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Efficient approaches to source localization and parameter estimation

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Efficient approaches to source localization and parameter estimation
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Zheng, Dunmin
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Antennas ( jstor )
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Narrowband ( jstor )
Plane waves ( jstor )
Sensors ( jstor )
Signals ( jstor )
Statistical estimation ( jstor )
Supernova remnants ( jstor )
Waveforms ( jstor )
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Dissertations, Academic -- Electrical and Computer Engineering -- UF
Electrical and Computer Engineering thesis, Ph. D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1996.
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Includes bibliographical references (leaves 123-131).
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Typescript.
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Vita.
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by Dunmin Zheng.

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EFFICIENT APPROACHES TO SOURCE LOCALIZATION
AND PARAMETER ESTIMATION





By

DUNMIN ZHENG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR, OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1996


UNIVERSITY OF FL'O:.A Lr.RA';iES




EFFICIENT APPROACHES TO SOURCE LOCALIZATION
AND PARAMETER ESTIMATION
By
DUNMIN ZHENG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996
UNIVERSITY OF FLORIDA LIBRARIES


To the Memory of My Mother,
Yu-Xian Liu


ACKNOWLEDGMENTS
I would like to express my sincere gratitude to the chairman of my committee,
Dr. Jian Li, for her guidance, encouragement, and support during the course of my
studies. Special thanks are due to Dr. Scott L. Miller for his insightful comments
and careful proofreading on my work. I would like to convey my appreciation to Dr.
Fred J. Taylor, Dr. John M.M. Anderson and Dr. William W. Hager for serving
on my committee and the guidance they granted me. The insightful comments and
constructive suggestions from the committee have greatly helped to improve the
quality of this dissertation. Thanks also go to Dr. Jose C. Principe for attending my
oral exam, and to Dr. William W. Edmonson for attending my final exam.
I am very grateful to Dr. Petre Stoica of Uppsala University of Sweden for many
fruitful discussions and comments on my work.
It has been such a wonderful experience for me to be a student here at the
University of Florida. I benefitted from it not only academically but also socially. I
wish to thank all my friends in Gainesville for the friendship they extended me.
I am greatly indebted to my family for their constant love, support and patience
during the years of my graduate studies.
Finally, the financial support of the National Science Foundation Grant MIP-
9308302 is gratefully acknowledged.
iii


TABLE OF CONTENTS
ACKNOWLEDGMENTS in
KEY TO ACRONYMS vi
LIST OF FIGURES xi
ABSTRACT xii
1 BACKGROUND AND MOTIVATION 1
1.1 Background 1
1.2 Motivation and Contributions 2
1.3 Outline of the Dissertation 5
2 WAVE PROPERTIES AND SENSOR ARRAY PROCESSING 7
2.1 Introduction 7
2.2 Propagation of EM Waves 7
2.3 Polarization of EM Waves 9
2.4 Narrowband Signals 14
2.4.1 Array Data Model and Problem Formulation 14
2.4.2 Nonparametric Methods 20
2.4.3 Parametric Methods 22
2.5 Wideband Signals 35
2.6 Polarization Diversity 38
2.7 Other Applications 39
2.8 Summary 40
3 NARROWBAND ANGLE AND WAVEFORM ESTIMATION VIA RELAX 41
3.1 Introduction 41
3.2 Problem Formulation 42
3.3 Angle and Waveform Estimation Using RELAX 43
3.4 Results of Simulated and Experimental Data 47
3.5 Conclusions 53
4 ANGLE ESTIMATION OF WIDEBAND SIGNALS USING RELAX ... 59
IV


4.1 Introduction 59
4.2 Problem Formulation 59
4.3 Angle Estimation Using RELAX 62
4.4 Numerical Results 64
4.5 Conclusions 67
5 ANGLE AND POLARIZATION ESTIMATION WITH A COLD ARRAY 69
5.1 Introduction 69
5.2 COLD Array and Problem Formulation 70
5.3 Angle and Polarization Estimation using MODE 76
5.4 Statistical Performance Analysis 83
5.5 Numerical Results 84
5.6 Conclusions 88
6 PARAMETER ESTIMATION USING RELAX WITH A COLD ARRAY 98
6.1 Introduction 98
6.2 Problem Formulation 98
6.3 Parameter Estimation with RELAX 102
6.4 Numerical Results 107
6.5 Conclusions 110
7 CONCLUSIONS AND FUTURE WORK 114
7.1 Summary 114
7.2 Contributions 116
7.3 Future Work 117
APPENDICES 119
A THE CRAMER-RAO BOUND FOR PARAMETER ESTIMATES 119
B PROOF OF EQUATION (5.27) 121
REFERENCES 123
BIOGRAPHICAL SKETCH .
132


KEY TO ACRONYMS
AOA: Angle-Of-Arrival
AR: Autoregressive
ARMA: Autoregressive Moving Average
AP: Alternating Projection
ANPA: Alternating Notch-Periodogram Algorithm
CCD: Co-centered Crossed Dipoles
COLD: Co-centered Orthogonal Loop and Dipole
CRB: Cramer-Rao Bound
CSM: Coherent Signal-subspace Method
DFT: Discrete Fourier Transform
DS-CDMA: Direct-Sequence Code-Division Multiple Access
EM: ElectroMagnetic
ESPRIT: Estimation of Signal Parameters via Rotational Invariance Techniques
FFT: Fast Fourier Transform
IQML: Iterative Quadratic Maximum Likelihood
LS-ESPRIT: Least-Squares based ESPRIT
LSML: Large Sample Maximum Likelihood
vi


ME: Maximum Entropy
ML: Maximum Likelihood
MODE: Method Of Direction Estimation
MUSIC: Multiple Signal Classification
NLS: Nonlinear Least-Squares
NSF: Noise Subspace Fitting
RELAX: RELAXation algorithm for the minimization of the NLS criterion
RMSE: Root-Mean-Squared Error
SNR: Signal-to-Noise Ratio
SSF: Signal Subspace Fitting
TLS-ESPRIT: Total Least-Squares based ESPRIT
ULA: Uniform Linear Array
WSF: Weighted Subspace Fitting
Vll


LIST OF FIGURES
2.1 Polarization ellipse 10
2.2 Poincare sphere 12
2.3 One octant of Poincare sphere with polarization states 13
2.4 An arbitrary two-dimensional array 14
2.5 A uniform linear array 17
3.1 An example of using RELAX, (a) Modulus of FFT of y in step (1),
K 1. (b) Modulus of FFT of y2 in step (2), K = 2 (1st iteration).
(c) Modulus of FFT of yi in step (2), K = 2 (2nd iteration), (d)
Modulus of FFT of y2 in step (2), K = 2 (3rd iteration), (e) Modulus
of FFT of yi in step (2), K = 2 (4th iteration), (f) Modulus of FFT
of y2 in step (2), K = 2 (5th iteration) 55
3.2 RMSEs of the angle and waveform estimates of the first signal as a
function of SNR when N = 10, M = 8, and K = 2 correlated incident
signals with the correlation coefficient equal to 0.99 arrive from 6\
10 and 62 =2. (a) Angle estimation, (b) Waveform estimation. . 56


3.3 F3 vs. 6\ and 02 when N = 10, M = 8, the correlation coefficient
of the two incident signals is equal to 0.99, and the realization of the
noise is the one that gives the worst waveform estimates for ANPA in
Figure 1. (a) Mesh plot, (b) Contour plot 57
3.4 Estimated noise correlation coefficients between the first and the other
sensors. Figures (a) (d) are for the carrier frequencies 8.62, 9.76, 9.79,
and 12.34 GHz, respectively 58
4.1 RMSEs of the angle estimates of the second signal as a function of
SNR when K = 2 uncorrelated wideband signals arrive from 6\ 10
and #2 = 20, M = 8, and L = 33. (a) In the presence of white noise.
(b) In the presence of unknown AR noise 68
4.2 Angle estimates obtained from the experimental data, corresponding
to 64 observation intervals. The solid lines denote the means and the
dashed lines denote the means plus and minus the standard deviations
of the angle estimates. The true incident angles are believed to be
ex = -33 and 02 = -36. (a) RELAX, (b) CSM-ESPRIT 68
5.1 A linear COLD array 90
5.2 Root-mean-squared errors (RMSEs) of estimates versus A6 for the first
of the two signals when 6\ = A0/2, 02 = +A0/2, aq = a2 = 45,
Pi = f32 = 0, correlation coefficient = 0.99, N 400, and SNR = 10
dB. (The CRBs for the CCD array nearly coincide with those for the
COLD array.) (a) Direction estimates, (b) Polarization estimates. . 91
IX


5.3Root-mean-squared errors (RMSEs) of estimates versus A9 for the
second of the two signals when 9\ = 50, 92 = 50 + A9, a\ =2 = 0,
Pi fa = 0, correlation coefficient = 0.99, N 400, and SNR = 10
dB. (a) Direction estimates, (b) Polarization estimates 92
5.4 Root-mean-squared errors (RMSEs) of estimates versus Aa for the
second of the two signals when 9\ = 50, 92 = 70, ot\ = 45 Aa and
a2 = 45, Pi = f32 = 0, correlation coefficient = 0.99, N = 400, and
SNR = 10 dB. (a) Direction estimates, (b) Polarization estimates. . 93
5.5 Root-mean-squared errors (RMSEs) of estimates versus source correla
tion coefficient for the first of the two signals when 91 6, 92 6,
ax = a2 45, Pi = p2 = 0, N = 400, and SNR = 10 dB. (The CRBs
for the CCD array nearly coincide with those for the COLD array.) (a)
Direction estimates, (b) Polarization estimates 94
5.6 Root-mean-squared errors (RMSEs) of estimates versus SNR for the
second of the two signals when 9\ 50, 92 70, ai a2 = 0, Pi =
P2 0, correlation coefficient = 0.99, and N = 400. (a) Direction
estimates, (b) Polarization estimates 95
5.7 Root-mean-squared errors (RMSEs) of estimates versus N for the sec
ond of the two signals when 91 = 50 and 92 = 70, cti = a2 = 45,
Pi = P2 = 0, correlation coefficient = 0.99, and SNR = 10 dB. (a)
Direction estimates, (b) Polarization estimates 96
x


5.8 Root-mean-squared errors (RMSEs) of estimates versus N for the sec
ond of the two signals in the presence of contaminated Gaussian noise
when 0i 50 and 02 = 70, eq = a2 = 45, /?i = /?2 = 0, correlation
coefficient = 0.99, and SNR = 10 dB. (a) Direction estimates, (b)
Polarization estimates 97
6.1A uniform linear COLD array Ill
6.2 Root-mean-squared errors (RMSEs) of estimates versus SNR for the
first of the two signals in the presence of white noise when 6\ = 10,
02 = 22, ctq = o2 = 0, /?i = 0, (32 = 10, correlation coefficient =
0.99, and N = 2. (a) Direction estimates, (b) Polarization estimates.
(c) Waveform estimates 112
6.3 Root-mean-squared errors (RMSEs) of estimates versus SNR for the
first of the two signals in the presence of unknown AR noise when
6i = 10, 02 = 22, aq = a2 = 0, /?i = 0, /?2 = 10, correlation co
efficient = 0.99, and N = 2. (a) Direction estimates, (b) Polarization
estimates, (c) Waveform estimates 113
xi


