estimation of direction of arrival for adaptive …

ESTIMATION OF DIRECTION OF ARRIVAL FOR

ADAPTIVE BEAMFORMING

By

FAWAD ZAMAN Reg. No. 31-FET/PhD (EE)/F-09

A dissertation submitted to I.I.U.I in partial fulfillment of the

Requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Electronic Engineering

Faculty of Engineering and Technology

INTERNATIONAL ISLAMIC UNIVERSITY

ISLAMABAD 2013

ii

Copyright © 2013 by Fawad Zaman

All rights reserved. No part of the material protected by this copyright notice may

be reproduced or utilized in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval

system, without the permission from the author.

iii

Dedicated to my loving parents,

Whose dreams for me and prayers for me have always kept me encouraged

and directed towards my goals, despite so many hardships in life.

iv

CERTIFICATE OF APPROVAL

Title of Thesis: Estimation of Direction of Arrival for Adaptive Beamforming

Name of Student: FAWAD ZAMAN

Registration No: 31-FET/PHDEE/F-09

Accepted by the Department of Electronic Engineering, INTERNATIONAL ISLAMIC

UNIVERSITY, ISLAMABAD, in partial fulfillment of the requirements for the Doctor of

Philosophy Degree in Electronic Engineering.

Viva Voce Committee

Prof. Dr. Aqdas Naveed Malik

Dean, Faculty of Engineering & Technology

International Islamic University, Islamabad

Dr. Muhammad Amir

Chairman, Department of Electronic Engineering

International Islamic University, Islamabad

Prof. Dr. Abdul Jalil (External Examiner-I)

Professor, Pakistan Institute of Engineering and

Applied Science Nilore, Islamabad

Dr. Muhammad Usman (External Examiner-II)

Principle Scientist, AWC, Wah Cantt.

Dr. Ihsan Ul Haq (Internal Examiner)

Assistant Professor, Department of Electronic Engineering

International Islamic University, Islamabad.

Prof. Dr. Ijaz Mansoor Qureshi (Supervisor)

Department of Electrical Engineering

Air University, Islamabad

Friday, 26th Dec, 2013

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DECLARATION

I hereby declare that this research and simulation, neither as a whole nor as a part thereof,

has been copied out from any source. It is further declared that I have developed this

research, simulation and the accompanied report entirely on the basis of my personal effort

made under the guidance of my supervisor and teachers.

If any part of this report to be copied or found to be reported, I shall stand by the

consequences. No portion of this work presented in this report has been submitted in

support of any application for any other degree or qualification of this or any other university

or institute of learning.

Fawad Zaman

Reg# 31-FET-PhDEE/F-09

vi

ABSTRACT

Estimation of Direction of Arrival (DOA) of sources is a basic component of adaptive

beamforming. The objective is to steer the main beam in the desired direction, while nulls

are allocated in the direction of unwanted signals. It is an area of research which has got

direct applications in radar, sonar, seismic exploration, mobile communication etc.

Besides DOA estimation, amplitude, frequency and range are the other important

parameters that need to be estimated.

This dissertation is a contribution towards the above mentioned areas. These contributions

are mainly divided into two parts. In first part, our contribution is to develop efficient

schemes to jointly estimate the amplitude and DOA of the far field sources. Specifically,

we have targeted the joint estimation of amplitude and 2-D DOA (elevation & azimuth

angles) of far field sources impinging on 1-L and 2-L shape arrays. In the second part, we

deal with near field sources impinging on uniform linear and centro-symmetric cross

shape arrays. The basic tool applied to estimate these parameters are meta-heuristic or

nature inspired algorithms, which are tailored and trained to solve the problem in hand.

These techniques include Genetic algorithm, Particle swarm optimization, Differential

evolution and Simulated Annealing. In order to improve the performance, the global

search optimizers (meta-heuristic techniques) are hybridized with rapid local search

optimization methods such as Pattern search, Interior point algorithm and Active set

algorithm.

We have used two fitness functions for the far field, as well as, for the near field sources.

Initially, we have used Mean Square Error (MSE) as a performance evaluation criterion.

This fitness function is based on maximum likelihood principle. The second fitness

function is multi-objective, which is the combination of MSE and correlation between

desired and estimated vectors after normalization. Both of the fitness functions are easy to

implement and need a single snapshot to generate the results. They also avoid any

ambiguity among the angles that are supplement to each other. The proposed hybrid

schemes are compared with the individual responses of these algorithms and also with the

traditional classical techniques available in the literature. The comparison parameters are

chosen as the estimation accuracy, convergence, robustness against noise, MSE and

proximity effects. To get the near optimum statistics, a large number of Monte-Carlo

simulations are carried out for each scheme.

vii

LIST OF PUBLICATIONS

1. Fawad Zaman, I.M. Qureshi, ―5D parameter estimation of Near-field sources using

Hybrid Evolutionary Computational Technique,‖ The Scientific World Journal, Paper-

ID- 310875, 2013.

2. Fawad Zaman, I.M.Qureshi, Fahad Munir and Z.U. Khan, ―4D parameters

estimation of plane waves using swarming intelligence,‖ Chinese Physics B,

Paper-ID 132170, 2013.

3. Fawad Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, ―An Application of Artificial

Intelligence for the Joint Estimation of Amplitude and Two Dimensional Direction of

Arrival of far field sources using 2-L shape array,‖ International Journal of Antennas and

Propagation, Article ID 593247, 10 pages, Volume 2013.

4. Fawad Zaman, Ijaz Mansoor Qureshi, A. Naveed, Junaid Ali Khan and Raja

Muhammad Asif Zahoor ―Amplitude and Directional of Arrival Estimation: Comparison

between different techniques,‖ Progress in Electromagnetic research-B (PIER-B), Vol.

39, pp.319-335, 2012.

5. Fawad Zaman, I. M. Qureshi, A.Naveed and Z. U. Khan, ―Real Time Direction of

Arrival estimation in Noisy Environment using Particle Swarm Optimization with single

snapshot,‖ Research Journal of Engineering and Technology (Maxwell Scientific

organization), Vol. 4(13) pp. 1949-1952, 2012.

6. Fawad Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, ―joint estimation of amplitude,

direction of arrival and range of near field sources using memetic computing‖ Progress

in Electromagnetic research-C (PIER-C) ,Vol,31, pp. 199-213, 2012.

7. Fawad Zaman, Shahid Mehmood, Junaid Ali Khan and Ijaz Mansoor Qureshi ―joint

estimation of amplitude and direction of arrival for far field sources using intelligent

hybrid computing‖, Research Journal of Engineering and Technology (Maxwell

Scientific organization), pp. 3723-3728, 2013.

8. Fawad Zaman, J. A. Khan, Z.U.Khan, I.M.Qureshi, ―An application of hybrid

computing to estimate jointly the amplitude and Direction of Arrival with single

snapshot,‖ IEEE, 10th-IBCAST, pp-364-368, Islamabad, Pakistan, 2013.

9. Fawad Zaman, Shafqat Ullah Khan, Kabir Ashraf and I.M Qureshi, ―An application of

hybrid differential evolution to 3-D source localization‖ Accepted in IEEE, 11th,

IBCAST, 2013.

10. Ayesha Khaliq, Fawad Zaman, Kiran Sultan, I.M.Qureshi, 3-D near field source

localization by using hybrid Genetic Algorithm, ―Research Journal of Engineering and

Technology (Maxwell Scientific organization)”, pp, 4464-4469, 2013.

viii

11. Y.A.Shiekh, Fawad Zaman, I.M.Qureshi, A.U. Rehman, ―Amplitude and Direction of

Arrival Estimation using Differential Evolution,‖ IEEE, ICET, pp, 45-49, 2012.

12. Zafar Ullah Khan, Fawad Zaman, A. Naveed Malik, I M Qureshi, “Comparison of

adaptive beamforming algorithm robust against direction of arrival mismatch‖ Journal of

space technology (JST), vol.1, pp. 28-31, 2012.

13. A.U. Rehman, Fawad Zaman, Y.A.shiekh, , I.M.Qureshi, ―Null and sidelobe

adjustment in damaged antenna array,‖ IEEE, ICET, Islamabad, pp-21-24, 2012.

14. Y.A.Shiekh, Fawad Zaman, I.M.Qureshi, A.U. Rehman, ―Azimuth and

elevation angle of arrival estimation using differential evolution with single

snapshot,‖ Accepted in Recent development on signal processing (RDSP),

Istanbul, Turkey. 2013.

15. Shafqat Ullah Khan, I.M Qureshi, Fawad Zaman, Aqdas Naveed and Bilal Shoaib,

―Correction of faulty sensors in Phased Array Radars using Symmetrical Sensor Failure

Technique and Cultural Algorithm with Differential Evolution,‖ The Scientific World

Journal, Paper ID-852539.

16. S.Ullah Khan, I. M.Qureshi, Fawad Zaman, A. Naveed, ―Null placement and sidelobe

suppression in failed array using symmetrical element failure technique and hybrid

heuristic computation‖, PIER-B, pp, 165-184, vol 52, 2013.

17. Shahid Mehmood, Z.U Khan and Fawad Zaman ―Performance Analysis of the

Different Null Steering Techniques in the Field of Adaptive Beamforming,‖ Research

Journal of Engineering and Technology, pp, 4006-4012, vol, 2013.

18. Z. U. khan, A. Naveed, I. M. Qureshi, Fawad Zaman “Independent Null Steering by

decoupling complex weights‖ IEICE, Electron express, vol. 8, no. 13, pp. 1008-1013,

July 10, 2011.

19. Z.U. Khan, A. Naveed, I.M.Qureshi and Fawad Zaman, “Robust Generalized Sidelobe

Canceller for Direction of Arrival mismatch,‖ Archives Des Sciences, Vol 65, pp. 483-

497, 2012.

20. Z.U. Khan, A.Naveed, M.Safeer, Fawad Zaman, ―Diagonal Loading of Robust General-

Rank Beamformer for Direction of Arrival mismatch,‖ Accepted for publication in

―Research Journal of Engineering and Technology (Maxwell Scientific organization),

pp, 4257-4263, 2013.

21. S. Azmat Hussain, A. Naveed Malik, I. M. Qureshi, Fawad Zaman ―Spectrum sharing

in cognitive radio prop up the Minimum transmission Power and Maxi-min SINR

stratagem‖, Research Journal of Engineering and Technology, pp,4289-4296, 2013.

ix

22. Shafqat Ullah Khan, I.M Qureshi, Fawad Zaman, and Abdul Basit ―Application

of firefly algorithm to fault finding in linear arrays antenna,‖ World Applied

Science Journal, (In Press) paper ID- WASJ-2013-1387.

SUBMITTED PAPERS

1. Fawad Zaman, I.M.Qureshi, M. Zubair, and Z.U. Khan, ―Multiple target

localization with bistatic radar using heuristic computational intelligence‖ Paper

key: 13052307, PIER, 2013.

2. Fawad Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, ―Hybrid Differential

Evolution and hybrid Particle swarm optimization for the joint estimation of

amplitude and Direction of Arrival of far field sources using L shape arrays,”

Iranian journal of Science and Technology, Transaction of Electrical engineering,

Paper key: 1311-IJSTE, 2013.

3. Atif Elahi, I.M Qureshi, Fawad Zaman and Fahad Munir, ―An application of

Golay Complementary Sequences to Channel Estimation of OFDM System,‖ The

Scientific World Journal, Paper ID- 275781, 2013.

4. Shafqat Ullah Khan, I.M Qureshi, Fawad Zaman, Aqdas Naveed, ‗Computationally

Efficient Method for Finding the Faulty Element in Linear Arrays Antenna,‖ The

Scientific World Journal, Paper ID-681038.

5. Abdul Basit, I. M. Qureshi, Wasim Khan and Fawad Zaman, ―Design of a novel hybrid

cognitive phased array radar with transmit beamforming,‖ PIER, 2013.

6. Azmat Hussain Shah, Aqdas Naveed, and Fawad Zaman, ―Interference control in

cognative radio networks,‖ International Journal of distributed sensor networks, Paper ID-

862726.

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ACKNOWLEDGEMENTS

All thanks to Allah Almighty Who gifted me with knowledge, skills and strength to

carry out this tedious task of research successfully. I am highly indebted to my parents,

teachers, colleagues and friends who helped me accomplish this task.

This research would not have been possible without the encouragement and

scholarly guidance of my supervisor Prof. Dr. Ijaz Mansoor Qureshi. His personality and

scholarly stature has always been an inspiration. Dr. Qureshi has always been a fatherly

figure for me during my academic/research carrier, as he always dealt me as his own

child, and always paid special attention to grooming me for carrying out research in my

area of specialization. I cannot express my gratitude for the time he spared for me so

generously, despite his very busy and hectic schedule. His motivating and scholarly

guidance and beneficial suggestions have always remained with me throughout this

whole process of research project.

I owe my special thanks to my foreign evaluators, Dr. Tae Sun Choi and Dr.

Ibrahim Devili, and in-country examiners, Dr. Abdul Jalil and Dr. Muhammad Usman,

for their critical review of this research work and their useful inputs and suggestions.

Here, it would be injustice not to acknowledge the Higher Education Commission (HEC),

Pakistan, generous financial support, as without this support, it would have been very

difficult to carry out this demanding task of research. I am highly obliged to HEC for

awarding me Fellowship in my MS and PhD Degree Programs.

I am really very grateful to my colleagues, Mr. Zafarullah Khan, Syed Azmat

Hussain Shah, Mr. Kabir Ashraf, Mr. Fahad Munir and many others, whose friendly

guidance, inputs and all sort of help were always there for me in all the phases of this

research work. Some very dear friends, Mr. Shahid Mehmood Jammu and Mr. Zulfiqar

Ahmad Chillasi, encouragement and moral support provided me the required spirit to go

for this project of research. Their friendly support was really encouraging, especially in

difficult phases of this project.

I am highly indebted to Prof. Dr. Aqdas Naveed Malik, Dr. Muhammad Amir, Dr.

Ihsan-ul-Haq at the Department of Electronic Engineering, Faculty of Engineering &

Technology, and Mr. Tariq and other administrative staff here at the University who were

always very kind in providing the much needed administrative support.

Last but not the least, I am very grateful to my parents, my younger brother, Mr.

Jawad Zaman, my brothers-in law, Mr. Sardar Naeem, Mr. Usman Khan, Mr Adnan

Khan and other family members, for their affection, love and support that kept me moving

towards my goal. A big thanks to all of them.

(Fawad Zaman)

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TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ vi

List of Publications ............................................................................................................ vii

Table of Contents ................................................................................................................ xi

List of Figures .................................................................................................................. xvii

List of Tables .................................................................................................................... xix

List of Abbreviations ..................................................................................................... xxiii

Chapter 1 .............................................................................................................................. 1

Introduction .......................................................................................................................... 1

1.1 Problem Statement ................................................................................................... 2

1.2 Contributions Of The Dissertation ........................................................................... 3

1.3 Organization Of The Dissertation ............................................................................ 6

Chapter 2 .............................................................................................................................. 8

DOA Estimation Techniques: An Overview ...................................................................... 8

2.1 Data Model............................................................................................................... 9

2.2 DOA Estimation Techniques ............................................................................... 11

2.2.1 Conventional Beamforming Algorithms ......................................................... 11

2.2.1.1 The Conventional Beamformer Method .................................................. 11

2.2.1.2 Minimum Variance Distortionless Response Beamformer ...................... 13

2.2.2 Parametric or Maximum Likelihood Algorithms ............................................ 14

2.2.2.1 Unconditional or Stochastic Maximum Likelihood Algorithm ............... 14

xii

2.2.2.2 Conditional or Deterministic Maximum Likelihood Algorithm .............. 15

2.2.3 Signal Subspace Algorithms ............................................................................ 16

2.2.3.1 Covariance Based Methods ...................................................................... 16

2.2.3.1.1 Multiple Signal Classification Algorithm (MUSIC) ............................ 17

2.2.3.1.2 Estimation of Signal Parameter through Rotational Invariance

Technique (ESPRIT) .............................................................................................. 19

2.2.3.2 Direct Data Domain Method .................................................................... 22

Chapter 3 ............................................................................................................................ 25

Selected Optimization Techniques .................................................................................... 25

3.1 Genetic Algorithm ................................................................................................. 28

3.2 Particle Swarm Optimization ................................................................................. 33

3.3 Differential Evolution ............................................................................................ 36

3.4 Simulated Annealing .............................................................................................. 39

3.5 Pattern Search ........................................................................................................ 41

3.6 Interior Point Algorithm ........................................................................................ 42

3.7 Active Set Algorithm ............................................................................................. 43

Chapter 4 ............................................................................................................................ 45

DOA Estimation Including Amplitude And Frequency Of Far Field Sources .................. 45

Part- 1 ................................................................................................................................. 46

4.1 Data Model ................................................................................................................... 46

4.2 Signal Subspace Dimension ......................................................................................... 46

4.3 Joint Estimation Of 2-D Parameters Using GA-PS ............................................... 47

xiii

4.3.1 Results and Discussions ........................................................................................ 51

4.3.1.1 Estimation Accuracy ...................................................................................... 51

4.3.1.2 Convergence .................................................................................................. 52

4.3.1.3 Robustness ..................................................................................................... 53

4.3.1.4 Comparison with MUSIC and ESPRIT algorithms ....................................... 53

4.4 Joint Estimation Of 2-D Parameters Using PSO-PS................................................. 54

4.4.1 Results and Discussion ......................................................................................... 57

4.4.1.1 Estimation Accuracy ...................................................................................... 57

4.4.1.2 Convergence and MSE .................................................................................. 58

Part-II ................................................................................................................................. 60

4.5 Data Model ................................................................................................................... 61

4.5.1 1-L Shape Array .................................................................................................... 62

4.5.2 2-L Shape Array .................................................................................................... 64

4.6 Joint Estimation Of 3-D Parameters Using GA-PS And SA-PS ............................. 64

4.6.1 Result and Discussions ......................................................................................... 67

4.7 Joint Estimation Of 3-D Parameters Using DE-PS and PSO-PS ............................. 75

4.7.1 Differential Evolution Hybridized With Pattern Search (DE-PS) ........................ 75

4.7.2 Particle Swarm Optimization Hybridized With Pattern Search (PSO-PS) ........... 77


Part- III ............................................................................................................................... 87

4.8 Joint Estimation Of 4-D Parameters Using PSO-PS ......................................... 88

xiv

4.8.1 Data Model............................................................................................................ 88

4.8.2 Particle Swarm Optimization Hybridized With Pattern Search ............................ 91


4.8.3.1 Comparison with PSO, PS and GA-PS .......................................................... 93

4.8.3.1.1 Estimation Accuracy ................................................................................... 94

4.8.3.2 Convergence .................................................................................................. 95

4.8.3.3 Proximity Effect ............................................................................................. 96

4.8.3.3 Performance on Reference Axis .................................................................... 97

4.8.3.4 Comparison with Traditional Technique ....................................................... 97

4.9. Conclusion .................................................................................................................. 99

CHAPTER 5 .................................................................................................................... 100

DOA Estimation Including Range, Amplitude And Frequency Of Near Field Sources . 100

5.1 Data Model ................................................................................................................. 101

Part-I ................................................................................................................................ 104

5.2 Joint Estimation Of 3-D Parameters Using GA-IPA and SA-IPA ........................ 104

5.2.1 Simulation and Results ....................................................................................... 106

5.3 Joint Estimation Of 3-D Parameters Using DE-PS and PSO-PS ............................... 113

5.3.1 Results and Discussion ....................................................................................... 113

Case I ........................................................................................................................... 113

5.3.1.1 Estimation Accuracy: ................................................................................... 114

5.3.1.3 MSE and Convergence ................................................................................ 115

xv

5.3.1.4 DOA Proximity: ........................................................................................... 115

Case II .......................................................................................................................... 116

5.3.1.6 Robustness: .................................................................................................. 117

5.3.1.7 MSE and Convergence: ............................................................................... 117

5.3.1.7 DOA Proximity: ........................................................................................... 118

Case III ......................................................................................................................... 119

5.3.1.8 Estimation Accuracy: ................................................................................... 119

5.3.1.9. Robustness: ................................................................................................. 119

5.3.1.10 MSE and Convergence: ............................................................................. 120

5.3.1.11 DOA proximity: ......................................................................................... 120

Part-II ............................................................................................................................... 121

5.4 Data Model For 4-D Near Field Targets ............................................................. 123

5.5 Joint Estimation Of Amplitude, Range And 2D DOA Using DE-ASA And PSO-ASA

For Bi-Static Radar .......................................................................................................... 125


5.5.1.2 Convergence ................................................................................................ 130

5.5.1.3 Proximity Effects ......................................................................................... 131

5.5.1.4 Estimation Accuracy For DOA On Reference Axis .................................... 132

5.5.1.5 Comparison with Other Techniques Using Root Mean Square Error (RMSE)

.................................................................................................................................. 134

Part-III .............................................................................................................................. 136

5.6 Joint Estimation Of 5D Parameters Using GA-PS And GA-IPA ............................. 136

xvi

5.6.1 Signal Model For 5D Parameters Of Near Field Sources ....................................... 136

5.6.2 GA-PS and GA-IPA ................................................................................................ 138


5.7 Conclusion ................................................................................................................. 147

Chapter 6 .......................................................................................................................... 149

CONCLUSION AND FUTURE DIRECTIONS ............................................................. 149

6.1 Conclusion ................................................................................................................. 149

6.2 Future Directions ....................................................................................................... 151

6.2.1 Tracking Problem .......................................................................................... 152

6.2.2 Main Beam and Null Steering ....................................................................... 152

6.2.3 Noise Consideration ...................................................................................... 152

6.2.4 Array Miss Perfection .................................................................................... 153

References ........................................................................................................................ 154

xvii

LIST OF FIGURES

Fig. 1.1 Schematic diagram for adaptive beamforming along with DOA estimation ... 1

Fig. 2. 1 Signal Model for far field sources ................................................................. 10

Fig. 2. 2 Array Geometry for ESPRIT ......................................................................... 19

Fig. 3. 1 General Flow chart of Evolutionary Algorithms ........................................... 27

Fig. 3. 2 Generic Flow Diagram of Genetic Algorithm ............................................... 32

Fig. 3. 3 Generic Flow Diagram of Particle Swarm Optimization .............................. 35

Fig. 3. 4 Generic Flow diagram of Differential Evolution .......................................... 37

Fig. 3. 5 Generic Flow diagram of Simulated Annealing ............................................ 40

Fig. 3. 6 Generic flow chart for Pattern Search technique. .......................................... 42

Fig. 4. 1 Flow Diagram for Hybrid GA-PS ................................................................. 48

Fig. 4. 2 Convergence rate vs number of sources ........................................................ 53

Fig. 4. 3 MSE vs SNR .................................................................................................. 53

Fig. 4. 4 Generic flow diagram for Hybrid PSO-PS ................................................... 56

Fig. 4. 5 Performance analysis of MSE vs SNR .......................................................... 60

Fig. 4. 6 Geometry of 1-L shape array ......................................................................... 62

Fig. 4. 7 Geometry of 2-L shape array ......................................................................... 62

Fig. 4. 8 Convergence vs number of sources for 2-L shape array at 10 dB noise ....... 71

Fig. 4. 9 Convergence vs number of sources for 1-L shape array at 10 dB noise. ...... 71

Fig. 4. 10 Root- Mean-Square Error vs SNR ............................................................... 74

Fig. 4. 11 Flow chart of hybrid DE .............................................................................. 77

Fig. 4. 12 Convergence Rate Vs Number of sources using 1-L shape array ............... 83

Fig. 4. 13 Convergence Rate Vs Number of sources using 2-L shape array ............... 83

Fig. 4. 14 RMSE vs SNR ............................................................................................. 86

xviii

Fig. 4. 15 Convergence vs number of sources for 2-L shape array at 10 dB noise ..... 96

Fig. 4. 16 Comparison of the frequency estimate ........................................................ 98

Fig. 4. 17 Comparison of the elevation angle estimate ................................................ 98

Fig. 4. 18 Comparison of the azimuth angle estimate .................................................. 99

Fig. 5.1 Array Geometry for near field sources ......................................................... 102

Fig. 5. 2 Mean Square Error vs Signal to Noise ratio ................................................ 111

Fig. 5. 3 MSE vs SNR for 2 sources and 10 sensors ................................................. 115

Fig. 5. 4 MSE vs SNR for 4 sources and 12 sensors ................................................. 117

Fig .5. 5 MSE vs SNR for 4 sources and 14 sensors ................................................. 120

Fig. 5. 6 Schematic Diagram for bistatic radar .......................................................... 122

Fig. 5. 7 Convergence Rate versus Number of sources ............................................. 131

Fig. 5. 8 Elevation angle estimation on reference axis .............................................. 133

Fig. 5. 9 Azimuth angle estimation on reference axis ................................................ 133

Fig. 5. 10 Root Mean Square Error of Elevation angles versus SNR ........................ 134

Fig. 5. 11 Root Mean Square Error of Azimuth angles versus SNR ......................... 135

Fig. 5. 12 Root Mean Square Error of Ranges versus SNR ....................................... 135

Fig. 5. 13 Root Mean Square Error of Amplitudes versus SNR ................................ 135

Fig. 5. 14 Convergence Rate vs number of sources ................................................... 143

Fig. 5. 15 Convergence VS SNR ............................................................................... 145

Fig. 5. 16 Error estimation of the frequencies Vs SNR ............................................. 146

Fig. 5. 17 Error estimation of the Azimuth angles Vs SNR ...................................... 146

Fig. 5. 18 Error estimation of the elevation angles Vs SNR ...................................... 147

Fig. 5. 19 Error estimation of the ranges Vs SNR ..................................................... 147

Fig. 5. 20 Error estimation of the amplitudes Vs SNR .............................................. 147

xix

LIST OF TABLES

Table 4. 1 Parameter settings for GA and PS .............................................................. 51

Table 4. 2 Estimation accuracy for two sources .......................................................... 52

Table 4. 3 Estimation accuracy for three sources ........................................................ 52

Table 4. 4 Comparison with MUSIC and ESPRIT for three sources .......................... 54

Table 4. 5 Comparison with MUSIC and ESPRIT for four sources ............................ 54

Table 4. 6 Amplitudes and DOA estimation of two sources ....................................... 57

Table 4. 7 Amplitude and DOA estimation of 3 sources ............................................. 58

Table 4. 8 MSE and convergence for different numbers of element ........................... 58

Table 4. 9 MSE and convergence of all three schemes for different numbers of

element .................................................................................................... 59

Table 4. 10 MSE and convergence rate for different numbers of element .................. 59

Table 4. 11 parameters setting for SA ......................................................................... 67

Table 4. 12 Estimation accuracy of 2-L shape array for 2 sources .............................. 68

Table 4. 13 Estimation accuracy of 1-L shape array for 2 sources .............................. 69

Table 4. 14 Performance of 2-L type array for 3 sources ............................................ 69




Table 4. 18 Means, Variances and standard deviations at 10 dB noise for different

elevation angles and fixed azimuth angle by using PM with parallel

shape array .............................................................................................. 72


elevation angles and fixed azimuth angle by using GA-PS with 1-L

shape array .............................................................................................. 72

xx


elevation angles and fixed azimuth angle by using GA-PS with 2-L

shape array .............................................................................................. 72

Table 4. 21 Comparison among 2-L shape aray, 1-L shape array and parallel shape

array ........................................................................................................ 74

Table 4. 22 Estimation accuracy of 1-L-shape array for 2-sources ............................. 80



Table 4. 25 Estimation accuracy of 2L-shape array for 3-sources .............................. 81


Table 4. 27 Estimation accuracy of 2L-shape array for 4-sources .............................. 82

Table 4. 28 Proximity effect of Elevation angle .......................................................... 84

Table 4. 29 Proximity effect of Azimuth angles .......................................................... 84

Table 4. 30 Means, Variances and standard deviations using PM parallel shape

array ........................................................................................................ 85

Table 4. 31 Means, Variances and standard deviations using PM with L shape

array ........................................................................................................ 86

Table 4. 32 Means, Variances and standard deviations using DE-PS with 2-L

shape array .............................................................................................. 86

Table 4. 33 Comparison among 2-L shape array, parallel shape array and L shape

arrays ....................................................................................................... 87

Table 4. 34 Estimation accuracy for 3 sources ............................................................ 94



Table 4. 37 Proximity effect of elevation and azimuth angles .................................... 96

xxi

Table 4. 38 Comparison analysis on reference axis ..................................................... 97

Table 5.1 Parameters setting for GA, IPA and SA .................................................... 104

Table 5.2 Amplitude, DOA and Range of two sources ............................................. 107

Table 5.3 MSE and %convergence of two sources for different number of sensors . 108

Table 5. 4 Amplitude, DOA and Range of three sources .......................................... 108

Table 5.5 MSE and %convergence of three sources for different number of

sensors ................................................................................................... 109

Table 5. 6 Amplitude, DOA and Range of four sources ............................................ 109

Table 5. 7 MSE and %convergence of four sources for different number of sensors110

Table 5. 8 GA-IPA for Amplitude proximity ............................................................ 111

Table 5. 9 GA-IPA for DOA proximity ..................................................................... 112

Table 5. 10 GA-IPA for Range proximity ................................................................. 112

Table 5. 11 Estimation Accuracy of Amplitudes, Ranges & DOA for 2 Sources

and 4 sensors ......................................................................................... 114

Table 5. 12 MSE and Convergence rate of two sources for different number of

sensors ................................................................................................... 115

Table 5. 13 DOA proximity for two sources and 6 sensors ....................................... 116

Table 5. 14 Estimation accuracy of Amplitude, Ranges & DOA for 3 sources with

6 sensors ................................................................................................ 117

Table 5. 15 MSE and %convergence of three sources for different number of

sensors ................................................................................................... 118

Table 5. 16 DOA proximity for 3 sources and 8 sensors ........................................... 118

Table 5. 17 Accuracy of Amplitude, Ranges & DOA for 4 sources and 8 sensors ... 119

Table 5. 18 MSE and Convergence of four sources for different number of sensors 120

Table 5. 19 DOA proximity for four sources and 10 sensors .................................... 121

xxii

Table 5. 20 Parameters Setting for ASA .................................................................... 127

Table 5. 21 Estimation accuracy for 2-targets ........................................................... 129

Table 5. 22 Estimation accuracy for 3-targets ........................................................... 129

Table 5. 23 Estimation accuracy for 4-targets (continue) ......................................... 130

Table 5. 24 Estimation accuracy for 4-targets .......................................................... 130

Table 5. 25 Proximity effect of Elevation angles for 𝑠1 = 1, 𝑠2 = 3, 𝑠3 = 5

𝑟1 = 1.5𝜆, , 𝑟2 = 3𝜆, , 𝑟3 = 4𝜆 & 𝜙1 = 1300, 𝜙2 = 700, 𝜙3 =

1600. ..................................................................................................... 132

Table 5. 26 Proximity effect of azimuth angles for 𝑠1 = 1, 𝑠2 = 3, 𝑠3 = 5

𝑟1 = 1.5𝜆, , 𝑟2 = 3𝜆, , 𝑟3 = 4𝜆 & 𝜃1 = 300, 𝜃2 = 500, 𝜃3 = 850. ... 132

Table 5. 27 Estimation Accuracy of 2 sources using 9 sensors (continue) ................ 141

Table 5. 28 Estimation Accuracy of 2 sources using 9 sensors ................................. 141

Table 5. 29 Estimation Accuracy of 3 sources using 13 sensors (continue) .............. 141

Table 5. 30 Estimation Accuracy of 3 sources using 13 sensors ............................... 142

Table 5. 31 Estimation Accuracy of 4 sources using 17 sensors (continue) .............. 142

Table 5. 32 Estimation Accuracy of 4 sources using 17 sensors ............................... 143

Table 5. 33 Estimation Accuracy for 3 sources at SNR=5 dB (continue) ................. 144

Table 5. 34 Estimation Accuracy for 3 sources at SNR=5 dB .................................. 144

Table 5. 35 Proximity effect of DOA of three sources and 17 sensors at SNR=10

dB .......................................................................................................... 145

xxiii

LIST OF ABBREVIATIONS

ACO Ant Colony optimization

ASA Active set algorithm

AI Artificial intelligence

BCO Bee Colony optimization

CA Culture algorithm

CSCA Centro Symmetric cross array

DOA Direction of arrival

DE Differential Evolution

DE-PS Differential Evolution hybridized with pattern search

DE-IPA Differential Evolution hybridized with interior point algorithm

DE-ASA Differential Evolution hybridized with Active set algorithm

EC Evolutionary computation

GA Genetic algorithm

GA-IPA Genetic algorithm hybridized with Interior Point algorithm

GA-PS Genetic algorithm hybridized with Pattern search algorithm

IPA Interior Point algorithm

MSE Mean square error

MLP Maximum Likelihood principle

PSO Particle swarm optimization

PSO-PS Particle swarm optimization hybridized with Pattern search

PSO-IPA Particle swarm optimization hybridized with Interior point algorithm

PSO-ASA Particle swarm optimization hybridized with Active set algorithm

PS Pattern search

xxiv

PM Propagator method

RMSE Root mean square error

SA Simulated annealing

SNR Signal to noise ratio

SA-PS Simulated annealing hybridized with pattern search

SA-IPA Simulated annealing hybridized with interior point algorithm

ULA Uniform linear array

URA Uniform rectangular array

CHAPTER 1

INTRODUCTION

Beamforming is the signal processing technique used in conjunction with sensor array to

provide adaptable form of spatial filtering. The sensor array pulls together the spatial

samples of the impinging signals from space. The prime objective of the beamformer is

the directional signal transmission and reception i.e., to steer the main beam towards

desired direction in space, while placing nulls in the direction of unwanted signals or

jammers [1], [2], [3], [4], [5], [6], [7], [8]. In this context, Direction of Arrival (DOA)

estimation is a preliminary and indispensable requirement for adaptive beamformer [9].

DOA

Algorithm Beamforming

W

W

W

W

W

Fig. 1.1 Schematic diagram for adaptive beamforming along with DOA estimation

The DOA algorithms compute the direction of signals impinging on sensors array and

once the direction information is available, it is further fed to the beamformer network to

CHAPTER 1 . INTRODUCTION

2

calculate the complex weight vectors essential for beam steering. The beamforming setup

along with DOA estimation is shown in Fig. 1.1.

1.1 PROBLEM STATEMENT

DOA estimation of sources impinging on an array of sensors of array originated from

Fraunhofer zone (far field) or from Fresnel zone (near field) has numerous applications in

the field of radar, sonar, seismic exploration, wireless communication system etc. Along

with DOA estimation, the other parameters which are of significant importance are the

amplitude, range and frequency of the received signals. The major problems in joint

estimation of these parameters are the estimation failure, pair matching and computational

cost.

In literature, there exist two kinds of algorithms for DOA estimation depending on the

sources impinging on sensor array from far field or from near field. It is relatively easy to

deal with the DOA of far field sources as compared to that of near field. Obviously the

reason is that in case of far field, we have plane wave fronts which are easy to deal with,

whereas, in case of near field one has to deal with spherical wave fronts. There are several

algorithms available in literature to address this issue of DOA estimation. Majority of

them have considered only far field scenario. Though the contributions are there i.e. in the

domain of near field, however, they are quite limited. Moreover no one has tackled it with

the help of the algorithm that we have proposed. Few researchers have also worked on the

joint parameter estimation of far field, as well as, near field sources, but no one has paid

serious attention to the estimation of amplitude of these sources. By observing the

importance of these parameters of far field, as well as, of near field sources, it is the

requirement that some efficient schemes which are able to resolve the above mentioned

issues must be developed. In this context, we have targeted the near field, in addition to

the far field parameters estimation. We have used diversified arrangement of antenna


3

arrays in our algorithms and starting from the joint estimation of two parameters i.e. 2D

(Two dimension) estimation, have gone up to 5D (Five dimension) estimation i.e. the

joint estimation of five parameters. A detail of our contribution has been given in the

section to follow.

For these contributions, we have used the evolutionary heuristic computational

techniques. It is well acknowledged that in todays recent development, no one can decline

the importance of these techniques. These techniques have broadened the horizon of

optimization in every field of engineering. These techniques are quite successful, reliable

and efficient because of their ability in decision making and autonomous learning. Due to

ease in implementation, ease in concept, and most importantly avoiding getting stuck in

the presence of local minima, they have received special attention by the research

community. These techniques include mainly Genetic algorithm (GA), Particle Swarm

optimization (PSO), Differential Evolution (DE), Genetic Programming (GP) etc.

Another advantage is that in many problems, the efficiency and reliability of these global

optimization techniques increase even more when they are hybridized with any other

efficient local search optimizer such as Pattern Search (PS), Interior Point Algorithm

(IPA), Active Set Algorithm (ASA) etc. This hybridization is sometimes referred to as

memetic computing in the literature.

1.2 CONTRIBUTIONS OF THE DISSERTATION

In this dissertation, we have developed efficient hybrid algorithms based on evolutionary

computational techniques for the joint estimation of parameters that belong to far field

sources, as well as, near field sources. For this purpose, we have used two fitness

functions, initially Mean Square Error (MSE) is used, while at the end another new multi-

objective fitness function is developed which is the combination of MSE and correlation

between normalized desired and normalized estimated vectors. Moreover, different


4

sensors array structures are used which include Uniform Linear Array (ULA), 1-L shape

array, 2-L shape array and Centro Symmetric Cross Array (CSCA). A summary of

contributions included in this dissertation is given below.

1. 2-D PARAMETERS

Initially we started with 2D parameters estimations. The parameters we have taken are

specifically the amplitude and elevation angle of the incoming signal from far field and

impinging on ULA. Both the parameters are jointly estimated and relevant contributions

in this context are:

i. 2-D parameters (amplitude and elevation angle) of far field sources impinging on

ULA are jointly estimated using hybrid GA-PS scheme.

ii. 2-D parameters (amplitude and elevation angle) of far field sources impinging on

ULA are jointly estimated using hybrid PSO-PS scheme.

2. 3-D PARAMETERS

In this case, other than the above two parameters, an additional parameter i.e. azimuth

angle has been included for joint estimation of far field sources. Similarly 3D parameters

i.e. Amplitude, Range and Elevation angle are jointly estimated for near field sources. In

this case, different antenna arrays are used which include 1-L, 2-L and ULA accordingly.

