Design of Linear Array Geometry for High Resolution Array ...
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DESIGN OF LINEAR ARRAY GEOMETRY FOR HIGH RESOLUTION
ARRAY PROCESSING
By
XINPING HUANG, B.Sc.(Eng.), University of Science and Technology'of China
M.Eng., Nanjing A"ronautical Institute
A Thesis
Submitted to the School of Graduate Studies
in Partial Fulfilment of the Requirements
for the Degree
Doctor of Philosophy
McMaster University
September 9, 1993
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TO ALL MEMBERS IN MY LOVELY FAMILY
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DESIGN OF LINEAR ARRAY GEOMETRY
FOR HIGH RESOLUTION ARRAY PROCESSING
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DOCTOR OF PHILOSOPHY (September 9, 1993)
(Electrical and Computer Engineering)
II
MCMASTER UNIVERSITY
Hamilton, Ontario
TITLE:
AUTHOR:
SUPERVISOR(S):
Design of Linear Array Geometry for High Resolution
Array Processing
Xinping Huang
B.Sc.(Eng.), University of Science and Technology of China
M.Eng., Nanjing Aeronautical Institute
Dr. Kon Max Wong
Professor, Chairman of Department of Electrical and Com
puter Engineering
B.Sc.(Eng.), Ph.D., (University of London)
D.I.C. (University of London)
Fellow, I.E.E.
Fellow, Royal Statistical Society
Fellow, Institute of Physics
Dr. James P. Reilly
Professor, Department of Electrical and Computer Engineer
ing
B.A.Sc. (University of Waterloo)
M.Eng., Ph.D. (McMaster University)
NUMBER OF PAGES: xvii, 141
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Abstract
The linear array is one of the most important types of multi-element sensor arrays, being
extensively used in radar, sonar, telecommunications, radio astronomy and medical imaging
systems. Traditionally, the array assumes uniform geometry with an inter-sensor spacing
of %' which limits resolvability because of the axed aperture. Since the 1950's, much work
has been done on designing nonuniform arrays with focus on conventional beamforming
techniques. One of the typical results is the Minimum Redundancy (MR) arrays which
provide improved performance over the uniform array.
In this thesis, this issue is re-investigated from the viewpoint of high resolution
array processing. A new criterion (called DOBC), based on D-Optimality, is developed,
which yields a new array geometry by minimizing a measure of joint estimation error with
respect to the array geometry parameters. The sensor gain and phase calibration errors
and their effects on high resolution array signal processing are also examined, and formulae
are developed to evaluate such effects.
In addition, the Modified Forward-Backward Linear Prediction (MFBLP) method
is modified to substantially improve the low SNR performance without increase in compu
tationalload. A form of Cramer-Rao lower bound (CRLB) is derived for a reduced model
which facilitates performance comparisons in directions of arrival (DOA) estimation.
Compliter experiments are conducted to verify our analysis. We conclude that (1).
the DOBC design outperforms the conventional uniform array and the MR array; (2). the
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IY
formulae developed predict very well the behaviour of high resolution algorithms in the
presence or absence of calibration errors. The design criterion and formulae can be used by
the system designer to design a new array geometry given the performance requirement and
hardware specifications, to evaluate the expected performance of an array, given information
about hardware specifications, or to develop hardware specifications given the performance
requirement.
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Acknowledgement
The author wishes to express his deep appreciation to Professors K. M Wong and J. P. Reilly
for their encouragement, continued assistance, expert guidance and supervision throughout
the course ofthis work. He also thanks Professors. J. F. McGregor and Z. Q. Luo, members
of his Supervisory Committee, for their continuing interest and useful suggestions.
The author is very grateful to Drs. Q. WU, Q. Jin, and many other researchers
and fellow graduate students in the Communications Research Laboratory for providing
stimulating discussions, helpful suggestions and generous assistance.
He is also d.leply indebted to his wife, Ping He, and his son, Andy, for their under
standing, constant encouragement and support.
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Contents
Abstract
Acknowledgement
List of Tables
List of Figures
Glossary
1 INTRODUCTION
1.1 What Is A Sensor Array?
1.2 Array Signal Processing . .
1.3 Geometry of Sensor Arrays
1.4 Errors in Sensor Arrays
1.5 Organization of thesis .
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CO~TE~7S
2 SENSOR ARRAY SIGNAL PROCESSING
2.1 Introduction .
2.2 Model of Array Observations
2.3 Detection of Signals
!1
10
\~
2.3.1 Akaike's Information Theoretic Criterion \3
2.3.2 Minimum Description Length Criterion 1:1
2.3.3 Detection of Number of Signals Using Information Theoretic Criteria 14
2.3.4 Example......
2.4 Estimation of Parameters
2.4.1
2.4.2
Maximum Likelihood Estimator
MUSIC Estimator . . . . . .
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18
21
2.4.3 The FBLP Based Algorithms
2.4.4 Examples . . . . . .
2.5 Geometry of Sensor Array . . . .
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2.5.1
2.5.2
Co-array of Linear Arrays . . .
Minimum Redundancy Arrays
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35
2.6 Conclusions.................................... 36
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:I OPTIMAL DESIGN OF LINEAR SENSOR ARRAYS
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37
3.1 Introduction . 37
:1.3 Design Criterion ..
Formu!ation of Problem3.2
3.3.1
3.3.2
D-optimality ...
