Antenna Array Synthesis Using the Cross Entropy Method

224 Pages·2015·5.49 MB·English

by  Jeffrey David Connor

Florida State University Libraries
Electronic Theses, Treatises and Dissertations The Graduate School
2008
Antenna Array Synthesis Using the Cross
Entropy Method
Jeffrey D. (Jeffrey David) Connor
Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected]
FLORIDA STATE UNIVERSITY

COLLEGE OF ENGINEERING

ANTENNA ARRAY SYNTHESIS USING THE CROSS ENTROPY METHOD

By

JEFFREY D. CONNOR

A Dissertation submitted to the
Department of Electrical and Computer Engineering
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy

Degree Awarded:
Summer Semester, 2008

Copyright © 2008
Jeffrey D. Connor
All Rights Reserved
The members of the Committee approve the Dissertation of Jeffrey D. Connor defended
on June 19, 2008.

Simon Y. Foo
Professor Directing Dissertation

Mark H. Weatherspoon
Professor Co-Directing Dissertation

Amy Chan-Hilton
Outside Committee Member

Anke Meyer-Baese
Committee Member

Approved:

Victor DeBrunner, Chair, Electrical and Computer Engineering

C. J. Chen, Dean, College of Engineering

The Office of Graduate Studies has verified and approved the above named committee
members.

i i
This work is dedicated to anyone and everyone who helped me to get to where I am
today. Thank you all.

ii i
ACKNOWLEDGEMENTS

I would like to thank my major professor Dr. Simon Foo for his support throughout this
process. He truly made this experience an enjoyable one and through all the highs and lows of
the process kept me heading in the right direction.
I would also like to thank the other members of my committee, Dr. Mark Weatherspoon, Dr.
Amy Chan-Hilton and Dr. Anke Meyer-Baese. A special thanks is owed to Dr. Chan-Hilton in
particular since antenna array processing is not her expertise, yet she still provided valuable
feedback and support which I greatly appreciated.
I would also like to thank my parents and family for their love and support throughout my
life and particularly during these last 22 years of studying!
I would also like to thank Dr. Frank Gross for getting me interested in antenna arrays and
optimization problems for electromagnetics. In hind sight I am not sure if I should thank him or
hurt him for that, but all jokes aside he has been a tremendous influence on my educational
career and personal life.
I would like to thank my fellow members of the Machine Intelligence lab, particularly Jason
Isaacs, Erastus Ogunti and Dana Skinner.
I would also like to thank a number of people who helped me along the way, namely Joseph
Amato, Roger Bourgeois, William Bulger, Matt Elliott, Larry Marquis, Leah Marquis and
Margarita Mello.
For anyone I forgot to thank, I am sorry, just know that I could not have done this without
you.

iv
TABLE OF CONTENTS

List of Tables ................................................................................................ vii
List of Figures ................................................................................................ x
Abstract ...................................................................................................... xix

1. INTRODUCTION......................................................................................... 1
1.1 Antenna Array Synthesis......................................................................... 1
1.2 Optimization Algorithms Applied to Array Synthesis............................. 2
1.3 Cross-Entropy Method............................................................................. 3
1.4 Motivation For Work............................................................................... 4
1.5 Summary of Objectives............................................................................ 5
1.6 Organization of Work.............................................................................. 7
2. ANTENNA ARRAYS.................................................................................... 10
2.1 Introduction.............................................................................................. 10
2.2 Linear Array Factor.................................................................................. 11
2.3 Summary.................................................................................................. 23
3. CROSS ENTROPY METHOD..................................................................... 24
3.1 Introduction.............................................................................................. 24
3.2 The General Cross Entropy Method........................................................ 27
3.3 Cross Entropy Method For Continuous Optimization............................. 29
3.4 Cross Entropy Method For Combinatorial Optimization........................ 33
3.5 Convergence Properties of the Cross Entropy Method............................ 35
3.6 Modifications of the Cross Entropy Method........................................... 36
3.7 Summary.................................................................................................. 43
4. APERIODIC ARRAY DESIGN..................................................................... 44
4.1 Introduction.............................................................................................. 44
4.2 Single Objective Aperiodic Array Design By Element Thinning........... 48
4.3 Study of CE Parameter Choice on Outcome of Thinning Scenario......... 55
4.4 Multi-Objective Aperiodic Array Design By Element Thinning............ 58
v
4.5 Single Objective Aperiodic Array Design By Non-Uniform Element
Positions................................................................................................... 66
4.6 Introducing Pattern Null with Non-Uniform Element Positions............. 82
4.7 Non-Uniform Element Spacing for Grating Lobe Cancellation.............. 85
4.8 Multi-Objective Aperiodic Array Design By Non-Uniform Element
Spacings................................................................................................... 91
4.9 Summary.................................................................................................. 99
5. PHASED ARRAY SYNTHESIS................................................................... 101
5.1 Introduction.............................................................................................. 101
5.2 Sector Beam Pattern Synthesis................................................................ 103
5.3 Array Failure Correction.......................................................................... 122
5.4 Phase-Only Array Synthesis.................................................................... 138
5.5 Switched-Beam Array Design................................................................. 153
5.6 Multi-Objective Array Design By Amplitude-Only Array Weighting.... 162
5.7 Summary.................................................................................................. 168
6 . CONCLUSIONS AND RECOMMENDATIONS........................................ 169
6.1 Summary of Work.................................................................................... 169
6.2 Specific Technical Contributions............................................................. 170
6.3 Recommendations for Future Work......................................................... 171