The contributions in this setup are listed below.

i. 3-D parameters (amplitude, elevation angle, azimuth angles) of far field

sources impinging on 1-L shape array are jointly estimated using hybrid GA-

PS scheme.

ii. 3-D parameters (amplitude, elevation angle, azimuth angles) of far field

sources impinging on 2-L shape array are jointly estimated using hybrid GA-

PS scheme.


5

iii. 3-D parameters (amplitude, elevation angle, azimuth angles) of far field

sources impinging on 1-L and 2-L shape arrays are jointly estimated using

hybrid PSO-PS scheme.

iv. 3-D parameters (amplitude, elevation angle, azimuth angles) of far field

sources impinging on 1-L and 2-L shape arrays are jointly estimated using

hybrid DE-PS scheme.

v. 3-D parameters (amplitude, range, and elevation angle) of near field sources

impinging on ULA are jointly estimated using hybrid GA-IPA scheme.

vi. 3-D parameters (amplitude, range, and elevation angle) of near field sources

impinging on ULA are jointly estimated using hybrid PSO-PS and DE-PS

schemes.

3. 4-D PARAMETERS

Ultimately the complicated task at hand was the estimation of four parameters. The fourth

parameter was taken as frequency for far field sources, whereas, the fourth parameter in

case of near field included azimuth angle. For this we used 2-L and CSCA arrays. The

following contributions with this setup are given as.

i. 4-D parameters (amplitude, frequency, elevation angle, azimuth angles) of far

field sources impinging on 2-L shape arrays are jointly estimated using hybrid

PSO-PS scheme along with multi-objective fitness function.

ii. 4-D parameters (amplitude, range, elevation angle and azimuth angle) of near

field sources impinging on CSCA are jointly estimated using hybrid PSO-

ASA scheme.

iii. 4-D parameters (amplitude, range, elevation angle and azimuth angle) of near

field sources impinging on CSCA are jointly estimated using hybrid DE-ASA

scheme.


6

4. 5-D PARAMETERS

Finally the most complicated problem included in this dissertation has been taken into

account i.e. the estimation of five parameters of near field sources. The fifth parameter in

this case has been taken as the frequency in addition to the already considered four

parameters of previous setup. The contributions are given as follows.

i. 5-D parameters (amplitude, frequency, range, elevation angle and azimuth

angle) of near field sources impinging on CSCA are jointly estimated using

hybrid GA-PS and GA-IPA schemes along with a new multi-objective fitness

function.

1.3 ORGANIZATION OF THE DISSERTATION

The dissertation is organized as follows. In chapter 2, a brief literature review regarding

DOA is provided. In chapter 3, we have discussed the proposed global optimization

techniques (GA, PSO, DE, SA), as well as, the local search optimizers (PS, IPA, ASA).

Their brief introduction, flow diagrams and steps in the form of pseudo code are

provided.

Chapter 4 is divided into three parts. In part one, we have developed schemes based on

GA, PSO and SA hybridized with PS for the joint estimation of amplitude and elevation

angles of far field sources impinging on ULA. In the second part, we have hybridized

GA, PSO, DE and SA with PS to estimate jointly 3-D parameters (amplitude, elevation

and azimuth angles) of far field sources impinging on 1-L and 2-L shape arrays. In part

one and part two, MSE is used as a fitness evaluation function. In the third part, PSO is

hybridized with PS for the joint estimation of 4-D parameters (Amplitude, frequency,

elevation and azimuth angles) of far field sources and a new multi-objective fitness

function is developed. A comprehensive statistical analysis is given to test the validity of

each scheme in terms of estimation accuracy, convergence, robustness against noise etc.


7

In Chapter 5, near field sources are dealt with and it is also divided into three parts. In

part one, GA and SA are hybridized with IPA, while PSO and DE are hybridized with PS

to jointly estimate the 3-D parameters (amplitude, range and elevation angle) impinging

on ULA. In the second part, we have linked our problem to bi-static radar and have

jointly estimated the 4-D parameters (Amplitude, range, elevation and azimuth angles) of

near field sources impinging on centro-symmetric cross array (CSCA). For this, PSO and

DE are hybridized with ASA. In part three, we have jointly estimated 5-D parameters

(amplitude, frequency, range, elevation and azimuth angles) by using PSO-PS along with

a new multi-objective fitness function. Again a comprehensive analysis is provided to

check the validity of each scheme.

Chapter 6 summarizes and concludes the dissertation along with some future work

directions and recommendations.

CHAPTER 2

DOA ESTIMATION TECHNIQUES: AN OVERVIEW

Recently, one of the dynamic research areas in electromagnetic and wireless

communication systems is smart antennas. The demand of smart antenna drastically

increases when dealing with multi-user communication system, which needs to be

adaptive, especially in unknown time varying scenarios. Adaptive or smart antennas

system consists of an array of radiating sensors which are able to steer the main beam in

any desired direction in space while, placing suitable nulls in the direction of unwanted

signals or jammers. In this connection, Direction of Arrival (DOA) estimation of received

signals is one of the fundamental and necessary steps to construct a smart or adaptive

receiver. DOA estimation in adaptive Beamforming, as well as, in wireless

communication, has been the area of great research interest for the last few decades. Thus

it has numerous applications in radar, sonar, seismic exploration, mobile communications

etc [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. The

sources impinging on sensors array can be mainly divided in two categories based on

their distance from the array i.e.

Far Field sources (Fraunhofer Zone)

Near Field sources (Fresnel Zone)

Near field sources lie in the radiating zone 2

/ 2 , 2 /D while, far field sources

exist beyond the range 2

2 /D where D is the dimension of sensors array and is

the wavelength of impinging signals (see e.g. [25], [26] for details).

CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW

9

It is relatively easy to estimate the DOA of far field sources because in this case, we deal

with plane waves, where the DOA is a function of angle only. Moreover, we make use of

the following assumptions about the amplitude, angle and phase variations of the

impinging sources [27],

1 2s s s For amplitude variations

1 2 For angle variations

1

2

cos2

cos2

ds s

ds s

For phase variations

However, on the other hand, it is comparatively difficult to estimate the DOA of near

field sources as the plane wave concept is no more valid and one has to deal with

spherical waves where the DOA is the function of angle, as well as, range of the sources.

For simplicity sake, in this chapter, we will confine ourselves only to the discussion of far

field sources impinging on ULA, whereas, far field sources impinging on L-Shape arrays

(1-L and 2-L shape arrays) and near field sources will be discussed in upcoming chapters.

In this chapter, we shell present a data model for far field sources impinging on ULA,

while data model for far field sources impinging on L shape arrays and near field sources

are given in the subsequent chapters.

2.1 DATA MODEL

Consider P narrow band sources impinging from Fraunhofer zone (far field) on ULA

composed of M sensors where the inter-element spacing between the two consecutive

sensors is d as shown in Fig. 2.1. The response of the th

m sensors in the ULA for

P M sources, can be mathematically modeled as,

( ) ( )exp( ( )) ( )1

P

y t s t j m n tm i mii

(2.1)


10

In (2.1) coskdi i is propagation delay between the reference and th

m sensor for thi

source where 2

k

is the wave number and

2d

. In matrix-vector form (2.1), can be

represented as,

1 10 01

cos cos11 2 1

( 1)cos( 1)cos 11 1

y nsjkd jkd p

e ey s n

jkd M Pjkd M ey s nPM Me

(2.2)

1 m M-10

Far Field ith source

where i= 1,2,… P

d

Si

θi

Si

θi

Si Si

θi θi

Fig. 2. 1 Signal Model for far field sources

Generally (2.2) can be represented as,

( ) ( )t t y Bs n (2.3)

In (2.3), B is called steering matrix which contains the steering vectors of P sources,

and s represents the elevation angle (with respect to broad side) and amplitude,

respectively, of the sources. Similarly, n(t) is additive white Gaussian Noise (AWGN)

vector added at the output of each sensor. It has zero mean, unit variance 2 and is

independent of source signals i.e.

2[ ( ) ( ) ]

HE n t n t (2.4)


11

( ) ( ) 0E n t s t (2.5)

Before discussing the new proposed algorithms, some existing methods for the estimation

of DOA are being presented.

2.2 DOA ESTIMATION TECHNIQUES

On the basis of data analysis and implementation, all the algorithms of DOA estimation

are broadly divided into three categories i.e.

Conventional or Beamforming Algorithms

Parametric or Maximum Likelihood Algorithms

Subspace Based Algorithms

2.2.1 Conventional Beamforming Algorithms

In these methods, we use the concept of Beamforming and null steering and do not make

use of the statistics of received data. Hence, the DOA of the impinging signals can be

found from the peaks of the output power spectrum attained by steering the beam in all

feasible directions. Examples of these algorithms are Conventional Beamformer and

minimum Variance Distortion-less response (MVDR) beamformer methods are discussed

below.

2.2.1.1 The Conventional Beamformer Method

The conventional beamformer developed by Barlett in 1950 [28] and is considered to be

one of the oldest techniques used for the DOA estimation of sources. This method

estimates the DOA by using the idea of steering the array in one direction and computes

the output power. Thus the direction which yields maximum power gives the true DOA of

the impinging sources. It makes use of the linear combinations of the sensors output to

perform the steering as,

( ) ( )H

x t t w y (2.6)


12

where 10 1[ , ,..., ]MT

w w w w is a weight vector used for steering and the output power

for L snapshots can be written as,

2

( ) ( )1

LP x t

t

w (2.7)

Now, to calculate the optimum weights which maximize the output power, we suppose

that a single source is impinging on sensor array from 1 direction. The response of the

sensors array for this single source is given as,

1 1( ) ( ) ( ) ( )t s t t y b n (2.8)

where ( )tn is AWGN having zero mean, unit variance 2

and is independent of the

sources. Now, the output power is,

2( ) ( )1P E x t

(2.9)

By putting the values of (2.6) and (2.8) in (2.9), we can write (2.9) after simplification as,

11 1 12

( ) ( ) ( )H H H H

P Rs w R w w b b w w w (2.10)

where ( ) ( )H

E t t

R y y is M M auto correlation matrix of the sensor array for 1

direction and1 1 1

( ) ( )H

R E s t s ts

. The maximum power can be obtained as,

1

2max ( )

Hw w b Subject to 1

Hw w (2.11)

By using the condition 1H

w w and Cauchy-Schwarz inequality, we have

1 11

2 2 2 2( ) ( ) ( )

H w b w b b (2.12)

The solution of the optimal weight vectors can be given as,

1

1 1

( )

( ) ( )opt H

b

w

b b

(2.13)


13

Now, the maximum power, which will give us the correct DOA1( ) , can be found by

putting (2.12) in (2.10) as,

1 11

1 1

( ) ( )( )

( ) ( )

H

PH

b R b

b b (2.14)

Generally, the DOA is attained from the highest peaks of the spatial spectrum,

( ) ( )ˆ ( )

( ) ( )

H

PH

b Rb

b b (2.15)

The major disadvantages of this method are well exposed when the number of sources is

greater than one and also when the sources are closely spaced to each other. It is handy

only for single source.

2.2.1.2 Minimum Variance Distortionless Response Beamformer

Minimimum Variance Distortion-less Response (MVDR) filter was first proposed by

Capon in 1969 [29] which was later on used by Lacoss as a dual of beamformer [30]. The

basic purpose of this method was to overcome the drawbacks of conventional

beamformer for multiple narrow band sources impinging from different directions

(DOAs). In this, the output power not only contains the contribution of desired signals,

but also of undesired ones. Therefore, this method maintains gain in the look direction as

constant, while minimizing the contribution of those DOAs which belong to undesired

signals by minimizing the output power. Mathematically, the optimization problem of

MVDR beamformer can be represented as,

( )

subject to ( ) 1H

Min P

ww

w b

(2.16)

By using equation (2.10), the solution of weight vector for positive definite covariance

matrix can be given as,


14

1

1

( )

( ) ( )H

swcs

R b

b R b (2.17)

Now, with this weight vector, the array output signal power leads to the subsequent

special spectrum,

1( )

ˆ( ) ( )

1H

Pc

b R b

(2.18)

Its peak value for a particular will give us the desired DOA.

Although MVDR beamformer has better resolution than its counterpart the classical

beamformer, but it is unable to handle the problem of DOA for correlated signals.

Moreover, it is also dependent on the SNR and the number of sensors in the array.

2.2.2 Parametric or Maximum Likelihood Algorithms

These methods are efficient and robust in a way that they utilize the complete

mathematical data model of the received signals as given in (2.1). However, in general

these approaches are considered to be computationally very expensive as they require

multi-dimensional search to get the DOAs. Maximum Likelihood (ML) algorithm is the

well known example of this category which is further classified into two methods i.e.

Stochastic or Unconditional ML algorithm and Deterministic or Conditional ML

algorithm. Both of them are briefly discussed below,

2.2.2.1 Unconditional or Stochastic Maximum Likelihood Algorithm

―Bohme [31] and Jaffer [32]‖ have derived the ML estimate which can be achieved by

representing the signal waveform as a Gaussian random process. In this, the output vector

of sensors array ( )ty is considered to be complex Gaussian vector. The joint probability

density function (pdf) for S independent snapshot is mathematically given as,

1 1... exp ( ) ( ) ( )(1), (2), , ( ) ( )

1( )

s HP t ts s

t

y R yy y y θ y

R θ

(2.19)


15

where [ , ,... ]1 2 P θ and by ignoring the constant term, the negative log likelihood

function of equation (2.19) can be written as,

2 ˆ( , , )L TrSML

BS Rθ (2.20)

where Tr represent the trace of a matrix, 2

is the noise variance while

{ ( ) ( )}H

E t tS s s (2.21)

and

1( )

H H I B B BB (2.22)

Now by minimizing L with respect to S and 2

for fixed θ , we get

12ˆ ( ) Tr

M I

θ B (2.23)

and

1 2 1ˆ ˆ ˆ( ) ( ) [ ( ) ] ( )

H H H

S B B B R I B B Bθ θ (2.24)

Now, using the results of (2.23) and (2.24) in (2.22), the stochastic maximum likelihood

(SML) estimates can be achieved by solving the subsequent minimization problem,

2ˆ ˆarg min log ( ) ( )H

SML BS B Iθ θθ

(2.25)

2.2.2.2 Conditional or Deterministic Maximum Likelihood Algorithm

In this algorithm, the signals are considered to be deterministic and having unknown

waveforms. Assuming that the noise is AWGN having variance2

, the joint pdf for S

independent snapshots can be written as,

(1) (2) ( )

1 1... exp { ( ) ( )} { ( ) ( )}, , , ( ) 22 1

s

sHP t t t t

st

y Bs y Bsy y y θ y (2.26)

and the negative log likelihood function of the above equation can be given as,


16

1 22 2

( , ( ), ) log ( ) ( )2 1

hL t M t t

th

s y Bsθ (2.27)

Now, minimizing L with respect to ( )ts and2

, we have

12 ˆˆ { }TrM

RB (2.28)

1ˆ( ) ( ) ( )

Ht t

s B B By (2.29)

By using (2.28) and (2.29) in (2.27), the conditional or deterministic maximum likelihood

(DML) estimates can be achieved by simplifying the subsequent minimization problem.

ˆ ˆarg min { }TrDML

RBθθ

(2.30)

2.2.3 Signal Subspace Algorithms

The signal subspace based algorithms are widely used for the DOA estimation of far field

sources. It has better resolution as compared to classical methods and comparatively less

computationally expensive then parametric methods. These methods utilize the subspace

of the data received (Signal/Noise) on the sensor array to estimate the signal parameters

and can be further divided into two categories.

Covariance based methods

Direct data domain method

2.2.3.1 Covariance Based Methods

These methods work on the spatial covariance matrix of the output data vector on sensors

array. The two well known techniques are Multiple signal classification (MUSIC)

algorithm and estimation of signal parameter through rotational invariant technique

(ESPRIT).


17

2.2.3.1.1 Multiple Signal Classification Algorithm (MUSIC)

Multiple Signal Classification (MUSIC) method was first developed by Schmidt [33] in

1979 which was further independently used by Bienvena and Kopp [34], [35]. MUSIC

algorithm is basically a subspace based technique that can be used for the DOA

estimation of narrow band, uncorrelated sources. Consider the data model in section 2.1

for far field sources, the array covariance matrix can be written as,

[ ( ) ( )]H H

E t t s R y y BR B D (2.31)

where D is M M noise covariance matrix while [ ( ) ( )]H

E t ts R s s is P P source

covariance matrix. The estimated covariance matrix R can be given as,

1ˆ ( ) ( )

1

L Hk k

L k

R y y (2.32)

Suppose the noise covariance matrix consists of uniform noise power on the diagonal as

2D I so (2.31) can be written as,

2[ ( ) ( )]

H HE t t s R y y BR B I (2.33)

By using eigen value decomposition, (2.33), in terms of eigen values and eigen vectors

can be written as,

1

M H Hi i i

i

R v v VΛV (2.34)

where

[ , ,... , ,... ]1 2 1P MP Λ (2.35)

and

[ , ,... , ,... ]1 2 1P MP V v v v v v (2.36)

where 2

...1 2 MP P and for any i P


18

2i i i i Rv v v (2.37)

By putting (2.33) in (2.31), we get,

2( )Hi s i Rv BR B I v (2.38)

By comparing (2.37) and (2.38), we get,

0H

s i BR B v (2.39)

Now, by using full rank property,(2.39) becomes,

0Hi v B (2.40)

where andi p p P .

The above equation (2.40) states, that the M P lowest eigenvectors of R are

orthogonal to the actual DOA and this observation is the core of many other eigen-based

techniques. The subspace correspond to the M P eigen vector is called noise subspace

which is orthogonal to the K independent impinging sources (signal subspace). The entire

space (noise plus signal space), can be given as,

H SS NS (2.41)

where

1 2[ ( ), ( ),... ( )]PSS span b b b (2.42)

represent the signal subspace and

1 2[ , ,..., ]P P MNS span v v v (2.43)

is the noise sub-space. The spatial spectrum of MUSIC can be given as,

1ˆ ( )

2( )

1

PM H

ii P

v b

(2.44)


19

The P largest peaks correspond to the estimates of DOA in the spectrum. In order to

improve further the performance of MUSIC algorithm, Root-Music method [36], and

unitary MUSIC methods [37], are also proposed.

2.2.3.1.2 Estimation of Signal Parameter through Rotational Invariance

Technique (ESPRIT)

ESPRIT is one of the well known and widely used algorithms for the DOA estimation of

narrow band sources. Like MUSIC, it is also a sub-space based algorithm which was

introduced by Roy and Kailath in 1989 [38]. This algorithm utilizes two same sub-arrays

consisting of equal number of antenna sensors. In both sub-arrays, each matched pair of

antenna sensors having equal displacement is called a doublet. Consider a ULA

consisting of M=2Nx sensors is composed of two sub-arrays as shown in Fig. 2.2.

-Nx +1 -1 1 m Nx -10

dd

Nx-Nx +2

Sub Array 1

Sub Array 2

Fig. 2. 2 Array Geometry for ESPRIT

The response of th

i doublet in both sub-arrays for P sources can be mathematically

modeled as,

1 1( ) ( ) ( ) ( )

1

P

p py t b s t n ti i ip

(2.45)

2 2

(2 / ) cos( ) ( ) ( ) ( )

1

P

p p

j d py t b e s t n ti i i

p

(2.46)

where 1y i and

2y i represents the response of th

i doublet in sub-array 1 and 2, respectively.

In matrix-vector form, (2.45) and (2.46) can be given as,


20

( ) ( ) ( )1 1t t t y Bs n (2.47)

2 2( ) ( ) ( )t t t γy B s n (2.48)

where 1( )tn and 2 ( )tn are the AWGN vectors added at the output of the sub-array 1 and 2,

respectively. The γ is P P , matrix containing the DOA of the sources. So, to estimate

the DOA, one requires calculating γ , where,

(2 / ) cos (2 / ) cos (2 / ) cos1 2, ,...,j d j d j d Pdiag e e e

γ (2.49)

Considering the contribution of both sub-arrays, the output vector can be expressed as,

( ) ( )1 1( ) ( ) ( ) ( )

( ) ( )2 2

t tt t t t

t t

y nBy s Bs n

γy B n (2.50)

The covariance matrix of the entire sub-array can be written as,

2[ ]

H HEyy ss R yy BR B I (2.51)

Similarly, the covariance matrices for the two sub-arrays can be expressed as,

2[ ]11 1 1

H HE ss R y y BR B I (2.52)

2[ ]22 2 2

H HE ss R y y BR B I (2.53)

Each of the full rank covariance matrices provided in (2.52) and (2.53) has a set of eigen

vectors related to the P sources present. Creating the signal sub-space for both sub-arrays,

yields two matrices E1 and E2 while for the whole array, the signal subspace gives one

matrix Ex. However, due to the invariant structure of the array, Ex can be divided into two

subspaces E1 and E2 both having dimensions M P . The columns of E1 and E2 contain

the P eigen vectors which belong to the largest eigen values of R11 and R22. As the arrays

are related transitionally, so the subspaces E1 and E2 can be related by a unique non-

singular transformation matrix ψ i.e.

1 2ψE E (2.54)


21

and there also exists a unique transformation non- singular matrix F such that

1 E BF (2.55)

2 γE B F (2.56)

now, by substituting (2.55) and (2.56) in (2.54), we get,

1ψ γF F (2.57)

or (2.40), can be rewritten as,

1ψ γF F (2.58)

According to equation (2.58), the diagonal values of γ are equal to the eigen values of the

subspace rotation operator ψ and the eigen-vectors of the ψ is equal to the columns of F.

In fact, this is the main involvement in the development of so called ESPRIT algorithm.

Now, the only problem left is to calculateψ . There are many methods available to

calculate it [39], [40] but the most popular among them is the TLS ESPRIT [41], [42].

Though MUSIC and ESPRIT are well accepted widely used methods and are less

computationally expensive as compared to parametric based methods, however, both of

them are also computationally intensive as the numbers of snapshots required are twice

that of the total number of sensors used in array. The other problem with both the

algorithms is their performance limitation towards the correlated signals. Both the

algorithms fail to generate useful results when the impinging signals are highly

correlated, because the rank of the covariance matrix is reduced. Similarly the

performance is drastically degraded for closely spaced sources. In this situation, some

pre-processing techniques are required such as spatial smoothing [43]. In order to

decrease the computational cost of ESPRIT, a popular alternate of ESPRIT called unitary

ESPRIT method is introduced in which unitary transformation is used to convert the


22

complex covariance matrix into a real matrix and reduce the computational complexity

[44], [45].

2.2.3.2 Direct Data Domain Method

To overcome the problems of covariance based methods, direct data domain method was

introduced in mid-nineties which is time-wise efficient and it uses only single snapshot

for the estimation of DOA in real time non-stationary environment. It requires minimum

computational burden for implementation in real time. Matrix Pencil (MP) method is a

well known example of direct data domain method which was first developed by Hua and

Sarkar [46]. To estimate the DOA of signals, the MP method works directly on sensors

data that performs well even for highly correlated signals and does not require the

additional step of pre-processing techniques [47]. According to MP method, the output of

thm sensor in the ULA for single snapshot can be written as,

exp( ( ))1

Py G j m nm i i i m

i

(2.59)

where i and Gi represents the signal phase and amplitude respectively at reference sensor

and nm is the noise added at the th

m sensor in the array. A complete data matrix which is

also called Hankel matrix can be formed from the above equation as,

y0 1 1

1 2

1 1

y y L

y y yL

y y yM L M L M

Y

(2.60)

where L is known as the pencil parameter whose values should lie in between M/3 and

M/2 [48]. In noise free environment, the above matrix Y can be decomposed as,

0a bY U G U (2.61)

where


23

2

1 1 1

1

( ) ( ) ( )1 2

P

P

j j je e e

a

j M L j M L j M Le e e

U

(2.62)

1 21 2

0 K

j j j Pdiag G e G e G e

G (2.63)

and

2

( 1)1 11

( 2) 21

( )1 P P

j j Le e

j Lje e

b

j j L Pe e

U

(2.64)

We can decompose the matrix Y into two matrices by deleting its first and last rows, i.e.

1a a o bY U G U (2.65)

2b a o bY U G U (2.66)

where

1 1a aU J U (2.67)

2 2a aU J U (2.68)

In (2.67) and (2.68), 1J and 2J are selection matrices which can be defined as,

1 J I o (2.69)

2 J o I (2.70)

where I is identity matrix while o is M L column vector with zero entries. Now,

consider the following

1 [ ]0 0e a a eb b Y Y Z G Z I Z (2.71)

where we have used the following relation,


24

2 1 0a aU U U (2.72)

In the above equation,

1 2[ ]0j j j Kdiag e e e

U (2.73)

It can be easily shown that for 1P L M , the rank of e ab Y Y is P but if

j pee

, the corresponding diagonal element in [ ]0 eU I become equal to zero and

thus reduces the rank of e ab Y Y by 1. So, it means that for ( 1,2,... )k P , Pje

are

the eigen values of matrix pair { , }a bY Y . For more detailed study about MP methods the

readers are advised to see [49], [50], [51].

The most expensive part of MP method is the singular value decomposition (SVD) of a

complex data matrix. In [52], a unitary matrix pencil method is introduced which reduces

the computational burden to one fourth by converting the complex data matrix into real.

The other method which is used for converting the complex data matrix into real is Beam

Space Method [53] and hence, further reduces the computational burden.

CHAPTER 3

SELECTED OPTIMIZATION TECHNIQUES

Optimization is a process of routine which takes place naturally in our daily lives. It deals

with the assignment of choosing the best out of possible choices available in a usual real-

life atmosphere. For example, finding a fastest or shortest path to our school or work

place is a part of our daily affairs which is basically optimization. Similarly, a

manufacturer faces an optimization problem, as he tries to speed up the production rate

while keeping the cost manageable. In the same way, optimization can be observed

everywhere, as everyone in this world struggles to maximize the production, quality,

profit etc but on the other hand he wants to minimize the required energy, budget and

time. According to Yuqi He [54], one of the famous members of US National Academy

of Engineering (Harvard University) said, ―Optimization is a corner stone for the

development of civilization‖.

In this context, no one can disagree with the contributions of evolutionary computing

techniques in the field of optimization. Evolutionary computation (EC), also known as

computational intelligence is a sub field of artificial intelligence which can be employed

for combinatorial, as well as, for continuous optimization problems. All the EC

algorithms have stochastic or meta-heuristic optimization nature and they are considered

to be global optimization methods. The Evolutionary Algorithms (EAs) can mainly be

applied to black box or derivative free problems which do not require the gradient of the

problem. EAs are based on the principles of biological evolution such as genetic

inheritance and natural selection. Due to ease in concept, ease in implementation,

robustness against noise and having less probability of getting stuck in the local minima,

CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES

26

EAs are being successfully applied to a mixture of problems ranging from handy

applications in industry and commerce to leading-edge scientific research [55], [56], [57].

By having such importance, the researchers are taking great interest in EC, especially for

their applications in science and engineering. Actually, EC makes use of the iterative

progress (growth or development) in a population and then to achieve the desired goal,

this population is selected in random manner using parallel processing. This kind of

mechanism is inspired by biological evolution. Some of the major advantages of EC over

traditional optimization methods are described below [58].

(I) Wide-Ranging Applicability: Virtually, EAs can be applied to any problem that can

be characterized as function of optimization task. It requires performance index and data

structure to assess solution while, production new solution from old needs variation

operators. The variation operators are used in such a way that a behavioral link is

established between the parents and their children i.e. a small variation in parents must

produce small effects on children and vice versa. A range of expected changes should be

done in such a way that under consideration ―step size‖ of the algorithm may be tuned, in

fact in a self adaptive mode. This elasticity permits for application essentially the same

process to mixed-integer problems, discrete combinatorial problems, continuous- valued

parameter optimization problems, and so forth [59].

(Ii) Hybridization: The simple exact methods or even the approximate EC are failed to

produce an acceptable solution for many real-world optimization problems when applied

alone or independently [60]. Therefore, in order to get more efficient and flexible

solutions, the interest of the researchers have been towed recently towards the concept

which is not only limited to a single traditional method but also to the combination of

different schemes with one another which belong to diverse fields such as computer

science, artificial intelligence, operation research etc [61]. In this regard, the EAs have


27

already got remarkable achievements and their reliability and efficiency increase

significantly when they are hybridized with any other technique [62]. We shall discuss the

application of this property in more detail in the upcoming chapters.

(III) Conceptual Simplicity: One of the most important advantages of EC is that it is

easy to be understand and easy to be implemented. Every EC algorithm starts up with the

random initialization phase. The next step involves some iterative variation and selection

to improve the performance index. Similarly, each EC algorithm requires fitness function

which is basically the core of any EC technique. The general flow diagram of EC is

shown in Fig.3.1

Vary

Individuals

Calculate

Fitness

Function

Apply

Selection

Randomly

Initialize

Population

Fig. 3. 1 General Flow chart of Evolutionary Algorithms

(IV) Adaptive To Dynamic Changes: Most of the traditional algorithms are not

adaptive to dynamic changes in environment and they need a complete re-start of the

system to settle the problem e.g. dynamic programming. On the other hand, EC

techniques are adaptive enough to perform well to dynamic changes and do not require

the re-initialization of the population at random [63].

Similarly, one can find the other advantages of EC over traditional techniques which

include the ability of self optimization, capability of solving problems which have no

solutions, parallelism etc [64]. To get more knowledge about EC, the readers are referred

to [65], [66], [67]. In literature various EC techniques are available, some of which are

listed below.

Genetic Algorithm


28

Particle Swarm optimization

Ant Colony Optimization

Bee Colony Optimization

Differential Evolution

Self-Organizing Migrating Genetic Algorithm

Fly Algorithm

Evolutionary Programming

Artificial Immune systems

Parallel Simulated Annealing

Evolution Strategy

Cuckoo Algorithms

Learnable Evolution Model

Cultural Algorithm etc

However, our discussion is limited to Genetic Algorithm (GA), Particle Swarm

Optimization (PSO), Differential Evolution (DE) and Simulated Annealing (SA). We

have also hybridized these global search techniques with local search optimizers which

include Pattern Search (PS), Interior Point Algorithm (IPA) and Active Set Algorithm

(ASA).

3.1 GENETIC ALGORITHM

In this universe, all breathing beings grow from beginning to end through natural

selection and only those will proliferate their genetic legacy that are fit enough to endure

till the reproductive period. In simple words, only those can survive who have high

robustness and their children have a high possibility of inheriting excellent characters

after the partial amalgamation of the sexed reproduction. These kinds of features are put

into practice in the so called Genetic Algorithm which belongs to a large family of


29

evolutionary computation and was first proposed by John H. Holland in 1975 in his work

to present a simple solution of natural selection [68]. A complete history of GA can be

found in [69]. The handy features of GA can be well appreciated and can produce

satisfactory results when the problem under consideration is not easy to be tackled by

means of classical methods e.g. in the case where the analytical model does not exist or

may be highly complex, or the number of parameters are so many that a mathematical

approach would be much expensive in terms of time [70]. In the recent years, GA has got

great success in solving different complex optimization problems in the field of signal

processing [71], communication [72], soft computing [73], network design and synthesis

[74] etc. GA is an iterative method which starts with fixed number of candidate solutions

called population. This candidate solution represents a possible solution to the problem

under consideration. The individual candidate solution is generally called as chromosome

where each chromosome consists of finite number of genes. The number of genes in each

chromosome is problem dependent (which may be of same or different nature). All these

chromosomes can be easily randomly generated through simple heuristic constructions.

Once the population is generated, then a stochastic selection operator based on evaluation

process is applied to choose the best solution during each iteration or generation. All the

left behind chromosomes after the selection process constitute a new set of chromosomes

called parents. Now the parent chromosomes will take part in the remaining evolution

process. In order to find better solution, the parent chromosomes uses mutation, elitism,

cross over etc and as a result a new set of chromosomes are produced called children or

off springs. This process continues until a termination is reached. (The termination

criterion generally depends upon the total number of iterations is complete or the best

chromosome has been achieved). All the above discussion about GA can be given in the

form of Pseudo step as under.


30

Step I Initialization: This step is required almost for every Evolutionary algorithm. In

this, the Chromosomes (population) are randomly generated where each chromosome

consists of genes. There is no specific rule about the size of population and it depends

upon the choice of user for problem under consideration.

Step II Fitness Function: The fitness or objective evaluation function is considered to be

the core of every evolutionary algorithm. One can easily achieve the required result for

any optimization problem if the fitness function is modeled properly and correctly for it.

In this step, the fitness of each chromosome is calculated and the type of fitness function

varies from problem to problem. We have used a fitness function based on Mean Square

Error (MSE) that defines an error between the desired response and estimated response. It

can be mathematically represented as,

1

1Fitness Function

MSE

(3.1)

This fitness function is basically derived from Maximum Likelihood Principle (MLP),

which will be discussed in the next chapter.

Those chromosomes which have high fitness values are awarded high rank while the ones

having less fitness values are placed in the bottom of the population. The entire

chromosomes are called parents.

Step III Termination Criteria: Terminate if any of the following conditions is being

satisfied else go to the next step,

a) The desired MSE is attained

b) The maximum number of iterations is completed.

Step IV Selection of Parent: In order to create next generation (offspring), some of the

parent chromosomes are selected. The purpose of this selection is to provide better

opportunity for individuals to survive in the subsequent generation. It is unwise to kill all

those genes that have less fitness as they may also produce something useful. In many


31

problems it has been observed that the combination of high fitness chromosomes and low

fitness chromosomes produce better reproduction.

Step V Reproduction: In order to improve the fitness function, a sub population

(offspring) is generated from the parents by using the GA operators. These operators

include crossover and mutation.

V a) Crossover: In this step, the information among the different chromosomes is

exchanged. It can be done either by using single point cross over or multi point cross

over. To select the cross over point, in most cases a random function is used. In single

point cross over, we select randomly a single number within the bounds of a

chromosome. For example, we have two parent chromosomes where both of them

consists of twenty genes and to perform a single point cross over, we need to select a

random numbers within length of chromosome#1, as shown below,

Chromosome # 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Chromosome # 2: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Cross over point= rand (1, length (chomosome#1)), e.g. gene # 11 is selected. As a result

of this single point cross over, the two newly born offspring will be

Offspring#1: 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0

Offspring#2: 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1

By using the same procedure one can easily do the multi points cross over.

V b) Mutation: It is the process of changing gene randomly within a individual

chromosome and is mainly used in a situation when the under consideration fitness

function is not improving satisfactorily. It avoids GA from getting stuck in local minima.

A simple mutation for binary encoded scheme can be done as,

Original Offspring #1: 1 1 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1

Original Offspring #2: 1 1 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 1


32

Mutated Offspring #1: 1 1 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 0 0 1

Mutated Offspring #2: 1 1 0 0 0 0 0 1 1 1 1 0 0 1 0 1 0 0 0 1

The number of genes to change is the choice of the user.

Randomly

Initialize

Population

Fitness Evaluation

Termination

Criterion

Selection of

Parents

Crossover

MutationBest

Individual

Stop

Start

No

YesSelection of

Offsprings

Fig. 3. 2 Generic Flow Diagram of Genetic Algorithm

Step VI Selection of offspring: The selection of offspring for next generation can be done

through several ways. The most widely used methods include, elitism, roulette wheel

method, generation replacement and fight for survival.

VI a) Elitism: In this method, we select some chromosomes from the parents while some

from the offspring. For example, in some problems sixty percent chromosomes are

selected from the parents while the remaining forty percent from the offspring. However,

there is no specific rule to select these ratios and it also depends on the choice of users.


33

VI b) Generation Replacement: This method of selection has also shown good results, in

which the entire parent chromosomes are killed and discarded. They are completely

replaced by offspring.

VI c) Fight of survival of fittest: It is the most widely used technique. In this, a common

merit is built under which the fitnesses of parent and offspring are sorted. As a result, the

best fitness value goes to top while, the low will be placed at the end. Now, only the

required and desired are kept, while the overflow is discarded. Again the fitness function

of each chromosome is calculated by using the step II and will continue this process until

every condition in termination criteria is being satisfied. The generic flow diagram of GA

is shown in Fig.3.2.

3.2 PARTICLE SWARM OPTIMIZATION

Particle swarm optimization (PSO) is comparatively new global optimization method and

is considered to be an alternate to GA. It is inspired from bird flocking and fish schooling

to search food in random manner and was first proposed by Kennedy and Eberhart in

1995 [75]. PSO is a type of optimization technique, which is also based on iterative

process, in order to obtain the best solution [76], [77], [78], [79], [80], [81]. It is highly

stochastic in nature and can be used for the optimization of highly non-linear, noisy,

differentiable and non-differentiable data as it does not require gradient. It has received

considerable attention by the research community during the last one and a half decade

because it is easy to be implemented and has less computational complexity as compared

to GA [82], [83], [84], [85], [86], [87]. The idea for food search is heuristic, because, all

the birds know the distance from the food but do not possess the knowledge of exact

location. They find their food by sharing their search information in cooperative behavior,

unlike in GA, by crossover and kids production method.


34

The action plan, upon PSO works consist of mutual information sharing procedure.

Initially, the entire candidate‘s solutions are randomly generated and then their fitness is

calculated. The entire fitness‘s are stored in one table which represents the personal

fitness of each particle. Among them the particle having the best fitness value is taken as

a global best and all the remaining particles (called as local best) will be following that

global best. The PSO differs from GA as in the second stage it uses to update velocity and

position of each particle instead of using the parameters such as selection, cross over and

mutation and thus is computationally less expensive. For more detail about the PSO, one

should study [88]. The generic flow diagram of PSO is shown in Fig. 3.3 while its steps in

the form of pseudo code can be given as,

Step I Initialization: Similar to GA, the first step of PSO is to initialize the swarm

randomly.

Step II Fitness Function: Calculate the fitness of each particle by using (3.1) and store

each particle as local best ( )Lbest while the one having maximum fitness function be

stored as global best (Gbest) . Now, instead of using selection, cross over, mutation etc,

PSO simplifies the things as it only requires updating the velocity and position of each

particle.