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EFFICIENT APPROACHES TO SOURCE LOCALIZATION
AND PARAMETER ESTIMATION
By
DUNMIN ZHENG
August 1996
Chairman: Dr. Jian Li
Major Department: Electrical and Computer Engineering
This dissertation considers the problem of source localization and parameter
estimation with antenna arrays. The problem is to estimate the parameters of the
incident electromagnetic plane waves with antenna arrays. Our focus is on array
geometry, sensor characteristics, narrowband and wideband signals, and estimation
algorithms for the estimation problem.
In particular, the RELAX algorithm, recently proposed for temporal spectral
analysis, is extended to solve the spatial problem of angle and waveform estimation for
both narrowband and wideband plane waves arriving at a uniform linear array. Unlike
Xll


most existing high resolution algorithms, the narrowband and wideband RELAX al
gorithms are robust against the presence of unknown spatially colored noise. Further,
the wideband RELAX algorithm does not need the initial angle estimates that exist
ing wideband algorithms require to construct focusing matrices, which causes biases
in angle estimates. The wideband RELAX algorithm naturally focuses the narrow-
band components in the spatial frequency domain. Both numerical and experimental
examples are used to demonstrate the performance of the RELAX algorithm and com
pare the performance of RELAX with that of other well-known algorithms including
ESPRIT with forward/backward spatial smoothing, MODE/WSF, and AP/ANPA
for narrowband signals; and CSM-ESPRIT for wideband signals. We also show that
better parameter estimates can be obtained by using RELAX as compared to using
those other algorithms by means of both numerical and experimental results.
To exploit the advantages of array geometry and antenna sensors, a polariza
tion sensitive linear array that consists of Co-centered Orthogonal Loop and Dipole
(COLD) pairs are proposed for the estimation of the parameters of completely polar
ized narrowband electromagnetic plane waves. The performance of both angle and
polarization estimation using the COLD array are shown to be greatly improved as
compared to using a crossed dipole array. A MODE algorithm is presented for both
angle and polarization estimation of correlated (including coherent) or uncorrelated
incident signals with a COLD array. Numerical example are given to show the better
estimation performance of the MODE algorithm as compared to those of the MUSIC
and NSF algothrithms. Finally, we devise a RELAX approach for angle, polarization


and waveform estimation of narrowband signals with a COLD array. We also use nu
merical examples to demonstrate the superior performance of RELAX as compared
to MODE when the additive noise is spatially colored and unknown.
xiv


CHAPTER 1
BACKGROUND AND MOTIVATION
1.1 Background
Many theoretical studies on parameter estimation with an array of sensors have
been carried out and deep insight has been achieved in the past two decades. The re
search in sensor array processing was originally motivated by its applications in source
localization and interference suppression in radar and sonar. Many algorithms have
appeared in the literature for estimating signal parameters from the measurement
output of a sensor array.
The methods in sensor array processing can be classified into two categories:
nonparametric methods and parametric methods. The nonparametric methods do
not make any assumption on the statistical properties of the data. Spatial filtering
techniques [1, 2] are the early approaches to perform a space-time processing of data
sampled at an array of sensors. Beamforming and Capons methods [3, 4, 5, 6, 7]
are the typical nonparametric methods. Their idea is to form some spectrum-like
function of the parameter(s) of interest, and then take the locations of the highest
peaks of the function as the estimates. All these approaches have an inherent limi
tation of poor resolution. They are usually used in situations where the information
about the statistical properties are not available. However, spatial filtering methods
with an increasing number of novel applications inspired much of the subsequent
1


2
efforts in statistical signal processing. The well-known Maximum Entropy (ME)
spectral estimation method in geophysics by Burg [8] and the Yule-Walker autore
gressive estimation method are the early parametric approaches. The introduction of
subspace-based techniques [9, 10, 11] provided a new geometric interpretation for the
sensor array processing problem. The vector space formulation of the sensor array
problem resulted in a large number of algorithms [12, 13, 14, 15]. The Maximum
Likelihood (ML) parameter estimates can also be derived for the sensor array prob
lem in an appropriate statistical framework. ML estimation is a systematic approach
to many parameter estimation problems, and has been studied by many researchers
[16, 17, 18, 19, 20, 21, 22]. Unfortunately, the ML method usually requires a multidi
mensional non-linear optimization search at a considerable computational complexity.
Reduced computational cost is generally achieved by the use of a suboptimal esti
mator. Much work on devising a family of suboptimal estimators and the analysis
of the performance of the estimators has been done [23, 24, 25, 26, 27, 28]. Most
of these algorithms were motivated by the subspace based technique introduced in
the MUSIC algorithm. The subspace-based approach results in a resolution that is
not limited by the array aperture, provided that the number of data samples or the
signal-to-noise ratio (SNR) is sufficient large. A review of the development of array
signal processing algorithms will given in the next chapter.
1.2 Motivation and Contributions
Array signal processing was centered on the ability to fuse data from data acqui
sition systems, such as an array of sensors, to carry out a given estimation task. Many


3
sophisticated estimation algorithms were the results of an attempt of researchers to
go beyond the classical Fourier-limit. However, many high resolution array process
ing algorithms are usually very sensitive to the presence of unknown spatially colored
noise. Thus the performance of those algorithm is frequently very poor when used
in practical applications. To derive more robust algorithms, we need to relax the
additive white noise assumption made by most existing high resolution algorithms.
We propose a RELAX (RELAXation algorithm for the minimization of the NLS
criterion) algorithm for angle and waveform estimation of narrowband plane waves
arriving at a uniform linear array. The RELAX algorithm is robust against the
presence of unknown spatially colored noise. Since most of the algorithms in array
processing are devised only for narrowband signals, the algorithm development for
wideband signals has only received some limited attention. Although some subspace-
based methods [29, 30] were introduced for the wideband case, most of them use
focusing matrices, which may result in biased estimates. We propose a wideband
RELAX algorithm for angle estimation of wideband plane waves arriving at a uniform
linear array. Unlike other wideband algorithms, the wideband RELAX does not
need initial angle estimates and naturally focuses the narrowband components in the
spatial frequency domain.
It is also important to take advantage of array geometries and receiving proper
ties of antenna elements. Although many algorithms have been developed for array
signal processing recently, the characteristics of specific antenna sensors are only


4
beginning to attract more attention. Previous work on angle and polarization es
timation using crossed dipoles [31, 32] and orthogonal dipoles and loops [33] are
examples of using specific antenna sensors to estimate the angles and polarizations
of incident narrowband electromagnetic plane waves. To exploit the advantages of
array geometry and antenna sensors, a polarization sensitive linear array that con
sists of Co-centered Orthogonal Loop and Dipole (COLD) pairs are investigated for
the problem of parameter estimation of completely polarized electromagnetic (EM)
plane waves. The performance of both angle and polarization estimation using the
COLD array are shown to be greatly improved as compared to using a crossed dipole
array. We propose an efficient MODE (Method Of Direction Estimation) algorithm
for both angle and polarization estimation of correlated (including coherent) or un
correlated completely polarized narrowband signals with a COLD array. We also
devise a RELAX algorithm that can be used with a uniform linear COLD array
for angle, polarization and waveform estimation of completely polarized narrowband
plane waves.
The high resolution parameter estimation approaches we develop in this disser
tation can be used for source localization in radar and sonar as long as incident waves
can be modeled as having discrete angles of arrival. For example, in radar applica
tions, the radar returns of targets impinging on the antenna array of a radar receiver
enable estimation of the directions of targets with these high resolution parameter
estimation algorithms. The parameters of interest include angles-of-arrival (AOAs),
waveforms, and polarization states of the incident signals.


5
1.3 Outline of the Dissertation
Chapter 2 provides the background material for the remainder of the disserta
tion. The material includes brief reviews of electromagnetic wave propagation and
polarization, and the sensor array processing algorithms. This chapter also presents
the fundamental sensor array data models for the dissertation.
Chapter3 describes how the RELAX algorithm can be used for angle and wave
form estimation of narrowband signals with a uniform linear array for the case of
multiple snapshots. To evaluate the performance of the RELAX algorithm, we ap
ply it to both simulated and experimental data, and compare its performance with
that of other well-known algorithms including ESPRIT (Estimation of Signal Parame
ters via Rotational Invariance Techniques) with forward/backward spatial smoothing,
MODE/WSF (Weighted Subspace Fitting), and AP (Alternating Projection)/ANPA
(Alternating Notch-Periodogram Algorithm) for narrowband signals.
In Chapter 4 we extend the RELAX algorithm to the case of wideband sources
and also present numerical and experimental examples to illustrate the performance
of the wideband RELAX algorithm and compare its performance with that of CSM
(Coherent Signal-subspace Method) based ESPRIT.
In Chapter 5 an arbitrary linear array that consists of Co-centered Orthogonal
Loop and Dipole (COLD) pairs is proposed for the problem of statistically efficient es
timation of the parameters of completely polarized electromagnetic waves. A MODE


6
algorithm is presented for both angle and polarization estimation of correlated (in
cluding coherent) or uncorrelated incident signals with a COLD array. The perfor
mance of both angle and polarization estimation using the COLD array are shown to
be greatly improved as compared to using a co-centered crossed dipole (CCD) array.
Several numerical examples are presented to compare the MODE algorithm with the
MUSIC algorithm and the NSF algorithm for both angle and polarization estimation.
The Cramer-Rao bound (CRB) for the COLD array is also compared with that for
the CCD array.
Chapter 6 presents a RELAX approach for parameter estimation of narrowband
signals arriving at a uniform linear COLD array. The statistical performance of
this RELAX estimator is compared with that of the MODE estimator described in
Chapter 5 via numerical examples.
Finally, Chapter 7 gives the conclusions and future work.
Parts of the original work presented in this dissertation have already been doc
umented in the publications [34, 35, 36, 37].


