University of Moratuwa 
7
, ' \ r V 
WAVELET PACKET BASED 
ANTENNA RADIATION PATTERN 
ANALYSER 
Submitted in partial fulfilment for the degree of Master of Engineering in 
Electronics & Telecommunication Engineering 
University of Moratuwa 
82720 
Wipula Wimalshanthi 
January 2005 
8 2 7 2 0 
Thesis 
&Q720 
The work presented in this dissertation has not been submitted for the 
fulfillment of any other degree. 
W Wimalshanthi 
(Candidate) 
Prof. J A K S Jayasinghe 
(Supervisor) 
ABSTRACT 
Analysis of antenna radiation patterns, especially in respect of antennas 
with complex shapes and sizes, require the adoption of numerical 
methods of obtaining solutions to electro-magnetic equations. Method of 
Moments (MoM) being one of such proven methods, still poses the 
problem of manipulation of large matrices. 
Objective of this exercise is to investigate the possibility of using wavelet 
transform techniques in obtaining fast solutions for the matrix equations 
resulting from MoM method. Specific attention has been given to 
Discreet Wavelet Transform (DWT) and Discreet Wavelet Packet (DWP) 
transform methods in order to sparsify the large impedance matrices 
generated by MoM method. 
Wavelet transform being a recently developed technique, the 
mathematical background and related theoretical aspects have been 
illustrated prior to analysing several examples of thin wire centre fed 
antennas. 
Examples have been selected to demonstrate effective adaptation of 
Discrete Wavelet Transform and Discrete Wavelet Packet Transform 
techniques in obtaining solutions for the matrix equations in the analysis 
of thin wire antennas. Comparisons have been made with the 
conventional method of solving these matrix equations illustrating the 
improvement in the computation times as a result of sparsification of 
matrices using Wavelet transform methods with the extensive assistance 
of MatLab Wavelet Tool Box. 
Having indicated the advantages of wavelet transform techniques over the 
conventional methods of solving large matrix equations, several 
suggestions have been made towards optimising the results obtained to be 
taken up as further research work. 
i 
• 
LIST OF FIGURES 
No. Title Page 
2.1 Cylindrical conductor of radius a, with current density 02 
measured along its perimeter K(Am') 
2.2 Conductor replaced by current filament / = 2mK (A) 03 
at a distance a from the z-axis 
2.3 Source point on current filament with field dE, at a 04 
distance r on the z-axis 
2.4 Segmentation of wire-antenna 07 
2.5 Illustration of Fourier Analysis 09 
2.6 Illustration of Short-Time Fourier Analysis 10 
2.7 Illustration of Wavelet Analysis 11 
2.8 Comparison of Wavelet Analysis with other Signal 11 
Analysis techniques 
2.9 Comparison of a sine wave and a wavelet 12 
2.10 Illustration of "Mexican hat' wavelet 12 
2.11 Illustration of Continuous Wavelet Transform 13 
2.12 Scaling of a Wavelet 14 
2.13 Illustration of shifting of Wavelet function 14 
2.14 Step-by-Step illustration of CWT 15 
2.15 A plot of the Wavelet Transform Coefficients vs Time 16 
2.16 Decomposition of Signal in DWT 17 
2.17 Single stage DWT of a Noisy Sine wave 18 
2.18 Wavelet Decomposition Tree 18 
2.19 Two Dimensional DWT Decomposition 19 
2.20 Wavelet Reconstruction in IDWT 20 
2.21 Decomposition and Reconstruction Filters 20 
2.22 Reconstruction of First Level Approximation (A i) 21 
2.23 Reconstruction of First Level Detail (D|) 21 
2.24 Reconstruction of Signal from Multilevel Details and 22 
Approximations 
2.25 Two-Dimensional Inverse DWT Reconstruction 23 
2.26 Wavelet Packet Decomposition Tree 24 
3.1 Multistage Decomposition and Reconstruction of the 25 
signal X(n) 
4.1 Segmentation of short thin-wire antenna 29 
4.2 Current distribution plots of Example (4.1.a) using dbl 31 
Wavelet with moderate thresholding 
4.3 Current distribution plots of Example (4.1 .b) using dbl 33 
Wavelet with moderate thresholding 
4.4 Current distribution plots of Example (4.1 .c) using db2 35 
Wavelet with excessive thresholding 
4.5 Segmentation of Thin-wire antenna 37 
4.6 Current distribution plots of Example (4.2.a) 39 
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ft 
LIST OF FIGURES (Continued ....) 
4.7 Comparison of computation times between conventional 40 
and DWP methods 
4.8 Current distribution plots of Example (4.2.b) 41 
4.9 Current distribution plots of Example (4.2.c) 43 
B.l Graphical illustration of 'dbl ' and 'db2' wavelets 67 
B.2 Graphical illustration of four filters of 'dbl ' wavelet 67 
B.3 Graphical illustration of four filters of 'db2' wavelet 68 
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LIST OF ABBREVIATIONS USED 
AIM Adaptive Integral Method 
C W T Continuous Wavelet Transform 
D W P Discrete Wavelet Packets 
D W T Discrete Wavelet Transform 
FMM Fast Multipole Method 
FT Fourier Transform 
IDWT Inverse Discrete Wavelet Transform 
IML Impedance Matrix Localization Method 
MoM Method of Moments 
M V M Matrix Vector Multiplication 
STFT Short Time Fourier Transform 
W T Wavelet Transform 
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CONTENTS 
Abstract i 
1. Introduction 01 
2. Relevant Concepts 02 
2.1 Method of Moments Developed for a Wire Antenna 02 
2.2 Wavelet Transform 08 
2.2.1 Introduction to Wavelets 08 
• 2.2.2 Wavelet Analysis in Comparison with several other 09 
Signal Analysis methods [7] 
2.2.2.1 Fourier Analysis 09 
2.2.2.2 Short-Time Fourier Analysis 10 
2.2.2.3 Wavelet Analysis 10 
2.2.3 Wavelet Transform techniques 11 
2.2.3.1 An Example of a Wavelet 12 
2.2.3.2 Continuous Wavelet Transform [7] 13 
2.2.3.2.1 Scaling 14 
2.2.3.2.2 Shifting 14 
2.2.3.2.3 Steps involved in Continuous Wavelet 15 
Transform (Figure 2.14) 
2.2.3.3 Discrete Wavelet Transform [7] 16 
2.2.3.3.1 Multiple-level Decomposition 18 
• 2.2.3.3.2 2-Dimensional DWT 19 
2.2.3.3.3 Wavelet Reconstruction 19 
2.2.3.3.3.1 Reconstruction Filters 20 
2.2.3.3.3.2 Reconstructing Approximations 20 
and Details 
2.2.3.3.4 2-Dimensional Inverse DWT 23 
2.2.3.4 Discrete Wavelet Packet Transform 24 
3. Mathematical Analysis 25 
3.1 Application of Discrete Wavelet Transform in Solving Large 25 
Matrix Equations 
4. Numerical Examples 29 
• 4.1 Computation of Current Distribution along a Centre-fed 29 
thin-wire Antenna 
4.1.1 Using MoM method and assuming uniform current 29 
distribution per segment 
4.1.2 Using MoM method and assuming piecewise uniform 37 
current distribution per segment for improved accuracy 
5. Discussion, Conclusions and Suggestions for further research work 45 
5.1 Discussion 45 
5.2 Conclusions and Suggestions for future research work 46 
5.3 Acknowledgements 47 
v 
CONTENTS (Continued ....) 
List of References 
ANNEX - (A) MatLab Program Listings 
ANNEX - (B) Description of Specific MatLab Commands 
and Wavelets used 
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