Step III Update Particle Velocity and Position: The velocity and position of each particle

can be updated by using the following relations,

1 1 1 1 11 1( ) 2 2( )

k k k k k kw b rand b randi i i i i i

V V Pbest x Gbest x (3.2)

1k k ki i i

x x V (3.3)

In the above relations, kiv and

kix are the velocity and position of

thi particle at time k

while 1k

i

Pbest and 1k

i

Gbest represent the local best and global best particles

respectively. The first term in the right hand side of (3.2), represents the previous velocity


35

which is called as momentum or inertia. The second term is called as cognitive term

which shows the private thinking of the particle. This term is also some times known as

remembrance or memory. The last term is called a social component which explains the

collective behavior of the population. Moreover, 1b and 2b are stochastic acceleration

constants which try to drag each particle towards global best ( )Gbest and local best

( )Pbest positions respectively. Similarly, rand1 and 2rand are two randomly generated

vectors in the range of [0,1].

Initialized

Swarm

Initial Swarm with randomly

Taken Position and Velocity

Fitness

Evalution

Termination

Criteria

Present

Better than Lb

Lb = Present

Present

Better than Gb

Gb = Present

Update Velocity

and Position

Global best

Particle

Stop

Start

No

Yes

No

No

Yes

Yes

Fig. 3. 3 Generic Flow Diagram of Particle Swarm Optimization


36

Step IV Choose Local Best And Global Best Particle: Replace the previous local best

and global best particles by the new local and global best particles if their fitness is

greater as compared to the previous particles.

Step V Termination: Terminate, if any of the following conditions is satisfied, otherwise

go to step II,

i) The required maximum fitness function value is achieved

ii) The desired MSE is attained.

Step VI Storage: Store all the results for later discussion and comparison.

3.3 DIFFERENTIAL EVOLUTION

Differential Evolution (DE) was first introduced by Stone and Price in 1996 [89] and it

has been proved to be a powerful and impressive global optimization technique as

compared to the other EC techniques. It has attracted the researchers due to its ease in

implementation, fewer parameters are required to be tuned and they are highly random in

nature. It can handle easily and efficiently nonlinear, multimodal and non-differentiable

cost functions. It has been also consistent and excellent convergence towards global

minimum in successive independent runs [90]. Moreover, it has been successfully applied

to the solution of discrete, as well as, constrained problems [91] and thus it has direct

applications in every field of science and engineering [92], [93]. DE is a kind of scheme

which iteratively searches large spaces of candidate solution and tries for the

improvement of candidate solution with respect to specified measurement of quality. In

simple words, DE optimizes a problem in such a way that first it maintains the candidate

solution and then by using its simple formulae it creates a new solution by combining the

existing ones. Now, it will keep only those candidate solutions which have best score or

fitness of the under consideration optimization problem. DE is basically the combination

of GA, Genetic programming [94] and evolutionary programming [95]. Like GA, DE is


37

also mainly based on the three operators i.e. mutation, crossover and selection but its way

of incorporating these operators is different from that of GA. Among these operators,

mutation has the central role in the performance of DE algorithm and the kind of DE

strategies to be constituted depends upon the mutation variants. Unlike GA, the average

fitness function of DE monotonically decreases or increases without the requirement of

elitism as the struggle between parents and children started after the cross over. The

generic flow diagram of DE is shown in Fig.3.4, while its steps in the form of pseudo

code can be given as,

Initialize

Population

Update the Generations

Calculate the next

generation chromosomes

Termination Criterion

Mutation

Cross Over

Selection Best Individual

YesNo

Stop

Start

Fig. 3. 4 Generic Flow diagram of Differential Evolution

Step 1 Initialization: This step is exactly the same as the one discussed for GA and PSO

i.e. create random population of N chromosomes. Let, the entire population represented

by a matrix" "C .


38

Step 2 Updating: In this step, update all the chromosomes of the current generation

" ".ge Now, select any chromosome from the randomly generated population e.g. choose

,i gek

c where " "i ( 1,2,... )i N represent the position of that particular chromosome in the

population while " "k ( )k any real number is its respective length. The goal is to find

the chromosome of next generation i.e. , 1i gec by using the following steps,

a) Mutation: Due to this step, we have the name DE as it works on the difference of the

vectors. Pick up any three different numbers (chromosomes) from 1 to N i.e. ( , , )1 2 3n n n

under the following conditions,

1 , ,1 2 3n n n N

where

, 1,2,3n n i ki k

1,2,3n ii i

now,

31 2 ,, , ,( )

n gei ge n ge n geF d c c c (3.4)

where ‗F‘ is a constant whose values usually lie in the range 0.5 to 1.

b) Crossover: The crossover can be performed as,

,(),

,/

i geif rand CR or k krandi ge k

k i geo w

k

do

c

(3.5)

where CR is cross over rate which is 0.5 1.CR

c) Selection Operation: The selection operation for the chromosome of next generation is

performed as,

, , ,( ) ( ), 1

,/

i ge i ge i geif err erri ge

i geo w

o o cc

c

(3.6)


39

Repeat this for all chromosomes.

Step 3 Termination: The termination criterion of DE is based on the following results

achieved,

(I) If , 1( ) ,

i geerr

c where is a very small positive number

(II) Total number of generation has reached,

Else go back to step 2.

Step 5 Storage: Store the entire results for later discussion and statistical analysis.

3.4 SIMULATED ANNEALING

The well known optimization techniques such as Conjugate-Gradient method, Qusai

Newton method, Golden search, steepest descent method, Davidon- Fletcher-Powell

method [96], [97] etc are all considered to be aggressive in a way that they make use of

the local approximation of cost function to move rapidly towards minimum. But the

minimum search out by these schemes is a local minimum. However, to deal with the

global minima, we need a global optimizer such as Simulated Annealing (SA). It uses

stochastic searching technology to search out the global minima. In literature, various

types of SA such as hybrid simulated annealing (HSA) [98], clustering algorithm, parallel

SA [99] and division algorithm have been applied to different optimization problems.

Simulated annealing (SA) method was first of all introduced in 1950 by Metropolis, in

which the process for crystallization model is explained. However, proper research on SA

has been carried out by Kirkpatric et al [100]. Simulated Annealing (SA) is a probabilistic

computational technique which is used for the local and global optimization problems

based on modeling of materials having controlled cooling and heating properties. The

core objective of SA is to find out the candidate solution efficiently and effectively in

fixed amount of time. In many optimization problems, the condition of differentiability,

convexity and continuity is required, while SA technique does not need it, which is its


40

main advantage. Many researchers have used SA in diverse field of engineering like the

transmission network expansion planning problem [101], 3D face recognition [102] and

unit commitment problems [103]. Some other applications of SA can be found in the

optimal reconfiguration of distribution networks [104], allocation of capacitors in

distribution feeders [105], reactive planning [106] and phase balancing [107]. Although,

SA is simple, efficient and performs well in the presence of local minima but not like the

other global optimizers such as GA, PSO and DE. The drawback of SA becomes well

exposed when the problem is highly convex and singular as it diverges for these

problems. The flow diagram of SA is shown in Fig.3.5.

Randomize according to

the current temperature

Better than current

solution?

Yes

No

Initialization

Stop

Replace current with

new solution

Lower temperature

bound reached ?

Reached maximum

tries for the

temperature ?

Decrease temperature by specified

rate

Yes

Yes

No

No

No

Start

Fig. 3. 5 Generic Flow diagram of Simulated Annealing


41

3.5 PATTERN SEARCH

Pattern Search (PS) is a numerical optimization technique which also does not need the

gradient of the problem. It was introduced by Hookes and Jeeves in 1961 and they

successfully applied it to statistical problems [108].The main goal of PS technique is to

compute a sequence of points that reach an optimal point. In each step, the technique tries

to find out a set of points called mesh around the optimal point of previous step. The

mesh can be obtained by adding the current point to a scalar multiple of vectors called a

pattern [109]. The new point becomes the current point in the next step of algorithm, if

the PS finds out the point in the mesh that improves the objective function at the current

point. In order to achieve, the optimum value of objective function, the PS is repeated

again and again until any of the following condition is satisfied.

The size of mesh become less than its tolerance,

The gap between successful and sub-sequent successful pole is lower than the set

tolerance,

The total number of iteration has reached at pre-defined value.

PS method is not only useful for optimization problems but also for parallel computing

[110]. In [111], PS method, based on the theory of positive basis is proposed by YU,

while in [112] the convergent capabilities of other PS techniques are discussed by

Torczon using the same positive bases. PS can also be used as a global optimizer alone

and as a global optimizer it has got applications in thermal control theory [113], character

and pattern recognition [114], optimal control theory [115] etc. However, as a global

optimizer the PS is not as accurate as GA, PSO and DE and better to use it as a local

search optimizer with any global search optimizers. The generic flow diagram of PS is

shown in Fig. 3.6.


42

Is objective function is

Achieved?

Read Data

Set Starting Point

Create mesh point

Evaluate objective function

Double mesh sizeReduce mesh size

Stopping Criteriayes

No

yesNo

Stop

Fig. 3. 6 Generic flow chart for Pattern Search technique.

3.6 INTERIOR POINT ALGORITHM

The Interior Point Algorithms (IPA), which is also called barrier methods, is a certain

class of algorithms which can be used for linear and non-linear convex optimization

problems [116]. Unlike simplex method, it reaches the optimum solution of the problem

by going through the feasible region of the problem rather than its surrounding [117].

This algorithm has been deduced from the well known Karmarkar's algorithm [118] and

is basically a constrained minimization solving technique. In order to get solution of the

approximated problems during each iteration, it uses either conjugate gradient step

through trust region or Newton step by using linear programming [119]. The whole

working procedure of IPA during each iteration can be summarized in the following two

steps,

a. Conjugate Gradient Step: A conjugate gradient step, via trust region.


43

b. Newton Step: It is a direct step which uses linear approximation so as to handle the

approximate problem by solving the Karush-Kuhn-Tucker (KKT) equations.

In fact, the IPA uses a Newton Step by default and if it does not then it tries to use the

conjugate gradient step. The case where it uses conjugate gradient step instead of Newton

step is that when the approximated problem is not locally convex near the ongoing

iteration. During each iteration, the algorithm tries to minimize the problem specific merit

function and in case where the attempted step is unable to get success in minimizing the

merit function, the algorithm rejects the present attempted step and tries a new step. The

most expensive step in terms of computation is the one when the algorithm computes the

Lagrangian equation. The purpose of this step is to find out whether the projected Hessian

is positive definite or not. If not, then the algorithm uses the other step called conjugate

gradient. For detail applications and derivation of the algorithm, it is recommended that

the reader should see [120]. The IPA performs well especially in the presence of less local

minima. However, in the presence of strong and more local minima, its performance is up

to the mark when it is used as local search optimizer with any global search optimizers.

3.7 ACTIVE SET ALGORITHM

Active set algorithm (ASA) is local search optimization method. It is mainly used in

constrained optimization problems. Its basic purpose is to transform the problem into an

easier solvable problem [121]. The solution of ASA is not guaranteed to be on one of the

edges of polygon since it uses quadratic programming. Actually, the ASA provides us a

subset of inequalities which basically decreases the complexity of the search during

searching of the solution hence it maintains a fairly high level precision in small

computational time. The ASA has been successfully used for many optimization

problems, such as, sparse linear programming problems [122], Box constrained

optimization problem [123], mathematical programming with equilibrium constraint


44

[124], problems in modern physics and astrophysics [125] etc. However, it performs even

better when it is hybridized with any other efficient global optimization method and in

this dissertation it is used as a local search optimizer with PSO and DE.

CHAPTER 4

DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF

FAR FIELD SOURCES

In this chapter, we have considered that the sources are impinging on sensors array from

the far field. This chapter contains three major parts. In part one, we have developed

efficient techniques based on GA and PSO for the joint estimation of 2-D parameters

(amplitude and elevation angle) of far field sources impinging on the ULA. For further

improvement in efficiency and reliability, GA and PSO are hybridized with PS. MSE is

used as fitness evaluation function which is basically derived from Maximum likelihood

Principle (MLP) to be discussed in detail in the subsequent sections. The result of the

hybrid (GA-PS) and (PSO-PS) schemes are compared with the individual responses of

GA, PSO and PS in terms of estimation accuracy, convergence rate and robustness

against noise.

In part two, we have developed optimization techniques based on GA, PSO, DE and SA

for the joint estimation of 3-D parameters (amplitude, elevation and azimuth angles) of

far field sources impinging on 1-L and 2-L shape arrays. To improve the results, the

global search optimizers are hybridized with PS. The MSE based fitness function is again

used and the results of these optimization techniques are compared to each other and also

with traditional techniques available in literature.

In part three, we have also included the frequency estimation of far field sources

impinging on 2-L shape array. It is in addition with the 3-D parameters of part two i.e.

amplitude, elevation and azimuth angles, the resultant problem is the estimation of four

parameters which is termed as 4-D. This time, we have used PSO-PS approach with new

CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES

46

multi-objective fitness function. This multi-objective fitness function is the combination

of MSE and correlation between the normalized desired and normalized estimated

vectors. It has shown better performance as compared to the previous one which were

based solely on MSE.

Most of the data presented in this chapter is taken from the publications [126], [127],

[128], [129] [130], [131], [132], [133], [134].

PART- 1

In this part, we have proposed hybrid schemes based on GA-PS and PSO-PS for the joint

estimation of amplitude and elevation angle of far field sources impinging on ULA.

4.1 DATA MODEL

In this section, the data model used is same as developed in section 2.1. As given in (2.2),

the unknown parameters are the elevation angle ( ) and amplitude ( )s so, clearly the

problem in hand is to jointly estimate them accurately and efficiently.

4.2 SIGNAL SUBSPACE DIMENSION

Before going to estimate the unknown parameters of the sources, it is important to first

know the dimension of signal sub-space. For this purpose, we have used non- parametric

approach.

y Bs n (4.1)

where s is a 1P source vector and B is our M P array manifold matrix. n is a

AWGN vector. The covariance matrix can be given as,

2Hn s ny S BSB I S S (4.2)

where I is the identity matrix and

[ ]H

ES ss (4.3)


47

We expect that the signals are incoherent, so that the rank of sS is equal to the number of

signals. Let the rank of sS be P, then eigen-decomposition of yS is given as,

H Hy s s s N N N S Q Λ Q Q Λ Q (4.4)

where

1

2 2 2

2[ . . . ]

Ps s s sdiag Λ (4.5)

2 2 2

[ . . . ]N n n ndiag Λ with M P sensors. (4.6)

and sQ has column vectors which are eigen vectors of sS and nQ has column vectors

which are eigen vectors of nS . We expect the last M P eigen values representing noise

to be the smallest and also equal. For this, we can use the following hypothesis [135],

1( )

1( )

1

1

lnp

M

iM P i P

L M P

M M Pi

i P

(4.7)

This numerator is the arithmetic mean of M P smallest eigen values while

denominator is their geometric mean. We start with 1p and then increase .p When p

is correct, then the last M P eigen values are the smallest and are equal thus making

( ) 0L Pp . After having found P by this test, we know exactly the number of signals.

Whether any of these signals is friend, foe, or indifferent, is not the topic of concern for

this dissertation.

4.3 JOINT ESTIMATION OF 2-D PARAMETERS USING GA-PS

In this section, a hybrid (GA-PS) technique is developed for the joint estimation of

elevation angle and amplitude of far field sources impinging on ULA. The flow diagram

of the hybrid scheme is shown in Fig 4.1, while its steps in the form of pseudo code are

given below.


48

Initialize

Population

Randomly vary

Individuals

Fitness Evaluation

Termination

Criterion

Crossover

Selection

Mutation

Best Individual

Refinement

(Local search)

Global Best

Individual

Stop

Start

Fig. 4. 1 Flow Diagram for Hybrid GA-PS

Step1 Initialization: The first step is to initialize GA, that is, to produce randomly Q

number of chromosomes. In the current problem, the length of each chromosome is 2*P,

where P is the total number of sources. In each chromosome, the first P gene represents

the elevation angles while the next P genes correspond to the amplitude of sources. All

the Q chromosomes can be represented as,


49

1,1 1,2 1, 1, 1 1, 2 1,2

2,1 2,2 2, 2, 1 2, 2 2,2

,1 ,2 , , 1 , 2 ,2

P P P P

P P P P

Q Q Q P Q P Q P Q P

s s s

s s s

s s s

C

(4.8)

In (4.8),

, ,

: 01,2,..., , 1,2,...

:

qi qi

q P i b q P i b

Rq Q i P

s R L s U

where bL andUb are the lower and upper bounds of signals amplitude.

Step2 Fitness Function: One of the core step involved in every meta-heuristic technique

is the design of fitness function over which the solution for finding the most optimized

chromosome is directly dependent. For this purpose, we have used a fitness function

based on MSE which defines an error between the desired response and estimated ones.

For thq chromosome, it can be given as,

2

ˆ( ) (1 / )1

qm m

MMSE q M y y

m

(4.9)

Derivation: This fitness function is basically derived from the Maximum likelihood

principle (MLP) as,

2

22

1 1 ˆ ˆexp22 n

n

pN

y/s,θ y-Bs (4.10)

where the probability of y is to be maximized conditioned on .s,θ It is very obvious, that

to maximize ( / , ),p y s we need to minimize 2

ˆ ˆy Bs which is actually our MSE

(fitness function).

The most fascinating features of this fitness function are that it is easy to be understood,

easy to be implemented and it requires single snapshot to achieve the optimum results. It


50

avoids any ambiguity among the angles that are supplement to each others. It has the

advantage of linking automatically the DOA estimated in the previous snapshot to a

current estimated DOA which is the main issue in multiple targets tracking system. N

targets imply N! possible combination which requires some computations [136] while

using this MSE as fitness function, the new estimation of DOAs is automatically linked

with old estimation of angles from previous snapshot which obviously decreases the

computational complexity [137]. Moreover, it provides fairly good results even in the

presence of low Signal to Noise ratio (SNR).

In (4.9), ym is defined in (2.1) while ˆm

yq is defined as,

ˆ ˆ ˆexp( ( 1) cos( )1

Pqy c j m cm iP i

i

(4.11)

where ic and ˆP ic are defined in (4.8).

Step 3 Termination Criteria: The termination criterion is based on the fulfillment of any

of conditions. If fulfilled then go to step 5, else go to step 4.

The pre-defined MSE has been achieved which is12

10

The total number of iteration has been completed.

Step 4 Reproduction: We have used a MATLAB built-in optimization tool box for which

the parameters setting are provided in Table 4.1. Reproduce the new population by using

the parameters of elitism, mutation, crossover and go to step 2.

Step 5 Hybridization: Once we have found the results through GA, we need to improve

them further. The best individual chromosome of GA is given to PS as a starting point.

We have used the same MATLAB built-in optimization tool box for PS as well, for which

the parameters setting are also provided in Table 4.1.

Step 6 Storage: Store all the results for discussion, comparison and analysis.


51

Table 4. 1 Parameter settings for GA and PS

GA PS

Parameters Settings Parameters Setting

Population size 240 Poll method GPS Positive basis 2 N

No of Generation 1000 Polling order Consecutive

Migration Direction Both Way Maximum iteration 800

Crossover fraction 0.2 Function Evaluation 16000

Crossover Heuristic Mesh size 01

Function Tolerance 10-12 Expansion factor 2.0

Initial range [0-1] Contraction factor 0.5

Scaling function Rank Penalty factor 100

Selection Stochastic uniform Bind Tolerance 10-03

Elite count 2 Mesh Tolerance 10-06

Mutation function Adaptive feasible X Tolerance 10-06

4.3.1 Results and Discussions

In this section, we have evaluated the performance of hybrid GA-PS approach. Initially,

the comparison of GA-PS is made with GA and PS alone and finally it is compared with

MUSIC and ESPRIT algorithms. Throughout the simulations, the distance between the

two consecutive sensors in the ULA is kept same i.e. / 2d and all the values of

DOA are taken in radians. Each result is averaged over 100 independent runs.

4.3.1.1 Estimation Accuracy

To check the estimation accuracy of each scheme, three cases are considered on the basis

of different number of sources and no noise is added to the system in this sub-section.

Case 1: In this case, two far field sources are impinging on ULA where the ULA is

composed of five sensors. The desired values of amplitudes and elevation angles are

1 1 2 2( 7, 0.5236 ),( 9, 1.9199)s rad s . In this simple case, all the three

techniques have produced fairly good estimation accuracy as provided in Table 4.2.

However, it can be seen that GA becomes more accurate when hybridized with PS and

produce better results as compared to GA and PS alone. The second best scheme in this

archive is GA alone.


52

Table 4. 2 Estimation accuracy for two sources

Scheme 1S 1( )rad 2S 2 ( )rad

Desired Values 7.0000 0.5236 9.0000 1.9199

GA 7.0011 0.5251 9.0012 1.9187

PS 7.0051 0.5286 9.0065 1.9113

GA-PS 7.0000 0.5235 9.0000 1.9198

Case 2: In this case, the estimation accuracy is discussed for 3 sources and this time the

ULA is composed of 6 sensors. The desired values of amplitudes and elevation angles for

this case are 1 1 2 2 3 3( 3, 0.6109),( 8, 1.1345),( 4, 1.6581)s s s . Due to the

increase in sources (unknown), we faced few local minima due to which the performance

of each scheme is degraded as provided in Table 4.4. However, one can see again that the

hybrid scheme (GA-PS) has produced the most accurate results as compared to GA and

PS alone. The second best is again GA alone.

Table 4. 3 Estimation accuracy for three sources

Scheme 1S 1( )rad 2S 2 ( )rad 3S 3( )rad

Desired Values 3.0000 0.6109 8.0000 1.1345 4.0000 1.6581

GA 2.9976 0.6085 7.9976 1.1368 4.0023 1.6605

PS 2.9908 0.6201 8.0091 1.1252 4.0092 1.6673

GS-PS 2.9991 0.6118 7.9991 1.1354 4.0010 1.6590

4.3.1.2 Convergence

In this sub-section, we have discussed the convergence of all the above mentioned

schemes for two, three and four sources in the absence of noise. From convergence, we

mean total number of times a particular algorithm attained its required results. All these

results are taken for the same MSE of 210 . As shown in Fig 4.2, the hybrid scheme

(GA-PS) has high convergence rate for all number of sources as compared to GA and PS

alone. For two sources, the PS and GA converged 86 and 93%, respectively, while the

hybrid GA-PS technique converged 100%. For three and four sources, the convergence of

PS and GA significantly reduced while GA-PS has maintained fairly good convergence

i.e. above 90%.


53

Fig. 4. 2 Convergence rate vs number of sources

4.3.1.3 Robustness

In this sub-section, the MSE of each scheme is evaluated in the presence of noise. For this

simulation, we have assumed the same scenario as discussed in case-1, while the values

of SNR are ranging from 5 to 15dB. As shown in Fig 4.3, the GA-PS technique is robust

enough to produce fairly good MSE even in the presence of low SNR. The second best

MSE is given by GA alone.

Fig. 4. 3 MSE vs SNR

4.3.1.4 Comparison with MUSIC and ESPRIT algorithms

In this section, the estimation accuracy of GA-PS is compared with MUSIC and ESPRIT

algorithms for three and four sources respectively. One can see from Table 4.4 and 4.5

that estimation accuracy of GA-PS is better as compared with MUSIC and ESPRIT

2 3 40

20

40

60

80

100

[Number of Sources]

% C

onverg

ence R

ate

PS

GA

GA-PS

5 10 1510

-7

10-6

10-5

10-4

10-3

10-2

MSE vs SNR

[SNR in dB]

MS

E

PS

GA

GA-PS


54

algorithms. The second best results are produced by ESPRIT. Moreover, MUSIC and

ESPRIT fail to estimate amplitude along with DOA.

Table 4. 4 Comparison with MUSIC and ESPRIT for three sources

Scheme 1S 1( )rad 2S 2 ( )rad

Desired Values 7.0000 0.5236 9.0000 1.9199

MUSIC ------ 0.5414 ------ 1.9375

ESPRIT ------ 0.5245 ------ 1.9455

GA-PS 7.0000 0.5235 9.0000 1.9198

Table 4. 5 Comparison with MUSIC and ESPRIT for four sources

Scheme 1S 1( )rad 2S 2 ( )rad 3S 3( )rad 4S 4 ( )rad

Desired values 2.0000 0.6981 6.0000 1.3963 3.0000 2.2689 1.0000 2.7925

MUSIC ------ 0.7278 ------ 1.4242 ------ 2.3003 ------ 2.8205

ESPRIT ------ 0.7156 ------ 1.4155 ------ 2.2846 ------ 2.8081

GS-PS 1.9981 0.6910 5.9980 1.3982 3.0018 2.2708 1.0019 2.7943

4.4 JOINT ESTIMATION OF 2-D PARAMETERS USING PSO-PS

In this section, we have developed an approach based on PSO hybridized with PS for the

joint estimation of amplitude and DOA of far field sources impinging on ULA. The

generic flow diagram of the hybrid PSO-PS scheme is shown in Fig 4.4, while its steps in

the form of pseudo code are given as follows,

Step 1 Initialization: This step is exactly the same as the one developed in (4.8) for GA.

Step Ii Fitness Function: Find the fitness function of each particle by using the relation

given below,

1

1FF

MSE

(4.12)

This fitness function approaches 1 when the MSE approaches zero, where the MSE is

defined in (4.9). Now, in this step, store each particle as local best ( )l while the one

having maximum fitness function be stored as global best ( )bg .

Step Iii Termination Criteria: Terminate, if any of the following condition is satisfied

and go to step-VI, otherwise go to step IVa,


55

i) The required maximum fitness function value is achieved which is 1

ii) The desired MSE is attained which is 10-12

Step Iva Update Particle Velocity: To update the velocity of each particle, use the

following relation,

1 1 1 1 1(1 )( ) ( )1 2

n n n n n nb bq q q q q q

v v l c g c (4.13)

In (4.13), the first term in the right hand side represents the previous velocity which is

called as momentum or inertia. The second term is called as cognitive term which shows

the private thinking of the particle. This term is also known as remembrance or memory,

whereas, the last term is called as a social component that explains the collective behavior

of the population. Initially the value of 0.1 which means that more weightage is given

to local intelligence in the beginning. Then there is gradual increase in the value of

towards 0.9 which means that more weightage is given to collective intelligence in the

end. In (4.13), both 1 and 2 are positive constants and for the ongoing problem

1.1 2

Here, the velocity is doubly bounded i.e.

.max maxv v vqm

if ( ) , ( )max maxv n v v n vqm qm

& if ( ) , ( ) .max maxv n v v n vqm qm

Step IVB Update Particle Position: The position of each particle is updated as,

( ) ( 1) ( )c n c n v nqm qm qm (4.14)

Step Va Choose Local Best Particle: Replace the previous local best particle with the

new local best particle if the fitness of new local best particle is greater than that of the

previous one.


56

Step Vb Replace Global Best Particle: Similarly replace the previous global best particle

with the new one if the fitness of the new one is greater than that of the old one.

Initialized

Parameters

Initial Swarm with randomly

Taken Position and Velocity

Fitness

Evalution

Termination

Criteria

Present

Better than Lb

Lb = Present

Present

Better than Gb

Gb = Present

Update Velocity

and Position

Global best

Particle

Best Individual

Stop

Start

No

Yes

No

No

Yes

Yes

Refine by

Local Search

Fig. 4. 4 Generic flow diagram for Hybrid PSO-PS

Step VI Hybridization: In this step, the best particle achieved through PSO is given to PS

as starting point for further improvement. For PS, we have used MATLAB built-in

optimization tool box for which the parameters setting is provided in Table 4.1.

Step VII Storage: Store all the results for later discussion and comparison.


57

4.4.1 Results and Discussion

In this section, we have carried out several simulations to validate the performance of

PSO-PS. The results of PSO-PS are compared to that of PSO and PS alone. Initially all

the results are described without adding any noise, while at the end the robustness of each

scheme is examined in the presence of noise. All the values of DOA of sources are taken

in radians and each result is averaged over 100 independent runs.

4.4.1.1 Estimation Accuracy

In this case, the estimation accuracy of PSO-PS, PSO and PS is discussed for different

number of sources.

Case 1: This case discusses the estimation accuracy of two sources impinging on ULA.

The ULA consists of seven sensors and the desired values of amplitudes and DOA are

1 21.0000, 2.0000,S S 1 20.5236( ), 1.9199( )rad rad . One can perceive from

Table 4.6, that all the three schemes produced fairly precise results, however, among all

these techniques, the hybrid PSO-PS technique is proven to be the best. The next to best

performance is that of PSO alone.

Table 4. 6 Amplitudes and DOA estimation of two sources

Scheme 1S 2

S ( )1 rad ( )2

rad

Desired Values 1.0000 2.0000 0.5236 1.9199

PSO 1.0016 2.0016 0.5253 1.9215

PS 1.0028 2.0028 0.5264 1.9227

PSO-PS 1.0003 2.0003 0.5238 1.9201

Case II: In this section, the estimation accuracy is discussed for three sources. The ULA

consists of ten sensors, while the desired values of amplitudes and elevation angles for

these sources are1 2 3( 1, 2, 3)s s s , 1 2 3( 0.6109, 1.1345, 1.6581). Although

with the increase of unknowns (Sources), the accuracy of all schemes degraded slightly as


58

compared to case-I, but still the accuracy of PSO-PS is remarkable as compared to PSO

and PS alone as provided in Table 4.7.

Table 4. 7 Amplitude and DOA estimation of 3 sources

Scheme 1S 2

S 3

S 1( )rad ( )

2rad ( )

3rad

Desired Values 1.0000 2.0000 3.0000 0.6109 1.1345 1.6581

PSO 1.0044 2.0044 3.0043 0.6153 1.1388 1.6625

PS 1.0092 2.0091 3.0092 0.6201 1.1437 1.6673

PSO-PS 1.0009 2.0009 3.0009 0.6118 1.1354 1.6590

4.4.1.2 Convergence and MSE

In this sub-section, the convergence and MSE of each scheme is discussed.

Case 1: In this case, we considered two sources. Initially the ULA is composed of three

sensors. As provided in Table 4.7, the convergence and MSE of hybrid PSO-PS is better

than that of PSO and PS alone. The hybrid PSO-PS converged 98% with average MSE as

610 .

Table 4. 8 MSE and convergence for different numbers of element

No of sensors Scheme MSE %Convergence

3 PSO 510 90

PS 310 82

PSO-PS 610 98

5 PSO 610 93

PS 410 84

PSO-PS 710 100

7 PSO 610 97

PS 410 87

PSO-PS 810 100

The convergence of PSO alone is 90% with MSE as 510 , while the PS technique

converged 82% with MSE as 310 .The convergence and MSE of all schemes are also

checked for subsequent increase of sensors in the array and as a result, we can see the

improvement in terms of MSE and convergence as provided in Table 4.8.


59

Case II: In this case, the convergence and MSE are discussed for three sources. Initially

the ULA consists of four elements. As given in Table 4.9, the PSO-PS is more reliable

and it has a convergence of 95% with MSE as 6

10

. The second reliable technique is PSO

which has convergence of 85% with MSE as 5

10

. Again with the increase of sensors in

the ULA, improvement can be observed in terms of reliability and MSE especially for

PSO-PS technique.

Table 4. 9 MSE and convergence of all three schemes for different numbers of element

No of sensors Scheme MSE % convergence

4 PSO 710 85

PS 410 75

PSO-PS 910 95

6 PSO 810 88

PS 510 78

PSO-PS 1010 98

8 PSO 810 92

PS 610 82

PSO-PS 1110 100

Case III: In this case, we have discussed the convergence and MSE of all above

mentioned schemes for four sources. Initially we considered five elements in the ULA.

Due to increase of unknowns (sources), we faced few local minima due to which the

reliabilty of all the schemes are degraded.

Table 4. 10 MSE and convergence rate for different numbers of element

No of Sensors Scheme MSE % Convergence

5 PSO 610 78

PS 310 60

PSO-PS 810 85

7 PSO 710 82

PS 410 62

PSO-PS 910 88

9 PSO 710 85

PS 510 65

PSO-PS 910 92


60

As provided in Table 4.10, again the PSO-PS is fairly reliable as compared to PSO and

PS alone. The reliability of all three schemes improved slightly with the increase of

elements in the ULA.

Case 1V: In this case, we have evaluated the performance of PSO, PS and PSO-PS

against noise. All the values of noise is taken in dB which ranges from 5dB to 30dB .

This experiment is performed for two sources and seven elements in the array. It has been

shown in Fig 4.5, that the robustness against noise of PSO-PS is better as compared to

that of PSO and PS alone for all values of SNR. Second best result is given by PSO itself.

Fig. 4. 5 Performance analysis of MSE vs SNR

PART-II

In part 1, we have discussed the 1-D DOA estimation of far field sources impinging on

ULA which was comparatively easy, as the DOA was the function of elevation angle

only. On the other hand 2-D DOA estimation of sources is relatively complicated because

in this case, the DOA is a function of both elevation and azimuth angles. 2-D DOA

estimation is very important and has direct applications in radar, sonar, wireless

communication system etc. The main problems involved in 2-D DOA estimation are

estimation failure, pair matching between elevation and azimuth angles, computational

complexity and higher MSE. In literature, several algorithms have already been proposed

to address the issue of 2-D DOA estimation [138], [139], [140], [141] but they have one

or the other aforementioned problems. In [142], Wu and Lioa proposed an algorithm

5 10 15 2010

-8

10-6

10-4

10-2

100

[SNR in dB]

MS

E

PSO

PS

PSO-PS


61

based on Propagator Method (PM) to overcome the computational load of [141] but it

failed to avoid the pair matching problem and estimation failure problem in the range of

1.2217 (radians) to 1.5708 (radians). This range is very important in mobile

communication. Besides, parallel shape array is used in [142] which needs not only more

sensors, but also requires a large number of snap-shots to achieve the goal (at least 200

snap-shots per sensor are required). In [143], the same PM is used with L shape arrays (1-

L & 2-L shape arrays) which tried to surmount the drawbacks of [142] but it also needs a

large number of snap-shots and sensors. Moreover, all of them failed to estimate the

amplitude of sources which is also an important parameter to be estimated.

In order to overcome these problems, we use meta-heuristic techniques and L-shape

arrays (1-L & 2-L shape). In this current section, GA, PSO, DE and SA are hybridized

with PS for the joint estimation of amplitude and 2-D DOA estimation of far field sources

impinging on L shape arrays. All the proposed hybrid schemes have used MSE as fitness

evaluation function as discussed in the previous section.

4.5 DATA MODEL

In this section, a data model is developed for P independent sources impinging on 1-L

and 2-L shape arrays from far field. The 1-L shape array consists of two sub-arrays which

are placed along X- axis and Z-axis as shown in Fig 4.6. The 2-L shape array is composed

of three sub-arrays that are placed along X-axis, Y-axis and Z-axis as shown in Fig 4.7.

The number of antenna elements (sensors) in each sub-array is M-1, while the reference

element is common for all sub-arrays in both arrays. The distance “𝑑” between the two

consecutive sensors in each sub-array is kept same i.e. / 2 . All the signals are

considered to be narrow band with known frequency (𝜔0) and having different amplitude

si , elevation angles i and azimuth angles i for 1,2,...i P .


62

. .

.

.

y

x

z

.

.

0

1

2

M-1

1 2

M-1 ∅i

θi

=1,2,…,p

Fig. 4. 6 Geometry of 1-L shape array

1 m M-10

Far Field ith source

where i= 1,2,… P

d

Si

1

1

m

m

M-1

M-1

Z-axis

Y-axis

X-axis

d

d

m

i

i

Fig. 4. 7 Geometry of 2-L shape array

4.5.1 1-L Shape Array: The sub-array along z-axis is used to estimate the elevation

angle, while the sub-array along x-axis is used to estimate the azimuth angle. The output

of th

m sensor in the z-axis sub-array is given as,

(1 ) ( )1

Py L s a nzm i zm i zm

i

(4.15)


63

In (4.15), (1 )y Lzm represents the output of thm sensor placed in z-axis sub array in 1-L

shape array whereas 2 cos

( ) exp( ).md ia jzm i

For / 2d and

0,1,2... 1,m M the output of complete sub-array in matrix-vector form can be written

as,

1 10 01

cos cos11 2 1

...

( 1) cos( 1) cos 11 1

y nsj j p

e ey s n

j M Pj M ey s nPM Mez z

(4.16)

Generally, it can be represented as,

( )z z z y B s n (4.17)

where zB is called the steering matrix which contains the elevation angles of the received

signals while s is a vector of signals amplitude. zn is an additive white Gaussian noise

(AWGN) vector added at the output of each sensor along z-axis.

Similarly, the output of x-axis sub-array at th

m sensor can be written as,

(1 ) ( , )1

Py L s a nxm i xm i i xm

i

(4.18)

where 2 sin cos

( , ) exp( ).md i ia jxm i i

For / 2 & 0,1,2,... 1,d m M the

output in matrix-vector form can be written as,

1 10 01sin cos sin cos

1 11 2 1

...

( 1) sin cos( 1) sin os1 11 1

y nsj j

P Pe ey s n

j MP Pj M c ey s nPM Mx xe

(4.19)


64

It can be represented as,

( , )x x x y B s n (4.20)

where the steering matrix ( , )x B contains the elevation and azimuth angles of the

received signals, while xn is AWGN added at the output of each sensor along x-axis.

4.5.2 2-L Shape Array: The 2-L shape array consists of three sub-arrays. The output of

x-axis and z-axis sub-arrays are exactly similar as discussed above for 1-L-shape array,

however, the output of y-axis sub-array at th

m sensor can be represented as,

(2 ) ( , )1

Py L s a nym i ym i i ym

i

(4.21)

where 2 sin sin

( , ) exp( ),md i ia jym i i

and thus the output in matrix-vector

form can be written as,

0 011 1

21 1

1 11 1

1 1

sin sin sin sin

...

( 1) sin sin( 1) sin sin

P P

P PPM M

y nsj j

e e sy n

j Mj M e sy n

y ye

(4.22)

which can be written in vector form as,

( , )y y y y B s n (4.23)

4.6 JOINT ESTIMATION OF 3-D PARAMETERS USING GA-PS AND SA-PS

The general settings consisting of population size, number of generations of the

algorithm, function evaluations and stoppage criteria is defined in Table 4.1 and Table

4.10, along with some specific parameter setting, values of the three algorithms based on


65

GA, PS and SA. The logical steps for GA and GA-PS in the form of pseudo code are

given in the following steps.

