estimation of direction of arrival for adaptive …
TRANSCRIPT
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.
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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.
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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
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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.
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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.
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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.
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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
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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
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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
4.7.3 Results and Discussion ......................................................................................... 79
Part- III ............................................................................................................................... 87
4.8 Joint Estimation Of 4-D Parameters Using PSO-PS ......................................... 88
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4.8.1 Data Model............................................................................................................ 88
4.8.2 Particle Swarm Optimization Hybridized With Pattern Search ............................ 91
4.8.3 Results and Discussion ......................................................................................... 93
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
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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 Results and Discussion ....................................................................................... 128
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
5.6.3 Results and Discussion ....................................................................................... 140
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. 15 Performance of 1-L type array for 3 sources ............................................ 70
Table 4. 16 Performance of 2-L type array for 4 sources ............................................ 70
Table 4. 17 Performance of 1-L type array for 4 sources ............................................ 70
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
Table 4. 19 Means, Variances and standard deviations at 10 dB noise for different
elevation angles and fixed azimuth angle by using GA-PS with 1-L
shape array .............................................................................................. 72
xx
Table 4. 20 Means, Variances and standard deviations at 10 dB noise for different
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. 23 Estimation accuracy of 2-L-shape array for 2-sources ............................. 80
Table 4. 24 Estimation accuracy of 1-L-shape array for 3-sources ............................. 81
Table 4. 25 Estimation accuracy of 2L-shape array for 3-sources .............................. 81
Table 4. 26 Estimation accuracy of 1-L-shape array for 4-sources ............................. 82
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. 35 Estimation accuracy for 4 sources ............................................................ 95
Table 4. 36 Estimation accuracy for 4 sources ............................................................ 95
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
CHAPTER 1 . INTRODUCTION
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
CHAPTER 1 . INTRODUCTION
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.
CHAPTER 1 . INTRODUCTION
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.
CHAPTER 1 . INTRODUCTION
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.
CHAPTER 1 . INTRODUCTION
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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,
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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,
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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).
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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,
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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)
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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,
CHAPTER 2 DOA ESTIMATION TECHNIQUES: AN OVERVIEW
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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.
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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.
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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.
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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 .
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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)
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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.
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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.
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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
CHAPTER 3 SELECTED OPTIMIZATION TECHNIQUES
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)
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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,
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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%.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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,
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
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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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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 .
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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)
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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)
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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)
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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 ˆexp[ cos( )],,21
Pqy c j m czm q iq P i
i
(4.28)
ˆ ˆ ˆ ˆexp[ 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 ˆexp[ 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.
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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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
70
Table 4. 15 Performance of 1-L type array for 3 sources
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.
Table 4. 16 Performance of 2-L type array for 4 sources
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
Table 4. 17 Performance of 1-L type array for 4 sources
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
Table 4. 19 Means, Variances and standard deviations at 10 dB noise for different elevation angles and fixed
azimuth angle by using GA-PS with 1-L shape array
𝜃 in radians for
∅ = 0.6109 (rad)
Mean of 𝜃 (rad) Variance of 𝜃 (rad) Standard Deviation of 𝜃
(rad)
1.2392 1.2394 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
Table 4. 20 Means, Variances and standard deviations at 10 dB noise for different elevation angles and fixed
azimuth angle by using GA-PS with 2-L shape array
𝜃 in radians for
∅ = 0.6109 (rad)
Mean of 𝜃 (rad) Variance of 𝜃 (rad) Standard Deviation of 𝜃
(rad)
1.2392 1.2392 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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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,
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
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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
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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.
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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 .
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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)
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
4.7.3 Results and Discussion
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
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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
Table 4. 23 Estimation accuracy of 2-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 0.5036 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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
Table 4. 24 Estimation accuracy of 1-L-shape array for 3-sources
Scheme 𝑠1 𝜃1∘ ∅1
∘ 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
Scheme 𝑠1 𝜃1∘ ∅1
∘ 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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
82
performance in terms of estimation accuracy becomes excellent when they are hybridized
with PS.
Table 4. 26 Estimation accuracy of 1-L-shape array for 4-sources
Scheme 𝑠1 𝜃1∘ ∅1
∘ 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
Scheme 𝑠1 𝜃1∘ ∅1
∘ 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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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.
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Table 4. 31 Means, Variances and standard deviations using PM with L shape array
𝜃 in radians for 𝜙 = 500 Mean of 𝜃 Variance of 𝜃 Standard Deviation of 𝜃
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
𝜃 in radians for 𝜙 = 500 Mean of 𝜃 Variance of 𝜃 Standard Deviation of 𝜃
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
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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.
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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)
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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)
CHAPTER 4 DOA ESTIMATION INCLUDING AMPLITUDE AND FREQUENCY OF FAR FIELD SOURCES
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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
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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.
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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,
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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.
4.8.3 Results and Discussion
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-
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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.
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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.
Table 4. 35 Estimation accuracy for 4 sources
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
Table 4. 36 Estimation accuracy for 4 sources
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.
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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
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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
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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
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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
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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)
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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,
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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)
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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
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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)
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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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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.
5.3.1 Results and Discussion
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
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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
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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.
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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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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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: 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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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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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5.5.1 Results and Discussion
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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.
Table 5. 22 Estimation accuracy for 3-targets
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
Table 5. 24 Estimation accuracy for 4-targets
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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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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
PSO-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
PSO-ASA at 180 degree
Desired at 180 degree
Desired at 0 degree
PSO-ASA at 0 degree
DE-ASA at 0 degree
Desired at 90 degree
DE-ASA at 90 degree
PSO-ASA at 90 degree
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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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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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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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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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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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:
: 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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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.
5.6.3 Results and Discussion
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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
Table 5. 29 Estimation Accuracy of 3 sources using 13 sensors (continue)
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
SOURCES
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.
Table 5. 30 Estimation Accuracy of 3 sources using 13 sensors
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.
Table 5. 31 Estimation Accuracy of 4 sources using 17 sensors (continue)
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
SOURCES
143
Table 5. 32 Estimation Accuracy of 4 sources using 17 sensors
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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( 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.
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
SOURCES
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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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
CHAPTER 5 DOA ESTIMATION INCLUDING RANGE AMPLITUDE AND FREQUENCY OF NEAR FIELD
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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
CHAPTER 6 CONCLUSION AND FUTURE DIRECTION
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
CHAPTER 6 CONCLUSION AND FUTURE DIRECTION
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.
CHAPTER 6 CONCLUSION AND FUTURE DIRECTION
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.
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