CHAPTER 2
WAVE PROPERTIES AND SENSOR ARRAY PROCESSING
2.1 Introduction
In this chapter we provide the groundwork necessary for the material to be de
veloped in the subsequent chapters. After briefly reviewing the important properties
of electromagnetic (EM) waves including wave propagation and wave polarization, we
introduce the central problem of the dissertation: the problem of locating radiating
sources by using an array of passive sensors. Once the data model for the output
signal of the receiving sensor array is formed, the source location problem is turned
into a parameter estimation problem. The estimation problem has been investigated
under both nonparametric and parametric approaches.
2.2 Propagation of EM Waves
Electromagnetic wave propagation is referred to the phenomena that a time
changing electric field produces a time-varying magnetic field, which in turn generates
an electric field, and so on with a resulting propagation of energy. The direction of
the electric field E and magnetic field H are everywhere perpendicular. The most
important and most fundamental electromagnetic waves are the transverse plane
waves. In a plane wave E and H lie in a plane. A wave of this type with both E and
H transverse to the direction of propagation is called Transverse ElectroMagnetic
(TEM) wave. In a medium with spatially constant permeability and permittivity
7


8
t and with no free charges and currents, the Maxwells equations are
VB = 0,
(2.1)
VE = 0,
(2.2)
<9H
V*E =
(2.3)
<9E
VxH = ear
(2.4)
By combining the two curl equations and making use of vanishing divergences, we
can find easily that each cartesian component of E and H satisfies the wave equation:
V2u-//e^ = . (2.5)
The wave equation has the well-known plane wave solutions
ti(,x) = e', (2.6)
where the temporal frequency u> and the magnitude of the wave vector k are related
by
k = Uly/Jii, (2.7)
and x is the position vector.
Clearly, the phase variation of the wave signal u(t, x) includes both temporal and
spatial variations. Spatial far-field receiving conditions, which implies plane waves,
are assumed throughout the dissertation. The superposition principle is valid if more


9
than one wave travel through a linear medium. Usually these propagating waves
carry information from their sources. The information may include source related
signal parameters, such as angle-of-arrival (AOA), signal waveform, signal polariza
tion state, propagation delay, etc. These signal parameters are very important in
many applications. For source localization, AOA, signal polarization state and wave
form are of special interest. The efficient estimation of these parameters, which is
the essence of sensor array signal processing, will be the topic of our main interest in
this dissertation.
2.3 Polarization of EM Waves
Polarization describes the orientation of the electric field of a wave. It is advan
tageous to employ an array of diversely polarized antennas since multiple signals can
be resolved on the basis of polarization as well as AOA. To specify the polarization
of the waves, we consider sinusoidal waves of the same frequency. In a plane wave
traveling along the positive 2 direction, the electric field generally has both x and y
components. The general expression for the electric field of such a wave is then given
by
E = (Elx + E2ejry)e~jkz, (2.8)
where Ei and E2 are real and 7/ ( 180 < ij < +180) is the phase angle between x
and y components. The corresponding magnetic field is
{-E2ex + Ex y)e~jkz.
H =
(2.9)


10
The phase and relative amplitudes E\ and E2 determine the state of polarization. In
the most general case of elliptical polarization the polarization ellipse described by E\
and E2e^v, as time progresses, may have any orientation as shown in Fig. 2.1 The
line segment OA is the semimajor axis, and the line segment OB is the semiminor
axis. The axial ratio is
AR =
OA
OB'
(1 < AR < oo).
(2.10)
Figure 2.1. Polarization ellipse.
We define f3 to be the tilt angle of the ellipse (0 < /? < 180), and a to be the
ellipticity angle, which is given by
a = tan_1(=FAR), (-45 < a <+45).
(2.11)


11
The ellipticity angle a is negative for right-handed and positive for left-handed po
larization. For the case shown in Fig. 2.1a is positive. The parameter 7 is defined
as
7 = tan-'(|i), (O'<7 < 90). (2.12)
The geometric relation of a, f3 and 7 to the polarization ellipse is illustrated in Fig.
2.1. The trigonometric interrelations of a, (3,7 and 7 are given by [38, 39]
COS 27
= cos 2a cos 2/3,
(2.13)
tan 7
tan 2a
sin 2/3
= tan 27 cos 7,
(2.14)
tan 2/3
(2.15)
sin 2a
= sin 27 sin 7.
(2.16)
The Poincare sphere representation of wave polarization [40] in Fig. 2.2 clearly shows
the relationship among the four angular variables a, ¡3,7 and 7. The polarization state
is described by a point on a sphere where the longitude and latitude of the point are
related to parameters of the polarization ellipse as:
Longitude
= 2/?,
(2.17)
Latitude
= 2a.
(2.18)
The polarization state described by a point on a sphere can also be expressed
in terms of the angle subtended by the great circle drawn from a reference point on


12
Figure 2.2. Poincare sphere.
the equator and angle between the great circle and the equator as:
Great-circle angle = 2q, (2.19)
Equator-to-great-circle angle = rj. (2.20)
Thus, it is convenient to describe the polarization state by either of the two sets
of angles (a, ¡3) or (7,7/). The case when a 0 corresponds to linear polarization.
The case when a = 45 corresponds to circular polarization, with left circular
polarization (cc = +45) at the upper pole. One octant of the Poincare sphere and
polarization states at specific points are shown in Fig. 2.3. In the general case any
point on the upper hemisphere describes a left elliptically polarized wave ranging


13
from pure left circular at the pole to linear at the equator. Likewise, any point on
the lower hemisphere describes a right elliptically polarized wave ranging from pure
right circular at the pole to linear at the equator.
EM waves may be either completely polarized or partially polarized, with the
former being the most common. A completely polarized EM wave is a special case of
a more general type of EM wave, i.e., a partially polarized EM wave. In other words,
the polarization state of a partially polarized EM wave is a function of time while a
completely polarized wave has a fixed state of polarization. In practical applications
such as radar and ionospheric radio [41, 42], the state of polarization of a returning
wave received by a radar with polarization diversity can vary even though the original
transmitted wave is completely polarized.


14
2.4 Narrowband Signals
2.4.1 Array Data Model and Problem Formulation
A general two-dimensional isotropic sensor array system is shown in Figure 2.4.
The wave field of the sources travels through space and is sampled, in both space
and time, by the sensor array. Assume that the array is planar, each of which has
coordinate rj = (x¡,y¡) and an impulse response
hi(t,rt) = at(6)S(t)S(ri), (2.21)
Y
Source 1
Source 2
Sensor Array
X
Figure 2.4. An arbitrary two-dimensional array.
For K signals impinging on an array of L sensors, we can define an L x K
impulse matrix H(f, 0) from the impinging emitter signals with parameter 6 =
[ 0.2_ ... f)K], to the sensor outputs. The Ikth element of H(i,0) takes the


15
form
H¡k(t, 9) = a:(6k)8(t)6(r,), I = 1, , L, k = 1,2, , K. (2.22)
From a convolution operation, the sensor outputs can be written as
k=1
(2.23)
for purely exponential signals and
y/(<) = E o/(^fc)ej(u,-k*r|)5jt() (2.24)
k=l
for narrowband signals, where is the common center frequency of the signals and
sk(t) represents the complex amplitude (waveform) of the fcth signal. The carrier eJwi
is usually removed from the sensor output before sampling.
It is clear that the geometry of a given array determines the relative delays
of the various angles-of-arrival (AOA). The formulation can be straightforwardly
extended to arrays where additional dimensions provide the flexibility for more signal
parameters per source, such as a polarization sensitive sensor array for both AOA
and polarization estimation. We will introduce a polarization sensitive sensor array
later in this dissertation.
The received signal plus noise gives the outputs of the sensor array in the form
of
y(0 =
a(0O, a {62), , a(9K)
s() + n (t) = As (t) + n(i),
(2.25)


16
where
a(Ok) =
n T
ai{Ok)e
a2(0fc)e
Jkfcr2
jkfcri
(2.26)
s(i)
T
Si(t), S2(t),---, SK(t)
(2.27)
and n(i) is the noise vector. The vector a{9k) is referred to as the kth array propa
gation steering vector. It describes via kf r; how a plane wave impinges at the array
(i.e. AOA) and via a¡{9k) how the sensors affect the signal amplitude and phase.
For polarization sensitive sensors, the a¡(9k) is related to the polarization state of
the incident electromagnetic plane wave. For omnidirectional isotropic sensors, the
a/(dfc) is a constant. We consider both isotropic sensor arrays (Chapters 3 and 4)
and polarization sensitive sensor arrays (Chapters 5 and 6) in this dissertation. The
reviews of previous work presented below are for an isotropic sensor array unless we
point out a polarization sensitive array.
For a uniform linear array (ULA) shown in Figure 2.5, the steering vector a(£4)
has the form
a(9k) =
where
= isin(0fc) (2.29)
c
1
1,
J(L-l)Qk
(2.28)
is called the spatial frequency.


17
Figure 2.5. A uniform linear array.
We can see from (2.29) that the vector a(0jt) is uniquely defined if and only if
fIk is constrained as < tt. The condition is satisfied if
6
(2.30)
The collection of these steering vector over the parameter space of interest
A = {a(0*r)|0jfc e 0}
(2.31)
is often called the array manifold. The parameterization of A is assumed known. Let
us define the set AK as the collection of all distinct array manifold vectors
AK = {A|A = [ a(^) ... a(0x) ]A < 92 < < 0K}.
(2.32)


18
Hence, AK is parameterized by the parameter vector 6 [ qk }t. The
array is assumed to be unambiguous. In other words, any A G Ah has full rank.
The sensor outputs are appropriately sampled at t = 1,2, , N time instances
and these snapshots y(l), y(2), , y(N) can be viewed as a multichannel ran
dom process, which is assumed Gaussian in this dissertation. The characteristics of
Gaussian processes can be well understood from its first and second order statistics
determined by the underlying signals as well as noise. The problem of central interest
for source localization is to estimate the AOAs, waveforms (and polarization states
if a polarization sensitive array is used) of emitter signals impinging on a receiving
array when a set of sample data {y(l), y(2), , y(N)} is given.
We first make some assumptions for the additive noise n(f) and signal waveforms
s(t). The noise vector n(t) is assumed to be a stationary, temporally white, zero-mean
and circularly symmetric with unknown covariance matrix Q:
E{n(i)nH(2)} = Q Stl,ta, (2.33)
E{n(f1)nT(<2)} = 0, (2.34)
where Stl and (-)r denotes the transpose. Note that the problem of angle estimation is ill-
defined for an arbitrary noise field without knowledge of the signal waveform. The
eigenstructure-based estimation methods assume the case of spatially white noise,
i.e., Q = cr2I.