Step I Initialization: Generate randomly Q number of chromosomes (Particles) where the

length of each chromosome is 3*P. In each chromosome the first P genes represent

elevation angles, the next P genes represent the azimuth angles, while the last P genes

represent the amplitudes as given below,

1,1 1,2 1, 1, 1 1, 2 1,2 1,2 1 1,2 2 1,3

2,1 2,2 2, 2, 1 2, 2 2,2 2,2 1 2,2 2 2,3

,1 ,2 , , 1 , 2 ,2 ,2 1 ,2 2 ,3

s s sP P P P P P P

s s sP P P P P P P

s s sQ Q Q P Q P Q P Q P Q P Q P Q P

C

(4.24)

In the above matrix ,2:qj a q P j as R L s U where aL

and aU are the lower and

upper bounds of signal amplitudes 1,2,...q Q and 1,2,... .j P In the same way, the

lower and upper bounds for elevation and azimuth angles are,

: 0 / 2,qj qjR 1,2,... &q Q 1,2,...,j P

,: 0 2 ,qj q P jR 1,2,... &q Q 1,2,... .j P

Step II Fitness Function For 1-L Shape Array: The same MSE is used as fitness

function as discussed above. In case of 1-L shape array, for th

q chromosome, it can be

given as,

1( ) ( ( ) ( ))

2MSE q E q E qx z

M (4.25)

where

21ˆ( )

0m

M qE q y yx x xm

m

(4.26)


66

21ˆ( )

0m

M qE q y yz z zm

m

(4.27)

In (4.26) and (4.27), yzm and yxm are defined in (4.15) and (4.18), respectively, whereas

ˆq

yzm and ˆq

yxm are defined as,

ˆ ˆ êxp[ cos( )],,21

Pqy c j m czm q iq P i

i

(4.28)

ˆ ˆ ˆ êxp[ sin( )cos( )], ,,21

Pqy c j m c cxm q i q P iq P i

i

(4.29)

where c is defined in (4.24).

Step III Fitness Function For 2-L Shape Array: In case of 2-L shape array, the MSE for

thq chromosome can be given as,

1( ) ( ( ) ( ) ( ))

3MSE q E q E q E qx y zM

(4.30)

where Exq and Ezq are similar as defined in (4.26),(4.27) while Eyq is defined as,

21ˆ( )

0m

M qE q y yy y ym

m

(4.31)

In (4.31), yym is defined in (4.21) while ˆq

yymis defined as,

2ˆ ˆ ˆ êxp[ sin( )sin( )], , ,1

P i P iqym i

Py c j m c cq q q

i

(4.32)

One feature of 2-L shape array is that it can be used for the elevation angle beyond / 2,

so the range of elevation angle is defined as,

: 0 ,Rqj qj 1,2,... &q Q 1,2,...j P

Step IV Termination Criteria: The termination criteria depends on the following

conditions being achieved.


67

a) If the objective function value is achieved which is pre- defined i.e.1210

b) Total number of iterations has been completed

Step V Reproduction: New population is generated by using the operators of Crossover,

Elitism and Mutation Selection as provided in Table 4. 1.

Step VI Hybridization: In this step, the best results got through GA for L shape arrays (1-

L and 2-L shape arrays) are further given to PS for more improvement. The setting used

for PS is also provided in Table 4. 1.

Step VII Storage: Store global best of the current iteration which will be used for

comparison and better statistical analysis and repeat step II to V for enough numbers of

independent runs.

We have also used the same MATLAB optimization tool box SA and to improve further,

the best individual results of SA are given to PS as starting point. The parameters setting

for SA is listed in Table 4.11.

Table 4. 11 parameters setting for SA

SA

Parameters Setting

Annealing Function Fast

Reannealing interval 100

Temperature update function Exponential temperature update

Initial temperature 100

Data type Custom

Function Tolerance 10-12

Max iteration 2000

Max function evaluations 3000*number of variables

Temperature update function Exponential Temperature update

Hybrid function call interval End

4.6.1 Result and Discussions

In this section, simulations are performed to assess the performance of proposed schemes.

These simulations are mainly divided into two parts. In the first part, GA, PS, SA, GA-PS

and SA-PS are examined for 2-L shape array in terms of estimation accuracy,

convergence rate and the results are compared with 1-L shape array of [130]. In [130], the

same five techniques have been used for the joint estimation of amplitude and 2-D DOA


68

impinging on 1-L shape array. In the second part, only the performance of the best

scheme among the above five mentioned schemes is discussed and is compared with

Propagator Method that has used parallel shape array [142]. Throughout the simulations,

the distance ―d‖ between the two consecutive sensors in each sub-array is kept 𝜆/2. The

entire results are comprehensively examined for 100 independent runs.

Case 1: In this case, the estimation accuracy of GA, PS, SA, GA-PS and SA-PS are

discussed for two sources impinging on 2-L type array. For better comparison and

analysis all the values of amplitudes and DOA are taken to be same as in [130]. Hence,

1 1 1( 1, 0.5236 , 1.2217 )s rad rad and 2 2 22, 0.8720 , 1.9199 )(s rad rad

where 1 1 1( , , )s , 2 2 2( , , )s represent the amplitudes, elevation and azimuth angles of

first and second source respectively. The 2-L-shape array is composed of 4 sensors that is

1-sensor is placed along each sub-array while the reference sensor is common for them.

As provided in Table 4.12, all the five techniques have produced fairly good estimate of

the desired values. However, among them, the hybrid GA-PS approach has produced

better results as compared to the other four techniques. The second and third best results

are given by GA and PS respectively. The results of SA are also improved when

hybridized with PS.

Table 4. 12 Estimation accuracy of 2-L shape array for 2 sources

Scheme 𝑠1 𝜃1(rad) ∅1(rad) 𝑠2 𝜃2(rad) ∅2(rad)

Desired 1.0000 0.5236 1.2217 2.0000 0.8727 1.9199

GA 1.0003 0.5240 1.2221 2.0003 0.8731 1.9203

PS 1.0020 0.5259 1.2239 2.0021 0.8748 1.9220

SA 1.0977 0.5385 1.2361 2.0178 0.8889 1.9284

SA-PS 1.0047 0.5278 1.2260 2.0047 0.8770 1.9242

GA-PS 1.0000 0.5235 1.2216 2.0000 0.8726 1.9198

The results obtained in [130] for the same 2 sources are provided in Table 4.12 that has

used 1-L shape array composed of 7 sensors. One can clearly deduce from the

comparison of Table.4.11 and Table 4.13 that in case of 2-L shape array all the schemes


69

produced better results than 1-L shape array. In addition, the 2-L shape array requires less

number of sensors as compared to 1-L shape array to achieve the required goals.

Table 4. 13 Estimation accuracy of 1-L shape array for 2 sources

Scheme 𝑠1 𝜃1(rad) ∅1(rad) 𝑠2 𝜃2(rad) ∅2(rad)

Desired 1.0000 0.5236 1.2217 2.0000 0.8727 1.9199

GA 1.0008 0.5243 1.2217 2.0008 0.8735 1.9207

PS 1.0032 0.5268 1.2225 2.0033 0.8759 1.9231

SA 1.0196 0.5432 1.2249 2.0195 0.8923 1.9395

SA-PS 1.0063 0.5299 1.2413 2.0063 0.8790 1.9263

GA-PS 1.0003 0.5233 1.2281 2.0002 0.8723 1.9195

Case II: In this case, the estimation accuracy of all above mentioned five techniques are

discussed for three sources impinging on 2-L shape array. This time the 2-L shape array

consists of 7 sensors i.e. 2 sensors are placed along each sub-array while the reference

sensor is common for them. For better comparison with 1-L type array, same values of

amplitudes and DOA are used as given in [130]. In this case, few local minima are

observed due to which the performance of all five techniques, especially SA, SA-PS and

PS are significantly degraded as given in Table 4.14. However, again the hybrid GA-PS

showed excellency in accuracy even in the presence of local minima. The second best

result is given by GA alone.

Table 4. 14 Performance of 2-L type array for 3 sources

Scheme 𝑠1 𝜃1(rad) ∅1(rad) 𝑠2 𝜃2(rad) ∅2(rad) 𝑠3 𝜃3(rad) ∅3(rad)

Desired 1.0000 0.1745 0.5236 2.0000 0.8727 1.9199 3.0000 1.3090 2.4435

GA 1.0043 0.1788 0.5278 2.0043 0.8769 1.9243 3.0042 1.3132 2.4478

PS 1.0189 0.1934 0.5427 2.0190 0.9005 1.9389 3.0189 0.8917 2.4624

SA 1.0509 0.2254 0.5744 2.0508 0.9237 1.9708 3.0509 1.3598 2.4944

SA-PS 1.0342 0.2087 0.5581 2.0342 0.9069 1.9541 3.0343 1.3432 2.4778

GA-PS 1.0003 0.1748 0.5240 2.0004 0.8730 1.9202 3.0003 1.3094 2.4439

The results of 1-L type array are provided in Table 4.15, which required thirteen sensors

[130]. One can observe that the proposed schemes along with 2-L shape array produced

better accuracy by using less number of sensors as compared to 1-L shape array.


70


Scheme s1 θ1 ∅1 s2 θ2(rad) ∅2(rad) s3 θ3(rad) ∅3(rad)

Desired 1.0000 0.1745 0.5236 2.0000 0.8727 1.9199 3.0000 1.3090 2.4435

GA 1.0073 0.1818 0.5309 2.0073 0.8800 1.9272 3.0073 1.3163 2.4508

PS 1.0278 0.2023 0.5514 2.0277 0.9005 1.9477 3.0277 1.3368 2.4713

SA 1.0610 0.2355 0.5846 2.0611 0.9337 1.9809 3.0610 1.3700 2.5045

SA-PS 1.0432 0.2177 0.5668 2.0432 0.9159 1.9631 3.0432 1.3522 2.4867

GA-PS 1.0011 0.1756 0.5247 2.0011 0.8738 1.9210 3.0011 1.3101 2.4446

Case III: In this case, the estimation accuracy is examined for 4-sources. The 2-L shape

array consists of 10 sensors i.e. 3 elements are placed along each sub-array, while the

reference element is common for them. As provided in Table 4.16, again the hybrid GA-

PS leads the edge over the remaining four techniques in terms of estimation accuracy.


Scheme s1 θ1 ∅1 s2 θ2(rad) ∅2(rad) s3 θ3(rad) ∅3(rad) s4 θ4(rad) ∅4(rad)

Desired 1.0000 0.2618 1.6581 2.0000 0.6109 2.1817 3.0000 1.0472 2.7925 4.0000 1.4835 3.4034

GA 1.0102 0.2721 1.6683 2.0103 0.6212 2.1920 3.0103 1.0576 2.8028 4.0102 1.4937 3.4137

PS 1.0313 0.2932 1.6895 2.0312 0.6423 2.2132 3.0313 1.0787 2.8239 4.0314 1.5148 3.4348

GA-PS 1.0040 0.2659 1.6624 2.0042 0.6151 2.1858 3.0043 1.0514 2.7966 4.0040 1.4877 3.4077

SA 1.1011 0.3628 1.7594 2.1010 0.7122 2.2829 3.1012 1.1483 2.8937 4.1010 1.5846 3.5044

SA-PS 1.0787 0.3405 1.7369 2.0786 0.6895 2.2606 3.0785 1.1260 2.8713 4.0787 1.5624 3.4821


Scheme s1 θ1 ∅1 s2 θ2(rad) ∅2(rad) s3 θ3(rad) ∅3(rad) s4 θ4(rad) ∅4(rad)

Desired 1.0000 0.2618 1.6581 2.0000 0.6109 2.1817 3.0000 1.0472 2.7925 4.0000 1.4835 3.4034

GA 1.0163 0.2781 1.6744 2.0163 0.6272 2.1980 3.0162 1.0635 2.8088 4.0163 1.4998 3.4197

PS 1.0425 0.3043 1.7006 2.0425 0.6534 2.2242 3.0426 1.0897 2.8350 4.0426 1.5260 3.4468

GA-PS 1.0083 0.2701 1.6664 2.0083 0.6192 2.1900 3.0083 1.0555 2.8008 4.0083 1.4918 3.4117

SA 1.1263 0.3881 1.7844 2.1263 0.7372 2.3080 3.1263 1.1735 2.9188 4.1164 1.6098 3.5306

SA-PS 1.0932 0.3550 1.7513 2.0932 0.7041 2.2749 3.0932 1.1404 2.8857 4.0932 1.5767 3.4966

The results for 1-L shape array for same schemes and same number of sources are

provided in Table 4.17 which needs fifteen sensors [130]. One can again make out the

advantages in terms of accuracy and number of sensors by using 2-L shape array instead

of 1–L shape.

Case 1V: In this case, the convergence is evaluated for 2-L shape array against different

number of sources in the presence of 10dB noise. For this experiment, the MSE is kept

same i.e. 2

10

. As shown in Fig 4.8, the convergence rates of all schemes are degraded


71

with the increase of sources. However, one can notice that the convergence of GA is

remarkable in case of hybridization with PS. The second best convergence is produced by

GA alone. The convergence of 1-L shape array is shown in Fig 4.9 and one can conclude

that all the five schemes have better convergence for 2-L shape array.

Fig. 4. 8 Convergence vs number of sources for 2-L shape array at 10 dB noise

Fig. 4. 9 Convergence vs number of sources for 1-L shape array at 10 dB noise.

Till the performance of GA, PS, SA, SA-PS and GA-PS is discussed for both 1-L and 2-L

type arrays and it has been shown through various cases that GA-PS produced fairly good

results for both arrays. So, from now onwards our focus will be limited only to the GA-PS

technique. In the upcoming second part of simulations, we compared GA-PS technique

using 2-L shape array with the same GA-PS technique using 1-L shape array [130] and

also with PM that has used parallel shape array [142].

Case V: In this case, GA-PS technique using L shape arrays (1-L and 2-L) is compared

with propagator Method that has used parallel shape array. In this regard, Table 4.18,

2 3 40

10

20

30

40

50

60

70

80

90

100

[Number of sources]

% C

onverg

ence

GA-PS

PS

SA-PS

SA

GA

2 3 40

10

20

30

40

50

60

70

80

90

100

Number of sources]

% C

onverg

ence

GA-PS

PS

SA-PS

SA

GA


72

Table 4.19 and Table 4.20 have listed the variances, means and standard deviations for

parallel shape array with PM method and L shape arrays with hybrid GA-PS approach

respectively. For this, the elevation angle is varied in the range of 1.2217 (rad) and 1.5708

(rad) for fixed azimuth angle of 0.6109 (rad) in the presence of 10 dB noise.

Table 4. 18 Means, Variances and standard deviations at 10 dB noise for different elevation angles and fixed

azimuth angle by using PM with parallel shape array

𝜃 in radians for

∅ = 0.6109 (rad)

Mean of 𝜃 (rad) Variance of 𝜃 (rad) Standard Deviation of 𝜃

(rad)

1.2392 1.2227 0.0123 0.0146

1.2915 1.2728 0.0171 0.0167

1.3439 1.3207 0.0228 0.0200

1.3963 1.3696 0.0403 0.0265

1.4486 1.4155 0.0822 0.0379

1.5010 1.4552 0.1250 0.0467

1.5533 1.4784 0.1631 0.0534


azimuth angle by using GA-PS with 1-L shape array

𝜃 in radians for

∅ = 0.6109 (rad)


(rad)

1.2392 1.2394 2.0963e-006 1.9128e-004

1.2915 1.2913 2.2918e-006 2.0000e-004

1.3439 1.3436 2.4819e-006 6.5816e-004

1.3963 1.3965 3.8608e-006 2.5959e-004

1.4486 1.4493 2.1907e-006 1.9554e-004

1.5010 1.5012 3.6759e-006 2.5329e-004

1.5533 1.5535 2.3424e-006 2.0219e-004


azimuth angle by using GA-PS with 2-L shape array

𝜃 in radians for

∅ = 0.6109 (rad)


(rad)

1.2392 1.2392 2.6389e-007 6.7866e-005

1.2915 1.2915 1.9565e-007 5.8436e-005

1.3439 1.3439 4.0858e-007 8.4446e-005

1.3963 1.3964 2.5674e-007 6.6940e-005

1.4486 1.4487 2.3073e-007 6.3459e-005

1.5010 1.5011 5.9533e-007 1.0193e-004

1.5533 1.5534 2.0944e-007 6.0460e-005

The performance of PM method with parallel shape array is getting worse especially

when the elevation angle approaches 1.5708 (rad). On the other hand the GA-PS


73

technique using 2-L shape array has produced better results for this range of elevation

angles. The second best result is given by GA-PS using 1-L shape array. This range of

elevation angles is of practical importance in mobile communication, so, 2-L shape array

with GA-PS technique is a good choice to be used.

Case VI: In this case, we discussed the computational complexity of GA-PS using L

shape arrays and PM with parallel shape array. The PM required O(3 x M x T x K)

computations where M, K and T represent the total number of sensors, sources and

snapshots respectively. The total number of snapshots required for PM is 200 [142]. On

the other hand, the major computations involved in GA-PS using 2-L shape array are the

total number of multiplication in fitness function (Q2 (3+16 x K) plus the multiplications

involved in cross over which is approximately 16 x Q2

and the multiplication required for

PS which is 16 x K. Here, 12Q , which is the number of chromosomes. So, the total

number of major multiplications are O(Q2(3+32 x K) +16 x K). In the same way, the

major computations required for GA-PS technique using 1-L shape array are O(Q2 (3+ 20

x K) +20 x K). The GA-PS technique using 2-L shape array is computationally less

expensive than PM using parallel shape array [142], but computationally more expensive

as compared to GA-PS using 1-L shape array [130].

Case VII: In this simulation, we compared the Root-Mean Square Error (RMSE) of GA-

PS technique using L shape arrays with PM method [142]. In this, single source is

considered which has elevation and azimuth angles 1.0472(rad) and 1.9199(rad),

respectively. The SNR is ranging from 5 dB to 25 dB. It is quite obvious from Fig 4.10,


74

Fig. 4. 10 Root- Mean-Square Error vs SNR

that GA-PS technique using 2-L shape array maintained minimum values of RMSE for all

values of SNR, while the second best scheme is the other GA-PS technique with 1-L

shape array.

Case VIII: In Table 4.21, some general properties of parallel shape array [142], 1-L

shape array [130] and 2-L shape array are listed. The main draw backs of parallel shape

array with PM include estimation failure in the range of 1.2217 (rad) to 1.5708 (rad),

computational complexity, pair matching problem and requirement of more sensors.

Some of the drawbacks have been covered up by 1-L shape array using GA-PS technique

[130], however, the main disadvantages of 1-L shape array [130] is the range limitation of

elevation angles beyond / 2 .

Table 4. 21 Comparison among 2-L shape aray, 1-L shape array and parallel shape array

Property Parallel Shape array

[142]

1-L shape array [130] 2-L shape array

Scheme used PM GA-PS GA-PS

Range of elevation and

azimuth angles

(0, / 2), (0, 2 ) (0, / 2), (0, 2 ) (0, ), (0, 2 )

Number of estimated

sources

2 2 2

Number of sensors

required

33 7 4

Number of snap-shots

required

M T 1 1

Pair matching Required Not required Not required

Estimation failure 1.2217rad to 1.5708rad No failure No failure

5 10 15 20 25-35

-30

-25

-20

-15

-10

-5

0

[SNR in dB]

RM

SE

(dB

)

PM using parallel shape array [142]

GA-PS using 1-L shape array [130]

GA-PS using 2-L shape array


75

Moreover, it also requires more sensors. The 2-L shape array with GA-PS technique is

more effective and requires not only minimum number of sensors, but also covers the

range of elevation angle between / 2 to which is of great practical importance in

mobile communication.

4.7 JOINT ESTIMATION OF 3-D PARAMETERS USING DE-PS AND PSO-PS

In this section, PSO and DE are hybridized with PS to jointly estimate the amplitude and

2-D DOA of far field sources impinging on L-shape (1-L & 2-L) arrays. Initially, their

results are compared with each other and then with propagator method that has used

parallel and L shape arrays.

4.7.1 Differential Evolution Hybridized With Pattern Search (DE-PS)

The flow diagram of hybrid DE-PS is shown in Fig 4.11 while the algorithm steps in the

form of Pseudo code are given as,

Step I Initialization: The initialization step is similar to the one developed for GA in the

previous section as given in (4.24).

Step II Updating: In this step, we update all chromosomes (particles) from 1 to Q of the

current generation ‗ge‘. Suppose we select th

i chromosome i.e. ,i gek

c from (4.24), where

1,2,... & 1,2,...3i Q k P and ‗ge‘ represent the particular generation. Now the goal

is to find the chromosome of next generation i.e. , 1i ge

c by using the following steps,

A) Mutation: In this step, one can pick up any three different numbers (chromosomes)

from 1 to Q i.e. 1 2 3( , , )n n n under the following conditions,

1 , ,1 2 3n n n Q

where

, 1,2,3n n i ki k


76

1,2,3n ii i

now,

31 2 ,, , ,( )

n gei ge n ge n geF d c c c (4.33)

where ‗F‘ is a constant whose values usually lie in the range 0.5 to 1.

B) Crossover: The crossover can be performed as,

,(),

,/

i geif rand CR or k krandi ge k

k i geo w

k

do

c

(4.34)

where 0.5 1CR and krand is between 1 and 3*P chosen at random.

C) Selection Operation: The selection operation for the chromosome of next generation

is performed as,

, , ,( ) ( ), 1

,/

i ge i ge i geif err erri ge

i geo w

o o cc

c (4.35)

where the ,( )

i gec and ,

( )i ge

o are defined in (4.24). Repeat this for all chromosomes.

Step III Termination: The termination criterion of DE is based on the following results

achieved,

(I) If , 1( ) ,

i geerr

c where is a very small positive number,

(II) Total number of generation has reached,

else go back to step 2.

Step IV Hybridization: In this step, the best results achieved through DE are given to PS

for further refinement. Use Table 4.1, for the parameters setting of PS.

Step V Storage: Store all the results for later on discussion and comparison.


77

Initialize

Population

Update the Generations

Calculate the next

generation chromosomes

Termination

Criterion

Mutation

Cross Over

Selection Best Individual

Local search technique

Yes

No

Stop

Start

Fig. 4. 11 Flow chart of hybrid DE

4.7.2 Particle Swarm Optimization Hybridized With Pattern Search (PSO-PS)

The generic flow diagram of PSO-PS is shown in Fig. 4.4, while its step in the form of

pseudo code are given as,

Step I Initialization: This step is exactly similar to the one discussed above in (4.24).

We

have produced randomly Q particles for both L shape arrays. The main difference

between the particles generated for both L- shape arrays is the range difference of

elevation angles. For 1-L shape array, the particles are generated in the range of

0 / 2 while in case of 2-L shape array, the range of elevation angles is 0 .


78

The range of amplitudes and azimuth angles is the same for both L -shape arrays. The

lower and upper ranges of , and s are defined as,

(1 ) : 0 / 2, ,

(2 ) : 0, ,, 1,2,... & 1,2,...

: 0 2, ,

:,2 ,2

L Ri k i k

L Ri k i kfor i Q k P

Ri P k i P k

s R L s Ui P k b i P k b

where Lb and Ub represent the lower and upper bounds of signals amplitude.

Step Ii Fitness Function: MSE is used as fitness evaluation function for both L-shape

arrays. The goal is to minimize the MSE to get maximum fitness function. By using the

following relation, the fitness of each particle for both L-shape arrays can be found as,

1( )

(1 ( ))FF i

i

(4.36)

where ( )i defines the MSE between desired and estimated response. For 1-L-shape

array, it can be given as,

1( )1 22M

(4.37)

and for 2-L-shape array, it can be defined as,

1( )1 2 33M

(4.38)

where

21ˆ( )1

0

M ii y yzl zl

l

(4.39)

21ˆ( )2

0

M ii y yxl xl

l

(4.40)

21ˆ( )3

0

M ii y yyl yl

l

(4.41)


79

where ,,y y yzl xl yl are defined in (4.18), (4.20) and (4.21) while ˆ ˆ ˆ, ,i i i

y y yzl xl yl are

defined as,

ˆ (exp( cos( ))21

Pi i iy j l c czl k P kk

(4.42)

ˆ exp( sin( )cos( )) 21

Pi i i iy j l c c cxl k P k P k

k

(4.43)

ˆ exp( sin( )sin( )). 21

Pi i i iy j l c c cyl k P k P k

k

(4.44)

where i

c is defined in (4.24). Now, store the particle as a global best bg which has

maximum fitness function while mark each ic as a local best il for this step where

1,2,... .i Q

The remaining steps of PSO are similar as discussed above in part-1.


This section is also mainly divided into two sections. In the first section, various

simulations are performed to compare the estimation accuracy and reliability of PSO,

PSO-PS, DE and DE-PS for the joint estimation of amplitudes and DOA (elevation and

azimuth) of far field sources impinging on 1-L and 2-L shape arrays. In the second part of

simulation, the comparison is carried out with PM that has used parallel shape array [142]

and L-shape arrays [143]. We have used 60 particles and 60 generations for PSO and DE

respectively. Each result is averaged over 100 independent runs.

Case 1: In this case, the estimation accuracy of PSO, DE, PSO-PS, and DE-PS are

examined for 1-L and 2-L shape arrays without having any noise in the system. Two

sources are considered which have amplitudes and DOA are

1 1 1( 0.5, 30 , 110 ),s

( 2, 70 , 210 )2 2 2s

. The 1-L Shape array


80

consists of 9 sensors i.e. 4-sensors are placed along X-axis and Z-axis sub-array

respectively, while the reference sensor is common for them. As provided in Table 4.22,

one can clearly observe that the estimation accuracy of DE and PSO increases when they

are hybridized with PS. Originally the accuracy of PSO is less than that of DE but as the

PSO is hybridized with PS, the estimation accuracy of PSO becomes even better than that

of DE. However, the more accurate scheme is DE hybridized with PS ( DE-PS).

In Table.4.23, the estimation accuracy of the same four schemes is provided for 2-L shape

array. The 2-L shape array consists of 4 sensors i.e. each sub-array consists of 1 sensor

while the reference sensor is common for them. As listed in Table.4.22, again the hybrid

DE-PS approach created fairly accurate results as compared to the other three schemes.

The other hybrid approach (PSO-PS) has produced the second best accurate results.

Now by comparing Table.4.21 and Table 4.22, it can be deduced very easily that each

scheme has produced better estimation accuracy in case of 2- L shape array by using less

sensors as compared to 1-L shape array.

Table 4. 22 Estimation accuracy of 1-L-shape array for 2-sources

Scheme 𝑠1 𝜃1∘ ∅1

∘ 𝑠2 𝜃2∘ ∅2

∘

Desired 0.5000 30.0000 110.0000 2.0000 70.0000 210.0000

PSO 0.4951 30.0061 109.0038 2.0050 70.0060 210.0061

DE 0.5022 30.0044 110.0044 1.9977 69.0058 209.9955

PSO-PS 0.5018 29.9967 110.0033 2.0019 70.0036 210.0033

DE-PS 0.4991 30.0015 110.0015 1.9988 69.0082 209.9984



∘ 𝑠2 𝜃2∘ ∅2

∘

Desired 0.5000 30.0000 110.0000 2.0000 70.0000 210.0000

PSO 0.5036 29.9950 110.0050 1.9963 70.0049 210.0050

DE 0.4964 30.0034 110.0034 2.0014 69.9966 209.9965

PSO-PS 0.4998 30.0022 109.9978 1.9991 70.0021 209.9979

DE-PS 0.5001 29.9998 110.0002 2.0001 70.0001 210.0001

Case 2: In this sub-section, the estimation accuracy is discussed for 3 sources impinging

on L-shape arrays without having any noise added to the system. This time the 1-L and 2-

L shape arrays are composed of 13 and 7 sensors respectively. The desired values of


81

amplitudes, elevation and azimuth angles of these 3 sources are

1 1 1( 2, 60 , 15 ),s

( 4, 25 , 85 ),2 2 2s

3 3 36, 40 , 170( )s . As

we increase the number of sources, few local minima are observed due to which the

performance of each scheme is degraded for both L-shape arrays as listed in Table 4.24

and Table 4.25. Even in this case, again one can notice that the estimation accuracy of DE

and PSO increases when they are hybridized with PS technique. The best estimation is

produced by DE-PS for both L-shape arrays, while the second best result is given by the

other hybrid PSO-PS technique. However, one can also see that over all the entire

schemes have produced better estimation accuracy by using 2-L shape array with less

number of sensors as compared to 1-L shape array.



∘ 𝑠2 𝜃2∘ ∅2

∘ 𝑠3 𝜃3∘ ∅3

∘

Desired 2.0000 60.0000 15.0000 4.0000 25.0000 85.0000 6.0000 40.0000 170.0000

PSO 2.1789 60.3843 15.3847 3.8209 24.6158 84.6157 5.8210 39.6156 169.6153

DE 1.9028 58.8028 15.1979 3.9026 25.1974 85.1972 6.0971 40.1973 169.8022

PSO-PS 2.0191 59.9264 14.9261 4.0192 24.9261 84.9263 6.0193 40.0738 170.0737

DE-PS 1.9931 60.0268 15.0269 3.9932 24.9733 85.0260 5.9932 39.9731 170.0272

Table 4. 25 Estimation accuracy of 2L-shape array for 3-sources


∘ 𝑠2 𝜃2∘ ∅2

∘ 𝑠3 𝜃3∘ ∅3

∘

Desired 2.0000 60.0000 15.0000 4.0000 25.0000 40.0000 6.0000 40.0000 170.0000

PSO 1.8447 60.1791 15.1791 3.8445 25.1790 39.8207 6.1556 39.8207 170.1792

DE 2.0481 60.0977 14.9020 4.0480 24.9024 40.0979 6.0483 40.0979 169.9020

PSO-PS 2.0137 60.0328 15.0327 3.9862 25.0331 40.0330 6.0137 40.0330 170.0329

DE-PS 1.9980 59.9923 14.9925 4.0021 25.0077 39.9925 5.9980 39.9925 169.9926

Case 3: In this case, we have taken 4 sources which have the desired values of

amplitude, elevation and azimuth angles are 1 1 1( 1, 30 , 40 ),s

( 3, 50 , 65 ),2 2 2s

3 3 3( 5, 85 , 255 ),s

4 4 4( 7, 70 , 315 ).s

The 1-L and 2-L shape arrays consist of 15 and 10 sensors respectively. The estimation

accuracy of PSO and DE is degraded more due to increase of sources but their


82

performance in terms of estimation accuracy becomes excellent when they are hybridized

with PS.



∘ 𝑠2 𝜃2∘ ∅2

∘ 𝑠3 𝜃3∘ ∅3

∘ 𝑠4 𝜃4∘ ∅4

∘

Desired 1.0000 30.0000 40.0000 3.0000 50.0000 65.0000 5.0000 85.0000 255.0000 7.0000 70.0000 315.0000

PSO 0.5211 31.3274 41.3276 2.5212 51.3276 66.3278 5.4792 86.3282 256.4006 7.4789 69.6723 316.4102

DE 1.2988 29.0324 40.9678 2.7009 50.9678 64.0320 5.2987 85.9681 256.0673 7.2990 70.9679 316.0874

PSO-PS 0.8143 30.5741 39.4252 2.8140 50.5745 65.5748 5.1858 84.4252 255.6775 7.1858 70.5746 315.8776

DE-PS 1.0989 30.2468 40.2470 2.9908 50.2469 65.2473 4.9010 85.2471 254.7431 7.0990 69.7530 315.3571

Again, the top best result is produced by DE-PS, while the second best scheme in this

scenario is PSO-PS for both L-shape arrays. Overall, each scheme produced better results

using 2-L shape array with less number of antenna sensors as compared to 1-L-shape

array as listed in Tables 4.26 and 4.27.

Table 4. 27 Estimation accuracy of 2L-shape array for 4-sources


∘ 𝑠2 𝜃2∘ ∅2

∘ 𝑠3 𝜃3∘ ∅3

∘ 𝑠4 𝜃4∘ ∅4

∘

Desired 1.0000 30.0000 40.0000 3.0000 50.0000 65.0000 5.0000 85.0000 255.0000 7.0000 70.0000 315.0000

PSO 0.7202 31.1284 41.1282 3.2799 51.1285 66.1284 5.2797 86.1286 256.1385 7.2711 71.1283 316.1369

DE 1.0988 30.6656 40.6656 3.0990 49.3375 65.6659 5.0989 85.6654 254.2342 6.9010 70.6655 315.6766

PSO-PS 1.0477 30.2941 40.2941 3.0479 50.2942 65.2946 4.9522 84.7056 255.2944 7.0480 69.7060 315.3041

DE-PS 1.0109 30.0869 40.0870 3.0111 50.0870 64.9125 5.0100 85.0867 254.9125 7.0112 70.0869 314.9082

Case 4: In this case, the convergence of each scheme is discussed for 2, 3 and 4 sources

impinging on 1-L and 2-L shape array. The MSE is kept 2

10

for this simulation. The

number of sensors, values of amplitudes, elevation and azimuth angles are same as

discussed in the previous cases. As shown in Fig.4.12, the convergence of PSO and DE

has increased when both are hybridized with PS for all number of sources in case of 1-L

shape array. However, among all of them the DE-PS scheme got fairly good convergence.

The second best convergence is achieved by the other hybrid PSO-PS approach.

Similarly, for 2-L shape array, the DE-PS scheme has got best convergence, while the

second best convergence is achieved by PSO-PS as shown in Fig 4.13. From the

comparison of Fig.4.12 and Fig 4.13, one can easily verify that each scheme produced


83

good convergence in case of 2-L shape array as compared to their counterpart schemes

used for 1-L shape array.

Fig. 4. 12 Convergence Rate Vs Number of sources using 1-L shape array

Fig. 4. 13 Convergence Rate Vs Number of sources using 2-L shape array

Till now, we have discussed PSO, DE, PSO-PS and DE-PS for both L-shape arrays and

we have observed that DE-PS scheme performed well in case of both L- shape arrays. So,

in next case, we shall focus only on the performance of DE-PS scheme using both L-

shape arrays.

Case 5: In this sub-section, the proximity effect of elevation and azimuth angles are

discussed using DE-PS approach for both L shape arrays. This experiment is performed

for three sources and 7 antenna elements are placed in both L shape arrays. In Table 4.28,

we provided the proximity of elevation angles for fixed amplitudes and azimuth angles.

Although due to proximity of the elevation angles, the estimation accuracy and

convergence of DE-PS technique is degraded for both L-shape arrays but still it is robust

2 3 405

10152020253035404550556065707580859095

100

[Number of sources]

Converg

ence R

ate

PSO

DE

PSO-PS

DE-PS

2 3 405

101520253035404550556065707580859095

100

[Number of sources]

Converg

ence R

ate

PSO

DE

PSO-PS

DE-PS


84

enough to produce fairly accurate results. In addition, DE-PS has performed better in case

of 2-L-shape array as compared to 1-L- shape array.

Table 4. 28 Proximity effect of Elevation angle

Scheme 𝜃1∘ 𝜃2

∘ 𝜃3∘ Convergence (%)

Desired Values 30.0000 80.0000 50.0000 ---

DE-PS(1-L) 30.3843 80.3842 50.3844 90

DE-PS(2-L) 30.1791 80.1790 50.1793 94

Desired Values 30.0000 65.0000 75.0000 ---

DE-PS(1-L) 30.3846 65.9832 75.9834 84

DE-PS(2-L) 30.1792 65.4301 75.4302 92

Desired Values 30.0000 40.0000 50.0000 ---

DE-PS(1-L) 31.3965 41.4011 51.4013 70

DE-PS(2-L) 30.7692 40.7694 50.7690 88

Desired Values 30.0000 35.0000 40.0000 ---

DE-PS(1-L) 32.3417 37.3518 42.3519 64

DE-PS(2-L) 31.1105 36.1107 41.1105 82

Table 4. 29 Proximity effect of Azimuth angles

Scheme 𝜙1∘ 𝜙2

∘ 𝜙3∘ Convergence (%)

Desired Values 15.0000 80.0000 230.0000 ---

DE-PS(1-L) 15.3841 80.3840 230.3845 90

DE-PS(2-L) 15.1790 80.1791 230.1793 94

Desired Values 15.0000 80.0000 70.0000 ---

DE-PS(1-L) 15.3844 80.9830 70.9832 83

DE-PS(2-L) 15.1792 80.4301 70.4302 91

Desired Values 60.0000 70.0000 80.0000 ---

DE-PS(1-L) 61.3966 71.4012 81.4014 72

DE-PS(2-L) 60.7694 70.7696 80.7692 86

Desired Values 60.0000 65.0000 70.0000 ---

DE-PS(1-L) 62.3419 67.3519 72.3523 66

DE-PS(2-L) 61.1103 66.1105 71.1104 80

Similarly, the proximity of azimuth angles are discussed for fixed values of amplitudes

and elevation angles. As given in Table 4.29, again the DE-PS technique acted well and

has shown good estimation accuracy and convergence for both L shape arrays. However,

the results for 2-L-shape array are better than that of 1-L shape array.

From so for discussion, we reached at conclusion that PSO, DE, PSO-PS and DE-PS

produced better results in terms of estimation accuracy and convergence rate for 2-L

shape array as compared to 1-L shape array. However, among all of them, the DE-PS

technique has proved to be the most efficient technique. For the sake of simplicity and to


85

summarize the discussion, only the DE-PS technique using 2-L shape array will be

compared with the PM using parallel and L shape arrays.

Case 6. In this second part of simulations, the comparison of DE-PS using 2-L shape

array is carried out with PM using parallel shape array [142] and L shape arrays [143] in

the presence of 10 dB noise. The Table 4.30 and Table 4.31 listed the mean, variance and

standard deviations for PM using parallel shape array and L shape array respectively

[142], [143], while Table 4.32 provided the same calculation for DE-PS technique using

2-L shape array. For PM, 11 sensors are used for both parallel and L shape arrays

configuration [142], [143] while for the proposed DE-PS scheme only 4-sensors are used

in 2-L shape array. The range of elevation angle is varied from 700 to 90

0 for fixed

azimuth angle of 500.

Table 4. 30 Means, Variances and standard deviations using PM parallel shape array

𝜃 in radians for 𝜙 = 500 Mean of 𝜃 Variance of 𝜃 Standard Deviation of 𝜃

720 72.0681 0.7021 0.8379

760 74.9583 0.9895 0.9947

790 77.8421 1.4082 1.1867

820 80.4967 2.3482 1.5323

860 83.3760 4.7215 2.1729

890 84.7062 9.4567 3.0752

As obvious from Table 4.30, the PM method with parallel shape array has failed to

produce accurate results, as soon as, the elevation angle is getting close to 900 but the

same PM method using 1-L shape array configuration has got accurate results for the

same range of elevation angle as listed in Table 4.31. However, at the same time, if we

look at Table 4.32, one can observe that the proposed DE-PS scheme has produced even

better results for the same range of elevation angles by using 2-L shape array

configuration with less number of sensors as compared to PM using parallel and L-shape

arrays.


