ] ^ α h (w) exp [ jw ( t – T )] dw [4.3.2.a] -∞ ∞

and h(w) = ∫ h(t) exp (-jwt) dt -∞

where h(t) = Incident waveform.

α ∈ {0, ±1/2, ±1, ±3/2, ±2, ±5/2}.

T = Continuous shift parameter. [4.3.2.b]

For equation [3.1.2.b], it can be assumed that h(w) is the Fourier transform of the spectrum of a

rectangular pulse. To further illustrate the Fourier Transform of the spectrum of a rectangular

pulse, example 2.5 taken from the text “Digital and Analog Communication Systems (Sixth

Edition)”, by Leon W. Couch II [23] is being used. This example is illustrated as on the

following page.

The spectrum is obtained by taking the Fourier transform of w(t) = ∏ ( t / T ).

W(f) = ∫ 1 exp (-jwt) dt

= [exp (-jwt/2) - exp (jwt/2)] / -jw

= T sin (wT/2) / (wT/2)

= T Sa (πTf) [4.3.2.c]

32

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Thus, ∏ ( t / T ) ↔ T Sa (πTf) [4.3.2.d]

Figure 4.4 shows the rectangular pulse and figure 4.5 shows the Sa(x) function.

∏ ( t / T )

2

Figure 4

Figure 4

33

t

- T / 2

T /

.4 Rectangular Pulse

Sa (x) = (sin x) /x

x

.5 Sa (x) Function

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Thus, from equation [3.1.2.b], a Fast Fourier Transform (FFT) can be assumed being carried out

on the equation.

∞

gw ( t; T, α ) = ∫ [ w exp ( jπ / 2) ] ^ α h (w) exp [ jw ( t – T )] dw [4.3.2.a] -∞

where h (w) represents the FFT of a rectangular pulse. The dictionary vector will be represented

by α in this case. The positive α corresponds to the α differentiations and the negative α

corresponds to the α integrations. T will be the continuous shift parameter in this case.

Both figure 4.2 and 4.6 show portions of how the Mat lab source codes are being presented out.

Figure 4.2 shows the “Pursuit” Mat lab coding.

34 Figure 4.2 “Pursuit” Mat lab coding.

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Figure 4.6 shows the “Wave front Dictionary” Mat lab coding.

Figure 4.6 “Wave front Dictionary” Mat lab coding.

From the “Pursuit” Mat lab coding (figure 4.2), line 17 to 22 show how both the dictionary

vector, α and the continuous shift parameter, T are being generated at random for the wave front

signal.

35

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

From the “Wave front Dictionary” Mat lab coding (figure 4.6), line 8 to 22 show how the actual

dictionary is created, by performing a summation calculation on the equation [4.3.2.c].

From the “Wave front Dictionary” Mat lab coding (figure 4.6), line 24 to 28 shows how the

dictionary is being orthonormalized.

(4.4) Matching Pursuit Algorithm Figure 4.7a and 4.7b show the “Pursuit Projection” Mat lab coding, in which the Matching

Pursuit operations are actually being carried out.

Figure 4.7a “Pursuit Projection” Mat lab coding.

36

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Figure 4.7b “Pursuit Projection” Mat lab coding.

To give the reader a better idea on how the whole “Pursuit Projection” Mat lab coding

works, figure 4.8 shows the pseudo code for the implementation of the Matching Pursuit

Algorithm.

37

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Spectrum = Pursuit_ Projection (F, D, threshold) where, F = Input signal. D = Dictionary. Threshold = threshold for error tolerance specified by User.

Figure. 4.8

Take note

the array s

R =F For i =1…. Numbers of dictionary elements. n = arg max ⟨ R , φj ⟩ spectrum (n) = ⟨ R , φbest_element ⟩ R = R - ⟨ R , φbest_element ⟩ φbest_element if ||R || / ||F|| ≤ t stop end (if) end (for)

that the “spectrum (n)” listed in the above pseudo code refers to the “nth” element of

pectrum.

38

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Chapter5

(5.1) Discussion on Simulation Results In this chapter, simulation results will be presented based on both the Resonance Dictionary

(RD) and the Wave front Dictionary (WD). Altogether, there will be eight different simulation

results. Four results will be based on RD and the other four will be based on WD. A short

discussion for each of the simulated results will also be included.