19
The signal waveform s(t) is assumed to be deterministic unknown or random.
For the latter case, the source covariance matrix is defined as
R, = E{s{t)sH(t)}. (2.35)
For the former case,
R- = > (*)"() (2-36)
/V t=\
For the case of spatially white noise, the array covariance matrix has the form
R = E{y(t)yH{t)} = ARSAH+ a2I. (2.37)
The eigendecomposition of R results in the representation
L
R = £ A,e,e? = E,AsEf + EnAnE* (2.38)
t=i
where Ai > > Ak > Aft-+i = XL = a2. The matrix Es = [ ej5 ... ? eR. ]
contains the K eigenvectors corresponding to the largest eigenvalues. The range space
of Es is called the signal subspace. Its orthogonal complement is the noise subspace
and is spanned by the columns of En = [ eA-+1, . ]
The eigendecomposion of the sample covariance matrix R is given by
R = 4 yy"W = ..f + ".
/v t=i
(2.39)


20
The number of signals K is assumed known throughout the dissertation. Signal
enumeration methods can be found in, e.g., [43, 44, 45, 46, 47, 48].
In the next two subsections, we introduce the most well known estimation tech
niques, classified as nonparametric and parametric methods.
2.4.2 Nonparametric Methods
The nonparametric methods do not make any assumption on the statistical
properties of the data. The fundamental idea of the nonparametric methods is to
form some spectrum-like function of the parameter(s) of interest, and then take the
location of the highest peaks of the function as the estimates.
The beamforming is to steer the array in one direction at a time and measure
the array output power. The locations of maximum power yield the AOA estimates.
The array is steered by forming a linear combination of the sensor outputs
yF(t) = Y w*iy<(t) = wHy(0- (2-4)
1=1
Given {?/f(£)}1i? ^e output power is measured by
T7 Y MOI2 = T7 Y Hy(t)yH(t) = w"Rw. (2.41)
iV t=i /v t=1
Different choices of weighting vector w leads to different beamforming approaches.
There are two types of beamformers.


21
Conventional Beamformer
The weighting vector for conventional beamformer is given by
w bf = a(0),
(2.42)
which can be interpreted as a spatial filter matched to the impinging signal.
Substituting the weighting vector (2.42) into (2.41), the beamforming AOA
estimates are given by the locations of the largest peaks of the spatial spectrum
PBF{6 = aH(0)Ra(9). (2.43)
Clearly, for a ULA, the beamforming is a spatial extension of the classical peri-
odogram in temporal time series analysis [7]. Thus, the spatial spectrum suffers
from the same resolution problem as does the periodogram. This sets a limit on the
resolution achievable by beamforming, which is 27r/T for a ULA.
Capons Beamformer
A well-known method was proposed by Capon [3], whose beamformer attempts
to minimize the power received from noise and any signals coming from other direc
tions than 0, while maintaining a fixed gain in the direction 9. The Capon spatial
filter design problem is posed as
minw^Rw subject to wH a(9) 1.
(2.44)


22
The optimal w is given by
w cp
R-'ajO)
a^(0)R-1a(0)
(2.45)
which leads to the following spatial spectrum upon insertion into (2.41)
Pep
1
a^R-^fl)'
(2.46)
Thus, the Capon AOA estimates are obtained by the locations of the largest peaks
of the spatial spectrum given by (2.46).
Capons beamformer outperforms the conventional one because the former uses
every single degree of freedom to concentrate the received energy along the direc
tion of interest. However, the resolution capability of the capon beamformer is still
dependent on the array aperture. Since nonparametric methods do not assume any
thing about the statistical properties of the data, they can be used in situations
where we lack information about these properties. Several alternative methods for
beamforming have been proposed for addressing various issues [4, 5, 6].
2.4.3 Parametric Methods
When some of the statistical information of the data is available, the use of
a nonparametric approach is often associated with a degradation in performance
as compared to the use of model-based parametric approach. The most important
model-based approach is the Maximum Likelihood (ML) method. The ML estimates
are those values of the unknown parameters that maximize the likelihood function.


23
There are two different ML approaches in the sensor array problem, depending on
the model assumption about the signal waveform. The unknown, deterministic signal
model leads to the deterministic ML method [18, 17, 49, 50, 51], while the Gaus
sian random signal model results in the stochastic ML method [19, 52, 53]. Al
though the ML approaches are optimal to the sensor array problem, these techniques
are often deemed exceedingly complex. For the case of spatially white noise, the
subspace-based techniques provide many high resolution and computationally effi
cient algorithms. These algorithms include MUSIC [10, 11] and its modified versions
[54, 55, 56, 57], and ESPRIT [58, 59] as well as root-MUSIC [60, 61] for a uniform
linear array. The subspace-based techniques also lead to the subspace-based ML ap
proximation approaches, which include Signal Subspace Fitting (SSF) [23, 62, 63]
and related, yet more efficient Method Of Direction Estimation (MODE) [25, 26], as
well as Noise Subspace Fitting (NSF) [25, 64, 65]. Compared to the exact ML ap
proach, the subspace-based ML approximation approaches are computationally more
attractive. However, the subspace-based ML approximation approaches are usually
sensitive to the violation of white Gaussian noise assumption, and the number of
incident signals. In the next three subsections we will briefly review these important
parametric methods.
ML Methods
The ML methods include the deterministic ML and stochastic ML.
Deterministic ML


24
The deterministic ML method maximizes the conditional likelihood function
(given the signal waveforms and the incident angles) of the data. It is also called
the nonlinear least-squares (NLS) method. This leads to the minimization of the
following function [18, 17]:
NLF = [y(n) A(0)s(n)]w[y(n) A(0)s(n)].
n=1
(2.47)
Minimizing the cost function with respect to s(l), s(2), , s(N) gives
s{t) = (AHA)-1AHy(t), t = 1,2, ,N. (2.48)
By substituting (2.48) in (2.47), we get
8 = Ell{I-A(A'A)-1A)y(t)||J (2.49)
iV <=1
= argmaxTr{A(A/A)-1AR}, (2.50)
0
where R is the sample covariance matrix.
Note that a nonlinear A'-dimensional optimization problem must be solved for
the DML estimator. A good initial estimate is important to guarantee the desired
global minimum since the criterion function often possesses a large number of local
minima. A spectral-based method is a natural choice for an initial estimator provided
all sources can be resolved. The alternating projection (AP) [66] and alternating
notch-periodogram algorithm (ANPA) [67] are two similar efficient approaches to


25
minimize the nonlinear least-squares criterion. The IQML algorithm [49, 50, 51]
is an iterative procedure to minimize the deterministic ML criterion in (2.50) for
a uniform linear array. The RELAX algorithm presented in the next chapter is a
FFT-based method to minimize the NLS criterion in (2.49).
Stochastic ML
The Gaussian signal model assumption not only is a way to obtain a tractable
ML method, but also is often motivated by the Central limit Theorem. Assume the
signal waveforms are zero-mean with second-order properties
E{s(i)s'(2)} = R.ii,,,,, E{s((i)sr(i2)} = 0.
(2.51)
Under the assumption, the observation process y(t) is a white, stationary, zero-mean
and circularly symmetric Gaussian random vector with covariance matrix
R = A(0)R3AN(0) + (2.52)
The negative log-likelihood function (ignoring constant terms) of the complete
data set y(l), , y(N) is given by
l{6, Rs, o2) = log|R| + tr{R_1R}.
(2.53)
Minimization with respect to a2 and Rs leads to the following expressions:
R,(fl) = A^(R (t2(6)1)A^h
(2.54)


26
= J^trPiR}. (2.55)
where
At = (AhA)~1Ah, (2.56)
= I A [AHA) _1 Ah. (2.57)
Substituting (2.54) and (2.55) into (2.53) results in the following compact form
of the criterion:
0 = arg min {log |AR,(0)Aff + 0
The dimension of the parameter space is reduced substantially for the SML
estimator. However, the criterion function in (2.58) is a highly non-linear function of
6 and the minimization 6 cannot be found analytically in general. A Newton-type
implementation of numerical search may result in a very good accuracy when the
global minimum is attained. In fact the stochastic ML estimator has been shown
to have a better large sample accuracy than the corresponding deterministic ML
estimator [20, 62] regardless of the actual distribution of the signal waveforms.
Subspace-Based Methods
The tremendous interest in the subspace approach is mainly the result of the in
troduction of the Multiple Signal Classification (MUSIC) algorithm [68, 60], although
Pisarenkos work (a special case of MUSIC) [69] in time series analysis was published
in early 70s. These methods assume that the additive noise is either spatially white


27
or its covariance matrix is known. Our discussions on the subspace-based methods
below assume that the noise is spatially white.
MUSIC
The MUSIC algorithm is based on the fact that the noise eigenvectors in En are
orthogonal to A, i.e.,
E?a(0) = O, 6e{01, ,#*}. (2.59)
For unique AOA estimates, the array is usually assumed to be unambiguous. In
particular, a ULA is unambiguous if 90 < 6 < 90 and d < A/2. In practice, the
MUSIC spatial spectrum is defined as
a"(0)EnE"a (0)'
The MUSIC AOA estimates are given by the locations of the largest peaks of
the spatial spectrum in (2.60) provided EE^ is close to EnE^.
In contrast to the beamforming techniques, the MUSIC algorithm provides es
timates of an arbitrary accuracy if the data samples are sufficiently large or the SNR
(signal-to-noise ratio) is adequately high. A number of modifications of MUSIC have
been proposed to improve or overcome some of its shortcoming in various specific
scenarios. The Min-Norm algorithm [54, 55, 56, 57], a weighted MUSIC, exhibits a
better resolution than the original MUSIC algorithm [70].
ESPRIT


28
A uniform linear array steering matrix has the structure of a Vandermonde
matrix as follow
A =
pj^K
(2.61)
ej(L-l)Hi . ej(L-l)QK
Making use of this special structure results in several computationally and statis
tically efficient subspace-based algorithms. This type of algorithm may include
ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques)
[58, 59], and Root-MUSIC [60] that will be introduced later.
The ESPRIT algorithm exploits a so-called shift structure of the Vandermonde
matrix A. let Ai and A2 be the sub-matrices by deleting the first and last rows from
A respectively. Then, Ai and A2 are related by
A2 = A!$, (2.62)
where
$ = diag{ejl,ejV--,ej1*}. (2.63)
Consider the structure of the eigendecomposition of the array covariance matrix R
given in (2.38). If the signal covariance matrix Rs has rank K', then the matrix Es
will span a A' -dimensional subspace of A. This observation implies that there exists


29
a full-rank K X K' matrix such that
Es = AT. (2.64)
Deleting the first and last rows of (2.64), respectively, gives
Esl = AiT, Es2 = A2T. (2.65)
Combining (2.62) and (2.65) leads to
Es2 = Esltf, (2.66)
where = T-1$T. Clearly, and $ have the same eigenvalues. The eigenvalues
are given by e^k, k = 1,2, , K, which are related to AOAs. There are two different
ESPRIT algorithms, depending on how to approximate the relation
s2 = Esl#. (2.67)
Solving the approximation relation (2.67) in a least-squares sense results in LS-
ESPRIT, while in a Total-Least-Squares sense leads to TLS-ESPRIT.
Root-MUSIC


30
The idea of the Root-MUSIC dates back to Pisarenkos method [69]. If the
signals covariance matrix has full rank, then the polynomial
p,(z) = ef a(*), l = K + l,---,L, (2.68)
where a(z) = [1, z, , zi-1]r, has K zero-roots at e-'0*, k 1,2,---, A'. Consider
the zeros of the MUSIC function
||a(2)||2 = aff(z)a(z). (2.69)
The search for zeros is complicated since the function is not a polynomial in z. In
fact, we are only interested in values of z on the unit circle. This suggests that we
can use aT{z~1) to substitute for aH(z), which gives the Root-MUSIC polynomial
p(z) = iaT(z-)a(z). (2.70)
The polynomial p(z) has 2K roots with K roots inside the unit circle and K mirrored
images outside the unit circle. Among those inside, the phases of the K closest to
the unit circle gives the AOA estimates.
The Root-MUSIC has empirically been found to have a significant better per
formance than MUSIC in the small sample case according to [28, 71].
Coherent Signals