86

Table 4. 31 Means, Variances and standard deviations using PM with L shape array


720 72.3528 0.0151 0.1228

760 76.2237 0.0111 0.1053

790 79.0999 0.0068 0.0825

820 82.0998 0.0037 0.0608

860 86.0566 0.000632 0.0251

890 89.0146 0.000371 0.0192

Table 4. 32 Means, Variances and standard deviations using DE-PS with 2-L shape array


720 71.9998 0.00002415 4.91 x 10-03

760 76.0003 0.00001871 4.32 x 10-03

790 78.9996 0.00006854 8.27 x 10-03

820 81.9998 0.00003278 5.72 x 10-03

860 86.0006 0.00006132 7.83 x 10-03

890 89.0008 0.00001371 3.70 x 10-03

Case 7: In this case, the Root Mean Square Error (RMSE) of DE-PS using 2-L shape

array is compared with PM using parallel shape array [142] and L shape arrays [143].

Fig. 4. 14 RMSE vs SNR

Only one source is considered which has elevation and azimuth angles 400 and 65

0. The

range of signal-to-noise ratio (SNR) is taken from 5 dB to 30 dB. As shown in Fig 4.14,

the DE-PS technique maintained minimum RMSE for all values of SNR. The second best

RMSE is maintained by PM using L shape arrays.

Case 8: In Table 4.32, some general properties are listed for PM with parallel shape array,

L shape arrays and DE-PS technique using 2-L shape array. As given in Table 4.33, the

5 10 15 20 25 30-35

-30

-25

-20

-15

-10

-5

0

SNR (dB)

RM

SE

(dB

)

PM with Parallel shape array [142]

PM with L shape array [143]

DE-PS with 2-L shape array


87

PM using I-L shape array has estimation failure problem in the range of elevation angles

from 00 to 20

0.

Table 4. 33 Comparison among 2-L shape array, parallel shape array and L shape arrays

Property Parallel Shape Array 1-L shape array 2-L shape array 2-L shape array

Scheme used PM PM PM DE-PS

Range of elevation &

azimuth angles

(00,900)&(0,3600) (00,900)&(0,3600) (00,1800)&00,3600) (00,1800)&(00,3600)

Number of e sources 1 1 1 1

Number of sensors 15 11 10 4

Number of snapshot 200 200 200 1

Pair matching Required Not, required Not, required Not, required

Failure estimation From 700 to 900 From 00 to 200 No failure No failure

Amplitude

estimation

Cannot estimate Cannot estimate Cannot estimate Yes, can estimate

The other drawback with 1-L shape and parallel shape arrays is their limitation for

elevation angles beyond 900. Moreover, the parallel shape array has also the pair

matching problem. As most of the draw backs have been removed by using PM with 2-L

shape arrays but it not only require more sensors but also need a large number of

snapshots. At least it requires 200 snapshots which obviously increase the computational

burdens. On the other hand the DE-PS technique removed the flaws of PM by using less

number of sensors as compared to 2-L shape array with PM. The other advantages of DE-

PS technique include the estimation of sources amplitude which is also an important

parameter to be estimated. Moreover, it requires only single snapshot and hence,

decreases the computational cost.

PART- III

In this part, PSO is hybridized with PS to jointly estimate the amplitude, frequency,

elevation and azimuth angles of far field sources impinging on 2-L shape array. This time

the proposed hybrid scheme has used a new multi-objective function as a fitness

evaluation function. The proposed hybrid scheme (PSO-PS) is compared with GA-PS, as

well as, with existing traditional techniques.


88

4.8 JOINT ESTIMATION OF 4-D PARAMETERS USING PSO-PS

This section discusses an easy and efficient approach for four dimensional (4-D)

parameters estimation of plane waves impinging on 2-L shape array. The 4-D parameters

include amplitude, frequency and 2-D direction of arrival namely azimuth and elevation

angles. The proposed approach is based on memetic computation, in which the global

optimizer (Particle Swarm Optimization) is hybridized with a rapid local search technique

(Pattern Search). For this purpose, a new multi-objective fitness function is used. This

fitness function is the combination of Mean Square Error and correlation between the

normalized desired and estimated vectors. The proposed hybrid scheme is not only

compared with Particle Swarm Optimization and Pattern Search alone but also with the

hybrid Genetic algorithm and traditional approach. A large number of Monte- Carlo

simulations are carried out to validate the performance of the proposed scheme. It gives

promising results in terms of estimation accuracy, convergence, proximity effect and

robustness against noise.

4.8.1 Data Model

In this section, we have developed a data model for P narrow band sources existing in the

Fraunhofer zone (far field). For this purpose, we have considered 2-L shape sensor array

that has already shown better performance [144] as compared to linear array [145], planar

array [146], [147], parallel shape array [148] and 1-L shape array [149]. This 2-L shape

array consists of 3-ULA placed along x-axis, y-axis and z-axis. Each ULA is composed of

1M sensors while the reference sensor is common for all of them as shown in Figure

4.7. For P M , the response of the th

m sensor placed in the z-axis sub-array can be

represented as,

( , )1

Py s b f npzm z p zmp

p

(4.45)


89

In (4.45),

( , ) exp( cos ); 0 1pp pb f jqk d m Qz p (4.46)

d is separation between any two adjacent sensors in the array and can be given as,

/ 2mind (4.47)

where / maxmin c f , c is the speed of light and maxf represent the maximum

frequency possible to be used. Besides, in (4.45), pk is the wave number of th

p source,

given by,

2 2pk f pcp

(4.48)

. By using (4.46)-(4.48) in (4.45), we get,

exp cos1 max

p

P f py s j m nzm zmpfp

(4.49)

In matrix-vector form, (4.49) can be generally written as,

( , )fz zz y B s n (4.50)

where zB is steering matrix which contains the frequencies and elevation angles of P

sources received on z-axis sub-array. It can be written as

( , ) [ ( , ), ( , ),..., ( , )]1 1 2 2 P Pf f f fz z z z B b b b (4.51)

Similarly, " "s contains the sources amplitude and can be given as,

[ , ,..., ]1 2 PT

s s ss (4.52)

The response of th

m sensor in the sub-array along x-axis for P sources can be given as,

( , , )1

p

P

x py s b f nxm xmp pp

(4.53)

( , , ) exp( sin cos )px pb f jmk dpp p p (4.54)


90

So, (4.53) can be expressed as,

exp sin cos1 max

p

P f py s j m nxm xmp pfp

(4.55)

In matrix-vector form (4.55) can be given as,

( , , )x xfx y B s n (4.56)

where xB is steering matrix which contains the frequencies, elevation and azimuth angles

of P sources received on x-axis sub-array. i.e,

( , , ) [ ( , , ), ( , , ),..., ( , , )]1 1 2 21 2x x x x p Pf f f f P B b b b

(4.57)

In the same way, the response of th

m sensor in the sub-array along y-axis for P sources

can be given as,

( , , )1

p

P

y py s b f nym ymp pp

(4.58)

( , , ) exp( sin sin )py p pb f jmk dp p p (4.59)

and in simplified form (4.58) can be expressed as,

exp sin sin1 max

p

P f py s j m nym ymp pfp

(4.60)

In matrix-vector form it can be given as,

( , , )y yfy y B s n

(4.61)

In (4.50), (4.56) and (4.61) ,z xn n , yn are AWGN vector added in particular sub-array

and can be given as,

1 2[ , ,.., ]Mw wT

n n nw wn (4.62)

where , , .w x y z


91

From (4.49), (4.55) and (4.60), it is obvious that the unknown parameters are the

amplitudes ( )ps , frequencies ( )pf , elevation angles ( )p and azimuth angles ( )p . So,

clearly the problem in hand is to efficiently and jointly estimate these unknown

parameters for 1,2,3,... .p P

4.8.2 Particle Swarm Optimization Hybridized With Pattern Search

The generic flow diagram of hybrid PSO is shown in Fig 4.4, while its steps in the form

of pseudo code are given as,

Step 1 Initialization: In this step, the swarm is randomly initialized and Q particles are

generated. The length of each particle is 4 P where P is the total number of far field

sources. It can be given as,

, , ... , , ... , , , ... , , , ...1,1 1,2 1, , 1, 1 1, 2 1,2 1,2 1 1,2 2 1,3 1,3 1 1,3 2 1,41

, , ... , , ... , , , ... , ,2,1 2,2 2, , 2, 1 2, 2 2,2 2,2 1 2,2 2 2,3 2,3 1 2,32

3...

s s s f f fP P P P P P P P P Ps s s f f fP P P P P P P P P

Q

e

e

e

e

, ...2 2,4, , ... , , ... , , , ... , , , ...3,1 3,2 3, , 3, 1 3, 2 3,2 3,2 1 3,2 2 3,3 3,3 1 3,3 2 3,4

.

.

., , ... , , ... , , , ... , , , ...,1 ,2 , , , 1 , 2 ,2 ,2 1 ,2 2 ,3 ,3 1 ,3 2 ,4

Ps s s f f fP P P P P P P P P P

s s s f f fQ P Q P Q P Q P Q P Q P Q P Q P Q P Q P Q P

The above matrix can be represented as,

1 2 3, , , ... Q

T E e e e e (4.63)

The lower and upper bounds of , , ,s f are defined as

max

:, ,

:, ,min, 1,2,..., & 1,2,...,

: 0 / 2,2 ,2

: 0 2,3 ,3

s R l s uq p q pb b

f R f f fq P p q P pfor q Q p P

Rq P p q P p

Rq P p q P p

where lb and ub are the lower and upper bounds of amplitudes while minf and maxf are

lower and upper range of frequencies.


92

Step II Fitness Function: In this part, we have used a new multi-objective fitness

function which is the combination of MSE and correlation between the normalized

desired and normalized estimated vectors. For thq particle, it can be expressed as,

1( )

1 ( )FF q

err q

(4.64)

where

1( ) ( ( ) ( ) ( ))1 2 33

err q err q err q err qM

(4.65)

In (4.65),

1

2ˆ ˆ( ) . 1

H q

N N

qzerr q y yzm zm z y y (4.66)

where yzq is defined in (4.49), while ˆq

yzq is defined as,

ˆ

ˆ ˆ ˆexp cos( )21 max

p

P eP pqy e j m ezm P pfp

(4.67)

( )2err q in (4.65), can be given as,

2

2ˆ ˆ( ) . 1

HN N

q qerr q y yxm xm x x y y

(4.68)

where yxm is defined in (4.55) while ˆq

yxm is defined as,

ˆ

ˆ ˆ ˆ ˆexp sin( )cos( )2 31 max

p

P eP pqy e j m e exm P p P pfp

(4.69)

and ( )3err q can be defined as,

3

2ˆ ˆ 1q qH

err y yym yp yN yN y .y

(4.70)

where yym is defined in (4.58) while y yqis defined as,


93

ˆ

ˆ ˆ ˆ ˆexp sin( )sin( )2 31 max

P eM my e j m e emym P p P pfp

(4.71)

In (4.66), (4.68) and (4.70), ˆ ˆ ˆ, , , , ,zN zN xN xN yN yNy y y y y y can be defined as,

Nw

ww

y

yy

(4.72)

and

ˆˆ

ˆNw

ww

y

yy

(4.73)

where , ,w x y z .

In this step, store each particle as a local best ( )bel and the one having maximum fitness

function be stored as global best ( )beg . The remaining steps are same as discussed above

for PSO.


In this section, we have carried out several simulations to assess the performance of the

proposed (PSO-PS) scheme. This section is mainly divided into two parts. In first part,

the results are not only compared with PSO and PS alone but also with the other hybrid

GA-PS technique discussed in [131]. In the second part, the proposed scheme is

compared with traditional non-heuristic technique [150] using an error as a figure of

merit. Throughout the simulations, only single snapshot is used. The value of maxf is

taken to be 90MHZ. All the values of DOA and frequencies are taken in radians (rad) and

Hertz (Hz) respectively and each result is averaged over 100 independent trials.

4.8.3.1 Comparison with PSO, PS and GA-PS

In this subsection, we have compared PSO-PS with PSO, PS and GA-PS in terms of

estimation accuracy, convergence and proximity effect. In [131], a hybrid scheme (GA-


94

PS) using 2-L shape array is developed for the joint estimation of amplitude, elevation

and azimuth angles using MSE as a fitness evaluation function. As the GA-PS in [131]

has shown better performance than GA alone and propagator method [142], [143], so, we

will compare our results only to GA-PS. Moreover [131] has not discussed the frequency

estimation, so, the comparison of the proposed approach is done only with the amplitude,

elevation and azimuth angles.

4.8.3.1.1 Estimation Accuracy

In this sub-section, two cases are considered based on the number of sources. No noise is

added to the system.

Case 4.1.1a: In this case, three sources are taken which have the desired values are

1 1 1 1( 1, 80 , 1.0472 , 3.8397 ),s f MHz rad rad ( 4, 55 , 0.5236 , 1.6581 )2 2 2 2s f MHz rad rad

( 7, 70 , 0.7854 , 2.1817 ).3 3 3 3s f MHz rad rad The 2-L shape array consists of seven

sensors i.e. each ULA is composed of two sensors, whereas the reference sensor is

common for them.

Table 4. 34 Estimation accuracy for 3 sources

Scheme 𝑠1 𝑓1(MHz) 𝜃1(rad) ∅1(rad) S2 f2(MHz) 𝜃2(rad) ∅2(rad) S3 f3(MHz) 𝜃3(rar) ∅3(rad)

Desired 1.0000 80.0000 1.0472 3.8397 4.0000 55.0000 0.5236 1.6581 7.0000 70.0000 0.7854 2.1817

PS 1.0187 79.9810 1.0660 3.8590 3.9812 54.3807 0.5425 1.6771 7.0189 70.0195 0.7666 2.1627

PSO 1.0034 80.0037 1.0629 3.8361 4.0035 54.9963 0.5271 1.6619 6.9965 69.9960 0.7888 2.1852

GA-PS 0.9996 ------- 1.0664 3.8393 4.0004 ------- 0.5240 1.6585 6.9997 ------ 0.7858 2.1813

PSO-PS 1.0000 80.0001 1.0473 3.8397 4.0000 55.0002 0.5236 1.6582 7.0000 70.0001 0.7855 2.1817

As listed in Table 4.34, one can observe, the advantage of hybridization of global

optimizer with local search optimizer. The PSO alone is less accurate than that of GA-PS

but when we hybridized PSO with PS, it produced even better results as compared to GA-

PS. So, in this case, the proposed PSO-PS technique proved to be the most accurate

scheme.


95

Case 4.1.1b: In this, four far field sources are taken which have the desired values are

( 5, 40 , 0.4363 , 1.9199 ), ( 3, 50 , 0.8772 , 2.4435 ),1 1 1 1 2 2 2 2s f MHz rad rad s f MHz rad rad

( 8, 60 , 1.1345 , 0.6981 ), ( 2, 45 , 0.2618 , 2.6180 ).3 3 3 3 4 4 4 4s f MHz rad rad s f MHz rad rad

Due to increase of unknowns, the performance of PSO and PS alone are degraded a lot.

However, the hybrid schemes especially the PSO-PS scheme produced fairly good

estimation accuracy. The second best scheme is GA-PS as given in Table 4.35 and 4.36.


Scheme 𝑠1 𝑓1

(MHz) 𝜃1(rad) ∅1

(rad) S2 f2(MHz) 𝜃2(rad) ∅2

(rad) S3 f3(MHz) 𝜃3(rar) ∅3

(rad)

Desired 5.0000 40.0000 0.4363 1.9199 3.0000 50.0000 0.8720 2.4435 8.0000 60.0000 1.1345 0.6981

PS 4.9688 40.0316 0.4050 1.8879 2.9689 49.9684 0.9033 2.4121 8.0316 60.0317 1.1031 0.6667

PSO 5.0091 40.0094 0.4271 1.9282 3.0094 49.9905 0.8630 2.4522 8.0095 59.9903 1.1436 0.6889

GA-PS 5.0040 ------- 0.4404 1.9231 3.0042 ------- 0.8679 2.4476 8.0043 ------- 1.1305 0.6940

PSO-PS 4.9983 39.9980 0.5381 1.9208 2.9282 49.9979 0.8739 2.4416 8.0020 60.0022 1.1364 0.6997


Scheme 𝑠4 𝑓4(MHz) 𝜃4(rad) ∅4(rad)

Desired Values 2.0000 45.0000 0.2618 2.6180

PS 1.9689 45.0318 0.2306 2.6494

PSO 2.0094 45.0097 0.2526 2.6090

GA-PS 2.0043 ------- 0.2577 2.6221

PSO-PS 1.9982 39.9978 0.2601 2.6164

4.8.3.2 Convergence

In this sub-section, we have performed several simulations to check the convergence of

our proposed hybrid scheme (PSO-PS). This experiment is done in the presence of 10 dB

noise for two, three and four sources respectively where the array consists of ten sensors.

As shown in the bar graph of Fig 4.15, the hybrid PSO-PS scheme converged for the most

number of times as compared to the other schemes for all number of sources. The PSO-

PS scheme converged 98%, 95 % and 93% for two, three and four sources respectively.

The second best scheme is the other hybrid GA-PS scheme. The convergence is degraded

for each scheme with the increase of sources.


96

Fig. 4. 15 Convergence vs number of sources for 2-L shape array at 10 dB noise

4.8.3.3 Proximity Effect

In this simulation, we have checked the proposed scheme for the sources located close to

each other. In this experiment, the PSO-PS is compared only to GA-PS of [131] as from

the above discussion, we reached at a conclusion that the PSO-PS and GA-PS schemes

are the two most promising algorithms as compared to PSO and PS alone. For this

simulation, the 2-L shape array consists of ten sensors and 10 dB noise is added.

Table 4. 37 Proximity effect of elevation and azimuth angles

Scheme 𝜃1(rad) 𝜃2(rad) 𝜃3(rad) ∅1(rad) ∅2(rad) ∅3(rad)

Desired 0.4363 0.6981 1.0472 2.0944 2.7925 1.3963 -------------

GA-PS 0.0467 0.6977 1.0477 2.0948 2.7921 1.3967 97

PSO-PS 0.0462 0.6981 1.0471 2.0944 2.7926 1.3962 100

Desired 0.4363 0.5236 1.0472 2.0944 2.7925 1.3963 -------------

GA-PS 0.4389 0.5209 1.0477 2.0948 2.7921 1.3967 91

PSO-PS 0.4372 0.5251 1.0471 2.0944 2.7926 1.3962 97

Desired 0.4363 0.5236 0.6109 2.0944 2.7925 1.3963 -------------

GA-PS 0.4332 0.5268 0.6141 2.0948 2.7921 1.3967 81

PSO-PS 0.4346 0.5252 0.6125 2.0944 2.7926 1.3962 90

Desired 0.4363 0.6981 1.0472 2.0944 2.1817 1.3963 ---------------

GA-PS 0.0467 0.6977 1.0477 2.0973 2.1847 1.3967 90

PSO-PS 0.0462 0.6981 1.0471 2.0929 2.1831 1.3962 97

Desired 0.4363 0.6981 1.0472 2.0944 2.1817 2.2689 ---------------

GA-PS 0.0467 0.6977 1.0477 2.0901 2.1859 2.2649 80

PSO-PS 0.0462 0.6981 1.0471 2.0926 2.1835 2.2671 90

2 3 40

10

20

30

40

50

60

70

80

90

100

Number of sources

Converg

ence R

ate

(%

)

PS

PSO

GA-PS

PSO-PS


97

In Table 4.37, first the proximity of elevation angles are checked and then the proximity

of azimuth angles are provided for fixed values of amplitudes and frequencies. The

estimation accuracy, as well as, convergence rate of both hybrid schemes is degraded

when the elevation and azimuth angles are brought close to each other. However, the

PSO-PS scheme has produced better results as compared to GA-PS [131].

4.8.3.3 Performance on Reference Axis

In this sub-section, the proposed scheme is checked for more practical scenario where the

DOA of the sources is taken on reference axis. For this purpose, the 2-L shape array is

composed of 13 sensors, while 4 sources are considered which have the desired values are

( 2, 40 , 0 , 1.5708 ), ( 5, 50 , 1.5708 , 0 ),1 1 1 2 2 21 2s f MHz rad rad s f MHz rad rad

( 1, 60 , 1.2217 , 3.1416 ), ( 4, 70 , 1.0472 , 4.7124 )3 3 3 4 4 43 4s f MHz rad rad s f MHz rad rad .

As provided in Table 4.38, the estimation accuracy of both hybrid schemes degraded for

the elevation angles on reference axis and produced significant errors. However, the PSO-

PS scheme is less degraded as compared to GA-PS. On the other hand, the azimuth angles

on reference axis have produced negligibly small error especially in the case of PSO-PS

Table 4. 38 Comparison analysis on reference axis

Scheme 𝜃1(rad) ∅1(rad) 𝜃2(rad) ∅2(rad) 𝜃3(rad) ∅3(rad) 𝜃4(rad) ∅4(rad)

Desired 0.0000 1.5708 1.5708 0.0000 1.2217 3.1416 1.0472 4.7124

GA-PS 0.1065 1.5832 1.4653 0.0375 1.2300 3.1486 1.0555 4.7195

PSO-PS 0.1040 1.5757 1.6748 0.0165 1.2245 3.1446 1.0491 4.7152

4.8.3.4 Comparison with Traditional Technique

In [150] hierarchical space-time decomposition method is used for the joint estimation of

frequencies and 2-D DOA of far field sources impinging on uniform rectangular array

(URA). It basically makes use of the 1-D ESPRIT algorithm along with MSE as cost

function. We have compared the error achieved by our proposed scheme with the MSE of

[150]. As [150], has not estimated the amplitudes, so, we will not consider it in this


98

section. For this experiment, we have considered the same two cases having the same

values of frequencies, elevation and azimuth angles as discussed in section 4.8.3.1.1. The

2-L shape array and URA are composed of 13 and 36 sensors respectively. The values of

signal to noise ratio (SNR) are ranging from -5 to 25 dB. As shown in Fig 4.16, Fig 4.17,

and Fig 4.18, the proposed scheme maintained minimum error for frequencies, elevation

angles and azimuth angles as compared to [150] at all values of SNR. More importantly,

our proposed scheme utilized less number of sensors as compared to [150] and thus

requires less hardware budget to implement.

Fig. 4. 16 Comparison of the frequency estimate

Fig. 4. 17 Comparison of the elevation angle estimate

-5 0 5 10 15 20 2510

-3

10-2

10-1

100

SNR (dB)

Err

or

(Deg

rees

)

source-1[150] source-1Proposed source-2[150] source-2 proposed

-5 0 5 10 15 20 2510

-3

10-2

10-1

100

SNR (dB)

Err

or

(Deg

ree)

Case-1[150] Case-1-Proposed Case-2[150] Case-2-Prposed


99

Fig. 4. 18 Comparison of the azimuth angle estimate

4.9. CONCLUSION

This chapter was divided into three parts, In part one ,we have developed hybrid schmes

based on GA-PS and PSO-PS for the joint estimation of amplitude and 1-D DOA of far

field sources impinging on ULA. MSE was used as a fitness evaluation function. It has

been shown through different simulations that the hybrid schemes produced better results

as compared to their individual responses.

In Part two, GA-PS, PSO-PS, SA-PS and DE-PS were developed for the joint estimation

of amplitude and 2-D DOA estimation of far field sources impinging on L-Shape arrays

(1-L & 2-L). For this again MSE was used as fitness function. It has been shown through

different experiments that the hybrid schemes produced better results as compared to the

individual responses of GA, PSO, SA and DE. Moreover, it has been also shown, that the

proposed hybrid schemes are also better than that of the traditional technques.

In part three, amplitude, frequency and 2-D DOA are jointly estimated by using PSO-PS

along with a new multi-objective fitness function. This multi-objective fitness function

has not only estimated the 4-D parametes accurately, but has also shown supermacy over

the previously used MSE as a fitness function.

-5 0 5 10 15 20 2510

-3

10-2

10-1

100

101

SNR [dB]

Err

or

[degre

e]

source1[150] Case1-Proposed source2 [150] source2-Proposed

CHAPTER 5

DOA ESTIMATION INCLUDING RANGE, AMPLITUDE AND

FREQUENCY OF NEAR FIELD SOURCES

In previous chapter, we have discussed the parameters estimation, specifically DOA

of multiple electromagnetic waves impinging on sensors array with ULA and L-shape

arrays when sources are in the far field zone. The situation however, becomes more

complicated when the sources are in the Fresnel zone or near field of array aperture.

In such situations, the wave-front is no longer planar, but is spherical and the source

location cannot be solely found by simply estimating the angle. In this case, in

addition with the angle, we also need to estimate correctly the range of sources [151],

[152], [153], [154], [155], [156]. Hence, the techniques developed for the estimation

of far field sources cannot be applied directly to estimate the DOA of near field. This

scenario may appear quite rottenly, while dealing with electronic surveillance, seismic

exploration, ultrasonic imaging, under-water source localization, speech enhancement

etc with microphone arrays e.g. see reference [10]. For joint estimation of ranges and

DOA, Maximum Likelihood (ML) method was proposed first [157]. Later on, an

effort was made by using least squares ESPRIT like algorithm based on fourth order

cummulants, which is computationally heavy [158]. G. Emmanuele in [159]

proposed, a weighted linear prediction method which needs additional computation to

CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD

SOURCES

101

solve pairing problem in case of multiple sources. This may generate inaccurate

pairing at low Signal-to Noise (SNR), when arrival angles are close enough.

In this chapter, we have developed an efficient algorithms based on hybrid meta-

heuristic techniques for the estimation of DOA combined with other parameters such

as range, frequency and amplitude of near field sources. The hybrid meta-heuristic or

memetic computing techniques are used again which are the combination of global

and local search optimizers. GA, PSO and DE are used as global search optimization

methods while PS, IPA and ASA are used as local search optimizers. This chapter has

also three major parts. In part one, we have developed the hybrid techniques for the

joint estimation of 3-D parameters (DOA, range and amplitude) of near field sources

impinging on ULA, where MSE is used as fitness evaluation function. In part two, we

have linked our problem to bi-static radar and placed the centro symmetric cross

shape (CSCS) array on the receiver side. In part three, we have jointly estimated 5-D

parameters i.e., elevation angle, azimuth angle, amplitude, range and frequency of

near field sources. For this, we have used the multi-objective fitness function. Most of

the data presented in this chapter is taken from the publications [160], [161], [162],

[163], [164], [165].

5.1 DATA MODEL

Consider 𝑃 near field sources impinging on a passive ULA. This linear array consists

of 2M Mx sensors and having the same inter-element spacing d between the two

consecutive elements as shown in Fig 5.1. For this, our signal model for P narrow

band sources on th

l sensor can be given as,

, 1,...,0,1,...,1

lll

P j ix s e n l M Mi x xi

(5.1)


SOURCES

102

where 0,l be the phase reference point of our co-ordinate system and li represent

the phase difference of th

i source received at th

l sensor and reference sensor. Due to

the assumption of narrow band, the phase difference can be defined as,

2( )l lr ri i i

(5.2)

where the distance between the th

l sensor and th

i source can be given as,

2 2( ) 2 sinlr r ld r ldi i i i

2( ) 21 sin

2

ld ldr ii rr ii

(5.3)

-Mx +1 -1 1 m Mx

θ

ri

r

0

Near Field ith source

where i= 1,2,… P

dd

Fig. 5.1 Array Geometry for near field sources

In the Binomial expansion of (5.3), one can get the far field approximation by

maintaining terms up to the first power of /ld ri , however, to get near field

approximation, we should maintain terms up to the second power of /ld ri .

Therefore, by Fresnel approximation, we get,


SOURCES

103

2 2 2 2(1 sin )

2 2 22 2 sin

l d ld l dr ril i irr rii i i

2 22

cos sin22

l dr ldi i i

ri

(5.4)

So, the phase difference can be given as,

22 2 2( sin ) ( cos )

l

d dl li i iri

2l li i (5.5)

By using (5.5) in (5.1), we get,

2exp( ( ))

1

Px s j l l ni i il l

i

(5.6)

The parameters i and i in (5.6) are the function of elevation angle i and range ir

respectively for the thi source, where,

2sin( )

di i

(5.7)

22

cos ( )d

i iri

(5.8)

In vector form, (5.6) can be written as,

x Bs n (5.9)

where

[ ,... ... ]1 MM

Tx x xo xx

x (5.10)

[ ,..., ,..., ]1 MM

Tn n no xx

n (5.11)


SOURCES

104

1 2[ , ,..., ]PT

s s ss (5.12)

[ , ,..., ]1 2 PB b b b (5.13)

where

2

2

( , ) [exp( ( 1) ( 1) ),...,exp( ( )), 1,

exp( ( )),...exp( ( ))]

x i x i i i

i i x i x i

ri i i

T

j M j M j

j j M M

b

(5.14)

is the steering vector. The goal is to estimate jointly the unknown parameters i.e. the

amplitude ( )si , elevation angle ( )i and range ( )ri of the waves for 1,2,...,i P .

PART-I

5.2 JOINT ESTIMATION OF 3-D PARAMETERS USING GA-IPA AND SA-IPA

To jointly estimate the unknown parameters, we have used GA and SA as global

optimizers and IPA as a rapid local search optimizer. Again we have used a

MATLAB built-in optimization tool box for GA, SA and IPA, where the parameters

setting are provided in Table 5.1.

Table 5.1 Parameters setting for GA, IPA and SA

GA IPA SA

Parameters Settings Parameters Setting parameters setting

Population size 240 Chromosome

size

30

No of Generation 1000 Sub problem

algorithm

Idl

factorization

Annealing

Function

Fast

Migration

Direction

Both Way Maximum

perturbation

0.1 Reannealing

interval

100

Crossover fraction 0.2 Minimum

perturbation

1e^-8 Temperature

update

function

Exponential

temperature update

Crossover Heuristic Scaling Objective &

Constraint

Initial

temperature

100

Function Tolerance 10-12 Hessian BFGS Data type Double

Initial range [0-1] Derivative

type

Central

difference

Function

Tolerance

10-12

Scaling function Rank Penalty factor 100 Max iteration 1000

Selection Stochastic

uniform

Maximum

function

evaluation

50000 Max function

evaluations

3000*number of

variables

Elite count 2 Maximum

Iteration

1000 Annealing

Function

Fast

Mutation function Adaptive

feasible

X Tolerance 10-15 Reannealing

interval

100


SOURCES

105

The steps for GA and hybrid GA-IPA are given as follows,

Step I Initialization: Generate Q number of chromosomes at random where each

chromosome consists of unknown (genes) i.e. amplitude, DOA and range. The length

of each chromosome is 3 P where P is number of near field sources.

Mathematically, it can be written as,

1 , 1 2 2 1 3[ ,... , ,... , ,... ], , , , ,P P P P Pq q q q q qs s r rq

c (5.15)

where

:

: 0 , 1,2,..., , 1,2,...,, ,

:,2 ,2

s R L s Hqj s qj s

R q Q j Pq P j q P j

r R L r Hr rq P j q P j

where Ls and Hs are the lowest and highest possible limits of the signal amplitudes

while, rL and rH are the lowest and highest possible limits of the source ranges.

Step II Fitness Evaluation: Calculate the MSE of each chromosome by using the

following relation,

2

( ) (1 / )1

M qD q M x xl l

l

(5.16)

where ( )D q represents the MSE between desired and estimated response for thq

chromosome. In (5.16) lx is given by (5.6) while q

xl is defined as,

2ˆˆexp( ( )1

Pqx c j l li i il

i

(5.17)

where

2ˆ ˆsin( )

dci P i

(5.18)


SOURCES

106

22ˆ cos ( )

,2P i

dci

rc P i

(5.19)

where 1,2,...,i P

Step III Termination Criteria: The termination criteria of the algorithm are made on

the following results achieved,

The pre-defined fitness value is achieved i.e. 10−12 OR

The maximum numbers of cycles have reached.

Step IV Reproduction: Use the operators of elitism, crossover, and mutation selection

as given in Table 5.1, to mimic the new population.

Step V Refinement: IPA is used for further refinement of the results (Call

FMINCON Function of MATLAB). The best individual of GA and SA has been set

as a preliminary point for IPA algorithm.

Step VI Storage: Store the global best of this cycle and repeat the steps 2 to 5 for

sufficient number of independent runs, which will ultimately be used for better

statistical analysis.

5.2.1 Simulation and Results

In this section, the accuracy and reliability of GA, IPA, SA, GA-IPA, and SA-IPA are

discussed for joint estimation of amplitudes, DOA and ranges of near field sources. A

uniform aperture array having 2M Mx sensors is used in which the inter-element

spacing ―d‖ between the two consecutive sensors is taken to be / 4 . MSE is setup as

a fitness evaluation function which is given by (5.16). Different cases are discussed

on the basis of different number of sources and different number of sensors in the

array. The proximity in terms of angular separation, distance and signal level is also

examined for GA-IPA. All the values of DOA are taken in radians, while the values


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of ranges are taken as a multiple of wavelength )( . A MATLAB built-in toolbox

―optimization of population‖ based algorithm with the setting provided in Table 5.1

and a MATLAB version 7.8.0.347 is used. Throughout the simulations, only a single

snapshot is used and each result is averaged over 100 independent runs.

Case I: In this case, the performance of all techniques is discussed for two sources

and eight sensors in the (ULA). The amplitudes, DOA and ranges of these two

sources are denoted by 𝑠1 , 𝑠2, 𝜃1 , 𝜃2, 𝑟1 , 𝑟2, respectively. Desired values are taken as

𝑠1 = 1, 𝑠2 = 2, 𝜃1 = .6981(𝑟𝑎𝑑), 𝜃2 = 1.2217(𝑟𝑎𝑑), 𝑟1 = .3𝜆, 𝑟2 = 4𝜆 where

𝑠1 , 𝜃1, 𝑟1 correspond to the first source, while 𝑠2 , 𝜃2, 𝑟2 correspond to the second

source. As listed in Table.5.2, all the five schemes produced fairly good estimates,

however, among these techniques, the hybrid GA-IPA gives better results. The second

and third best results are given by GA and IPA, respectively for the same said case.

Table 5.2 Amplitude, DOA and Range of two sources

Scheme s1 s2 θ1(rad) θ2(rad) r1(λ) r2(λ)

Desired values 1.0000 2.0000 0.6981 1.2217 0.3000 4.0000

GA 1.0024 2.0026 0.7006 1.2243 0.3025 4.0027

IPA 1.0088 2.0089 0.7069 1.2306 0.3088 4.0089

GA-IPA 1.0015 2.0015 0.6996 1.2232 0.3015 4.0015

SA 1.0206 2.0205 0.7187 1.2423 0.3206 4.0207

SA-IPA 1.0104 2.0105 0.7089 1.2326 0.3104 4.0106

Now, the MSE and convergence (reliability) is discussed for increasing number of

sensors in the array. For this purpose, 10−2 is used as a threshold MSE value.

Initially, the array consists of four sensors for which the GA has converged 90 % with

MSE as 10−5 as provided in Table.5.3. The convergence and MSE of GA have got

improvement when hybridized with IPA i.e. it has convergence of 93% with MSE as

10−6. Similarly, one can see that convergence and MSE of SA algorithm has also

improved when hybridized with IPA. Moreover, the convergence rate and MSE of all


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schemes have got improvement when the number of sensors are increased in the

array.

Table 5.3 MSE and %convergence of two sources for different number of sensors

No.of Elements Scheme MSE %Convergence No. of Elements Scheme MSE %Convergence

4 GA 10−5 90 8 GA 10−7 92

IPA 10−3 40 IPA 10−4 45

GA-IPA 10−6 93 GA-IPA 10−8 95

SA 10−3 10 SA 10−4 13

SA-IPA 10−4 30

SA-IPA 10−5 34

6 GA 10−6 92 10 GA 10−8 93

IPA 10−3 42 IPA 10−4 48

GA-IPA 10−7 94 GA-IPA 10−9 96

SA 10−3 11 SA 10−4 14

SA-IPA 10−4 32

SA-IPA 10−6 35

Case II: In this case, the performance of all five techniques is evaluated for three

sources. As given in Table 5.4, the desired values are𝑠1 = 1, 𝑠2 = 2, 𝑠3 = 3, 𝜃1 =

0.872781(𝑟𝑎𝑑), 𝜃2 = 1.3963(𝑟𝑎𝑑), 𝜃3 = 1.9199(𝑟𝑎𝑑), 𝑟1 = 4𝜆, 𝑟2 = 5𝜆, 𝑟3 = 6𝜆.

In this case, with the increase of sources (unknown), we faced few local minima due

to which the performance of all schemes was degraded slightly. However, the hybrid

GA-IPA proved to be the best among all techniques even in the presence of local

minima.

Table 5. 4 Amplitude, DOA and Range of three sources

Scheme s1 s2 s3 θ1(rad) θ2(rad) θ3(rad) r1(λ) r2(λ) r3(λ)

Desired 1.0000 2.0000 3.0000 0.8727 1.3963 1.9199 4.0000 5.0000 6.0000

GA 1.0084 2.0083 3.0083 0.8811 1.4047 1.9283 4.0083 5.0084 6.0083

IPA 1.0548 2.0548 3.0547 0.9275 1.4511 1.9747 4.0548 5.0548 6.0548

GA-IPA 1.0058 2.0057 3.0058 0.8785 1.4021 1.9257 4.0058 5.0057 6.0058

SA 1.0883 2.0883 3.0884 0.9610 1.4846 2.0082 4.0883 5.0884 6.0883

SA-IPA 1.0810 2.0811 3.0811 0.9537 1.4773 2.0009 4.0810 5.0810 6.0811

Now, the reliability and MSE of all schemes are discussed for three sources. As given

in Table.5.5, the GA-IPA converged many times and has minimum MSE as compared

to the other schemes. It converged 85% times with MSE as 10−5. The second best is


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GA which converged 80% times with MSE as 10−4. The effect of increasing the

sensors in the array is also provided due to which the convergence and MSE of all

these schemes got improvement .