APPENDICES ................................................................................................ 173
REFERENCES ................................................................................................ 198
BIOGRAPHICAL SKETCH .............................................................................. 204
vi
LIST OF TABLES

Table 4.1. CE Design Parameters for Symmetric, Linear Array Scenario......... 50
Table 4.2. Solution of Array Thinning Procedure for Symmetric, Linear Array 52
Table 4.3. Average Number of Iterations For Convergence............................... 56
Table 4.3. Comparison of CE Results to Literature............................................ 53
Table 4.4. Average Sidelobe Powers Achieved By Best & Mean Population
Scores................................................................................................................... 57
Table 4.5. MOCE Design Parameters for Symmetric, Linear Array Scenario... 62
Table 4.6. Composite Pareto Optimal Solutions of Multi-Objective Thinned Array
Scenario................................................................................................................ 64
Table 4.7. Mean Pareto Optimal Solutions of Multi-Objective Thinned Array
Scenario................................................................................................................ 65
Table 4.8. CE Design Parameters for Symmetric, Linear Array Scenario......... 71
Table 4.9. Sidelobe Power and Beamwidth of Best Array Designs for Each Score
Function............................................................................................................... 81
Table 4.10. Absolute Element Positions From Origin and Element Spacing for Best
Performing Samples of Non-Uniform Array Scenario........................................ 81
Table 4.11. Absolute Element Positions From Origin and Element Spacings for Best
Performing Samples of Non-Uniform Array Scenario Including Null Positions In
Score.................................................................................................................... 83
Table 4.12. Absolute Element Positions From Origin and Element Spacing for Best
Performing Samples of Grating Lobe Scenario................................................... 90
Table 4.13. Sidelobe Power and Beamwidth of Best Array Design of Grating Lobe
Scenario................................................................................................................ 90
Table 4.14. MOCE Design Parameters for Symmetric, Linear Array Scenario. 94
vii
Table 4.15. Composite Pareto Optimal Solutions of Multi-Objective Aperiodic
Array Scenario..................................................................................................... 96
Table 4.16. Mean Pareto Optimal Solutions of Multi-Objective Aperiodic Array
Scenario................................................................................................................ 97
Table 5.1. Parameters for Sector Beam Patterns from [59], [63]...................... 104
Table 5.2. CE Design Parameters for Sector Beam Scenario of [63]................. 107
Table 5.3. Complex Sector Pattern Weights at Final Iteration.......................... 112
Table 5.4. CE Design Parameters For Sector Beam Scenario of [59]................ 109
Table 5.5. Complex Sector Pattern Weights at Final Iteration........................... 118
Table 5.6. CE Design Parameters for Double Element Failure Dolph-Chebyshev
Array Scenario..................................................................................................... 125
Table 5.7. Measurables for Double Element Chebyshev Failure Scenario........ 127
Table 5.8. Original and Corrected Weights for Double Element Chebyshev Failure
Scenario................................................................................................................ 128
Table 5.9. Measurables for Double Element Taylor Failure Scenario............... 133
Table 5.10. Original and Corrected Weights for Double Element Taylor Failure
Scenario................................................................................................................ 134
Table 5.11. Measurables for Convergence Curves of Double Element Taylor Failure
Scenario................................................................................................................ 135
Table 5.12. CE Design Parameters for Null Placement Scenario....................... 141
Table 5.13. Measurables for Convergence Curves of Null Placement Scenario for
Score Function #1................................................................................................ 143
Table 5.14. Measurables of Null Placement Scenario for Score Function #1.... 143
Table 5.15. Measurables for Convergence Curves of Null Placement Scenario for
Score Function #2................................................................................................ 143
Table 5.16. Measurables of Null Placement Scenario for Score Function #2.... 143
Table 5.17. Phase-Only Weights (in deg) for Best Performing Trials of Null
Placement Scenario.............................................................................................. 152
Table 5.18. Parameters for Switched-Beam Array Design From [62]............... 154
viii
Table 5.19. CE Design Parameters for Switched-Beam Array Design.............. 156
Table 5.20. Complex Weights of Switched-Beam Array Design....................... 160
Table 5.21. MOCE Design Parameters for Symmetric, Linear Array Scenario. 164
Table 5.22. Polynomial Coefficients For 6th Order Polynomial Fit Using
MATLAB’s Polyfit Command............................................................................ 166
Table 5.23. Accuracy of Multi-Objective Methods Quantified by Error Estimates167
Table A.1. Progression of CE Parameters, Score and Positions for Continuous CE
Example............................................................................................................... 175
Table C.1. Progression of CE Parameters and Score For Combinatorial
Optimization Example......................................................................................... 184
Table D.1. Optimal Solutions to (D.2)................................................................ 187
Table E.1. Average Number of Iterations For Convergence.............................. 192
Table E.2. Average Scores Achieved By Best & Mean Population Scores....... 193

ix