31
For highly correlated or coherent signals, a rank deficiency occurs in the source
covariance matrix Rs. This results in a divergence of a signal eigenvector into the
noise subspace. Thus the property (2.59) no longer holds and the subspace method
fails to yield consistent estimates. The forward-backward (FB) averaging and spatial
smoothing techniques can be used with subspace methods for ULAs in the limiting
case of coherent signals [6, 72, 44, 73]. The idea of the FB averaging is to form
a so-called backward array covariance matrix Rb and then average the usual array
covariance R and Rb- The backward array covariance matrix has the form
Rb = JR*J = A$(L-1)R,$-(L"1)A,i + where J is an L x L permutation matrix, whose components are zero excepts for ones
on the anti-diagonal. Then the FB array covariance matrix is obtained by
Rbb = (R + Rb) = ARjA^ + where Rs = (Rs + ^L~1'>Rs^L~^)/2 has full rank.
The idea of the spatial smoothing technique is to split the ULA into a number
of overlapping subarrays. The outputs of the subarrays can therefore be averaged
for computing the covariance matrix. Since the spatial smoothing induces a random
phase modulation similar to (2.72), the signals are decorrelated and results in a full
rank covariance matrix.


32
Subspace-Based ML Approximations
Subspace-based methods offer significant performance improvement in compar
ison to conventional beamforming methods. Although the MUSIC method yields
estimates with the same large sample accuracy as that of the deterministic ML
method provided the signals are uncorrelated [20], it usually suffer from a large bias
for finite samples and is unable to cope with coherent signals. Several parametric
subspace-based methods that practically have the same statistical performance as
the ML method but less computational complexity than ML have been proposed
[25, 26, 23, 63].
Signal Subspace Fitting
Recall the relation in (2.64). Since 9 and T are unknown, it appears there are
no values of 9 such that Es = AT when only an estimate Es of Es is available. A
natural solution is to find the best weighted least-squares fit of the two subspaces by
the following criterion:
{0, T} = arg min ||ES AT||w! (2.73)
9, T
where ||A||yv = tr{AWAf/} and W is a K' x K' positive definite weighting matrix.
This is a separable nonlinear least-squares problem [74]. Substituting the pseudo
inverse solution, T = A^Eg, into (2.73) leads to the separated criterion function
9ssf arg minTr{P^sW^}.
9
(2.74)


33
Different choices of W lead to different methods. The choice W = I yields Cadzows
method [75]. A statistical analysis suggests other weightings, such as
W = (As f2I)2
-i
S ?
(2.75)
leading to superior performance [25, 23, 76]. The estimator defined by (2.74) with
weights given by (2.75) is called the MODE/WSF(Weighted Subspace Fitting) method
For a ULA, the MODE algorithm [25, 26] can be implemented efficiently. The
idea is to reparameterize the projection matrix using a basis for the nullspace
of Alf. The projector above may be reparameterized in terms of the coefficients
b = [ 60 ... f,K ]r of a polynomial defined as
bkzK~k = b0 I] (z e^) ; b0 0. (2.76)
k=0 k=l
Let B/y be the following (L K) x L matrix
Bh -
bw
W
bo
i
bo
(2.77)
Then P^ = Pg, where Pjj denotes the orthogonal projector onto the range space of
B. Thus the reparameterized SSF criterion for MODE is given by
b = arg minTr{B(B/B)-1B/sWf}.
b
(2.78)


34
Clearly, the resulting non-iterative procedure includes:
1)Determine b by minimizing (2.78) with (B/7B) 1 set to I;
2)Determine an asymptotically efficient estimate b as
b = arg min Tr{B(HB) lBHsW},
b
(2.79)
where (B^B) is formed by the initial estimate of b obtained in Step 1;
3)Compute the roots of the estimated polynomial.
This MODE algorithm for a ULA is attractive because of the fact that it is
computationally and asymptotically efficient and it remains so for highly correlated
or coherent signals. Two-dimensional spatial frequency and angle estimation using
MODE has been presented in [46, 77].
Noise Subspace Fitting
Inspired by MUSIC, if signal covariance Rs has full rank, then the following
relation holds:
(2.80)
Given an estimate of En, a natural estimate of 6 can be obtained by minimizing
the following Noise Subspace Fitting (NSF) criterion:
@nsf = argimntr{UAHnf A},
0
(2.81)
where U is a K x K positive semi-definite weighting matrix. Different choices of
weighting matrix U result in signal parameter estimates with different asymptotic


35
properties. It has been shown in [76] that the estimates calculated by (2.81) and
(2.74) are asymptotically equivalent if the weighting matrix is given by
U = A/WfAK (2.82)
Compared with SSF, however, NSF has some inherent disadvantages, namely that
it may not cope with coherent signals, and a two-step procedure has to be adopted
because the weighting matrix depends on 6.
The parametric high resolution algorithms introduced in this subsection are very
sensitive to the presence of unknown spatially colored noise. The performance of these
algorithms is usually very poor when used in practical applications. To derive more
robust algorithms, we may need to relax the spatially white noise assumption made
by these algorithms. A robust RELAX algorithm will be proposed in Chapter 3 for
angle and waveform estimation of narrowband plane waves arriving at a ULA. The
RELAX algorithm does not assume that the noise is spatially white.
2.5 Wideband Signals
The methods introduced above are essentially limited to narrowband signals. For
wideband signals, the delay spread is close to 1 /BW, where BW is the bandwidth of
signals. The wideband signal is usually decomposed into narrowband components by
filtering or DFT (Discrete Fourier Transform), and then somehow combine the differ
ent frequency bins before using the narrowband algorithms. The question of how to
best combine the information spread across different frequencies has received much


36
attention. The algorithms proposed in [78, 79] process each frequency bin separately
by using MUSIC, and then averages the spatial spectrums, whereas [80] proposes
to use ESPRIT. These methods are incoherent in the sense that the directional in
formation is spread over a wideband of frequencies, and each frequency is processed
separately followed by some statistical average of the results. This average of the
results tends to be ineffective at low SNR and thus the resolution of these methods
tends to be limited by the resolution of a single frequency bin. A number of focusing
algorithms have been proposed including coherent signal subspace methods (CSM)
[29, 30], and the spatial resampling method [81]. These coherent methods are an
alternative to the incoherent methods. These methods preprocess the data, focusing
the different frequency components to a signal reference frequency, removing the fre
quency dependence of the data. Other wideband AOA estimation algorithms have
been developed in [82, 83].
We now briefly review the wideband array data model and the concept of CSM.
Assume that K incident wideband signals impinge at a uniform linear array with L
sensors. The data model in frequency domain has the form
y (/) = A(0, /)r(/B) + N(/), n = 1,2, , J, (2.83)
where y(/n) denotes the I x 1 data vector of the n-th frequency bin, A(#,/n) =
[ a{6\,fn) a(0K.fn) ] is a L X K matrix, whose A:-th column is the direction


37
vector for the -th signal, and r(/) = [ Ti(fn) T^(fn) ]Ti 1S ^ x 1 signal
vector, and N(/) = [ N\(fn) Ni(fn) ]T> is i x 1 noise vector.
The resulting spatial covariance matrix is
R(/n) = A(0, /)Ra(/n) A*(0, fn) + ^(/)Rn(/), (2.84)
where Rs(/n) and Rn(/n) are the signal covariance matrix and the noise covariance
matrix, respectively. The most important question here is how to combine the in
formation from different frequency bins. The idea of the coherent signal-subspace
method (CSM) [29, 30] is to use transformation matrices to focus the energy of the
wideband signals to a single reference frequency. The focusing removes the frequency
dependency of the data. These focusing transformation matrices T(/n) are designed
to align the signal-subspace at all frequencies with the signal subspace at the reference
frequency /0,
T(/n) A(0, /) A(0, /), n = 1, , J. (2.85)
Thus, a coherent averaging of the covariance matrices R(/n), n = 1, , J, may be
performed as below
R. = 7 E T(/)R(/)T"(/)
J n=l
w A(6J0)RsAH(OJ0) + Rny
(2.86)


38
where
1 J
Rs = 7 E Rs(fn), (2.87)
J n=l
and
R, = 4 E ^(/n)T(/)R(/n)T"(/). (2.88)
J n=1
Hence any narrowband subspace-based algorithms may be applied to R;/ after the
coherent averaging is performed.
It is apparent from (2.85) that the transformation matrix depends on the un
known AO As 6. To form the focusing transformation matrices, one needs to have
preliminary estimates of AOAs. The preliminary estimates may be obtained by other
methods, such as, FFT or Capons method [29, 30]. A major disadvantage of CSM
is that the focusing may cause biased estimates, depending on the accuracy of the
preliminary estimates. We will proposed a wideband RELAX algorithm for angle
estimation of wideband plane waves arriving at a uniform linear array. The wide
band RELAX does not need initial estimates as other wideband algorithms require
and naturally focuses the narrowband components in the spatial frequency domain.
Thus the wideband RELAX can avoid the bias problem encountered by CSM that
uses focusing matrices.
2.6 Polarization Diversity
We have considered an array of omnidirectional isotropic sensors so far. When
a polarization sensitive array is used, the output of sensor array is related to the
states of polarization of the incident plane waves. In other words, a¡(#fc) in (2.26) is


39
no longer constant, it depends on the polarization property of antenna sensors and
incident EM plane waves. The advantages of utilizing polarization sensitive arrays
have been discussed previously [10, 84, 85, 31, 86, 87, 33, 65, 88, 89, 90, 91]. In
[10, 84] the MUSIC algorithm was used for direction finding with a diversely polarized
antenna array. In [92, 31, 86, 87] the methods of angle and polarization estimation
using ESPRIT with a crossed dipole array were presented. A noise subspace fitting
method with diversely polarized antenna arrays was proposed in [65]. The discussion
on the performance analysis of diversely polarized antenna array can be found in
[88, 89]. A ML parameter estimator with a crossed dipole array was proposed in
[93] for partially polarized EM waves. However, there are many shortcomings in the
previous work. For example, the noise subspace fitting and MUSIC cannot work
well for coherent signals, the performance of the crossed dipole array is sensitive to
AOAs. New approaches will be proposed in Chapters 5 and 6 to overcome these
problems. We will consider the case where all incident narrowband EM plane waves
are completely polarized.
2.7 Other Applications
The research progress of parameter estimation algorithms in sensor array pro
cessing has resulted in a great diversity of applications, such as multiuser estimation
and detection, channel identification and spatial diversity in personal communica
tions.
The MUSIC algorithm was used in [94] for propagation delay estimation of the
DS-CDMA (Direct-Sequence Code Division Multiple Access) signals. In [95, 96] a


40
large sample maximum likelihood (LSML) algorithm was proposed for propagation
delay, carrier phase, amplitude estimation of DS-CDMA signals. These algorithms
have been shown to be robust against the multiuser near-far problem of DS-CDMA
systems. Array signal processing algorithms are expected to play an important role
in accommodating a multiuser communication environment, subject to severe multi-
path.
2.8 Summary
We have reviewed important properties of electromagnetic waves including wave
propagation and wave polarization. The previous work in sensor array processing
has been reviewed for both nonparametric and parametric methods. We have also
described the relations of our work herein with the previous work. A rather complete
list of references on sensor array processing can be found in [97].