Table 5.5 MSE and %convergence of three sources for different number of sensors

No.of Elements Scheme MSE %Convergence No. of Elements Scheme MSE %Convergence

6 GA 10−4 80 10 GA 10−5 84

IPA 10−3 25 IPA 10−4 35

GA-IPA 10−5 85 GA-IPA 10−7 88

SA 10−3 0 SA 10−3 0

SA-IPA 10−3 4

SA-IPA 10−4 8

8 GA 10−5 82 12 GA 10−6 85

IPA 10−3 28 IPA 10−4 36

GA-IPA 10−6 88 GA-IPA 10−7 90

SA 10−3 0 SA 10−3 5

SA-IPA 10−3 5

SA-IPA 10−4 10

Case III: In this case, the performance of four sources impinging on ULA is

discussed. The desired values of sources are 𝑠1 = 1, 𝑠2 = 2, 𝑠3 = 3, 𝑠4 = 4,𝜃1 =

0.6981(𝑟𝑎𝑑), 𝜃2 = 1.3090(𝑟𝑎𝑑), 𝜃3 = 2.0944(𝑟𝑎𝑑), 𝜃4 = 2.7925(𝑟𝑎𝑑), while

𝑟1 = 5𝜆, 𝑟2 = 6𝜆, 𝑟3 = 7𝜆, 𝑟4 = 8𝜆. In this case, we faced more strong local minima as

compared to the previous case. Due to this, the accuracy of all techniques decreased.

GA gets stuck little in these local minima which is the inherent ability of GA and its

performance improved even more when hybridized with IPA as given in Table 5.6.

Table 5. 6 Amplitude, DOA and Range of four sources

Scheme s1 s2 s3 s4 θ1(𝑟𝑎𝑑) θ2(𝑟𝑎𝑑) θ3(𝑟𝑎𝑑) θ4(𝑟𝑎𝑑) r1(𝜆) r2(λ) r3(λ) r4(λ)

Desired 1.0000 2.0000 3.0000 4.0000 0.6981 1.3090 2.0944 2.7925 5.0000 6.0000 7.0000 8.0000

GA 1.0183 2.0184 3.0183 4.0183 0.7161 1.3274 2.1127 2.8107 5.0183 6.0183 7.0184 8.0182

IPA 1.0445 2.0446 3.0445 4.0445 0.7427 1.3537 2.1391 2.8371 5.0445 6.0445 7.0445 8.0445

GA-IPA 1.0103 2.0104 3.0102 4.0103 0.7083 1.3194 2.1047 2.8028 5.0103 6.0104 7.0103 8.0103

SA 1.1273 2.1274 3.1272 4.1273 0.8254 1.4362 2.2218 2.9199 5.1273 6.1272 7.1273 8.1274

SA-IPA 1.0232 2.0234 3.0231 4.0235 0.7213 1.3322 2.1176 2.8157 5.0232 6.0232 7.0233 8.0232


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Now, the reliability (convergence rate) of these techniques is discussed for four

sources. Even in this case, the hybrid approach GA-IPA has high convergence rate. It

converged 70% times with MSE as 10−5 for eight sensors in the ULA. The GA

converged 60% times with MSE as 10−4. However, the performance of IPA, SA-PS

and SA is drastically degraded in the presence of local minima. As given in Table.5.7,

The IPA alone got out only four times from these local minima while the SA does not

avoid the local minima even for a single time. The effect of increasing elements is

also considered due to which GA-IPA, GA and SA-IPA improved slightly. All these

techniques failed when the number of sensors in the array is less than the number of

sources as it becomes an under-determined problem.

Table 5. 7 MSE and %convergence of four sources for different number of sensors

.No.of Elements Scheme MSE %Convergence No. of Elements Scheme MSE %Convergence

8 GA 10−4 60 12 GA 10−6 65

IPA 10−3 4 IPA 10−3 5

GA-IPA 10−5 70 GA-IPA 10−7 77

SA 10−3 0 SA 10−3 0

SA-IPA 10−3 1

SA-IPA 10−3 2

10 GA 10−5 62 14 GA 10−7 70

IPA 10−3 4 IPA 10−3 6

GA-IPA 10−6 74 GA-IPA 10−8 80

SA 10−3 0 SA 10−3 0

SA-IPA 10−4 1

SA-IPA 10−3 2

Case VIII: In this section, we examined the robustness of all schemes against noise.

In this case, two sources and eight sensors are considered. The MSE of all five

schemes is evaluated against the different values of SNR ranging from 5 dB to 30 dB.

As shown in Fig 5.2, the hybrid approach GA-IPA is fairly robust to produce better

results even in the presence of low SNR. The second best is GA which gives

minimum MSE against the different values of SNR.


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Fig. 5. 2 Mean Square Error vs Signal to Noise ratio

As obvious from the above discussion, GA-IPA proved to be the best technique as

compared to GA, IPA, SA and SA-IPA, so from now onwards, our discussion will be

limited only to GA-IPA. We discussed the proximity in terms of amplitudes, angular

separation and ranges of three sources and eight sensors in the ULA. The actual

values of amplitudes, DOA and ranges are 1 1 1

( 1 , 0 . 5 2 3 6 , 0 . 5 ) ,s r a d r

2 2 2( 3 , 1 . 2 2 1 7 , 4 ) ,s r a d r 3 3 35 , 2 . 2 6 8 9 , 7( ) .s r a d r

Case IV (Amplitude Proximity Check): In this case, the behavior of GA-IPA

technique is discussed for the amplitudes proximity. Every time, only the values of

amplitudes are changing, while the values of DOA and ranges are left unchanged.

Table 5. 8 GA-IPA for Amplitude proximity

𝑠1 𝑠2 𝑠3 𝜃1 𝜃2 𝜃3 𝑟1 𝑟2 𝑟3 MSE Convergence

Desired 1.0000 3.0000 5.0000 0.5236 1.2217 2.2689 0.5000 4.0000 7.0000

Estimated 1.0054 3.0053 5.0054 0.5290 1.2270 2.2743 0.5054 4.0053 7.0053 10−6 89%

Desired 1.0000 2.0000 5.0000 0.5236 1.2217 2.2689 0.5000 4.0000 7.0000

Estimated 1.0058 2.0058 5.0054 0.5291 1.2272 2.2744 0.5055 4.0054 7.0053 10−6 88%

Desired 1.0000 1.5000 5.0000 0.5236 1.2217 2.2689 0.5000 4.0000 7.0000

Estimated 1.0061 1.5062 5.0057 0.5293 1.2274 2.2745 0.5056 4.0056 7.0055 10−6 86%

Desired 1.0000 1.5000 2.0000 0.5236 1.2217 2.2689 0.5000 4.0000 7.0000

Estimated 1.0065 1.5067 2.0067 0.5295 1.2276 2.2747 0.5058 4.0059 7.0057 10−5 84%

As given in Table 5.8, the performance of GA-IPA in terms of accuracy, MSE and

convergence rate is degraded when the amplitudes are very close to each other.

5 10 15 20 2510

-8

10-6

10-4

10-2

100

102

SNR (dB)

MS

E

SA SA-IPA IPA GA GA-IPA


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However, the GA-IPA is robust enough to produce fairly good results for the

amplitude proximity.

Case V (DOA Proximity Check): In this case, the performance of GA-IPA is

examined for DOA proximity in terms of accuracy, MSE and convergence rate. Each

time, only the values of DOA are changed while keeping the values of amplitude and

ranges unchanged. The number of local minima increases, as soon as we brought the

DOA close to each other. Due to these local minima the accuracy, MSE and

convergence rate are degraded, however, the GA-IPA is robust enough to produce

fairly good results even in this case as provided in Table 5.9.

Table 5. 9 GA-IPA for DOA proximity s1 s2 s3 θ1 θ2 θ3 r1 r2 r3 MSE Convergence

Desired 1.0000 3.0000 5.0000 0.5236 1.2217 2.2689 0.5000 4.0000 7.0000

Estimated 1.0054 3.0053 5.0054 0.5290 1.2270 2.2743 0.5054 4.0053 7.0053 10−6 89%

Desired 1.0000 3.0000 5.0000 0.5236 0.8727 2.2689 0.5000 4.0000 7.0000

Estimated 1.0055 2.0054 5.0055 0.5291 0.8782 2.2744 0.5055 4.0054 7.0054 10−6 88%

Desired 1.0000 3.0000 5.0000 0.5236 0.6981 2.2689 0.5000 4.0000 7.0000

Estimated 1.0056 3.0056 5.0057 0.5296 0.7042 2.2747 0.5056 4.0056 7.0055 10−6 85%

Desired 1.0000 3.0000 5.0000 0.5236 0.6458 0.7854 0.5000 4.0000 7.0000

Estimated 1.0058 1.5058 2.0059 0.5303 0.6527 0.7924 0.5058 4.0059 7.0057 10−5 83%

Case VI (Range Proximity Check): In this sub-section, we examined the proximity

of ranges. Again one can see from Table 5.10, the performance of GA-IPA is less

affected in terms of accuracy, MSE and convergence when the values of ranges are

kept close to each other.

Table 5. 10 GA-IPA for Range proximity

s1 s2 s3 θ1 θ2 θ3 r1 r2 r3 MSE Convergence

Desired 1.0000 3.0000 5.0000 0.5236 1.2217 2.2689 0.5000 4.0000 7.0000

Estimated 1.0054 3.0053 5.0054 0.5290 1.2270 2.2743 0.5054 4.0053 7.0053 10−6 89%

Desired 1.0000 3.0000 5.0000 0.5236 1.2217 2.2689 0.5000 2.0000 7.0000

Estimated 1.0055 2.0054 5.0055 0.5292 1.2273 2.2744 0.5059 2.0057 7.0054 10−6 88%

Desired 1.0000 3.0000 5.0000 0.5236 1.2217 2.2689 0.5000 2.0000 2.5000

Estimated 1.0056 3.0055 5.0056 0.5293 1.2275 2.2745 0.5061 4.0063 2.5062 10−6 87%

Desired 1.0000 3.0000 5.0000 0.5236 1.2217 2.2689 0.3000 0.9000 1.6000

Estimated 1.0057 3.0056 5.0057 0.5295 1.2277 2.2749 0.3070 0.9071 1.6072 10−5 85%


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5.3 JOINT ESTIMATION OF 3-D PARAMETERS USING DE-PS AND PSO-PS

In this section, we present an efficient approaches based on hybrid evolutionary

heuristic computations to estimate jointly the amplitudes, ranges and elevation angles

of near field sources arriving on passive ULA. In this hybrid approaches, DE and PSO

are hybridized with PS technique. MSE is used as fitness evaluation function. The

results gotten from hybrid techniques are not only compared with each other but also

with DE and PSO alone. An extensive statistical analysis is employed to check the

validity and consistency of the proposed technique through large number of Monte

Carlo simulations.

The steps of PSO and DE in the form of pseudo code are same as discussed in the

previous chapter.


In this section, several simulations are performed to analyze the performance of DE,

DE-PS, PSO, and PSO-PS. We discussed estimation accuracy, convergence,

robustness against noise and proximity effect for different number of sources and

sensors. Throughout the simulations the distance between two consecutive sensors in

the ULA is taken / 4 . All the values of ranges and elevation angles are taken in

terms of wavelength ( ) and radians (rad) respectively. The entire results are carried

out for 2

10

as a threshold MSE value. A MATLAB version 7.8.0 is used and each

result is averaged over 100 independent trials.

Case I: In this case, the estimation accuracy, MSE, % convergence, robustness

against noise and proximity effect of DE, DE-PS, PSO and PSO-PS are discussed for

two sources. The desired values of amplitudes, ranges and elevation angles are

1 1 1( 3, 0.6 , 1 )s r rad , 2 2 2( 5, 1.5 , 2 ).s r rad


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5.3.1.1 Estimation Accuracy: In this sub-case, the ULA consists of 4 sensors and no

noise is added. As listed in Table 5.11, all the four mentioned schemes make good

estimates and produce fairly less error between desired response and estimated

response. As given, the results of DE and PSO are further improved when they are

hybridized with PS. However, among all techniques, the hybrid DE-PS has produced

better results and has less error between desired and estimated response. The second

best result is given by the PSO-PS, while the DE and PSO alone have produced the

third and fourth best results respectively.

Table 5. 11 Estimation Accuracy of Amplitudes, Ranges & DOA for 2 Sources and 4

sensors

Scheme 1s 2s ( )1r ( )2r ( )1 rad ( )2 rad

Desired values 3.0000 5.0000 0.6000 1.5000 1.0000 2.0000

DE-PS 3.0010 5.0010 0.6012 1.0011 1.0012 2.0011

DE-PS (error) 0.0010 0.0010 0.0012 0.0011 0.0012 0.0011

PSO-PS 3.0023 5.0025 0.6025 1.0024 1.0026 2.0025

PSO-PS(error) 0.0023 0.0025 0.0025 0.0024 0.0026 0.0025

DE 3.0035 5.0035 0.6038 1.0039 1.0037 2.0036

DE (error) 0.0035 0.0035 0.0038 0.0039 0.0037 0.0036

PSO 3.0047 5.0048 0.6048 1.5047 1.0051 2.0051

PSO (error) 0.0047 0.0048 0.0048 0.0047 0.0051 0.0051

5.3.1.2 Robustness: Fig 5.3, shows the robustness of each scheme against noise for

two sources. The ULA consists of 10 sensors. The MSE of each scheme is sketched

against Signal to Noise Ratio (SNR), where the values of SNR are ranging from 5dB

to 20 dB. One can see that the hybrid DE-PS and hybrid PSO-PS techniques are more

robust as compared to DE and PSO alone respectively. In addition, among all of them

the hybrid DE-PS technique is more robust against all the values of SNR, while the

PSO-PS is the second best robust scheme.


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Fig. 5. 3 MSE vs SNR for 2 sources and 10 sensors

5.3.1.3 MSE and Convergence: In this sub-case, MSE and convergence of each

scheme is discussed for two sources, where the ULA consists of 4 sensors without

adding any noise. In Table 5.12, it has been shown that among all schemes, the hybrid

DE-PS technique has less MSE i.e.6

10

and has maximum convergence i.e. 92%.

Table 5. 12 MSE and Convergence rate of two sources for different number of sensors

No.of

sensors

Scheme MSE (Power of 10) Convergen

ce

No. of

sensors

Scheme MSE (Power of

10)

Convergen

ce

4 DE-PS -6 92% 8 DE-PS -8 97%

PSO-PS -5 87% PSO-PS -7 92%

DE -4 84% DE -6 90%

PSO -3 78% PSO -5 84%

6 DE-PS -7 95% 10 DE-PS -9 98%

PSO-PS -6 90% PSO-PS -8 94%

DE -5 87% DE -7 92%

PSO -4 80% PSO -6 87%

The second best scheme is PSO-PS which has MSE 5

10

and convergence of 86%. In

the same Table.5.12, the effect of increasing elements is also mentioned due to which

the MSE and convergence of each scheme has improved.

5.3.1.4 DOA Proximity: In this sub-case, the proximity effect of elevation angles is

discussed for two sources and 6 sensors without having any noise in the system. For

this, the values of ranges and amplitudes are kept same as taken above but the values

of elevation angles are brought closed to each other i.e. 1 21 , 1.0996rad rad .

5 10 15 2010

-8

10-6

10-4

10-2

MS

E

SNR (dB)

PSO

DE

PSO-PS

DE-PS


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As given in Table 5.13, estimation accuracy, MSE and convergence of each scheme is

degraded. However, again the hybrid DE-PS and hybrid PSO-PS techniques produced

better result as compared to DE and PSO alone. However, the best result is given by

DE-PS.

Table 5. 13 DOA proximity for two sources and 6 sensors

Scheme 1( )rad 2 ( )rad MSE (Power of 10) Convergence (%)

Desired Value 1.0000 1.0096 ---- ----

DE-PS 1.0018 1.1016 -6 91

DE-PS(error) 0.0018 0.0020 ---- ----

PSO-PS 1.0039 1.1036 -5 84

PSO-PS(error) 0.0039 0.0040 ---- ----

DE 1.0048 1.1044 -4 81

DE (error) 0.0048 0.0048 ---- ----

PSO 1.0060 1.1058 -3 76

PSO (error) 0.0060 0.0062

Case II: In this case, the estimation accuracy, robustness against noise, MSE,

convergence and proximity effect of all four mentioned schemes are discussed for

three sources. The desired values are 1 1 1( 6, 1 , 0.5 )s r rad

2 2 2( 2, 4 , 1.7 )s r rad 3 3 3( 3.5, 0.5 , 2.6 ).s r rad

5.3.1.5 Estimation Accuracy: In this sub-case, the ULA consists of 6 sensors and

noise is not added at the output of any sensor. Due to the increase of unknown, we

faced few local minima, due to which the estimation accuracy of each scheme

despoiled. However, as given in Table.5.14, the hybrid DE-PS technique is smart

enough to perform well even in the presence of local minima and have made a close

estimate of desired values as compared to the remaining three techniques. Again the

second best is the other hybrid PSO-PS technique.


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Table 5. 14 Estimation accuracy of Amplitude, Ranges & DOA for 3 sources with 6 sensors

Scheme 1s 2s 3s ( )1r ( )2r ( )3r ( )1 rad ( )2 rad ( )3 rad

Desired 6.0000 2.0000 3.5000 1.0000 4.0000 7.0000 0.5000 1.7000 2.6000

DE-PS 6.0034 2.0035 3.0034 1.0037 4.0038 7.0033 0.5031 1.7038 2.6032

DE-PS (error) 0.0034 0.0035 0.0034 0.0037 0.0038 0.0033 0.0031 0.0038 0.0032

PSO-PS 6.0053 2.0057 3.5058 1.0058 4.0057 7.0054 0.5059 1.7058 2.6061

PSO-PS(error) 0.0053 0.0057 0.0058 0.0058 0.0057 0.0054 0.0059 0.0058 0.0061

DE 6.0083 2.0084 3.5087 1.0084 4.0085 7.0084 0.5087 1.7089 2.6087

DE (error) 0.0083 0.0084 0.0087 0.0084 0.0085 0.0084 0.0087 0.0089 0.0087

PSO 6.0189 2.0190 3.5189 1.0188 4.0188 7.0187 0.5191 1.7190 2.6194

PSO (error) 0.0189 0.0190 0.0189 0.0188 0.0188 0.0187 0.0191 0.0190 0.0194

5.3.1.6 Robustness: In this sub-case, the robustness of all four schemes is checked

against SNR. As shown in Fig 5.4, the MSE of all schemes have increased as

compared to the previous case. However, the hybrid DE-PS technique maintained

better value of MSE for all values of SNR as compared to the others. The second best

robust algorithm in this scenario is PSO-PS. All these curves are carried out for 12

sensors in the ULA.

Fig. 5. 4 MSE vs SNR for 4 sources and 12 sensors

5.3.1.7 MSE and Convergence: In this sub-case, the MSE and convergence of all

four mentioned schemes are discussed for three sources and 6 sensors in ULA without

adding any noise to the system. As listed in Table 5.15, the MSE and convergence of

all schemes are affected due to the presence of local minima. However, the hybrid

DE-PS technique maintained a very good value of MSE i.e. 5

10

and has a fairly good

5 10 15 2010

-7

10-6

10-5

10-4

10-3

10-2

SNR (dB)

MS

E

DE-PS PSO-PS DE PSO


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convergence of 86%. The second best is PSO-PS. Due to the increase of sensors in the

ULA, the MSE and convergence becomes better for each scheme.

Table 5. 15 MSE and %convergence of three sources for different number of sensors

No.of sensors Scheme MSE (Power of 10) Convergence No. of sensors Scheme MSE (Power of 10) Convergence

6 DE-PS -5 86% 10 DE-PS -7 92%

PSO-

PS

-4 80% PSO-

PS

-6 86%

DE -3 77% DE -5 83%

PSO -2 70% PSO -4 75%

8 DE-PS -6 89% 12 DE-PS -8 95%

PSO-

PS

-5 83% PSO-

PS

-7 88%

DE -4 80% DE -6 85%

PSO -3 72% PSO -5 78%

5.3.1.7 DOA Proximity: In this sub-case, the performance of all schemes is evaluated

for three sources when they are placed closed to each other. The values of amplitudes

and ranges are same as mentioned above while the values of elevation angles are

1.0472 , 1.1345 , 1.2217 ).1 2 3( rad rad rad The ULA consists of 8 sensors.

Due to proximity of elevation angles, we faced more local minima and as a result the

estimation accuracy, MSE and convergence of all schemes despoiled more as

compared to the previous case of two sources.

Table 5. 16 DOA proximity for 3 sources and 8 sensors

Scheme ( )1 rad ( )2 rad ( )3 rad

MSE (Power of 10) Convergence (%)

Desired values 1.0472 1.1345 1.2217 ---- ----

DE-PS 1.0518 1.1392 1.2263 -5 85%

DE-PS (error) 0.0046 0.0047 0.0046 ---- ----

PSO-PS 1.0551 1.1425 1.2296 -4 78%

PSO-PS (error) 0.0079 0.0080 0.0079 ---- ----

DE 1.0577 1.1451 1.2322 -3 74%

DE (error) 0.0105 0.0106 0.0105 ---- ----

PSO 1.0671 1.1544 0.7998 -2 67%

PSO (error) 0.0199 0.0199 1.2416 ----- -----

As provided in Table 5.16, again the performance of hybrid DE-PS and hybrid PSO-

PS is better than DE and PSO alone. However, the hybrid DE-PS technique proved to


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be the best technique among all of them in terms of estimation accuracy, MSE and

convergence.

Case III: In this case, the estimation accuracy, robustness against noise, MSE and

convergence is discussed for four sources. The desired values of amplitudes, ranges

and DOA for four sources are 1 1 11, 2 , 1.8s r rad 2 2 24, 3 , 2.3s r rad

3 3 35, 6 , 0.8s r rad 4 4 47, 8 , 1s r rad .

5.3.1.8 Estimation Accuracy: In this sub-case, the ULA consists of 8 sensors and no

noise is added to the system. We faced more local minima with the increase of

unknowns and as a result the performance of all schemes affected which was quite

expected. However, again the hybrid DE-PS is good enough to make a close estimate

of desired values even in the presence of strong local minima. The second best

technique is PSO-PS as provided in Table 5.17.

Table 5. 17 Accuracy of Amplitude, Ranges & DOA for 4 sources and 8 sensors

Scheme

1s 2s 3s 4s ( )1r ( )2r

( )3r ( )4r

( )1 rad

( )2 rad

( )3 rad

( )4 rad

Desired Values 1.0000 4.0000 5.5000 7.0000 2.0000 3.0000 6.0000 8.0000 1.8000 2.3000 0.8000 1.0000

DE-PS 1.0053 4.0057 5.5054 7.0057 2.0055 3.0052 6.0053 8.0054 1.8058 2.3060 0.8058 1.0059

DE-PS (error) 0.0053 0.0057 0.0054 0.0057 0.0055 0.0052 0.0053 0.0054 0.0058 0.0060 0.0058 0.0059

PSO-PS 1.0093 4.0094 5.5093 7.0090 2.0095 3.0094 6.0090 8.0093 1.8096 2.3094 0.8097 1.0092

PSO-PS (error) 0.0093 0.0094 0.0093 0.0090 0.0095 0.0094 0.0090 0.0093 1.0096 0.0094 0.0097 0.0092

DE 1.019 4.0178 5.5179 7.0181 2.0178 3.0183 6.0185 8.0180 1.8182 2.3184 0.8180 1.0181

DE (error) 0.0179 0.0178 0.0179 0.0181 0.0178 0.0183 0.0185 0.0180 0.0182 0.0184 0.0180 0.0181

PSO 1.0388 4.0387 5.5388 7.0383 2.0383 3.0385 6.0384 8.0382 1.8392 2.3389 0.8391 1.0390

PSO (error) 0.0388 0.0387 0.0388 0.0383 0.0383 0.0385 0.0384 0.0382 0.0392 0.0389 0.0391 0.0390

5.3.1.9. Robustness: In this sub-case, the ULA consists of 14 sensors. As shown in

Fig 5.5, the hybrid DE-PS approach is more robust for all values of SNR for four

sources. The second best is other hybrid PSO-PS technique.


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Fig .5. 5 MSE vs SNR for 4 sources and 14 sensors

5.3.1.10 MSE and Convergence: Due to more local minima, the convergence and

MSE are also affected as given in Table 5.18. However, the DE-PS approach again

proved to be the best technique as compared to the other techniques. It has maintained

minimum MSE i.e. 4

10

and good convergence of 83% for 8 sensors in the ULA. The

second best result is produced by PSO-PS, while the performance of PSO and DE

alone is despoiled more in this case. The performance in terms of MSE and

convergence rate of all techniques is improved for increasing number of sensors in the

ULA also provided in Table 5.18.

Table 5. 18 MSE and Convergence of four sources for different number of sensors

No.of sensors Scheme MSE (Power of 10) Convergence No. of sensors Scheme MSE (Power of 10) Convergence

6 DE-PS -4 83% 10 DE-PS -6 88%

PSO-PS -3 77% PSO-PS -5 83%

DE -2 72% DE -4 78%

PSO -2 64% PSO -3 70%

8 DE-PS -5 85% 12 DE-PS -7 91%

PSO-PS -4 79% PSO-PS -6 86%

DE -3 75% DE -5 82%

PSO -2 67% PSO -4 73%

5.3.1.11 DOA proximity: In this sub-case the angles proximity is described for four

sources. The ULA consists of 10 sensors. The values of amplitude and ranges are kept

same as taken at the beginning of this current case. However the elevation angles are

changed and considered that all the four sources are placed close to each other i.e.

5 10 15 2010

-6

10-5

10-4

10-3

10-2

SNR (dB)

MS

E

DE-PS PSO-PS DE PSO


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1.99199 , 2.0071 , 2.0944 , 2.1817 )1 2 3 4( rad rad rad rad . In this case,

the performance of PSO is despoiled a lot which becomes better when hybridized

with PS. However the DE-PS acted well even in this case and produced better results

as compared to their counterparts in terms of estimation accuracy, MSE and

convergence rate as given in Table 5.19.

Table 5. 19 DOA proximity for four sources and 10 sensors

Scheme ( )1 rad ( )2 rad ( )3 rad ( )4 rad

MSE (Power of 10) Convergence

Desired values 1.9199 2.0071 2.0944 2.1817 --- ---

DE-PS 1.9265 2.0139 2.1010 2.1884 -4 81%

DE-PS (error) 0.0066 0.0068 0.0066 0.0067 --- ---

PSO-PS 1.9298 2.0178 2.1053 2.1925 -3 75%

PSO-PS(error) 0.0108 0.0107 0.0109 0.0108 --- ---

DE 1.9387 2.0260 2.1131 2.2004 -2 70%

DE (error) 0.0188 0.0189 0.0187 0.0187 --- ---

PSO 1.9605 0.0476 2.1350 2.2225 -2 62%

PSO (error) 0.0406 0.0405 0.0406 0.0408 --- ---

PART-II

In this part, we have developed schemes for jointly estimating 4-D parameters

(amplitudes, range, elevation and azimuth angles) of near field sources impinging on

CSCA. To solve 3-D (range, elevation and azimuth angles) near field source

localization problem, several algorithms are presented in literature [166], [167], which

are not only computationally expensive, but also have the problem of pair matching

between elevation and azimuth angles. Though, a two-stage separated steering vector-

based algorithm in [168] solves the problem of pair matching, yet it has higher MSE

and is computationally expensive as it requires more than 400 snapshots to achieve

the results. Moreover, it also fails to estimate the amplitude of sources, which is also

some time an important parameter to be estimated. Clearly the goal is to develop a

scheme which must be able to jointly estimate the amplitude, range, elevation angle


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and azimuth angle, should provide an improved MSE and finally should be free of

pair matching problem.

ReceiverTransmitter

kth target ,where k=1,2,…,K

. . . . . .

k

0 -P

k

P Y

Z

X

P

rk

P

Sub array 1

Sub array 2

Fig. 5. 6 Schematic Diagram for bistatic radar

In this part, we have assumed bistatic phase multiple input multiple output (MIMO)

radar having CSCA on its receiver. Let the transmitter of this bistatic radar send

coherent signals using a sub-array that gives a fairly wide beam with a large solid

angle so as to cover up any potential relevant target in the Fresnel zone (near field) as

shown in Fig 5.6. We developed heuristic computational intelligence techniques to

estimate jointly the amplitude, range, elevation angle and azimuth angles of these

multiple targets impinging on the passive CSCA. In these computational techniques,

first PSO and DE are used alone and then to improve the results further, PSO and DE


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are hybridized with ASA. In the hybridized version, PSO and DE are used as global

search optimizer, while ASA is used as rapid local search technique. The performance

of PSO, DE, PSO-ASA and DE-ASA is not only compared with each other but also

with some traditional techniques available in literature using Mean Square Error as

figure of merit.

5.4 DATA MODEL FOR 4-D NEAR FIELD TARGETS

In this section, we developed a data model for P near field targets impinging on

CSCA placed on the receiver of bistatic radar. The CSCA is composed of two

symmetric sub-ULAs, placed along X-axis and Y-axis as shown in Fig.5.6. Each ULA

carries 2*M passive sensors while the reference sensor is common for them. For

4 1P M , the data model at th

m and th

n sensor in the x-axis and y-axis sub-array,

respectively, can be represented as,

,0 ,01

P jm xpw s epm m

p

(5.20)

0, 0,1

P jn ypw s epn n

p

(5.21)

where

2( )m m mxp xp xp (5.22)

2

( )n n nyp yp yp (5.23)

where

2 sin cosd p p

xp

(5.24)

2 2 2(1 sin cos )d p p

xprp

(5.25)


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2 sin sind p pyp

(5.26)

2 2 2(1 sin sin )d p p

yprp

(5.27)

In (5.20) and (5.21), ,0m and 0,n are additive white Gaussian noise (AWGN)

added at th

m and th

n sensors respectively. In vector-matrix form, the signal model

can be represented as,

w Bs η (5.28)

where

[ ... ...,0 1,0 1,0 1,0 2,0 1,0 ,0 0,0

... ... ]0, 0, 1 0, 1 0,1 0,2 0, 1 0,

w w w w w w w wM M M M

Tw w w w w w wM M M M

w

(5.29)

,0 1,0 1,0 1,0 2,0 1,0 ,0 0,0

0, 0, 1 0, 1 0,1 0,2 0, 1 0,

[

]

... ...

... ...

M M M M

M M M MT

η (5.30)

and ―s‖ is a vector containing signals amplitudes i.e.

[ . . . ]1 2T

s s sPs (5.31)

Similarly ―B‖ is a matrix containing steering vectors of the targets, i.e.,

[ ( , , , ) ( , , , ) . . . ( , , , )]1 1 1 2 2 21 2x x y y x x y y xP xP yP yP B b b b (5.32)

where


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125

2 2

2 2 2

2

[( ) ( ) ] [( 1) ( 1) ] [ ] [ ]

[( 1) ( 1) ] [( ) ( ) ] [( ) ( ) ]

[( 1) ( 1) ] [ ] [

( , , , )

[ . . . . . .

1

. . .

xp xp xp xp xp xp xp xp

xp xp xp xp yp yp

yp yp yp yp yp

j M M j M M j j

j M M j M M j M M

j M M j j

xp xp yp yp

e e e e

e e e

e e e

b

2

2

] [( 1) ( 1) ]

[( ) ( ) ]]

. . .yp yp yp

yp yp

j M M

j M M T

e

e

(5.33)

Now clearly the problem in hand is to accurately and jointly estimate the unknown

parameters (amplitudes, ranges, elevation and azimuth angles) of the reflected signals

from the targets. For this, we have used PSO-ASA and DE-ASA.

5.5 JOINT ESTIMATION OF AMPLITUDE, RANGE AND 2D DOA USING

DE-ASA AND PSO-ASA FOR BI-STATIC RADAR

Step 1 Initialization: The first step of PSO is to initialize the swarm randomly, i.e.,

produce randomly Q particles. In the current problem, the length of each particle is

4*P where P is the number of targets. Mathematically, the particles can be written as,

, , , ...1 2 3

T

Q B b b b b (5.34)

, , ... , , ... , , , ... , , , ...1,1 1,2 1, 1, 1 1, 2 1,2* 1,(2* 1) 1,(2* 2) 1,(3* ) 1,(3* 1) 1,(3* 2) 1,(4* )

, , ... , , ,... , ,2,1 2,2 2, 2, 1 2, 2 2,2* 2,(2* 1) 2,(2*

,1

2

3...

r r r s s sP P P P P P P P P P

r rP P P P P P

Q

b

b

b

b

, ... , , , ...2) 2,(3* ) 2,(3* 1) 2,(3* 2) 2,(4* )

, , ... , , ,... , , , ... , , , ...3,1 3,2 3, 3, 1 3, 2 3,2* 3,(2* 1) 3,(2* 2) 3,(3* ) 3,(3* 1) 3,(3* 2) 3,(4* )

.

.

.

, , ... , , , ...,1 ,2 , , 1 , 2 ,2*

r s s sP P P P

r r r s s sP P P P P P P P P P

Q Q Q P Q P Q P Q

, , , ... , , ,...,(2* 1) ,(2* 2) ,(3* ) ,(3* 1) ,(3* 2) ,(4* )r r r s s sP Q P Q P Q P Q P Q p Q P

where the lower and upper bounds of , , ,r s , are defined as


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126

: 0 / 2, ,

: 0 2, ,, 1,2,... & 1,2,...

:,2 ,2

:,3 ,3

Rq p q p

R rq P p q P pfor q Q p P

r R r r ruq P p q P pl

s R s s suq P p q P pl

where ru and rl are the upper and lower bounds of ranges while us and ls represent

the upper and lower bounds of amplitudes.

Step 2 Fitness Function: To calculate the fitness of each particle, we used the

following relation,

1

( )(1 ( ))

FF qq

(5.35)

where ( )q is called MSE which has been derived from maximum likelihood

principle as discussed in chapter 4. This MSE defines an error between estimated and

desired signals and for qth particle it can be given as,

( ) ( ) ( )1 2q q q (5.36)

where

21

ˆ( )1 , ,02

M x qq w wm o mM x m M x

(5.37)

and

2

1ˆ( )2 0, 0,2

M yq

q w wn nM y n M y

(5.38)


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127

where M x and M y are the number of sensors placed along x-axis and y-axis

respectively. In ,0wm and 0,w n are defined in (5.20) and (5.21) respectively, while

ˆ,0

qwm

and ˆ0,q

w ncan be defined as,

2 2 2 2ˆ ˆ ˆ ˆ2 sin( ) cos( ) (1 sin ( )) cos ( )ˆˆ exp[ ( )],0 3

ˆ12

q q q qm d b b m d b bP p pP p P pq q

w b jm P p qp b P p

(5.39)

2 2 2 2ˆ ˆ ˆ ˆ2 sin( ) sin( ) (1 sin ( )) sin ( )ˆˆ exp[ ( )]0, 3

ˆ12

q q q qK n d b b n d b bq q K K k k K k

w b jn K k qk b K k

(5.40)

Now, store each particle as local best ( )l while the one having maximum fitness

function be stored as global best ( )bg . The remaining steps of are the same as

discussed above. In the same way, we can develop steps for DE which has the same

initialization step as just discussed above for PSO. The remaining steps of DE are also

same as discussed in the previous section. The best individual result of PSO and DE

are given to ASA for further improvements. For ASA, we have used a MATLAB

built-in optimization tool box for which the parameter setting is given in Table.5.20.

Table 5. 20 Parameters Setting for ASA

Parameters Setting

Starting Point Particles achieved by DE and PSO

No. of Iteration 2000

No. of variables 4*K

Fitness Limit 10-15

Function tolerance 10-15

Nonlinear Constraints tolerance 10-15

Derivative approximate Finite central difference

X-Tolerance 10-15

Maximum function Evaluations 50000


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In this section, several simulations are carried out to validate the performance of the

proposed techniques. This section is basically divided into two parts, in part 1, we

compared the performances of PSO, DE, PSO-ASA, DE-ASA with each other in

terms of estimation accuracy and convergence for different number of targets. While

in part 2, the performances of the two best techniques among them are compared with

existing traditional algorithms [167]-[168] by using MSE as a figure of merit. Every

time the number of sensors in both sub-arrays is taken to be the same, whereas the

reference sensor is common for them. Throughout the simulations, the distance

between two consecutive sensors in both sub-arrays is taken same i.e. / 4.d All

the signals reflected back from targets are assumed to be statistically independent and

having constant frequency. The received data at the output of each sensor are polluted

by zero mean, unit variance AWGN. All the values of elevation and azimuth angles

are taken in degrees while the values of ranges are taken in terms of wavelength ( )

.5.5.1.1 Estimation Accuracy

In this sub-section, 3 cases are discussed on the basis of different number of targets in

order to evaluate the estimation accuracy of PSO, DE, PSO-ASA, DE-ASA. In this

case no noise is added to the system.

Case I: In this case, 2 targets are taken which are impinging on CSCA. The array

consists of 9 sensors i.e. each sub-array is composed of 4 sensors, while the reference

sensor is common for them. The two targets are located at

35 , 73 , 1.5 , 4)1 1 1 1( r s

, 52 , 105 , 3 , 1)2 2 2 2( r s

. As

provided in Table 5.21, all the four techniques produced fairly good estimation

accuracy and one can observe the advantages of hybridization. Basically, the PSO


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alone is less accurate then DE alone but as soon as the PSO is passed through ASA, it

becomes even better than that of DE alone. However, the most accurate scheme is the

DE hybridized with ASA (DE-ASA).

Table 5. 21 Estimation accuracy for 2-targets

Scheme 1

1

( )

1r

1s

2

2

( )

2r

2s

Desired 35.0000 73.0000 1.5000 4.0000 52.0000 105.0000 3.0000 1.0000

PSO 35.0061 73.0060 1.4951 4.0050 52.0060 105.0061 3.1032 1.0051

DE 35.0044 72.0058 1.5022 3.9977 51.0058 104.9955 3.1022 0.9976

PSO-ASA 34.9967 73.0036 1.5018 4.0019 52.0036 105.0033 3.1017 1.0020

DE-ASA 35.0015 72.0082 1.4991 3.9988 51.0082 104.9984 3.1008 0.9988

Case II: In this case, the number of targets is increased to 3. For this, the array

consists of 13 sensors and the 3 targets are located at

1 1 1 1( 80 , 40 , 7 , 2),r s

65 , 160 , 1 , 8),2 2 2 2( r s

25 , 120 , 3 , 6)3 3 3 3( r s

. As listed in Table 5.22, degradation in the

performance of the global search methods (PSO and DE alone) can be observed.