Section 5.2 will cover the simulated results on the RD. There will be four sets of results

presented. The first two results will be based on real data for the resonant frequencies being

provided into the program. The other two results will be based on simulated frequencies being

used. During wave matching processes, different error tolerances will be set for these four

results, to further illustrate the accuracy between the actual and the approximate signals’

comparison.

39

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Section 5.3 will cover the simulated results on the WD. There will also be four set of results

presented. The first two results will be based on providing the actual dictionary vectors (α), the

continuous shift parameters (T ) and the time parameter (t), into the program. The other two

results will be based on the generation of random numbers of dictionary vectors (α), the

continuous shift parameters (T ) and the time parameter (t) being used. Different error tolerances

will also be used for these four sets of results.

(5.2) Resonance Dictionary (RD) In this section, the results based on the RD will be shown.

(5.2.1) Natural Frequencies

i) For the first result, an input signal consisting of ten resonant frequencies components will be

input into the Mat lab coding. The functions of this input signal are as shown in Table 5.1:

N Sn

1 -0.0828 + j0.9251

2 -0.1212 + j1.912

3 -0.1491 + j2.884

4 -0.1713 + j3.874

5 -0.1909 + j4.854

6 -0.2080 + j5.845

7 -0.2240 + j6.829

8 -0.2383 + j7.821

9 -0.2522 + j8.8807

10 -0.2648 + j9.800 Table 5.1

40

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Figure 5.1 show the result obtained using zero error tolerance.

Figure 5.1 Resonant Frequencies used when 0 error tolerance is set

As shown from the result, both the actual and the approximate signals are well matched properly.

The number of samples (i.e. also known as the Dictionary elements) displayed in this case is 10.

In this case, all of the 10 samples are being matched properly to the dictionary. Thus, the

software program will generate a whole list of fully displayed spectrum from the result.

41

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

ii) For the second result, the same ten resonant frequencies are still being used. The only

difference in this case however, is that the error tolerance is set to 0.2. Figure 5.2 shows the

result obtained using 0.2 error tolerance.

Figure 5.2 Resonant Frequencies used when 0.2 error tolerance is set

As shown from the result above, both the actual and the approximate signals are not as well

matched as the first result. Due to the 0.2 error tolerance, not all of the 10 samples are matched to

the dictionary. Thus, only partial spectrum will be generated from the result.

42

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(5.2.2) Simulated Frequencies

i) For the third result, simulated frequencies are being used. The Mat lab program will generate

a series of random frequencies for the Resonance signal, depending on the number of the

samples used. Figure 5.3 shows the third result. In this case, the number of samples used will be

20 and the error tolerance is set to zero.

Figure 5.3 Simulated Frequencies used when 0 error tolerance is set

As shown from the third result, both the actual and the approximate signals are being well

matched properly. Out of the 20 samples, all of them are being matched efficiently to the

dictionary.

43

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

ii) For the fourth result, the same number of samples is still being used (i.e., 20). However, the

error tolerance is set to 0.1. Figure 5.4 shows the result.

Figure 5.4 Result shown when 0.1 error tolerance is set

As shown from the result, both the actual and the approximate signals not as well matched as the

third result. Not all of the samples will be matched efficiently to the dictionary.

44

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

(5.3) Wave front Dictionary (WD) In this section, the results for the WD will be shown.

(5.3.1) Results

i) The first result is based on the following settings:

The dictionary vectors (α) = 2; 3; 5

The time parameter (t) = 3; 5; 7

The continuous shift parameters (T ) = 5; 7; 9

Number of samples = 30

The error tolerance = 0

Figure 5.5 Result shown when zero error tolerance is set

45

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

As shown from the result, both the actual and the approximate signals are pretty matched. When

zero tolerance is being set, all of the 30 samples are matched accordingly to the dictionary.

ii) The second result shown is based on the same parameters setting as that similar to the first

result. However, the error tolerance is set to 0.2 this time round. Figure 5.6 shows the result.

Figure 5.6 Result shown when 0.2 error tolerance is set

As shown from the results, both the actual and the approximate signals are not as well matched

as the first one. Apparently, this is due to fact that the error tolerance is set to 0.2, instead of the

usual zero error tolerance, as that in the first result. Not all of the samples arematched to the

dictionary in this case. 46

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

iii) In the third result, the number of samples used is 40 and the error tolerance is set to zero.