Covariance Matrix
40
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42
45
3.3.3 The DOBC Criterion Applied to Array Signal Processing
3.3.4 Some Design Examples .....
3.4 Aperture Selection and Threshold Effect
3.5 Sensitivity of Design .
3.6 Relationship of New Criterion to CRLB
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63
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C-l Maximum Likelihod Estimator in Colored Gaussian Noise
3.7 Conclusions • . . . . . . . . . . . . . . . . . . . . . . . . . 67
69
C-2 Proof of Equation 3.30 ..
C·3 Proof of Equation 3.31 ...
C·4 Proof of Equation 3.32 . . .
C·5 Proof of Equation 3.37 . . .
C·6 Proof of Invari~nce Property ..
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CO~TESH
e·6.1 Inyariance property of E {yyT }
e·6.2 Inyariance property of T .
C· i Cramer·Rao Lower Bound . . . .
C· i.l Probability density function: model reduction.
·IX
, ,
iH
C-7.2 CRLB for Reduced ~lodel . . . . . . . . . . . . . . . . . . . . . . .. sa
4 ANALYSIS OF CALIBRATION ERROR 85
4.1 Introduction . 85
4.2 Calibration Error Model . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Covariance Matrix of DOA Estimates Given Calibration Error .... 90
4.4 The Second Order Approximation To Conditional Covariance Matrix. 92
4.4.1 Expanding E ...
4.4.2 Approximating F ....
4.4.3 Evaluating the Product
92
93
94
4.5 A Matrix Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
4.6 The Averaged Covariance Matrix In the Presence of Gain and Phase Error 96
4.7 Examples . . . . . 97
4.8 Two Special Cases . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102
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4.8.1
oJ.8.2
4.8.3
Gain Error ~Iodel .
Averaged Covariance Matrix in the Presence of Gain Error
Phase Error Model .
x
102
103
104
4.8.4 Averaged Covariance Matrix in Presence of Phase Error . . . . . .. 105
4.8.5 Examples
4.9 Conclusions ...
D-1 Proof of Matrix Lemma 1 . . . . . . . . . . . . . . .
. . . . . . . . . . . .. 106
. . . . . . . . . . . .. 106
110
D-2 Five Ensemble Averages In the Presence of Gain and Phase Error.
D-3 Analysis of Calibration Error: Amplitude Error Only.
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D-3.1 Matrix Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119
D-3.2 Ensemble Averaged Covariance Matrix of DOA estimates . . . . .. 123
D-4 Analysis of Calibration Error: Phase Error Only 127
D-4.1 Matrb: Lemma 3 . . . . . . . . . . • . . . . . . . . . . . . . . . . .. 128
D-4.2 Ensemble Averaged Covariance Matrix of DOA estimates . . . . .. 130
5 CONCLUSIONS
BIBLIOGRAPHY
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List of Tables
2.1
2.2
A uniform array of 4 elements and its coarray . . . .
A non-uniform array of 4 element and its coarray . 3·1
2.3 List of non-redundancy arrays .
2.4 Some Minimum-Redundancy Array Configurations .
34
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3.2 Optimal arrays of 6 elements with different apertures.
3.1 List of various array designs . . . . . . . . . . . . . . . 53
61
4.1 Comparison between designs of Chapter 3 and Chapter 4, Colored noise .. 101
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List of Figures
1.1 Functional relationship between the environment, the sensor array, and the
processor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 illustration of a linear array and signal environment. . . . . . . . . . . . 10
2.2 Valuer. of AIC(k) and MDL;,,) versus k. . . . . . . . . . . . . . . . . . . . . 17
2.3 Probability of detection error using AIC and MDL.. .......... 17
2.4 Mean-squared error of the MLE and MUSIC estimators. . . . . . . . . . . . 31
2.5 Mean-squared error of the FBLP-based techniques. . . . . . . . . . . . . .. 32
3.1 illustration of a linear array .
3.2 Grating lobes of various array configurations. . ......•.
3.3 Spatial autocorrelation of noise, the spatial lag is in unit of ~..
41
51
53
3.4 Normalized spatial spectum of noise. . . . • . . . • . . . . • . . . . . 54
3.5 Mean-squared error of various arrays, M = 6, N = 13, Colored noise. 56
3.6 Mean-squared error of various arrays, M =8, N =23, Colored noise. 57
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LIST OF FIGt'RES xiii
3.7 Mean-squared error of various arrays. M = 6. S = 13. White noise. 57
3.S slean-squared error of various arrays. M = 8. ;\' = 23. White noise..
3.9 :Vlean-squared error of various arrays. M = 6. N = 30, Colored noise.
3.10 Mean-squared error of various arrays, AI = 6, N = 30, White noise.
3.11 Effect of the aperture on heights of spurious peaks.
3.12 Mean-squared errors versus the array aperture. . .
3.13 Probability of erroneous estimation versus the array aperture.
3.14 Mean-squared error of various arrays for different DOAs...
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3.15 Comparison between the covariance matrix and the CRLB. .. . . . . . .. 67
4.1 Decomposition of actual gain and phase .
4.2 MSE of MLE in the presence of calibration error (-20dB), M=6.
4.3 MSE of MLE in the presence of calibration error (-20dB). M=8. .
4.4 Composition of MSE in the presence of Calibration error (-30dB).
89
98
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4.5 Theoretical value of MSE for different level of calibration errors. 100
4.6 Performance comparison between designs of Chapters 3 and 4. 101
4.7 illustration of the ideal and actual values of gain. . . . . . . . . . . . . . .. 103
4.8 illustration of Ideal and actual values of phase. . . . . . . . . . . . . . . .. 105
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