However, their estimation accuracy is significantly improved when their results are

passed through the local search optimizer (ASA). Once again, one can conclude that

the hybrid DE-ASA produced the most accurate estimation, while the second best

performance is shown by the other hybrid PSO-ASA technique.


Scheme 1

1

( )1r

1s

2

2

( )2

r 2

s 3

3

( )3r

3s

Desired 80.0000 40.0000 7.0000 2.0000 65.0000 160.0000 1.0000 8.0000 25.0000 120.0000 3.0000 6.0000

PSO 80.3843 39.6156 6.8209 2.1789 64.6158 159.6156 1.3847 8.6157 24.6158 120.1789 2.8209 5.8210

DE 79.8028 40.1973 6.9026 1.9028 65.1974 160.1973 1.1979 8.1972 25.1974 119.9028 2.9026 6.0971

PSO-ASA 79.9264 40.0738 7.0192 2.0191 64.9261 160.0738 0.9261 7.9263 24.9261 120.0191 3.0192 6.0193

DE-ASA 80.0268 39.9731 6.9932 1.9931 64.9733 159.9731 1.0269 8.0260 24.9733 119.9931 2.9932 5.9932


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Case III: In this case, 4 targets are considered which are located at

1 1 1 140 , 120 , 2.5 , 3)( r s

, 2 2 2 275 , 50 , 4 , 9)( r s

,

55 , 160 , 12 , 6)3 3 3 3( r s

, ( 30 , 220 , 6 , 4)4 4 4 4r s

.

As provided in Tables 5.23 and 5.24, in this case one can observe the significance of

hybridization in a better way as we have faced more local minima. Due these local

minima, PSO and DE alone are degraded, however, both of them got good estimation

accuracy once they are hybridized with ASA. Once again DE-ASA proved to be the

most accurate scheme while the second best scheme is PSO-ASA.

Table 5. 23 Estimation accuracy for 4-targets (continue)

Scheme 1

1

( )1r

1s

2

2

( )2r

2s

Desired 40.0000 120.0000 2.5000 3.0000 75.0000 50.0000 4.0000 9.0000

PSO 38.2202 121.8799 2.9797 3.8711 73.1884 48.1285 4.8686 9.9128

DE 41.2388 121.2490 2.8989 3.7010 73.8656 48.8375 4.6654 9.6655

PSO-ASA 40.4177 119.6279 2.6522 3.4805 75.4947 50.3942 4.4056 9.4060

DE-ASA 40.0709 120.0781 2.5791 3.0812 75.0869 50.0870 4.0867 9.0869


Scheme 3

3

( )3r 3

s 4

4

( )4r

4s

Desired 55.00000 160.0000 12.0000 6.0000 30.0000 220.0000 6.0000 4.0000

PSO 56.8282 161.8284 12.6582 5.2837 28.2814 218.1369 6.6182 4.8273

DE 53.8656 161.2659 12.4942 5.4766 28.7766 218.7751 5.6656 4.6659

PSO-ASA 55.4941 160.4658 12.2944 6.4041 30.5041 220.4041 6.32941 4.2946

DE-ASA 55.0870 160.0791 12.0912 6.0908 30.0988 220.0715 6.0870 4.0912

5.5.1.2 Convergence

In this section, we have discussed the convergence of each scheme for different

number of targets. For this, the MSE is kept same i.e. 2

10

. As shown in Fig 5.7, the

convergence of each scheme is despoiled with the increase of unknowns (targets) in

the problem. However, the convergence of hybrid schemes are less degraded and they

have maintained fairly good convergence every time. The convergence of PSO acted


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alone is less than that of DE alone but PSO got better convergence rate as compared

to DE alone, when it is hybridized with ASA. However, the better scheme among all

of them is DE-ASA since it maintained better convergence as compared to all of

them.

Fig. 5. 7 Convergence Rate versus Number of sources

So far, we have discussed the estimation accuracy and convergence rate of PSO, DE,

PSO-ASA and DE-ASA for different number of targets and one can conclude that

DE-ASA and PSO-ASA are proved to be the two best schemes among them. So, in

order to summarize the discussion, from now onward, we shall be limited only to the

discussion of these two hybrid schemes.

5.5.1.3 Proximity Effects

In this sub-section, we have discussed the proximity effects of elevation angles, as

well as, azimuth angles for DE-ASA and PSO-ASA. For this, we considered 3 sources

impinging on CSCA where CSCA is composed of 13 sensors. The received data is

polluted by 10 dB noise. We performed this simulation in two parts, in first part we

considered the azimuth angles to be constant and varied the elevation angles. In the

second part, we did the same in opposite manner i.e., for fixed elevation angles, the

proximity of azimuth angles are investigated. As shown in Table 5.25 and 5.26, that

even for fairly closely spaced targets, both the hybrid schemes still produced good

2 3 40

10

20

30

40

50

60

70

80

90

100

converg

ence r

ate

(%

)

Number of sources

PSO

DE

PSO-AS

DE-AS


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estimation accuracy, as well as, good convergence rate. However, between them, the

DE-ASA produced better results as compared to PSO-ASA.

Table 5. 25 Proximity effect of Elevation angles for 𝑠1 = 1, 𝑠2 = 3, 𝑠3 = 5 𝑟1 = 1.5𝜆, , 𝑟2 = 3𝜆, , 𝑟3 =

4𝜆 & 𝜙1 = 1300, 𝜙2 = 700, 𝜙3 = 1600.

Scheme 𝜃10 𝜃2

0 𝜃3

0 %Convergence

Desired Values 35.0000 75.0000 60.0000 ---

PSO-ASA 35.3843 75.3842 60.3844 89

DE-ASA 35.1791 75.1790 60.1793 95

Desired Values 30.0000 65.0000 75.0000 ---

PSO-ASA 30.3846 65.9832 75.9834 81

DE-ASA 30.1792 65.4301 75.4302 92

Desired Values 30.0000 40.0000 50.0000 ---

PSO-ASA 31.3965 41.4011 51.4013 68

DE-ASA 30.7692 40.7694 50.7690 87

Desired Values 30.0000 35.0000 40.0000 ---

PSO-ASA 32.3417 37.3518 42.3519 62

DE-ASA 31.1105 36.1107 41.1105 80

Table 5. 26 Proximity effect of azimuth angles for 𝑠1 = 1, 𝑠2 = 3, 𝑠3 = 5 𝑟1 = 1.5𝜆, , 𝑟2 = 3𝜆, , 𝑟3 =

4𝜆 & 𝜃1 = 300, 𝜃2 = 500, 𝜃3 = 850.

Scheme ∅10 ∅2

0 ∅3

0 %Convergence

Desired Values 25.0000 80.0000 240.0000 ---

PSO-ASA 24.6841 79.6170 239.6254 90

DE-ASA 25.1790 79.8210 240.1793 93

Desired Values 50.0000 80.0000 70.0000 ---

PSO-ASA 49.6154 80.9832 70.9832 81

DE-ASA 49.8210 80.4305 70.4303 90

Desired Values 50.0000 60.0000 70.0000 ---

PSO-ASA 48.6031 58.5981 71.4014 70

DE-ASA 50.7694 60.7696 70.7692 85

Desired Values 50.0000 55.0000 60.0000 ---

PSO-ASA 52.4420 57.3519 57.6468 64

DE-ASA 51.1104 56.1105 58.8967 82

5.5.1.4 Estimation Accuracy For DOA On Reference Axis

Some of the elevation angles ( 0 ,90 ) and azimuth angles (0 ,90 ,180 )

are

considered to be critical angles. In this case, we have checked the accuracy of both

hybrid schemes for these angles. The three targets are located at

1 1 1 10 , 90 , 3 , 1( )r s

, 2 2 2 290 , 0 , 1 , 2( )r s

,


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133

35 , 180 , 5 , 43 3 3 3( )r s

. In Fig 5.8, the estimation accuracy of both

hybrid schemes for elevation angles on reference axis (00 & 90

0) is evaluated and it

has been shown that both schemes produced significant error (60 to 8

0) especially for

900 whereas, at 35

0, the estimation accuracy of both schemes is up to the mark.

However, every time, the DE-ASA produced comparatively less error as compared to

PSO-ASA.

Fig. 5. 8 Elevation angle estimation on reference axis

Fig. 5. 9 Azimuth angle estimation on reference axis

In Fig 5.9, the estimation accuracy of both hybrid schemes is shown for azimuth

angles at 00, 90

0 and 180

0 and one can see a negligible effect on the estimation

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90

100

Number of Iteration

Ele

vation A

ngle

s (

Degre

es)

DE-ASA at 90 degree

PSO-ASA at 90 degree

Desired at 90 degree

Desired at 0 degree

PSO-ASA at 0 degree

DE-ASA at 0 degree

Dssired at 35 degree

DE-ASA at 35 degree


0 100 200 300 400 500 6000

50

100

150

200

Number of Iteration

Azim

uth

Angle

s (

Degre

es)

DE-ASA at 180 degree



Desired at 0 degree

PSO-ASA at 0 degree

DE-ASA at 0 degree


DE-ASA at 90 degree



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134

accuracy of both schemes. Once again the DE-ASA produced better results as

compared to PSO-ASA.

5.5.1.5 Comparison with Other Techniques Using Root Mean Square Error

(RMSE)

In this sub-section, we have evaluated the RMSE of DE-ASA and PSO-ASA in the

presence of noise and compared with existing traditional techniques [167]-[168]. We

considered two sources that have the desired values of angles, ranges and amplitudes

are 25 , 55 , 3.5 , 1),1 1 1 1( r s

60 , 170 , 1 , 3).2 2 2 2( r s

For

this simulation the array consists of 9 sensors. The SNR is ranging from 0dB to 20

dB. In Fig 5.10, Fig 5.11, Fig 5.12 and Fig 5.13, we can observe that the proposed

hybrid schemes have lower RMSE as compared to the algorithms described in [167]-

[168]. It can also be seen that the RMSE of both hybrid schemes is lower for the

target located near to the array as compared to the target located comparatively away

from the array. One can again observe that among all techniques (Heuristic or Non-

Heuristic), the hybrid DE-ASA scheme produced lower RMSE, while the second

lower RMSE is obtained by the other hybrid PSO-ASA scheme for elevation, azimuth

angles and ranges.

Fig. 5. 10 Root Mean Square Error of Elevation angles versus SNR

0 2 4 6 8 10 12 14 16 18 2010

-2

10-1

100

101

SNR (dB)

RM

SE

(D

egre

e)

Source2[167]

Source1[167]

Source2[168]

Source1[168]

Source1[DE-AS]

Source2[DE-AS]

Source2[PSO-AS]

Source1[PSO-AS]


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135

Fig. 5. 11 Root Mean Square Error of Azimuth angles versus SNR

Fig. 5. 12 Root Mean Square Error of Ranges versus SNR

One of the other advantages of the proposed hybrid schemes is that they can also be

used for the amplitude estimation. In Fig 5.13, RMSE is shown against SNR for the

amplitude of the targets. Once again the DE-ASA scheme maintained a lower RMSE

for both the targets.

Fig. 5. 13 Root Mean Square Error of Amplitudes versus SNR

0 2 4 6 8 10 12 14 16 18 2010

-2

10-1

100

101

SNR (dB)

RM

SE

(D

egre

e)

Source1[167]

Source2[167]

Source1[168]

Source2[168]

Source1DE-AS

Source2 DE-AS

Source2 PSO-AS

Source1 PSO-AS

0 2 4 6 8 10 12 14 16 18 2010

-4

10-2

100

102

SNR (dB)

RM

SE

(W

avele

ngth

)

Source2 [167]

Source1 [167]

Source2 [168]

Source1 [168]

Source1 DE-AS

Source2 DE-AS

Source1 PSO-AS

Source2 PSO-AS

0 2 4 6 8 10 12 14 16 18 2010

-3

10-2

10-1

100

SNR(dB)

RM

SE

Source1 DE-AS

Source2 DE-AS

Source1 PSO-AS

Source2 PSO-AS


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136

PART-III

5.6 JOINT ESTIMATION OF 5D PARAMETERS USING GA-PS AND GA-IPA

In this section, hybrid GA is utilized to jointly estimate the frequency, amplitude,

range, elevation angle and azimuth angle of near field sources impinging on CSCA.

Specifically, GA is used as a global optimizer, where as PS and IPA are used as rapid

local search optimizers. The same new multi-objective fitness function is used, as

given in the third part of chapter number 4.

5.6.1 SIGNAL MODEL FOR 5D PARAMETERS OF NEAR FIELD SOURCES

In this section, signal model for near field sources impinging on CSCA is

developed. All sources are considered to be narrow band and statistically independent

from each others. The amplitude ( )a , frequency ( )f , range ( )r , and 2D DOA ( , )

are different for different sources. The CSCA is composed of two sub arrays that are

placed along x-axis and y-axis respectively, as shown in Fig.5.6. If P is the total

number of sources then the signal received at th

m and th

n sensor in x-axis and y-axis

sub-arrays respectively, can be modeled as,

2( )

,0 ,01

P j m mxp xpw a epm m

p

(5.41)

2( )0, 0,

1

P j n nyp ypw a epn n

p

(5.42)

where ,0m and 0,n represent the Additive White Gaussian Noise (AWGN) added at

thm and

thn sensors in x-axis and y-axis sub-arrays respectively.

In (5.41), xp and xp can be given as,

sin cosk dxp p pp (5.43)


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137

In (5.43), 2 /k p p where /c fp p . Similarly, / 4mind

where

/ maxmin c f . So, (5.43), can be represented as,

(sin cos )2 max

f pxp p pf

(5.44)

In the same way,

2 2 2(1 sin cos )d p p

xprpp

(5.45)

where (5.45) can be further re-written as,

22 2

(1 sin cos )16 max

f pxp p p

f rp

(5.46)

Similarly, in (5.42), yp and yp can be given as,

(sin sin )2 max

f pyp p p

f

(5.47)

22 2

(1 sin sin )16 max

f pyp p p

f rp

(5.48)

In more compact form, (5.41) and (5.42), can be given as,

2 22 2(sin cos ) (1 sin cos )))

2 16

0 0

( (

max, ,

1

p pp p p p

p

mf m f

rj

P fw a em p m

p

(5.49)

2 22 2(sin sin ) (1 sin sin )))

2 16

0 0

( (

max, ,

1

p pp p p p

p

nf n f

rj

P fw a en p n

p

(5.50)


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138

where f p represent the frequency of thp while maxf is the maximum frequency to

be used. In Matrix-vector form (5.49) and (5.50) can be collectively represented as,

w Ba η (5.51)

where w,η,a,B are defined in the previous section.

From (5.49) and (5.50), one can see that the unknown parameters are

, , , ,a f rp p p p p where 1,2,...,p P . So, the problem in hand is to estimate these

5D parameters jointly and efficiently.

5.6.2 GA-PS AND GA-IPA

In this section, an approach based on GA-IPA and GA-PS is developed to estimate

the unknowns. We have used the MATLAB Built-in optimization tool box for

GA, IPA and PS for which the parameters settings are provided in Table 4.8.The

steps for GA, GA-PS, and GA-IPA in the form of pseudo code are given as while

their flow diagram is shown in Fig 4.1.

Step1 Initialization: In this step, we randomly generate M chromosomes where

the length of each chromosome is 5*P. In each chromosome the first P genes

represent amplitudes, the second P genes contains the frequencies, the next P

genes represent the ranges while the fourth and fifth P genes represent elevation

and azimuth angles respectively, of the sources as given below,

... ... ... ... ...1, 1 1, 2 1,21,1 1,2 1, 1,2 1 1,2 2 1,3 1,3 1 1,3 2 1,4 1,4 1 1,4 2 1,5

... ... ... ... ...1, 1 1, 2 1,22,1 2,2 2, 2,2 1 2,2 2 2,3 2,3 1 2,3 2 2,4 2,4 1 2,4 2

a a a f f f r r rP P PP P P P P P P P P P

a a a f f f r r rP P PP P P P P P P P P

C 2,5

... ... ... ... ...1, 1 1, 2 1,2,1 ,2 , ,2 1 ,2 2 ,3 ,3 1 ,3 2 ,4 ,4 1 ,4 2 ,5

P

a a a f f f r r rP P PM M M P M P M P M P M P M P M P M P M P M P

where


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139

:

: max, ,min

: 1,2,... & 1,2,...,2 ,2

: 0 / 2,3 ,3

: 0 2,4 ,4

a R L a Hin ipl l

f R f f fi P p i P p

r R L r H for i M p Pr ri P p i P p

Ri P p i P p

Ri P p i P p

where Lland l

rare the lower while Hl and Hr are the highest limits of signals

amplitude and range respectively.

step2. Fitness function: Our goal is to minimize the errors received for both sub-

arrays. For thi chromosome, it can be given as,

( ) ( ) ( )err i err i err ix y (5.52)

21

ˆ ˆ( ) . 1, ,02 1

Qx i H ierr i w wx m o m xN xNQx m Qx

w w (5.53)

2

1ˆ ˆ( ) . 10, 0,2 1

Qyi H ierr i w wy n n yN yNQy n Qy

w w (5.54)

where in (5.53) and (5.54), ,0wm and 0,w n are defined in (5.49) and (5.50)

respectively, while ˆ ,0i

wm and ˆ0,i

w n are given as,

2 2ˆ ˆ( ) ( )2 2ˆ ˆ ˆ ˆ( (( )sin( )cos( ) (1 sin ( )cos ( ))))

3 4 3 42 ˆ16( )max 2ˆ ˆ

,01

i im c m cP p P pi i i ij c c c c

P p P p P p P pif cPi i P pw c e

m pp

(5.55)

2 22 2

3 4 3 42

ˆ ˆ( ) ( )ˆ ˆ ˆ ˆ)sin( )sin( ) (1 sin ( )sin ( ))))

2 ˆ16( )

0

( ((

maxˆ ˆ,

1

i iP p P pi i i i

P p P p P p P piP p

n c n cc c c c

cj

P fi iw c en p

p

(5.56)


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140

Similarly, ˆ ˆ, , ,N N N N

ix x y yw w w w can be defined as,

N

zz

z

ww

w (5.57)

and

ˆˆ

ˆN

ii zz i

z

w

ww

(5.58)

where ,z x y .

The remaining steps are same as discussed in the previous sections.


In this section, several simulations are performed to validate the proposed schemes.

Initially, the comparison of proposed hybrid schemes are carried out with the

individual performance of GA, IPA and PS in terms of estimation accuracy,

convergence rate and proximity effects. At the end of this section, the comparison of

proposed schemes is made with the traditional existing technique [169] using error as

a figure of merit. All the values of frequencies, ranges and DOA are taken in terms of

Mega-Hertz (MHz), wavelength ( ) and radians (rad), respectively. Every time, we

have used same number of sensors in both sub-arrays where the reference sensor is

common for them. The inter-element spacing between the two consecutive sensors in

each sub-array is taken as / 4 . Each result is averaged over 100 independent runs.

Case 1: In this case, the estimation accuracy of IPA, PS, GA, GA-IPA and GA-PS are

discussed for 2 sources. The CSCA consists of 9 sensors that is each sub-array

composed of four sensors, while the reference sensor is common for them. The

desired values are 11 1 1 1( 6, 30 , 2 , 0.2618 , 2.0071 )a f MHz r rad rad and

22 2 2 2( 4, 60 , 0.6 , 1.1345 , 2.9671 ).a f MHz r rad rad Although in this

case, GA alone has produced fairly good estimation accuracy as provided in Table


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141

5.27 and 5.28. It becomes even more accurate when hybridized with IPA and PS.

Among all schemes, the GA-PS approach produced better results and maintained less

error between desired values and estimated values. The second best scheme is GA-

IPA while GA alone provided the third best results.

Table 5. 27 Estimation Accuracy of 2 sources using 9 sensors (continue)

Scheme 1

a ( )1f MHz

( )

1r ( )

1rad ( )

1rad

Desired Values 6.0000 30.0000 2.0000 0.2618 2.0071

IPA 6.0096 33.5342 2.0095 0.2714 2.0168

PS 6.0050 32.7908 2.0051 0.2668 2.0123

GA 6.0031 30.9765 2.0032 0.2649 2.0103

GA-IPA 6.0020 30.2289 2.0019 0.2638 2.0091

GA-PS 6.0007 30.1089 2.0008 0.2625 2.0078

Table 5. 28 Estimation Accuracy of 2 sources using 9 sensors

2a ( )

2f MHz

( )

2r ( )

2rad ( )

2rad

4.0000 60.0000 0.6000 1.1345 2.9671

4.0095 63.5376 0.6095 1.1442 2.9767

4.0049 62.7546. 0.6050 1.1396 2.9722

4.0032 60.9782 0.6030 1.1376 2.9703

4.0020 60.2415 0.6018 1.1367 2.9692

4.0008 60.1063 0.6007 1.1352 2.9678

Case 2: In this case, the estimation accuracy is discussed for 3 sources having values

11 1 1 1( 3, 40 , 2.5 , 0.4363 , 1.0472 ),a f MHz r rad rad

22 2 2 2( 1, 70 , 5 , 0.7330 , 2.1817 ),a f MHz r rad rad

33 3 3 3( 7, 50 , 0.2 , 1.3963 , 3.5779 ).a f MHz r rad rad


Scheme 1a ( )

1f MHz

( )1r ( )1 rad

( )1 rad

2a ( )2

f MHz

( )2r ( )2 rad

( )2 rad

Desired Values 3.0000 40.0000 2.5000 0.4363 1.0472 1.0000 70.0000 5.0000 0.7330 2.1817

IPA 3.0557 46.9871 2.5558 0.4920 1.1030 1.0557 64.3425 5.0558 0.7888 2.2376

PS 3.0338 45.1204 2.5338 0.4702 1.0810 1.0338 75.8734 5.0339 0.7668 2.2154

GA 3.0092 43.7894 2.5093 0.4456 1.0565 1.0092 66.5682 5.0093 0.7423 2.1908

GA-IPA 3.0065 41.2187 2.5066 0.4428 1.0538 1.0065 71.2654 5.0067 0.7396 2.1883

GA-PS 3.0024 40.8903 2.5022 0.4388 1.0497 1.0024 70.8931 5.0025 0.7356 2.1841


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142

This time the array is composed of 13 sensors. Due to the increase of sources, the

accuracy of IPA, PS and GA have significantly despoiled. However as listed in Table

5.29 and 5.30, the accuracy of GA has improved when hybridized with IPA and PS.


3a ( )3f MHz

( )3r ( )3 rad ( )3 rad

7.0000 50.0000 0.2000 1.3963 3.5779

7.0558 57.0123 0.2557 1.4522 3.6337

7.0339 55.8693 0.2339 1.4301 3.6117

7.0092 54.1298 0.2093 1.4057 3.5871

7.0066 51.2879 0.2067 1.4029 3.5847

7.0021 50.7969 0.2022 1.3986 3.5802

The hybrid GA-PS technique proved to be the most accurate approach for three

sources, while the second best approach is the other hybrid GA-IPA approach.

Case 3: In this case, the estimation accuracy of four near field sources is discussed,

where the CSCA is composed of 17 sensors. The desired values of the sources are

( 3.5, 65 , 1 , 0.4712 , 0.1745 ), ( 5, 30 ,1 1 1 1 21 2

6 , 0.8727 , 2.0420 ), ( 2, 85 , 10 , 1.2741 ,2 2 2 3 3 33

2.7925 ), ( 8, 25 , 4 , 1.5184 , 4.4506 ).3 4 4 4 44

a f MHz r rad rad a f MHz

r rad rad a f MHz r rad

rad a f MHz r rad rad

One can see from Tables 5.31 and 5.32, that the estimation accuracy of all schemes

are degraded as we faced more local minima in this case. However, even in this case,

the hybrid approaches especially the GA-PS performed well and made a close

estimate of desired response.


Scheme 1a

1( )f MHz

( )

1r ( )

1rad ( )

1rad 2a

2( )f MHz

( )

2r ( )

2rad ( )

2rad

Desired 3.5000 65.0000 1.0000 0.4712 0.1745 5.0000 30.0000 6.0000 0.8727 2.0420

IPA 3.5989 75.0947 1.0989 0.5702 0.2735 5.0989 19.8975 6.0990 0.9718 2.1409

PS 3.5687 73.8971 1.5688 0.5400 0.2432 5.0688 22.8912 6.0689 0.9414 2.1108

GA 3.5197 70.1879 1.0198 0.4908 0.1941 5.0195 24.9765 6.0196 0.8923 2.0616

GA-IPA 3.5166 67.3215 1.0164 0.4879 0.1913 5.0166 32.7957 6.0167 0.8894 2.0586

GA-PS 3.5105 66.0469 1.0103 0.4818 0.1850 5.0105 31.1201 6.0106 0.8834 2.0526


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143


3a ( )3f MHz

( )

3r ( )

3rad ( )

3rad 4a ( )

4f MHz

( )

4r ( )

4rad ( )

4rad

2.0000 85.0000 10.0000 1.2741 2.7925 8.0000 25.0000 4.0000 1.5184 4.4506

2.0990 95.8734 10.0990 1.3731 2.8916 8.0989 15.0081 4.0990 1.6176 4.5497

2.0686 92.8714 10.0688 1.3427 2.8613 8.0687 32.8145 4.0685 1.5873 4.5194

2.0194 90.0012 10.0195 1.2936 2.8121 8.0194 30.8156 4.0195 1.5379 4.4702

2.0166 87.0123 10.0167 1.2908 2.8091 8.0166 27.9099 4.0167 1.5350 4.4674

2.0107 86.3459 10.0108 1.2847 2.8030 8.0105 26.5562 4.0104 1.5289 4.4620

Case 4: In Fig 5.14, convergence is shown for each scheme against different number

of sources. The MSE is kept same i.e.2

10

. In this case, we have taken the same two

sources as given in case-1 but this time the CSCA consists of 17 sensors for each

number of sources. The bar graph shows, that the hybrid GA-PS technique has

converged many times as compared to the remaining approaches for all sources. The

second best convergence rate is maintained by GA-IPA while the third best scheme is

GA alone.

Fig. 5. 14 Convergence Rate vs number of sources

Case 5: In this case, the estimation accuracy is checked in the presence of low signal

to noise ratio (SNR). The value of SNR is 5 dB while the array has 13 sensors. The

desired values of amplitude, frequency, range, elevation & azimuth angles are

2 3 40

10

20

30

40

50

60

70

80

90

100Convergence rate vs number of sources

[number of Sources]

% C

onverg

ence

IPA

PS

GA

GA-IPA

GA-PS


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144

( 3, 70 , 6 , 0.2618 , 0.6109 ),1 1 1 1 1a f MHz r rad rad

( 1, 45 , 2.4 , 0.7854 , 2.4435 ),2 2 2 2 2a f MHz r rad rad

( 7, 30 , 4.3 , 1.4835 , 3.7525 ).3 3 3 33a f MHz r rad rad

As provided in Tables 5.33 and 5.34, due to low SNR the accuracy of all schemes

despoiled. However, the hybrid GA-PS scheme is robust enough to produce better

results even in the presence of low SNR. The second best result is produced by the

other hybrid GA-IPA scheme.

Table 5. 33 Estimation Accuracy for 3 sources at SNR=5 dB (continue)

Scheme

1a ( )1f MHz

( )1r ( )1 rad ( )1 rad 2a ( )

2f MHz

( )2r ( )2 rad ( )2 rad

Desired 3.0000 70.0000 6.0000 0.2618 0.6109 1.0000 45.0000 2.4000 0.7854 2.4435

IPA 3.3711 78.8790 6.1712 0.4329 0.7820 1.4711 54.9876 2.5712 0.9567 2.6145

PS 3.2047 77.2137 6.1047 0.3665 0.7156 1.3047 53.1124 2.1047 0.8901 2.5483

GA 3.1824 75.8711 6.0423 0.3142 0.6534 1.2422 49.8879 2.4425 0.8279 2.4860

GA-IPA 3.0584 72.3298 6.0385 0.3002 0.6494 1.1385 47.6675 2.4384 0.8239 2.4820

GA-PS 3.0357 71.1903 6.0156 0.2775 0.6266 1.0958 46.1290 2.4158 0.8013 2.4592

Table 5. 34 Estimation Accuracy for 3 sources at SNR=5 dB

3a ( )3f MHz

( )3r ( )3 rad ( )3 rad

7.0000 30.0000 4.3000 1.4835 3.7525

7.1710 39.8791 4.4711 1.6547 3.9235

7.1045 38.1236 4.4048 1.5884 3.8572

7.0426 35.4398 4.3424 1.5259 3.7950

7.0382 32.9983 4.3384 1.5220 3.7910

7.0158 31.6722 4.3158 1.4993 3.7684

Case 6: In Fig 5.15, the convergence of each scheme is evaluated against noise and it

has been shown that the convergence rates of all schemes are degraded at low values

of SNR. However, with the increase of SNR, the convergence rate of each scheme has

improved. Again, the hybrid GA-PS has shown fairly good robustness against all the

values of SNR.


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145

Fig. 5. 15 Convergence VS SNR

Case 7: In this case, the proximity effect of DOA of three sources is evaluated in

terms of estimation accuracy and Convergence rate in the presence of 10 dB noise. As

given in Table 5.35, due to proximity and low SNR, we have faced more local

minima. However, again one can see that the hybrid GA-PS produced fairly good

results in terms of accuracy and convergence even in this case while the second best

result is given by GA-IPA.

Table 5. 35 Proximity effect of DOA of three sources and 17 sensors at SNR=10 dB

Scheme ( )1 rad ( )1 rad ( )2 rad ( )2 rad ( )3 rad ( )3 rad

%Convergence

Desired Val 0.6981 1.9199 0.7679 1.9897 0.8378 2.0595 -

IPA 0.8203 2.0402 0.8901 2.1118 0.9599 2.1817 1

PS 0.7941 2.0351 0.8849 2.1049 0.9512 2.1712 4

GA 0.7400 1.9600 0.8116 2.0298 0.8796 2.1031 64

GA-IPA 0.7208 1.9408 0.7906 2.0141 0.8587 2.0857 70

GA-PS 0.7103 1.9303 0.7821 2.0019 0.8482 2.0717 80

Case 8: In this case, we have compared the proposed two hybrid schemes with

traditional technique given in [169]. Basically, in [169], Junli Liang et al, has

proposed a cumulant based technique to estimate the 4D parameters (frequency,

range, elevation angle and azimuth angle) of near field sources. In [169], Mean

5 10 150

10

20

30

40

50

60

70

80

90

[SNR in dB]

% C

onverg

ence

IPA

PS

GA

GA-IPA

GA-PS


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146

Square error (MSE) is used, while in the current work, the error is the combination of

MSE and correlation between the desired and estimated vectors as discussed in

section-3. For these, simulations, two sources are considered in the presence of noise.

The values of the two sources are exactly same as given above in case-1. Fig 5.16, Fig

5.17, Fig 5.18 and Fig 5.19, have shown the error for frequency, azimuth angle,

elevation angle and range of two near field sources by using [169] and the two

proposed hybrid schemes respectively. One can clearly observe that in each case

(especially for range estimation) the proposed schemes have maintained fairly

minimum error as compared to [169]. Besides, [169] is unable to estimate the

amplitude, while our proposed schemes have shown satisfactory error for amplitude

estimation as shown in Fig 5.20.

Fig. 5. 16 Error estimation of the frequencies Vs SNR

Fig. 5. 17 Error estimation of the Azimuth angles Vs SNR

0 5 10 15-60

-50

-40

-30

-20

SNR (dB)

Err

or

(dB

)

1st sources [169]

2nd source [169]

1st sources GA-IPA

2nd sources GA-IPA

1st sources GA-PS

2nd sources GA-PS

0 5 10 15-60

-50

-40

-30

-20

-10

SNR (dB)

Err

or

(dB

)

1st source [169]

2nd source [169]

1st source GA-IPA

2nd source GA-IPA

1st source GA-PS

2nd source GA-PS


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147

Fig. 5. 18 Error estimation of the elevation angles Vs SNR

Fig. 5. 19 Error estimation of the ranges Vs SNR

Fig. 5. 20 Error estimation of the amplitudes Vs SNR

5.7 CONCLUSION

This chapter was also divided into three parts. In part one, GA-IPA, PSO-PS and

DE-PS were applied for the joint estimation of amplitude, range and 1-D DOA.

For this MSE was used as fitness evaluation function and it was shown through

several experiments that the hybrid schemes produced better results as compared

to GA, PSO, DE, IPA and PS alone in terms of estimation accuracy, convergence,

0 5 10 15-60

-50

-40

-30

-20

-10

SNR (dB)

Err

or

(dB

)

1st source [169]

2nd source [169]

1st source GA-IPA

2nd source GA-IPA

1st source GA-PS

2nd source GA-PS

0 5 10 15-60

-40

-20

0

20

40

SNR (dB)

Err

or

(dB

)

1st source [169]

2nd source [169]

1st source GA-IPA

2nd source GA-IPA

1st source GA-PS

2nd source GA-PS

0 5 10 15-60

-55

-50

-45

-40

-35

SNR (dB)

Err

or

(dB

)

1st source GA-IPA

2nd source GA-IPA

1st source GA-PS

2nd source GA-PS


SOURCES

148

proximity effect etc.

In part two, 4-D parameters are estimated of near field sources and the problem

was linked to bi-static radar. This time the 4-D parameters included amplitude,

range, elevation angle and azimuth angles. Again MSE was used as a fitness

function and again the results of hybrid schemes were remarkable.

In part three, hybrid GA-PS and GA-IPA have been used along with the multi-

objective fitness function for the joint 5D parameters estimation of near field

sources. The 5-D parameters were amplitude, frequency, range, elevation angle

and azimuth angles. The proposed hybrid schemes have shown better

performance not only as compared to GA, IPA and PS alone but also from the

traditional techniques already available in the literature.

CHAPTER 6

CONCLUSION AND FUTURE DIRECTIONS

6.1 CONCLUSION

The main objective of this dissertation was to estimate the DOA of the far and near

field sources for adaptive beamforming. Several algorithms have been proposed

which were quite efficient and were very easy to be implemented. The proposed

algorithms were based on meta-heuristic evolutionary computing techniques such as

GA, PSO, DE and SA. These techniques have been further hybridized with PS, IPA

and ASA to generate improved results. The idea was to divide the search space in two

phases. The global phase was covering the maximum possible search region,

whereas, the local search space was a region in vicinity of the optimal solution. In

this context during the hybridization process, GA, PSO, DE and SA were used as

global search optimizers, while PS, IPA and ASA were utilized as rapid local search

optimizers for fine tuning of the results. Sometimes this is termed as memetic

computing in the literature. In this regard, the DOA was estimated using the sources

from far field initially. The other parameters estimated jointly with DOA include

amplitude, range and frequency. Amplitude and elevation angle were estimated

initially by using GA-PS and it has been proved that the performance of hybrid

scheme is better than GA and PS when applied separately. Similarly PSO was

hybridized with PS for the joint estimation of amplitude same parameters i.e.

amplitude and elevation angles of far field sources impinging on ULA. Again it has

been shown that the performance of PSO-PS is better than PSO and PS applied

CHAPTER 6 CONCLUSION AND FUTURE DIRECTION

150

separately. In the next phase, GA, PSO and DE were hybridized with PS for the joint

estimation of 3-D parameters i.e. amplitude, elevation and azimuth angles of far field

sources. The array structure in this case is considered as 1-L and 2-L shape. The

proposed hybrid schemes have not only shown better performance as compared to the

individual performance of these techniques, but it has outperformed in comparison

with the classical techniques available in the literature. Subsequently the 4-D

parameters estimation i.e. amplitude, frequency, elevation azimuth angles were

jointly estimated by using PSO-PS. In this case, we have used two fitness functions

for the sources coming from far field. The first one is based on MSE, which define an

error between the desired and estimated signals and has been derived from MLP. The

second fitness function is a combination of MSE and correlation taken between the

desired normalized and estimated normalized vectors, hence termed as multi-

objective function. Both the fitness functions required single snapshot and produced

fairly good results in terms of estimation accuracy, convergence, robustness against

noise, MSE etc. However, at the end, it has been proved that the multi-objective

fitness function has greater efficiency and reliability as compared to MSE alone.

In the second part of the dissertation, we have again developed efficient hybrid

schemes for near field source localization. Initially GA and SA are hybridized with

IPA for the joint estimation of 3-D parameters i.e. amplitude, range and elevation

angles of near field sources impinging on ULA. Similarly, PSO and DE are

hybridized with PS for jointly 3-D parameter estimation. After extensive simulations,

it has been shown once again that the hybrid schemes are more effective in

comparison with the individual responses of GA, PSO, DE, SA, IPA and PS. Later

on, the near field sources localization problem was linked with bi-static radar. In this

connection 4-D i.e. Amplitude, range, elevation and azimuth angles parameters are


151

estimated by using two different hybrid techniques termed as PSO-ASA and DE-

ASA. The near field sources in this case are supposed to be impinging on CSCA. The

results are promising in this case as compared to the existing classical techniques

indicating the utility of the proposed algorithm. Towards the end of dissertation, 5-D

parameters i.e. Amplitude, frequency, range, elevation and azimuth angles are

estimated by using PSO-PS hybrid approach. Once again we have used two fitness

functions which are the same as were taken in the far field scenario of previous

chapter. We have observed that the proposed hybrid schemes are efficient, reliable

and need less number of sensors in the array and hence less budget for

implementation.

The proposed schemes in this dissertation have also few limitations as well.

1) All the proposed algorithms fail when the number of sensors in the array is

less than the number of sources.

2) The performance of each algorithm degraded when the number of sources is

more than six.

3) They also fail in case of array imperfection and sensor failure in the array.

6.2 FUTURE DIRECTIONS

In the existing work we have used a limited set of evolutionary computing

techniques which include PSO, GA, DE and SA etc. However, there exist many other

techniques based on meta-heuristic platform. In future, one can look into utility of

these left out hybrid techniques for the same problem, as well as, for other related

problems. Some of these techniques are Ant colony optimization (ACO), Bee colony

optimization (BCO), Culture algorithm (CA) etc. These techniques can be taken

individually and then as hybrid cases with each other and also with the tools already

explored in the present work. In other words it will be a complete data base of


152

individual and then combination of two or more schemes to be analyzed in different

application setup and different environment.