The rest of the other parameters (i.e. α, t and T ) are randomly being generated by the Mat lab

coding. Figure 5.7 shows the result.

Figure 5.7 Result shown when zero error tolerance is set

As shown from the result, both the actual and the approximate signals are pretty matched. All the

40 samples are being matched properly to the dictionary.

47

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

iv) In the fourth result, the number of samples remains as 40. However, the error tolerance is set

to 0.3. The α, t and T parameters are still being generated randomly. Figure 5.8 shows the result.

Figure 5.8 Result shown when 0.3 error tolerance is set

As shown from the result, both the actual and the approximate signals are not as well matched as

the previous result. Not all of the 40 samples are being matched properly to the dictionary.

48

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

It is quite apparent that as the error tolerance increases, the number of the samples used

decreases. This could be due to the fact that the error tolerance setting is the actual threshold

setting for the comparison of both the actual and the approximate signals. Usually, this threshold

setting will fall in between the range of 0 to 1. Anything beyond that, the wave matching pursuits

will not be perform correctly.

In other words, as the error tolerance increases, the wave matching between the actual signal and

the approximate signal deteriorates.

49

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

Chapter 6

(6.0) Conclusions The aim of the Wave-based Matching Pursuits algorithm is meant mainly for target identification

purposes. Target identification would refer to the representation of any scattered waveform

information. This scattered waveform information will be presented compactly, in terms of the

different features that can be linked to the target.

For most electromagnetic scattering problems, the wave physics can be classified in terms of

Resonance, Wave fronts and Chirps. For the Wave front Dictionary, the knowledge for the

incident waveform, the dictionary vectors and the continuous shift parameters for the Wave front

signals are important. In the case of Resonance Dictionary, the knowledge on the possible

Resonant frequencies plays an important part in the generation of the resonance signal.

50

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

From the above results collected for both the Resonance and the Wave fronts signals, it is quite

apparent that as the error tolerance increases, the number of samples used decreases.

This usually happens when not all the samples are being used in the Wave Matching

Pursuit process. Hence, when both the actual and the approximate signals are not well

matched, variations of signals will be displayed on the same result. In any case however, if the

signals are totally well matched, only a clear distinct signal will be showed.

Some of the further recommendations that could be looked into are the development of the Chirp

Dictionary. For the present software program, the user will have to prompt the program which

dictionary (i.e. Resonance and Wave front) to use. Improvements could be made on the program

in the future, to allow both the Resonance and the Wave front Dictionaries to be incorporated

simultaneously together. By doing so, the software program will be able to better interpret the

the early times of the incoming signal for the Wave front and the late times of the same incoming

signal for the Resonance.

For the results presented here, all computations were computed efficiently on a Pentium personal

computer, with the help of the Mat lab software program. All these results were based on

computer simulations.

51

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

REFERENCES

[1]. Burns J.W., Subotnic N.S., “Adaptive decomposition in electromagnetics” Ch 10 in

Frontiers in Electromagnetics eds D.H. Werner & R. Mittra , IEEE Press, 2001.

[2]. I. Daubechies, “Time-frequency Localization Operators: A Geometric Phase Space

Approach,” IEEE Trans. Inf. Th., 34 P.605 April 1988.

[3] R. R. Coifman and M. V. Wickerhausen, “Entropy-based Algorithms for Best-basis

Selection,” IEEE Trans. Inf. Th., 38 P.753 February 1992.

[4] S. Chen and D. Donoho “Basic Pursuits,” Tech. Rep., Stanford University, CA, 1995.

[5] S. Mallet and Z. Zhang , “Matching Pursuits with Time Frequency Dictionaries,” IEEE

Trans. Sign. Proc., 41 pp.3397-3415, December 1993.

[6] McClure, M.R., Carin L., “Matching Pursuits with a Wave-Based Dictionary”, IEEE Trans.

on Signal Processing, Vol 45, No.12, pp 2912-2927, December, 1997.

[7] D. Gabor, Theory of communication, J. Inst. Elect. Eng., 93 (1946), pp. 429-457.

[8] H. Shirai and L. B. Felsen, “Wavefront and Resonance of scattering by a perfectly

conducting flat strip,” IEEE Trans. Antennas Propagat., vol. AP-34, pp. 1196-1207, Oct.