CHAPTER 3
NARROWBAND ANGLE AND WAVEFORM ESTIMATION VIA RELAX
3.1 Introduction
Many high resolution array processing algorithms have been devised to estimate
the incident angles of the signals arriving at an array of sensors. The success of those
algorithms is typically demonstrated by means of simulated numerical examples,
often under the additive white Gaussian noise assumption. The high resolution angle
estimation algorithms, however, are usually very sensitive to the violation of the data
model assumptions. As a result, the performance of those algorithms is frequently
much worse when they are used in practical applications. To derive more robust
array processing algorithms, we shall, for instance, relax the additive white noise
assumption.
In this chapter we describe how the RELAX algorithm, recently proposed in [45]
for mixed spectrum estimation, can be used for angle and waveform estimation of
narrowband plane waves arriving at a uniform linear array in the presence of spatially
colored noise, such as an autogressive (AR) or autogressive moving average (ARMA)
noise. The RELAX algorithm presented in this paper is devised for the multisnapshot
case. To evaluate the performance of the RELAX algorithm, we apply RELAX to
both simulated and experimental data, and compare the performance of RELAX
41


42
with that of other well-known algorithms including ESPRIT with forward/backward
spatial smoothing [12, 72], MODE/WSF [25, 23], and AP/ANPA [66, 67] under the
same scenarios. The experimental data was collected by the Muti-parameter Adaptive
Radar System (MARS) developed at McMaster University. More importantly, we
explain why RELAX outperforms other well-known algorithms including ESPRIT,
MODE/WSF, and AP/ANPA. For the experimental data, since we do not know the
true incident angles, we introduce a cross-validatory method to assess the quality of
the estimates obtained with the algorithms we consider.
3.2 Problem Formulation
Consider the problem of angle and waveform estimation of K narrowband plane
waves impinging on a uniform linear array (ULA) with M (M > 2K) elements.
Assume the number of incident plane waves K is already known. The array output
vector can be written as:
y(n) = A(0)s(n) + e(n), n = 1,2, , N.
(3.1)
In (3.1), N is the number of temporal snapshots, e(n) is a complex M x 1 noise
vector, s(n) is the complex K x 1 signal vector and
A(0) =
a(6h) a(02) a(dK)
(3.2)
where 0
9\ 62 9
K
and a(9k), k = 1,2,*--, /T, is the complex M x 1
direction vector for the A:th signal arriving from angle 9k relative to the array normal.


43
(Here (-)T denotes the transpose.) For the ULA, the kth. direction vector a(0k) has
the form:
a(0k) =
l
0j-p-sin6k
k = 1,2, - , A ,
(3.3)
where 6 is the spacing between the array sensors and A is the signal wavelength.
The noise vector e(n) is a spatially correlated random process that is assumed
to be temporally white. In particular, e(n) has zero-mean and unknown covariance
matrix Q:
E{e(ni)eH(n2)} = Qniln2, (3.4)
where (-)H denotes the complex conjugate transpose and >nii2 is the Kronecker delta.
T
The signal waveforms s(n)
, n = 1,2,,iV,
¡i(n) s2(n) sK(n)
are modeled as deterministic unknowns. This assumption is usually referred to as
the deterministic (or conditional) signal model [20, 76].
The problem of interest herein is to estimate the angles #i, $2, , Ok and the
complex waveforms si(n), s2(n), , sx(n) from y(n), n = 1,2,---, TV.
3.3 Angle and Waveform Estimation Using RELAX
The RELAX algorithm was derived in [45] as an estimator of sinusoidal param
eters in the presence of colored noise. We describe below how the RELAX algorithm
can be modified for angle and waveform estimation with a uniform linear array of
sensors.


44
The angle and waveform estimates {0, s(l), s(2), , s(N)} can be obtained via
RELAX by minimizing, with respect to 0 and s(l), s(2), , s(TV), the following
nonlinear least-squares (NLS) criterion:
Fx[0,s(l),s(2),---,s(AO] = X][y() A(0)s(n)]w[y(n) A(0)s(n)]. (3.5)
n1
Note that if the noise e(n) were spatially white and Gaussian, minimizing (3.5) would
have yielded the deterministic maximum likelihood estimates of the incident angles
and signal waveforms [28, 66].
We present below our approach to the minimization of the cost function in (3.5).
Assume for now that there are K signals, where K is an intermediate value ( K < K,
see the steps below). Let
R
yk(n) = y(n) £ a(0)S(n), (3.6)
where {0,-, s,(l), st(2), , J,(Af)}N1 are assumed to have been previously esti
mated. Then the cost function for estimating the th signal parameters becomes
N
F2(0k, s*(l), a*(2), , sk{N)) = J2 iyfc(n) a{9k)sk(n)]H [yk(n) a(0k)sk(n)],
71=1
(3.7)
The minimization of F2 with respect to 9k and 5fc(l), ,sk(N) gives
4(n) =
a (9k)yk(n)
M
, n 1,2,' ,N,
(3.8)
6k=Bk


45
and
6}¡ -- arg min
6k
N
£
71 = 1
I-
a(0fc)a"(4)
M
y k{n)
N
= arg max ^ \aH(Ok)yk(n) (3.9)
71=1
Hence Ok is obtained as the location of the dominant peak of the sum of the peri-
2
odograms aH(6k)yk{n) /M, over n = 1,2, , N, which can be efficiently computed
by using FFT (fast Fourier transform) with each of the data sequence y*(n) padded
with zeros. (Note that padding with zeros is necessary to determine Ok with high
accuracy.) Furthermore, Sk(n) is easily computed from the complex height of the
peak of aH(6k)yk{n)/M (for n = 1,2, , ./V).
We can now proceed to present our relaxation (RELAX) algorithm for the min
imization of the nonlinear least-squares cost function in (3.5) Note that we assume
the number of incident signals K is already known or estimated. The RELAX algo
rithm comprises the following steps:
Step (1): Let K = 1. Obtain 6\ and ii(rc) from yi(ra) as in (3.8) and (3.9),
n = 1,2, , AL
Step (2): Let K 2. Compute y2() with (3.6) by using 6\ and Si(n) obtained in
Step (1), n = 1,2, , AL Obtain 02 and S2(n) from y2(n) as in (3.8) and (3.9),
n = 1,2, , N.
Next, compute yi(n) with (3.6) by using 02 and s2(n) and redetermine and
si(n) from yi(n) as in (3.8) and (3.9), n = 1,2,---, iV.


46
Iterate the previous two substeps until practical convergence is achieved (to
be discussed later on).
Step (3): Let K = 3. Compute y3(n) with (3.6) by using {0, s,(l), s(2), , ,st(iV)}-=1
obtained in Step (2). Obtain 03 and s3(n) from y3(n) as in (3.8) and (3.9),
n = 1,2, , N.
Next, compute yi(n) with (3.6) by using s(l), s,(2), , J(A^)}f_2 and re'
determine §i and Si(n) from yi(n), n = 1,2, , N. Then compute y2(n) with
(3.6) by using
s,(l), s(2), , S(N)}=i and redetermine §2 and s2(n) from y2(n), n =
1,2, ,1V.
Iterate the previous three substeps until practical convergence is achieved.
Remaining Steps: Continue similarly until K is equal to the known number K or
the estimated number K of signal sources. (See [45] for a possible approach to
the estimation of K when it is unknown.)
The practical convergence in the iterated substeps of the above RELAX algo
rithm may be determined by checking the relative change of the cost function in (3.5)
between two consecutive iterations. In our simulated and experimental examples, to
be presented in the next subsection, we terminate the iterations when the relative
change is less than or equal to e = 10~3, i.e.,
lTi[£g+1,s,+i(l),---, sq+i(N)] Fi[0g, sg(l), , s?(Ar)]|
Fl[9q, s,(l), ,sq(N)\
(3.10)


47
where 09,s9(l), , sq(N) denote the estimates obtained after the qth iteration.
To show how RELAX works, we consider an example shown in Figure 3.1, where
N = 1, K = 2, 61 = 35.5, 02 = 38.5, si(l) 1, a2(l) = 1, and the ULA has M = 32
sensors with the spacing between two adjacent sensors equal to a half wavelength of
the incident plane waves. The frequency axis used in the figure is related to AOAs
by fk = |sin(0fc), k = 1,2, where /j and /2 are shown with the dashed line. The
FFTs of yk in Figure 3.1 is obtained when the array data are zero padded to 1024.
Note that the AO A separation 3 is less than the FFT resolution 3.6. However, the
AOAs can be estimated accurately with RELAX after a few of iterations.
3.4 Results of Simulated and Experimental Data
We first present a simulation example for estimating the angles and waveforms
of incident signals. The simulation results were obtained by using 50 Monte-Carlo
trials. The noise em(n) is assumed to be a spatial AR random process of order 1, i.e.,
em(n) = -aiem_i(n) + wm(n), (3.11)
where a\ 0.85 and u>m(n), m = 1,2, , M, are independently and identically
distributed zero-mean complex Gaussian random variables with variance a2. The
complex amplitudes of the waveforms are assumed to be 1. The SNR referred to
below is defined by [98]
(^)
SNR = 10 log10
[dBj.
(3.12)


48
We assume that there are K 2 correlated incident signals, with the correlation
coefficient equal to 0.99, arriving from angles 61 = 10 and d2 = 2. The number of
incident signals is assumed known. The array is assumed to have M = 8 sensors that
are uniformly spaced with the spacing between two adjacent sensors equal to a half
wavelength. The number of snapshots taken at the array output is N = 10. All array
output vectors are zero padded to 213 before using them with FFT in the RELAX and
ANPA algorithms. (Note that the higher the SNR, the more padded zeros are needed
so that the estimation accuracy is not affected by the FFT intervals.) The waveform
estimates for MODE, ANPA, and LS-ESPRIT (least squares based ESPRIT) [12] are
obtained by using
s(n) = [AH(0)A(0)]-1AH(0)y(n). (3.13)
The root-mean-squared error (RMSE) of the kth waveform estimate is defined as
|s(n) 5fc(n)|2. Note that LS-ESPRIT is used with forward/backward spa
tial smoothing and subarray length equal to 4 (that is, M/2) due to the highly
correlated signals.
Figures 3.2(a) and (b) show the RMSEs of the angle and waveform estimates,
respectively, for the first signal as a function of SNR. (The results for the second signal
are similar.) The estimator performances are also compared to the corresponding
Cramr-Rao bound (CRB). (The CRB matrix for the signal parameters is derived in
Appendix A.) Note that RELAX has the best performance among the four algorithms
for both angle and waveform estimation, especially for waveform estimation at low
SNR. RELAX outperforms MODE and ESPRIT since the noise is not spatially white