6.2.1 Tracking Problem

Tracking is also a challenging task in DOA estimation. In this case the DOA is

assumed to be changing, which is to be tracked by some sensors array in near and far

field cases. However, while going for tracking the side lobes usually get out of

control and increase in level. Hence it is another area of future work in which DOA is

to be tracked at the same time keeping the side lobe levels in control.

6.2.2 Main Beam and Null Steering

The main beam needs steering whenever the DOA of the source of interest changes.

Similarly the nulls are required to be allocated at new position whenever the

intruder/jammer or any unwanted signal, which is needed to be blocked, changes its

position. These are already active areas of research, however, with the given set of

meta-heuristic-computing algorithms, one may improve the performance of existing

methods. Subsequently the hybridization of these techniques can be taken into the

account along with the combination of existing ones.

6.2.3 Noise Consideration

Noise is assumed to be always there in all the practical systems which is taken as

white. However, in general it could be colored as well. The effect of colored noise

with its probable distributions in available practice systems can also be an area of

research in future. Moreover, we shall like to develop the algorithms which will

provide robustness against possible noises.


153

6.2.4 Array Miss Perfection

Array miss-perfection is another hot topic of research. Miss-perfection occurs when

one or more elements in an array become out of order. In this situation the complete

radiation pattern gets disturbed. The research is being carried out to identify the

malfunctioning elements and algorithms are being developed to regain the desired

radiation pattern by remainder elements. The same problem may be incorporated with

DOA estimation and adaptive beam steering. That will be another area of research.

REFERENCES

[1]. Z. U. Khan., A. Naveed, I. M. Qureshi, and F. Zaman, ―Independent null

steering by decoupling complex weights," IEICE Electronics Express, Vol. 8,

No. 13, 1008-1013, 2011.

[2]. M. Mouhamadou., P. Vaudon, and M. Rammal, ―Smart antenna array patterns

synthesis: Null steering and multi-user beamforming by phase control,"

Progress In Electromagnetics Research, Vol. 60, 95-106, 2006.

[3]. M. Mukhopadhyay., B. K. Sarkar, and A. chakrabarty, ―Augmen-tation of

anti-jam GPS system using smart antenna with a simple DOA estimation

algorithm," Progress In Electromagnetics Research, Vol. 67, 231-249, 2007.

[4]. A.U. Rehman, F. Zaman, Y.A.shiekh, I.M.Qureshi, ―Null and sidelobe

adjustment in damaged antenna array,‖ IEEE, ICET, Islamabad, pp-21-24,

2012.

[5]. S.U. Khan, I. M.Qureshi, F. Zaman, A. Naveed, ―Null placement and sidelobe

suppression in failed array using symmetrical element failure technique and

hybrid heuristic computation,‖ PIER-B, pp, 165-184, vol 52, 2013.

[6]. S. Mehmood., Z.U Khan and F. Zaman ―Performance Analysis of the

Different Null Steering Techniques in the Field of Adaptive Beamforming‖

―Accepted in Research Journal of Engineering and Technology, pp, 4006-

4012, vol, 2013.

[7]. Z.U. Khan, A. Naveed, I.M.Qureshi and F. Zaman, ―Robust Generalized

Sidelobe Canceller for Direction of Arrival mismatch,‖ Archives Des

Sciences, Vol 65, pp. 483-497, 2012.

[8]. Z.U. Khan, A.Naveed, M.Safeer, F. Zaman, ―Diagonal Loading of Robust

General- Rank Beamformer for Direction of Arrival mismatch,‖ Research

Journal of Engineering and Technology (Maxwell Scientific organization), pp.

4257-4263, 2013.

[9]. D. Barry., V. Veen and K. M. Buckley, ―Beamforming: A versatile approach

to spatial filtering, IEEE ASSP Magazine, pp. 4-24, April, 1988.

[10]. H. Krim., and M. Viberg, ―Two decades of array processing research: The

parametric approach,‖ IEEE Signal Processing Magazine, Vol. 13, No. 4, 67–

94, July 1996.

[11]. P.E. Green, E.J. Kelly and M.J. Levin, ―A comparison of seismic array

proceeding methods,‖ Geophy. J.Royal. Astron. Soc., Vol.11, pp.67-84, 1996.

[12]. D. Byrne., M. O. Halloran, M. Glavin, and E. Jones, ―Data independent radar

beamforming algorithms for breast cancer detection," Progress In

Electromagnetics Research, Vol. 107, 331-348, 2010.

[13]. W.vanderkulk, ―Optimum processing of acoustic array,‖ J. Br. Inst. Radio

Eng., Vol.34,pp.285-292,1963.

[14]. T. Nishiura., and S. Nakamura, ―Talker localization based on the combination

of DOA estimation and statistical sound source identication with microphone

array," IEEE Workshop Statistical Signal Processing, 597-600, Oct. 2003.

REFERENCES

155

[15]. J. W. R. Griffiths and J.E.Hudson, ―An introduction to adaptive processing in

a passive system,‖ Aspect of signal processing, pt.1, NATO, advanced study

institute series, D. Reides publishing Co, Pordrech, Holland, 1977.

[16]. A. I. Sotiriou., P. K. Varlamos, P. T. Trakadas, and C. N. Capsalis,

―Performance of a six-beam switched parasitic planar array under one path

rayleigh fading environment," Progress In Electromagnetics Research, Vol.

62, 89-106, 2006.

[17]. G. M. Wang., J. M. Xin, N. N. Zheng, and A. Sano, ―Two- dimensional

direction estimation of coherent signals with two parallel uniform linear

arrays," IEEE Statistical Signal Processing Workshop (SSP), 2011.

[18]. L.E. Brennan and I.S. Reed, ―Theory of adaptive radar,‖ IEEE Trans.

Aerospace and electronic systems. AES-9, pp.237-252, November 1973.

[19]. J. H. Winters, ―Optimum combinig in digital mobile radio with co-channel

interference,‖ IEEE J. Select, Area commun, vol.com-35, pp.1222-1230, 1987.

[20]. M. L. Bencheikh., and Y. Wang, ―Combined esprit-rootmusic for DOA-DOD

estimation in polarimetric bistatic MIMO radar,‖Progress In Electromagnetics

Research Letters, Vol.22,109-117, 2011.

[21]. P. Yang., F. Yang, and Z. P. Nie, ―DOA estimation with sub-array divided

technique and interporlated esprit algorithm on a cylindrical conformal array

antenna," Progress In Electromagnetics Research, Vol. 103, 201-216, 2010.

[22]. J. H. Kim., I. S. Yang, K. M. Kim, and W. T. Oh, ―Passive ranging sonar

based on multi-beam towed array,‖ Proc. IEEE Oceans, Vol. 3, 1495–1499,

September 2000.

[23]. B. suard, A. F. Naguib, G.Xu, and A. Paulraj, ―Performance of CDMA mobile

communication system using antenna arrays,‖ IEEE, Int. conf. Accoustics,

Speech and signal processing (ICASSP), Mineapolis, MN, pp. 153-156, 1993.

[24]. A. Flieller, p.Larzabal, and H. clergeot, ― Application of high resolution array

processing techniques for mobile communication systems,‖ In proc, IEEE

Intelligent Vehicles Symp., paris, France, pp.606-611, 1994.

[25]. R.C. Johnson., Antenna Engineering Handbook, 3rd Edition, 9-12, McGraw-

Hill, Inc., 2006.

[26]. L. J.Ziomek., Fundamentals of Acoustic Field Theory and Space-time Signal

Processing, 410-412, 532-534, CRC Press, Bocaraton, FL, 1995.

[27]. C. A. Balanis, Antenna Theory: Analysis and design CRC Press LLC, 2004.

[28]. M. S. Bartlett, ―Smoothing periodograms from time series with continuous

spectra," Nature, 161:686:687, 1948.

[29]. J. Capon, ―High resolution frequency-wavenumber spectrum analysis," Proc.

IEEE, vol. 57, pp. 1408-1418, August, 1969.

[30]. R. T. Lacoss, ―Data adaptive spectral analysis methods," Geophysics, vol. 36,

pp. 661-675, 1971.

[31]. J. F. Bohme, ―Separated estimation of wave parameters and spectral

parameters by maximum likelihood, "Proc. ICASSP, Tokyo, Japan, pp.

2819-2822, 1986.

REFERENCES

156

[32]. A. G. Jafer, ―Maximum likelihood direction finding of stochastic sources: A

separable solution, "Proc. ICASSP, New York, vol. 5, pp. 2893-2896, April

1988.

[33]. R. O. Schmidt, ―Multiple emitter location and signal parameter estimation,"

Proc. RADC Spectrum Estimation Workshop, Grifths AFB, Rome, New York,

pp. 243-258, 1979.

[34]. G. Bienvenu and L. Kopp, ―Adaptivity to background noise spatial coherence

for high resolution passive methods," Proc. ICASSP, vol. 1, pp. 307-310,

Denver,Colorado, April 1980.

[35]. G. Bienvenu and L. Kopp, ―Optimality of high resolution array

processingusing the eigen system approach," IEEE Trans. Acoust., Speech,

Signal Process. vol. Assp-31, pp. 1234-1248, October 1983.

[36]. B.D. Rao and K. V. S. Hari, ―Performance Analysis of Root-MUSIC," IEEE

Trans. Acoust., Speech, Signal Process. vol. 37, pp. 1939-1949, Dec. 1989.

[37]. M. Pesavento, A. B. Gershman and M. Haardt, ―Unitary root-MUSIC with a

real-valued eigendecomposition: A theoretical and experimental performance

study," IEEE Trans. Signal Process. vol. 48, pp. 1306-1314, May 2000.

[38]. R. Roy, ―ESPRIT: Estimation of signal parameters via rotational

invariancetechniques." Ph.D. dissertation, Stanford University, Stanford,

California, 1987.

[39]. A. Paulraj, R. Roy and T. Kailath, ―A subspace rotation approach to signal

parameter estimation," Proc. IEEE, vol. 74, pp. 1044-1045, July 1986.

[40]. R. Roy, A. Paulraj and T. Kailath, ―ESPRIT,‖ A subspace rotation approachto

estimation of parameters of cisoids in noise," IEEE, Trans. Acoust., speech,

Signal Process., vol. ASSP-34, pp. 1340-1342, Oct. 1986.

[41]. R. Roy and T. Kailath, ―ESPRIT: Estimation of signal parameters via ro-

tational invariance techniques," IEEE, Trans. Acoust., speech, Signal

Process.,vol. ASSP-37, pp. 984-995, July 1989.

[42]. B. Ottersten, M. Viberg and T. Kailath, ―Performance analysis of the

totalleast squares ESPRIT algorithm," IEEE Trans. Signal Process. vol. sp-39,

pp.1122-1135, May 1991.

[43]. M. Haardt and J. A. Nossek, ―Unitary ESPRIT: How to obtain

increasedestimation accracy with a reduced computational burden," IEEE

Trans. Signal Process. vol. SP-43, pp. 1232-1242, May 1995.

[44]. M. Haardt and M. E. Ali-Hackl, ―Unitary ESPRIT: How to exploit

additionalinformation inherent in the rotational invariance structure," Proc.

ICASSP, Adelaide, Australia, vol. IV, pp. 229-232, April 1994.

[45]. H. Karim and J. G. Proakis, ―Smoothed eigenstructure based parameter es-

timation," Automatica, Special Issue on Statistical Signal Process. and

Control, Jan. 1994.

[46]. Y. Hua and T. K. Sarkar, ―Matrix Pencil method for estimating parametersof

exponentially damped-undamped sinusoids in noise," IEEE Trans.

Accoust.Speech and Signal Process., vol. 38, no. 5, May 1990.

REFERENCES

157

[47]. T. K. Sarkar and O. Pereira, ―Using the Matrix Pencil method to estimatethe

parameters of a sum of complex exponentials," IEEE Anten. and Propag.Mag.

vol. 37, no. 1, Feb. 1995.

[48]. J. E. F. del Rio and T. K. Sarkar, ―Comparison between the matrix pencil

method and the fourier transform technique for high resolution spectral

estimation," Digital Signal Process. vol. 6, no. 2, pp. 108-125, 1996.

[49]. J. E. F. del Rio and M. F. Catedra-Perez, ―A comparison between matrix pen-

cil and Root-MUSIC for direction-of-arrival estimation making use of uniform

linear arrays," Digital Signal Process. vol. 7, no. 3, pp. 153-162, 1997.

[50]. N. Yilmazer, S. Ari, T. K. Sarkar, ―Multiple snapshot direct data

domainapproach and ESPRIT method for direction of arrival estimation,"

Digital SignalProcessing, vol. 18, no. 4, pp. 561-567, 2008.

[51]. N. Yilmazer, J. Koh and T. K. Sarkar, ―Utilization of a unitary transformfor

efficient computation in the matrix pencil method to find the direction of

arrival," IEEE Trans. Anten. and Propag. vol. 54, no. 1, pp. 175-181, 2006.

[52]. C. F. Van Loan, ―Generalizing the singular value decomposition," SIAM J.

Numer. Anal. vol. 13, no. 1, pp. 76-83, 1976.

[53]. K. C. Huarng and C. C. Yeh, ―Unitary transformation method for angle of

arrival estimation," IEEE Trans. Signal Processing, vol. 39, no. 4, pp. 975-

977,1991. [54]. W. Sun and Y.-X. Yuan, Optimization Theory and Methods: Nonlinear

Programming, ―Optimization and its Applications,‖ New York: Springer

Science+Business Media, 2006. [55]. D. B. Fogel and L. J. Fogel, ―Using evolutionary programming to schedule

tasks on a suite of heterogeneous computers,‖ Comp Oper Res vol. 23, pp.

527-534, 1996. [56]. P. J Angeline, G. M. Saunders and J. B. Pollack, ―An evolutionary algorithm

that construct recurrent neural networks,‖ IEEE Trans Neural Networks, vol. 5

pp. 54-65, 1994. [57]. S. B. Haffner and A. V. Sebald, ―Computer-aided design of fuzzy HVAC

controllers using evolutionary programming,‖ Evolutionary Programming

Society, pp.98-107, 1993. [58]. D. B. Fogel, ―Evolutionary computation: Toward a new philosophy of

machine intelligence,‖2nd

edition, Piscataway, NJ, IEEE Press, 2000. [59]. H. P. Schwefel, ―Evolution and optimum seeking,‖ New York, John Wiley &

Sons, 1995. [60]. M. Pelikan, Nk landscapes, problem di_culty, and hybrid evolutionary

algorithms,"2010. [61]. C. Blum, M. Aguilera, A. Roli, and M. Sampels, Hybrid Metaheuristics: An

Introduction, vol.114 of Studies in Computational Intelligence, pp. 1-30.

Springer Berlin-Heidelberg, 2008. [62]. J.A. Khan, ―applications of evolutionary computing techniques to non-linear

systems,‖PhD dissertation, IIUI, 2011. [63]. A.P.Wieland., Evolving controls for unstable systems. Connectionist Models:

Proceedings of the 1990 Summer School, D.S. Touretzky, J.L. Elman, T.J.

Sejnowski, and G.E. Hinton(eds.), Morgan Kaufmann, San Mateo, CA (1990)

91-102.

REFERENCES

158

[64]. B. Fogel, The Advantages of Evolutionary Computation ―Natural Selection,

Inc.3333 North Torrey Pines Ct., Suite 200,La Jolla, CA 92037. [65]. J. Reed, R. Toombs and N. A. Barricelli, ―Simulation of biological evolution

and machine learning,‖ J. Theor. Biol, vol. 17, pp. 319-342, 1967. [66]. A. E. Eiben, and J. E. Smith, ―Introduction of Evolutionary Computing,‖

ISBN 3-540-40184-9, Springer-Verlag Berlin Heidelberg, 2003. [67]. D. Ashlock, ―Evolutionary computation for modeling and optimization,‖

ISBN-10 0-387-22196-4, Springer+Business Media, Inc, New York, 2006. [68]. John H. Holland, ―Adaptation in Natural and Artificial systems,‖ Amazon,

1975. [69]. D. Fogel, Evolutionary Computation: The Fossil Record. Wiley-IEEE Press,

1998.Whitley, D., (1994). [70]. P. Aarabi, "Genetic sensor selection enhanced independent component

analysis and its applications to speech recognition," in Proc. 5th IEEE

Workshop Nonlinear Signal Information Processing, June 2001. [71]. B. Addad, S.Amari, and J-J.Lesage, ―Genetic algorithms for delays evaluation

in networked automation systems‖ Engineering Applications of Artificial

Intelligence, Elsevier, Vol. 24. 485–490, 2011. [72]. U. Maulik, ―Analysis of gene microarray data in a soft computing

framework,‖ Engineering Applications of Artificial Intelligence, Elsevier,

Signal Process, vol. 24, 485–490, 2011 [73]. L. M. O. Queiroz and C. Lyra, ―Adaptive hybrid genetic algorithm for

technical loss reduction in distribution networks under variable demands,"

Power Systems, IEEE Trans-actions on, vol. 24, no. 1, pp. 445-453,2009. [74]. J. B. Grimbleby, ―Hybrid genetic algorithms for analogue network synthesis,"

in Proceedings of the 1999 Congress on Evolutionary Computation, 1999.

CEC 99, vol. 3, pp. 178. [75]. J. Kennedy and R. C. Eberhart, ―Particle swarm optimization,‖ Proceedings of

IEEE International Conference on Neural Networks, pp. 1942–1948,

Piscataway, NJ, USA, 1995. IEEE Press. [76]. V. Tandon, ―Closing the gap between CAD/CAM and optimized CNC end

milling,‖ Master‘s thesis, Purdue School of Engineering and Technology,

Indiana University Purdue University Indianapolis, 2000. [77]. Y. Fukuyama et al, ―A particle swarm optimization for reactive power and

voltage control considering voltage security assessment,‖ IEEE Trans Power

Systems 2000, vol. 15, no. 4, pp.1232-1239. [78]. H. Yoshida, K. Kawata, Y. Fukuyama, and Y. Nakanishi, ―A particle swarm

optimization for reactive power and voltage control considering voltage

stability,‖ Proceedings of International Conference on Intelligent System

Application to Power Systems” Brazil, IEEE Press, pp. 117-121, 1999. [79]. J. Kennedy, and R. Eberhart, ―Swarm intelligence,‖ Academic Press, 1

st

edition, San Diego, CA, 2001. [80]. S. N. Sivanandam, and P. Visalakshi, ―Multiprocessor Scheduling Using

Hybrid Particle Swarm optimization with dynamically varying inertia,‖

International Journal of Computer Science & Applications, vol. 4, no. 3, pp.

95-106, 2007. [81]. J. Kennedy, and R. Eberhart, ―A discreate binary version of the particle swarm

optimization algorithm,‖ Proceedings of the 1997 conference on systems, man,

cybernetics (SMC’97) pp. 4104-4109, 1997.

REFERENCES

159

[82]. Y. Fukuyama et.al, ―A particle swarm optimization for reactive power and

voltage control considering voltage security assessment,‖ IEEE Trans Power

Systems, vol. 15, no. 4, pp. 1232-1239, 2000. [83]. V. Miranda, and N. Fonseca, ―EPSO-Best of two world of meat-heuristic

applied to power system problems,‖ Proceedings of the 2002 congress of

evolutionary computation IEEE Press, 2002. [84]. P. Angeline, ―Evolutionary optimization versus particle swarm optimization:

philosophy and performance differences,‖ Proceeding of the 7th

annual

conference on evolutionary programming: IEEE Press, 1998. [85]. M. L. Diego, J. R. P. Lopez, and J. Basterrechea, ―Synthesis of planner arrays

using a modified particle swarm optimization algorithm by introducing a

selection operator and elitism,‖ Progress in Electromagnetic Research, vol.

93, pp. 145-160, 2009. [86]. S. H. Park, H. T. Kim, and K. T. Kim, ―Stepped- frequency Isar motion

compression using particle swarm optimization with an Island model,‖

Progress in Electromagnetic Research, vol. 85, pp. 25-37, 2008. [87]. S. M. Mikki, and A. A. Kishk, ―Physical theory for particle swarm

optimization,‖ Progress in Electromagnetic Research, vol. 75, pp. 171-207,

2007. [88]. M. Zubair, Applications of Particle Swarm Optimization to Digital

Communication, 2009 [89]. R. Storn, K. Price. Differential Evolution - A Simple and Efficient Heuristic

for Global Optimization over Continuous Spaces. Journal of Global

optimization, 11: 341-359, 1996. [90]. K. V. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical

Approach to Global Optimization. Springer, 2005. [91]. T. Rogalsky, R.W. Derksen and Kocabiyik, ―Differential evolution in

aerodynamic optimization‖, Proc. of 46th annual conf. of Canadian

Aeronautics and Space Institute, pp 29-36, 1999. [92]. S. Das and A. Konar, ―Design of two dimensional IIR filters with modern

search heuristics: a comparative study‖, International Journal of

Computational Intelligence and Applications, Vol. 6, No.3, pp.329-355, 2006. [93]. R. Joshi and A. C. Sanderson, ―Minimal representation multisensory fusion

using differential evolution,‖ IEEE Trans. Syst. Man Cybern. A, Syst. Humans,

vol. 29, no. 1, pp. 63–76, Jan. 1999. [94]. L.J. Fogel, ―Autonomous Automata,‖ Industrial Research, vol.4, pp.14–19,

1962. [95]. J. Lampinen and I. Zelinka, ―On stagnation of the differential evolution

algorithm,‖ in Proc. 6th Int. Mendel Conf. Soft Comput., P. Oˇsmera, Ed., pp.

76–83.2002. [96]. M. Schmitt, ―On the Complexity of Computing and Learning with

Multiplicative Neural Networks,‖ Neural Computation, vol. 14, no. 2, pp. 241-

301, 2002. [97]. K. Okonkwo, P. Simacek, S. G. Advan, and R. S. Parnas ―Characterization of

3D fiber preform permeability tensor in radial flow using an inverse algorithm

based on sensors and simulation,‖ Applied Science and Manufacturing, In

Press, 2011. [98]. M. E. Petersen, D. D. Ridder, and H. Handels, ―Image processing with neural

networks—a review,‖ Pattern Recognition, vol. 35, no. 10, pp. 2279-2301,

2002.

http://www.sciencedirect.com/science/journal/1359835X

REFERENCES

160

[99]. E. Aarts, J. Korst, and L. Van, ―A quantitative analysis of the simulated

annealing algorithm: A case study for the traveling salesman problem,‖ J

Statistical Phys vol. 50, no. 1-2, pp. 187-206, 1988. [100]. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, ―Optimization by Simulated

Annealing,‖ Science, vol. 220, pp. 671–680. [101]. V. Cerny, ―A thermodynamical approach to the travelling salesman problem:

an efficient simulation algorithm,‖ Journal of Optimization Theory and

Applications, vol. 45, pp. 41-51, 1985. [102]. J.X Zhang, H. Wen, and Y. Liu, ―Monte carlo model in metal recrystallization

simulation,‖ Journal of Shanghai Jiaotong University vol.16, pp. 337-342,

2011. [103]. F. Rehab, and A. Kader, ―Fuzzy Particle Swarm Optimization with Simulated

annealing and Neighborhood Information Communication for Solving TSP,‖

International Journal of Advanced Computer Science and Applications, vol. 2,

no. 5, 2011. [104]. R. Gallego, A. Alves, A. Monticelli and R. Romero, ―Parallel simulated

annealing applied to long term transmission network expansion planning,‖

IEEE Trans Power systems, vol. 12, no. 1, pp. 181-188, 1997. [105]. E. Aarts, and J. Korst, ―Simulated annealing and Boltzmann machines,‖ New

York: John Wiley & Sons, 1989. [106]. C.R. Reeves., ―Modern heuristic techniques for combinatorial problems,‖ New

York: John Wiley & Sons, 1993. [107]. H. Chiang, J. Wang, O. Cockings and H.Shin, ―Optimal capacitor placement

in distribution systems. Part I: A new formulation and the overall problem,‖

IEEE Trans Power Delivery, vol. 5, no. 2, pp. 634-642, 1990. [108]. R. Hooke, and R. Jeeves, ―Direct search solution of numerical and statistical

problems,‖ Journal of Association for Computing Machinery, vol. 8, pp. 212-

229, 1961. [109]. E. D. Dolan, R. M. Lewis, and V. J. Torczon, ―On the local convergence of

pattern search,‖ SIAM Journal on Optimization, vol. 14, no. 2, pp. 567-583,

2003. [110]. W. C. Yu, ―Positive basis and a class of direct search techniques,‖ Scientia

Sinica, pp. 53-68, 1979. [111]. E. D. Dolan, R. M. Lewis, and V. J. Torczon, ―On the local convergence of

pattern search,‖ SIAM journal on Optimization, vol.14, no. 2, pp. 567-583,

2003. [112]. M. Pelillo, and A. Torsello, ―Game theorey in pattern recognition and machine

learning,‖ IEEE Industry Applications Magazine, vol. 16, no. 3, 2010. [113]. Z. Huang et.al, ―Applying SVM in license plate character recognition,‖

Journal of Computer Engineering, vol. 29, no. 5, pp. 192-194, 2003. [114]. M. Wetter, and E. Polak, ―A convergent optimization method using pattern

search algorithms with adaptive precision simulation,‖ Eighth international

IBPSA conference, Eindhoven Netherland, pp. 1393-1400, 2003. [115]. L. T. Fan, Y. H. Hwang, and C. L. Hwang, ―Applications of modern optimal

control theory to environmental control of confined spaces and life support

systems,‖ Part 3- optimal control of systems in which straight variables have

equally constraints at the final process time, vol. 5, no. 3-4, pp.125-136, 1970. [116]. R. H. Byrd, J. C. Gilbert, and J. Nocedal, ―A trust region method based on

interior point techniques for nonlinear programming,‖ Mathematical

Programming, vol. 89, no. 1, pp. 149-185, 2000.

http://www.springerlink.com/content/1007-1172/

http://www.academypublisher.com/ijrte/vol01/no01/ijrte0101020024.pdf

REFERENCES

161

[117]. R. H. Byrd, M. E. Hribar, and J. Nocedal, ―An interior point algorithm for

large-scale non-linear programming,‖ SIAM Journal on Optimization, vol. 9,

no. 4, pp. 877-900, 1999. [118]. R. A. Waltz, J. L. Morales, J. Nocedal, and D. Orban, ―An interior algorithm

for nonlinear optimization that combines line search and trust region steps,‖

Mathematical Programming, vol. 107, no. 3, pp. 391-408, 2006. [119]. S. Wright, ―Primal dual interior point methods, 1997 by the society for

industrial and applied mathematics Philadelphia,‖ 1997. [120]. L. Wang, ―Support Vector Machines: Theory and Application,‖ Springer-

Verlag Berlin Heidelberg, 2005. [121]. S. W. Sloan, ―A steepest edge active set algorithm for solving sparse linear

programming problems,‖ Int. J. Numerical Method in Engineering, vol. 26,

no. 12, pp. 2671-2685, 2005. [122]. W. W. Hager, and H. Zhang, ―A New Active Set Algorithm for Box

Constrained Optimization,‖ SIAM J. on Optimization, vol. 17, no. 2 pp. 526-

557, 2006. [123]. J. J. Judice, H. D. Sherali, I. M. Ribeiro, and A. M. Faustino,

―Complementarity active set algorithm for mathematical programming

problems with equilibrium constraints,‖ Journal of Optimization Theory and

Applications, vol. 134, no. 3, pp. 467-481, 2007. [124]. S. W. Sloan, ―A steepest edge active set algorithm for solving sparse linear

programming problems,‖ Int. J. Numerical Method in Engineering, vol. 26,

no. 12, pp. 2671-2685, 2005. [125]. L. Hanka, ―An effiecent quadratic programming optimization method for

deconvolution of gamma rays spectra,‖ J. Comp. Appl. Math, vol. 9, no.1, pp.

47-66, 2010. [126]. F. Zaman,

I. M. Qureshi, A.Naveed

and Z. U. Khan, ―Real Time Direction of

Arrival estimation in Noisy Environment using Particle Swarm Optimization

with single snapshot,‖ Research Journal of Engineering and Technology

(Maxwell Scientific organization), Vol. 4(13) pp. 1949-1952, 2012.

[127]. F. Zaman, J. A. Khan, Z.U.Khan, I.M.Qureshi, ―An application of hybrid

computing to estimate jointly the amplitude and Direction of Arrival with

single snapshot,‖ IEEE, 10th

-IBCAST, pp-364-368, Islamabad, Pakistan,

2013. [128]. F. Zaman, Shahid Mehmood, Junaid Ali Khan

and Ijaz Mansoor Qureshi

―joint estimation of amplitude and direction of arrival for far field sources

using intelligent hybrid computing‖, Research Journal of Engineering and

Technology (Maxwell Scientific organization), (In Press) 2013. [129]. Y.A.Shiekh, F. Zaman, I.M.Qureshi, A.U. Rehman, ―Amplitude and Direction

of Arrival Estimation using Differential Evolution,‖ IEEE, ICET, pp, 45-49,

2012. [130]. F., Ijaz Mansoor Qureshi, A. Naveed, Junaid Ali Khan and Raja Muhammad Asif

Zahoor ―Amplitude and Directional of Arrival Estimation: Comparison between

different techniques,‖ Progress in Electromagnetic research-B (PIER-B), Vol. 39,

pp.319-335, 2012. [131]. F. Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, ―An Application of Artificial

Intelligence for the Joint Estimation of Amplitude and Two Dimensional Direction of

Arrival of far field sources using 2-L shape array,‖ International Journal of Antennas

and Propagation, Article ID 593247, 10 pages, Volume 2013.

REFERENCES

162

[132]. Y.A.Shiekh, F. Zaman, I.M.Qureshi, A.U. Rehman, ―Azimuth and elevation

angle of arrival estimation using differential evolution with single snapshot,‖

Accepted in Recent development on signal processing (RDSP), Istanbul,

Turkey. 2013. [133]. F. Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, ―Hybrid Differential

Evolution and hybrid Particle swarm optimization for the joint estimation of

amplitude and Direction of Arrival of far field sources using L shape arrays,”

Submitted to Iranian journal of Science and Technology, Transaction of

Electrical engineering, Paper key: 1311-IJSTE, 2013. [134]. F. Zaman, I.M.Qureshi, Fahad Munir and Z.U. Khan, ―4D parameters

estimation of plane waves using swarming intelligence,‖ Submitted to Chinese

Physics B, Paper-ID 132170, 2013. [135]. T.w. Anderson, Asymtotic Theory of principal component analysis, Ann.

Math. Stat, 34:122-148, (1963). [136]. P. Smith and G. Bucchler, ―A branching algorithm for discriminating and

detecting multiple objects,‖ IEEE Trans. Automat. Contr., vol. Ac-20, pp. 101-

104, 1975.

[137]. C. R. Sastry, E. W. Kamen, and M. Simaan, ―An efficient algorithm for

tracking the angles of arrival of moving targets,‖ IEEE trans. Signal Process,

vol. 39, no. 1, 242-246, 1991.

[138]. Q. Cheng, Y. Hua, Further study of the pencil-MUSIC algorithm, IEEE Trans.

Aerospace Electronic Systems 32 (1) (1996) 284–299.

[139]. Y. Hua, T.K. Sarkar, D.D. Weiner, An L-shaped array for estimating 2-D

directions of wave arrival, IEEE Trans. Antennas Propagation 39 (2) (1991)

143–146.

[140]. T.H. Liu, J.M. Mendel, Azimuth and elevation direction 0nding using arbitrary

array geometries, IEEE Trans. Signal Process. 46 (7) (1998) 2061–2065.

[141]. V.S. Kedia, B. Chandna, A new algorithm for 2-D DOA estimation, Signal

Process. 60 (1997) 325–332.

[142]. Yuntao Wu, Guisheng Liao, H.C. Sob, A fast algorithm for 2-D direction-of-arrival

estimation, Signal Processing 83 (2003) 1827 – 1831. [143]. T, Nizar, M. K, Hyuck, L-Shape 2-Dimensional Arrival Angle Estimation

with Propagator Method. IEEE Transactions on antennas and propagation,

53:1622–1630. (2005).

[144]. M. Donelli, S. Caorsi, F. DeNatale, M. Pastorino, and A. Massa, ―Linear

antenna synthesis with a hybrid genetic algorithm,‖ Progress In

Electromagnetics Research, vol. 49, pp. 1–22, 2004.

[145]. K. Guney and S. Basbug, ―Interference suppression of linear antenna arrays

by amplitude-only control using a bacterial foraging algorithm,‖ Progress In

Electromagnetics Research, vol. 79, pp. 475–497, 2008.

[146]. A. I. Sotiriou, P. K. Varlamos, P. Trakadas, and C. N. Capsalis, ―Performance

of a six-beam switched parasitic planar array under one path rayleigh fading

environment,‖ Progress In Electromagnetics Research, vol. 62, pp. 89–106,

2006.

[147]. T.-H. Liu and J. M. Mendel, ―Azimuth and elevation direction finding using

arbitrary array geometries,‖ Signal Processing, IEEE Transactions on, vol. 46,

no. 7, pp. 2061–2065, 1998.

[148]. S. Marcos, ―Calibration of a distorted towed array using a propagation

operator,‖ The Journal of the Acoustical Society of America, vol. 93, p. 1987,

1993.

REFERENCES

163

[149]. T. K. S. Hua Y. and D. D. Weiner, ―An l-shaped array for estimating 2-d

directions of wave arrival,‖ IEEE Transactions on Antennas and Propagation,

vol. 39, no. 2, pp. 143–146, 2005.

[150]. C.-H. Lin,W.-H. Fang, J.-D. Lin, and K.-H.Wu, ―A fast algorithm for joint

two-dimensional direction of arrival and frequency estimation via hierarchical

space–time decomposition,‖ Signal Processing, vol. 90, no. 1, pp. 207–216,

2010.

[151]. J.H.Kim., I. S. Yang, K. M. Kim, and W. T. Oh, ―Passive ranging sonar based on

multi- beam towed array,‖ Proc. IEEE Oceans, Vol. 3, 1495–1499, September

2000.

[152]. A. Swindlehurst., Kailath, T. (1988). Passive direction-of-arrival and range

estimation for near-field sources. In Proceedings of the 4th Annual ASSP

Workshop on Spectrum Estimation and Modeling, USA, 123–128.

[153]. D.Starer., Nehorai, A. (1994). Passive localization on near-field sources by

path following. IEEE Transactions on Signal Processing, 42, 677–680.

[154]. J. Chen., X. Zhu, X. Zhang., A new algorithm for joint range-DOA-

frequency estimation of near-field sources. EURASIP Journal on Applied

Signal Processing, 2004, 386–392, 2004.

[155]. Y. Wu, Y, L. Ma, L. Hou, C., G. Zhang., J. Li., ―Subspace-based method

for joint range and DOA estimation of multiple nearfield sources.‖. Signal

Processing, 86, 2129–2133, 2006.

[156]. N. Kabaoglu, H. Cirpan, E. Cekli, S. Paker., Deterministic Maximum

likelihood Approach for 3-D near-field source localization," International

Journal of Electronics and Communications, 57, 345-350,2005.

[157]. I. Ziskind and M. Wax, "Maximum likelihood localization of multiple sources by

alternating projection," IEEE Trans. on Acoust., Speech,Signal Processing, Vol. 36,

No. 10, pp. 1553-1560,1988.

[158]. R.N. Challa, and S. Shamsunder, ―Higher-order subspace based algorithms for

passive localization of near-field sources,‖ Proceedings of the 29th Asilomar

Conference on Signals and System Computer, Pacific Grove, CA, 777–781 October

1995.

[159]. G. Emmanuele, A.M. Karim, and Y. Hua, ―A weighted linear prediction method for

near- Field source localization,‖ IEEE Trans. Signal Process. Vol. 10, 2005.

[160]. F. Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan, ―joint estimation of amplitude,

direction of arrival and range of near field sources using memetic computing‖

Progress in Electromagnetic research-C (PIER-C) ,Vol,31, pp. 199-213, 2012.

[161]. F. Zaman, I.M.Qureshi, A. Naveed,and Z.U. Khan , ―Joint Amplitude, Range and

Direction of Arrival Estimation of near Field Sources using Hybrid Differential

Evolution and Hybrid Particle Swarm Intelligence,‖ Archives Des Sciences, Vol 65,

pp. 671-685, 2012.

[162]. A. Khaliq, F. Zaman, K. Sultan, I.M.Qureshi, 3-D near field source localization by

using hybrid Genetic Algorithm, ―Research Journal of Engineering and Technology

(Maxwell Scientific organization)”, (In press), 2013.

REFERENCES

164

[163]. F. Zaman, S.U. Khan, K. Ashraf and I.M Qureshi, ―An application of hybrid

differential evolution to 3-D source localization‖ Accepted in IEEE, 11th, IBCAST,

2013.

[164]. F. Zaman, I.M.Qureshi, M. Zubair, and Z.U. Khan, ―Multiple target

localization with bistatic radar using heuristic computational intelligence‖

Paper key: 13052307, PIER, 2013.

[165]. F. Zaman, I.M. Qureshi, ―5D parameter estimation of Near-field sources using

Hybrid Evolutionary Computational Technique,‖ The Scientific World Journal,

Paper-ID- 310875, 2013.

[166]. N.Kabaoglu, H. A. Cirpan, E. Cekli, and S. Paker, ―Deterministic Maximum

likelihood Approach for 3-D near field source localization," International

Journal of Electronics and Communications. Vol. 57, No. 5, 345-350, 2003.

[167]. R.N.Challa, and S. Shamsunder, ―Passive near-field localization of multiple

non-Gaussian sources in 3-D using cumulant," Signal Processing, Vol. 65, 39-

53, 1998.

[168]. J. Liang, D. Liu, X. Zeng, W. Wang, J. Zhang, and H. Chen, ―Joint azimuth-

elevation-range estimation of mixed near-field and far-field sources using two-

stage separated steering vector- based algorithm,‖ Progress In

Electromagnetics Research, Vol. 113, 17-46, 2011.

[169]. J. Liang, S. Yang, J. Zhang, L. Gao and F. Zhao., ―4D Near-Field Source

Localization Using Cumulant‖ EURASIP Journal on Advances in Signal

Processing (Hindawi Publication),Volume 2007, Article ID 17820, 10

pages,doi:10.1155/2007/17820, 2007.

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