1986.

i

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

[9] C. E. Baum, “The singularity expansion method,” in Transient Electromagnetic Fields, L. B.

Felsen, Ed. New York: Springer-Verlag, 1976.

[10] E.Heyman and L. B. Felsen, “A wavefront interpretion of the singularity expansion

method,” IEEE Trans. Antennas Propagat., vol. AP-33, pp. 706-718, July 1985.

[11] K-M. Chen, D. P. Nyquist, E. J. Rothwell, L. L. Webb, and B. Drachman, “Radar target

discrimination by convolution of radar return with extinction-pulses and single-mode

extraction signals,” IEEE Trans. Antennas Propagat., vol AP-34, pp.896-904, July 1986.

[12] S. S. Chen, D. L. Donoho and M. A. Saunders,” Atomic Decomposition by Basis Pursuit,”

2001 Society for Industrial and Applied Mathematics, SIAM REVIEW Vol. No. 1,

pp. 129-159

[13] G. Davis, S. Mallet and Z. Zhang, “Adaptive Time-Frequency Decompositions with

Matching Pursuits,” New York University, Courant Institute 251 Mercer Street, New York,

NY 10012.

[14] A. J. Poggio, M. L. Van Blaricum, E. K. Miller, and R. Mittra, “Evaluation of a processing

technique for transient data,” IEEE Trans. Antennas Propagat., vol. AP-26, pp. 165-173,

Jan. 1978.

[15] C. E. Baum, “On the singularity expansion method for the solution of electromagnetic

interaction problems,” Air Force Weapons Lab. Interaction Notes, no. 88, 1971.

ii

ADAPTIVE DECOMPOSITION IN ELECTROMAGNETICS

[16] R. A. Atles, “Sonar for generalized target description and its similarity to animal echo-

location systems,” J. Acoust. Soc. Amer., vol. 59, pp. 97-105., Jan. 1976.

[17] M. P. Hurst and R. Mittra, “Scattering centre analysis via Prony’s method,” IEEE Trans.

Antennas Propagat., vol. AP-35, pp. 986-988, Aug. 1987.

[18] E. K. Walton, “Comparison of Fourier and maximum entropy techniques for high-resolution

scattering studies,” Radio Sci., vol. 22, pp. 350-356, May/June 1987.

[19] W. M. Steedly and R. L. Moses, “High resolution exponential modeling of fully polarized

radar returns,” IEEE Trans. Aerosp. Electron. Syst, vol. 27, pp. 459-469, May 1991.

[20] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ:

Prentice-Hall, 1973.

[21] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989.

[22] M. McClure, R. B. Qiu and L. Carin “On the superresolution identification of observables

from swept-frequency scattering data”, IEEE Trans. Antennas Propagat. Vol. 45. No. 4

April 1997.

[23] Leon W. Couch II “Digital and Analog Communication Systems (Sixth Edition)”, Chapter 2

Signal and Spectra, Example 2.5, pg 54. 2001, 1997 by Prentice-Hal, Inc.

[24] Erwin Kreyszig, Advanced Engineering Mathematics, 8th Edition: Wiley, 1999.

iii

APPENDICES

Software Source Codes Dictionary Matlab File (dictionary.m)

function D = dictionary(length)

% The first half of the matrix consists of all the cosines functions.

% Each row of this matrix indicates a cosine frequency at different time interval

row_index = 1;

for ir = 1 : length/2 + 1

for ic = 1 : length

t = ic -1; % start from time zero

frequency = 2 * pi * (ir-1) /length ; % compute the frequency for that row

D(row_index,ic) = cos (frequency * t ); % compute the phase for the particular time and

frequency

end

row_index = row_index + 1;

end

% The 2nd half of the matrix consists of all the sines functions.