49
as the latter methods assume. ANPA gives bad waveform estimates at low SNR
because using ANPA may yield almost identical angle estimates in some Monte-
Carlo trials. The ANPA angle estimates are obtained by maximizing the following
cost function:
ft() = £ l|PA(0|y()ll2, (3.14)
71 = 1
which can be obtained by first minimizing the cost function in (3.5) with respect to
s(l), s(2), , s(N) [66, 67], where Pa = A (a/7a) Ah. Although using (3.13)
for waveform estimation assumes that the angle estimates are not nearly identical,
maximizing i*3(0) does not guarantee this assumption. Figure 3.3 shows as a
function of 6\ and 02 when the realization of the noise is the one that gives the worst
waveform estimates for ANPA in Figure 1. It can be seen from Figure 3.3 that the
maximum of the cost function corresponds to almost identical angles. Hence maxi
mizing the cost function in (3.14) with respect to 9 (only) and using the estimated
9 to estimate s(n) with (3.13), instead of leading to a simpler problem, can actually
complicate the optimization of the cost function in (3.5) and lead to poorer estimates
when the incident angles are closely spaced.
We next apply the algorithms to the experimental data collected by the array
system known as the Multi-parameter Adaptive Radar System (MARS) [99]. The
array system was developed at the Communications Research Laboratory at McMas-
ter University. The data was collected by deploying the array at the west coast of
the Bruce Peninsula, Ontario, Canada, overlooking Lake Huron. MARS is a vertical
uniform linear array consisting of M 32 horizontally polarized horn antennas. The


50
spacing between adjacent antenna sensors is 5.715 cm. The four sets of data we use
below were collected when the array system was operated at frequencies 9.76, 9.79,
11.32 and 12.34 GHz. The data was recorded with 12-bit precision and sampled at
62.5 samples per second. For each carrier frequency, 127 snapshots were collected at
each antenna output. There are two incident signals, a fact assumed to be known to
the angle estimation algorithms. One of the incident signals is the direct path and
the other is the specular path, which is reflected from the lake. The direct incident
signal is a continuous wave (CW), whose amplitude is a constant and whose phase is
a linear function of time. Since the specular path is a delayed and reflected version
of the direct path, the phase difference between the two paths is a constant, which
is determined by the time delay between the direct and specular paths, the carrier
frequency of the waves, and the reflection coefficient of the lake. The incident signals
arrive from near the array normal, but the exact incident angles are unknown since
the vertical array structure may have been on a slight tilt. The parameter estimation
algorithms we consider below do not assume any a priori knowledge of the incident
angles and the signal waveforms.
In this application we use a cross-validatory criterion to assess the quality of
estimates. Consider the case of where the algorithms are applied to a single snapshot
at a time. Let Gs be the estimated angle vector obtained from the sth snapshot by
using one of the algorithms. Then our cross-validatory criterion is:
c
N
E E
A|e-
.y(n)
(3.15)


51
where is the number of those snapshots that do not give almost identical angle
estimates, and = IA (A/; A) A11. (Note that almost identical angle estimates
cause the matrix AH A in P1 ^ to be ill conditioned.) What the criterion does here
MO,)
may be explained by the following steps: 1) estimate AOAs from a single snapshot; 2)
obtain least squares estimate of waveforms by using the estimate of AOAs for every
other snapshot, i.e., s (n) = [AH (GS)A(6)]~1 A11 (6s)y(n),(n = 1, ,7V, and n / s);
3) obtain the residue by subtracting the signal part from the data with the data model
for each snapshot, i.e, residuein) = y(n) A(0s)s(n) = Px h y(n),(n = 1, , N,
A (Vs)
and n ^ s); 4) find the summation of residues for each of these snapshots as in (3.15).
Clearly, the smaller the summation, the better the performance of the algorithm. The
inner sum in (3.15) shows how well (or bad) the estimate obtained from snapshot s
can be used to predict the array outputs observed in the other available snapshots,
according to the assumed data model. The outer sum adds those prediction errors
together. The lower the C, the better the performance of the algorithm.
Table 1 shows the cross-validatory criterion C obtained by using RELAX, ANPA,
MODE, LS-ESPRIT, and TLS-ESPRIT (total least squares based ESPRIT) [12] at
frequencies 8.62, 9.76, 9.79, and 12.34 GHz. Note that LS-ESPRIT and TLS-ESPRIT
are used with forward/backward spatial smoothing and subarray length equal to 10.
In Table 1, the failure rate indicates how often an algorithm gives almost identical
angle estimates, and hence very poor waveform estimates. Note that for the data set
analyzed here, RELAX and ANPA have similar performances and are better than
the eigendecomposition based MODE and ESPRIT, which assume that the additive


52
noise is spatially white (note that the noise statistics are unknown). To see whether
the noise is actually spatially correlated, we estimate the noise covariance matrix as
follows:
Q =
N
E
N
E
y(n)yH(n)P
(JV-l)JV£.=w. A<->
A(0,f
(3.16)
Figure 3.4 shows the correlation coefficients between the first sensor noise and the
other sensor noises. The noise appears to be strongly spatially correlated, which
explains the poorer performance of the eigenstructure based methods. Finally, we
note from Table 1 that LS-ESPRIT and TLS-ESPRIT have similar performances for
this data set, but TLS-ESPRIT has a much higher failure rate, which indicates that
when waveform estimation is desired, LS-ESPRIT should be preferred over TLS-
ESPRIT.
Finally, we compare the computational complexities of these algorithms. The
amount of computations needed by RELAX depends on N, K, SNR and the num
ber of zero-paddings. For example, RELAX requires about 40% of the amount of
computations required by ANPA when N = 1, K = 2, SNR= 0 dB and the num
ber of zero-paddings is 212. Since ESPRIT and MODE do not need iterations, they
require much fewer computations than RELAX and ANPA. For example, ESPRIT
and MODE only need about 0.3% and 1.5%, respectively, of the amount of compu
tations required by RELAX when N = 1, K = 2, SNR= 0 dB and the number of
zero-paddings is 212. However, RELAX can still be attractive because it can be easily
implemented with simple FFT chips in parallel.


53
3.5 Conclusions
We have presented a RELAX algorithm for the multi-snapshot case of angle and
waveform estimation of narrowband plane waves arriving at a uniform linear array.
The RELAX algorithms are conceptually and computationally simple; their imple
mentations mainly require a sequence of fast Fourier transforms. We have evaluated
the performance of the RELAX algorithm by using both simulated and experimental
data, and compared it with the performances of other well-known algorithms such as
ESPRIT with forward/backward spatial smoothing, MODE/WSF, and AP/ANPA.
We have explained by means of results of both simulated and experimental data why
better signal parameter estimates can be obtained by using RELAX as compared to
using the other algorithms.


54
Table 1: Cross-validatory criterion for the different algorithms when used with the
experimental data collected by MARS.
Algorithms
8.62 GHz
9.76 GHz
9.79 GHz
12.34 GHz
RELAX
Criterion C
1.3578e+04
2.9153e+04
6.6079e+04
1.7177e+04
RELAX
Failure rate
0
0
0
0
ANPA
Criterion C
1.3323e+04
2.9209e+04
6.6260e+04
1.7172e+04
ANPA
Failure rate
0
0
0
0
MODE
Criterion C
4.5306e+04
3.8840e+04
7.2059e+04
2.6091e+04
MODE
Failure rate
38%
9%
5%
6%
LS-
Criterion C
1.5353e+04
3.3716e+04
11.0460e+04
1.8302e+04
ESPRIT
LS-
Failure rate
0
0
0
0
ESPRIT
TLS-
Criterion C
1.6723e+04
3.3712e+04
11.4600e+04
1.8302e+04
ESPRIT
TLS-
Failure rate
47%
0
9%
0
ESPRIT


55
Figure 3.1. An example of using RELAX, (a) Modulus of FFT of y in step (1),
K = 1. (b) Modulus of FFT of y2 in step (2), K = 2 (1st iteration), (c) Modulus of
FFT of yi in step (2), K = 2 (2nd iteration), (d) Modulus of FFT of y2 in step (2),
K = 2 (3rd iteration), (e) Modulus of FFT of yi in step (2), K = 2 (4th iteration),
(f) Modulus of FFT of y2 in step (2), K = 2 (5th iteration).


56
(a)
(b)
Figure 3.2. RMSEs of the angle and waveform estimates of the first signal as a
function of SNR when N = 10, M = 8, and K = 2 correlated incident signals with
the correlation coefficient equal to 0.99 arrive from 8i = 10 and d2 = 2. (a) Angle
estimation, (b) Waveform estimation.


57
(a)
(b)
Figure 3.3. F3 vs. 9\ and 62 when N = 10, M = 8, the correlation coefficient of
the two incident signals is equal to 0.99, and the realization of the noise is the one
that gives the worst waveform estimates for ANPA in Figure 1. (a) Mesh plot, (b)
Contour plot.


58
(a) (b)
Figure 3.4. Estimated noise correlation coefficients between the first and the other
sensors. Figures (a) (d) are for the carrier frequencies 8.62, 9.76, 9.79, and 12.34
GHz, respectively.


CHAPTER 4
ANGLE ESTIMATION OF WIDEBAND SIGNALS USING RELAX
4.1 Introduction
In this chapter we extend the RELAX algorithm to the case of wideband sources.
We concentrate on angle estimation herein. We show that the wideband RELAX
algorithm we devise naturally focuses the narrowband components in the spatial
frequency domain. We use both numerical and experimental examples to demonstrate
the performance of the wideband RELAX algorithm and compare its performance
with that of the well-known CSM [29] based LS-ESPRIT (CSM-ESPRIT). We also
explain why wideband RELAX can outperform the CSM-ESPRIT.
4.2 Problem Formulation
Consider the same uniform linear array (ULA) with M elements as before. As
sume that the incident wideband deterministic signals have a common bandwidth
B (Hz) with center frequency /0 (Hz). The kth bandpass signal Sk(t) observed at a
reference point can be written as
Skit) = 7 (4.1)
59


60
where 7fc() denotes the complex envelope. Let the signal be observed over a duration
[to, to + To]. The complex envelope can be written as
L
7k{t) = XI rjt(/;)ej2,r^<, to < t < to + To, (4-2)
i=i
where
1 fto+To
Mil) = ~ / 7k(t)e-Wdt, (4.3)
Jo Jio
with f¡ 1 l = 1,2,,//. L is the number of frequency components
symmetrically placed around 0 Hz with /;+i fi = = jr. Thus the bandpass
signal at the reference point (say the first sensor) can be written as
sk(t) = (4.4)
i=i
The Arth signal at the mth sensor has a propagation time delay TkyTn so that
sk(t + rkm)ej2*'*, (4.5)
i=i
where
Tfc.m ^ sin 6k, k = 1,2, , 7\, m = 1,2, , M, (4.6)
with being the spacing between sensors, C the propagation speed, and 9k the angle
of arrival of the kth incident signal relative to the array normal.