% Each row of this matrix indicates a sine frequency at different time interval

for ir = 1 : length/2

for ic = 1 : length

t = ic -1; % start from time zero

frequency = 2 * pi * ir /length ; % compute the frequency for that row

D(row_index,ic) = sin (frequency * t ); % compute the phase for the particular time and

frequency

end

row_index = row_index + 1;

end

Pg 1

% normalise the matrix by taking each row and divide by its magnitude

[max_row, max_col] = size(D);

for ir = 1 : max_row

D(ir , :) = D(ir , : )/ sqrt (D(ir, :)* D(ir, :)');

end

Pursuit Matlab File (pursuit.m)

function [F_approx, spectrum] = pursuit(F,threshold, dictype, s);

L = length(F);

if dictype == 1

[D] = dictionary(L);

elseif dictype == 2

rand('state',1);

factor = 23;

s = rand(factor*L,1) + i * rand(factor*L,1);

%random complex no.s

[D] = resonance_dictionary(L,s);

D = orthogonalise_qr(D);

elseif dictype == 3

h = rand(L,1);

alpha = [-6.5,-6,-5.5,-5,-4.5,-4,-3.5,-3,-2.5,-2,-1.5,-1,-0.5,0,0.5,1,1.5,2,2.5,3,3.5,4,4.5,5,6,6.5];

tau = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29];

D = wavefront_dictionary(L,h,tau,alpha);

end

Pg 2

spectrum = pursuit_projection(F,D,threshold);

% reconstruct

F_approx = zeros(1,L);

[dictionary_size, void] = size(D);

for i = 1:dictionary_size

F_approx = F_approx + spectrum(i) * D(i,:);

end

Pursuit Projection Matlab File (pursuit_projection.m)

function spectrum = pursuit_projection(F,D,threshold);

[max_row,max_column] = size(D);

spectrum = zeros(max_row , 1);

R = F ;

for i = 1 : max_row

% find the next best matching dictionary element:

abs_max_inner_product = 0;

for row = 1 : max_row

inner_product = D( row , :)* R' ;

if (abs(inner_product) > abs_max_inner_product)

max_inner_product = inner_product;

abs_max_inner_product = abs(max_inner_product);

best_element = row ; % the position of the coresponding dictionary element

end

end

Pg 3

% now we have: best_element = argmax_j <R , D(j,:)>

spectrum( best_element) = max_inner_product;

R = R - max_inner_product * D(best_element , :) ;

if norm(R) / norm(F) <= threshold

disp(['m was chosen to be ' num2str(i)]);

break

elseif i == max_row

disp('Used the full dictionary');

end

D(best_element,:) = zeros(1,max_column); % don't choose this element again

end

Test Pursuit Matlab File (test pursuit.m)

function test_pursuit(L, threshold);

t = 0:L-1;

F = 2*tan(t/L)+ 3*tan(2*t/L)+ 3*tan(4*t/3*L)+ 4*tan (3*t/5*L);

[F_approx, spectrum] = pursuit(F, threshold, 1 )%, s)

% 1 refer to fourier; 2 refers to resonance; 3 refers to wavefront

plot(F, 'b');

hold on;

title('original = blue, reconstruced = red');

plot(F_approx,'r--');

xlabel('Time interval')

ylabel('Amplitude')

Pg 4

Orthogonalize QR Matlab File (orthogonalize qr.m)

function D2 = orthogonalise_qr(D);

[D2, R] = qr(D');

indices = find(sum(abs(R),2));

D2 = D2(:,indices)';

Resonance Dictionary Matlab File (resonance dictionary.m)

function D = resonance_dictionary (L,s);

M = length(s);

D = zeros(M,L);

for m = 1:M

for t = 1:L

D(m,t) = real(exp(s(m)*t));

end

end

% normalise the matrix by taking each row and divide by its magnitude

for m = 1:M

D(m,:) = D(m,:)/ sqrt (D(m,:)* D(m,:)');

end

Pg 5

Wavefront Dictionary Matlab File (wavefront dictionary.m)

function D = wavefront_dictionary (L, h, tau, alpha);

n_tau = length(tau);

n_alpha = length(alpha);

D = zeros(n_tau*n_alpha,L);

h_hat = fft(h);

r = 1;

for i_tau = 1:n_tau

for i_alpha = 1:n_alpha

for t = 1:L

for w = 1:L

D(r,t) = D(r,t) + (w * exp(i*pi/2))^alpha(i_alpha)*h(w)*exp(i*w*(t-tau(i_tau)));

end

end

r = r+1;

end

end

D = real(D);

% normalise the matrix by taking each row and divide by its magnitude

for m = 1:n_tau*n_alpha

D(m,:) = D(m,:)/ sqrt (D(m,:)* D(m,:)');

end

Pg 6