61
Let Xfc(t)
Ekfti'Tk,l) *£fc(L A;,2)
T
^A:(L Tfe,M)
. Then
Xi(f) = a(0t,/ + /,)r*(/,) ;=i
(4.7)
where a (0*., fo + ft) is the M x 1 direction vector of the &th source and has the form
a($fc5 fo + fi)
ej27r(/0+/i)7->c,l gj27r(/o+/i)Tfe,2
T
ej27r(/o+/¡)nc
(4.8)
Hence if K wideband signals along with some noise simultaneously impinge upon
the sensor array, the received data vector has the form
x(i) = [A(8,/ + f,)T(I) + N(/)] e*"'1 = y(7)e***, (4.9)
i=l i=l
where
/o + //)
a(#i?/o + //) a(^JC)/o + /i)
(4.10)
and
r(0
iT
ri(/,) rA-(/i)
(4.11)
is the /i x 1 signal vector, and
N(7) =
7Vi(/j) iVM(//)
(4.12)


62
is the Mxl noise Fourier coefficient vector, and (y(/)} are by definition the Mxl
Fourier coefficient vectors of x(t):
y (0 = A(0, /o + /|)F(/) + N(0, / = 1,2, , L. (4.13)
We assume that the noise vector N(/) has zero-mean and
E{N(/1)N"(2)} = Q,iJj, (4.14)
where Q is unknown.
The problem of interest herein is to estimate ffi, 62, ,0k from y(/), l =
1,2, ,£. The main difference with respect to the narrowband problem treated
in Section 2 is that now the array transfer matrix A depends on the snapshot index
/.
4.3 Angle Estimation Using RELAX
The estimates {0, r(l), , r(L)} can be obtained via RELAX by minimizing
the following nonlinear least-square criterion with respect to 6 and r(l),- , T(L)
(See (3.5)):
Gx =
E
/=!
(y(0 A(0, / + /i))r(i)]" (y(i) A(fl, / + /,))r(i)|.
(4.15)


63
To estimate the parameters of the &th signal, consider
G2 = MO a+ /OMi)]" WO + /,))r(/)], (4.16)
1=1
where
yt(0 = y(0 E a(i,/o + /l)fi(l), (4.17)
and where K is as defined before, and {0,-, f,-(l), , fare assumed to be
given or estimated. Let
vm aH(ekJo + fi)yk(l)
**(*) = n
(4.18)
which can be obtained by using FFTs with zero padding. Then similar to the nar
rowband RELAX algorithm, Ok can be obtained by
L
Ok = argmax^) \Yk(l)\2 (4.19)
6k i=i
and
rfc(/) = Yk(l)\ok=k, 1,2, ,!. (4.20)
Hence 6k is obtained from the location of the dominant peak of the sum of the
focused periodograms |FT;(/)|2 / = 1,2, , L. Then Tk(l) is easily computed from
the complex height of the peak of Yk(l). Note that the focusing of the narrowband
components can be naturally achieved in (4.19) by expanding and compressing the
FFTs (with zero padding) of yk(l) according to the ratio (/0 + /;)//0 since the spacing


64
between two adjacent FFT samples are different for different l The steps of our
RELAX algorithm for minimizing (6.20) are summarized as follows:
Step (0): Obtain y(/) from x(t) via DFT (discrete Fourier transform).
Step (1): Assume K = 1. Obtain and fi(/) from y(/) as in (4.19) and (4.20),
/ = 1,2, ,!.
Step (2): Assume/? = 2. Compute y2(I) with (4.17) by using 9\ and Ti(/) obtained
in Step (1), l = 1,2, ,£. Obtain 02 and f2(/) from y2(/) as in (4.19) and
(4.20), 1 = 1,2, ,!.
Next, compute yi(/) with (4.17) by using 62 and f2(/) and redetermine 6\ and
fa(/) from yi(/), l = 1,2, ,L.
Iterate the previous two substeps until practical convergence is achieved. (See
Section 2 for details.)
Remaining Steps: Continue similarly until K is equal to the known number K or
the estimated number K of signals.
We refer to this algorithm as the wideband RELAX.
4.4 Numerical Results
We present below numerical and experimental examples showing the perfor
mance of the proposed algorithm for estimating the incident angles of wideband
signals in the presence of either white or unknown colored noise. We also compare
the performance of wideband RELAX with that of CSM based LS-ESPRIT (CSM-
ESPRIT) for angle estimation.


65
In both of the simulation examples below, the array is assumed to be a ULA of
M = 8 sensors with the spacing between two adjacent sensors equal to half of the
wavelength corresponding to the center frequency /0. The wideband sources have
the same center frequency /0 = 100 Hz and the same bandwidth B = 40 Hz. The
noise is a temporally stationary zero-mean white Gaussian process, independent of
the signals, and spatially either a white or an autoregressive (AR) process. The total
observation time is To = 0.8 seconds. The demodulated data is sampled at twice the
Nyquist rate. The array output is decomposed into L = 33 narrowband components
via the DFT (discrete Fourier transform). The signal-to-noise ratio (SNR) is defined
as the ratio of the power of each signal to the noise power. We assume that there
are K = 2 independent wideband signals with equal power impinging on the array
from 0\ = 10 and 92 = 20. Note that the two signals cannot be resolved by
the spatial periodogram. All sequences are zero padded to 4096 for FFT in the
RELAX algorithm. The simulation results were obtained by using 30 Monte-Carlo
simulations.
Figure 4.1(a) shows the root-mean-squared error (RMSE) of the second signal
as a function of SNR in the presence of white noise. (The results for the first signal
are similar.) We note that the RMSE of the wideband RELAX decreases with SNR,
but the RMSE of CSM-ESPRIT stops decreasing when SNR reaches a certain level.
This is because, due to focusing, CSM-ESPRIT provides biased angle estimates even
when the angle separation of the two sources is within the resolution of the spatial
periodogram.


66
In Figure 4.1(b), we consider the performance of both wideband RELAX and
CSM-ESPRIT in the presence of unknown AR noise. The noise Nm(f¡) is assumed
to be a complex AR process of order 1, i.e, Nm(f¡) = -f where
ai = 0.85e_J4 and Wm(fi) is a zero-mean complex white Gaussian random process
with variance equal to a2. The SNR is defined by 101og10 1_^|2 dB. Note that
wideband RELAX again performs better than CSM-ESPRIT.
Finally, we apply both RELAX and CSM-ESPRIT to the experimental data
collected by the sensor array testbed [100, 101] at the University of Minnesota. The
uniform linear array consists of 8 sensors. The spacing between adjacent sensors is
about 2.1 times the wavelength corresponding to /o. Two correlated sources arrive
from around 33 and 36. The SNR is 21 dB for each source. The center carrier
frequency for this data is 40 kHz. The bandwidth of the data is 4 kHz. The data was
sampled at a rate of 5 kHz, and was decomposed into L 5 frequency bins (38.125
kHz, 39.0625 kHz, 40 kHz, 40.9375 kHz, and 41.875 kHz). Figure 4.2 shows the angle
estimates obtained from several observation intervals. (Note that one of the angle
estimates does not show up in Figure 4.2 (b) because it is too small.) The means
and the standard deviations of the angle estimates in Figure 4.2 were calculated
by averaging the angle estimates obtained from all 64 observation intervals. Note
that wideband RELAX provides smaller standard deviations and biases than CSM-
ESPRIT.


67
4.5 Conclusions
A wideband RELAX algorithm for the angle estimation of wideband sources
has been presented. The wideband RELAX naturally focuses the narrowband com
ponents in the spatial frequency domain. Numerical and experimental examples have
shown that the wideband RELAX can perform better than CSM.


68
(a) (b)
Figure 4.1. RMSEs of the angle estimates of the second signal as a function of SNR
when K = 2 uncorrelated wideband signals arrive from 6\ = 10 and 02 = 20,
M ~ 8, and L = 33. (a) In the presence of white noise, (b) In the presence of
unknown AR noise.
Figure 4.2. Angle estimates obtained from the experimental data, corresponding
to 64 observation intervals. The solid lines denote the means and the dashed lines
denote the means plus and minus the standard deviations of the angle estimates. The
true incident angles are believed to be 9\ = 33 and 62 = 36. (a) RELAX, (b)
CSM-ESPRIT.
#


CHAPTER 5
ANGLE AND POLARIZATION ESTIMATION WITH A COLD ARRAY
5.1 Introduction
This chapter studies the advantages of an arbitrary linear array that consists
of Co-centered Orthogonal Loop and Dipole (COLD) pairs. By using the COLD
array, the performance of both angle and polarization estimation can be significantly
improved as compared to using a co-centered crossed dipole (CCD) array.
The case where all incident narrowband electromagnetic (EM) plane waves are
completely polarized is considered. A completely polarized EM wave is a limiting case
of a more general type of EM wave, viz. a partially polarized EM wave. The state of
polarization of a partially polarized EM wave is a function of time while a completely
polarized wave has a fixed state of polarization (see [102] and the references therein).
We present an asymptotically statistically efficient signal subspace-based MODE
algorithm [25, 26] for both angle and polarization estimation. Since the MODE algo
rithm is a signal subspace-based approach, it is asymptotically statistically efficient
for both correlated (including coherent) and uncorrelated incident signals. We show
with numerical examples that the estimation performance of MODE is better, espe
cially for highly correlated or coherent signals, than that of MUSIC and NSF (noise
subspace fitting) [65]. (We remark that the signal subspace eigenvector based MODE
69


70
algorithm and the NSF algorithm are asymptotically statistically equivalent whenever
the signals are non-coherent. For coherent signals, MODE remains asymptotically
statistically efficient, whereas NSF is no longer asymptotically statistically efficient
[76]. This observation suggests that when the correlation coefficient is close/very
close to one, the NSF may need a much larger number of data samples than MODE
to converge to the asymptotics, and hence for a given finite N, MODE is likely to
perform better than NSF in such a case of highly correlated signals.)
5.2 COLD Array and Problem Formulation
Consider a 2L-element linear array consisting of L Co-centered Orthogonal Loop
and Dipole (COLD) pairs as shown in Figure 5.1. The signal received from each
antenna sensor is to be processed separately for direction and polarization estimation.
The /th COLD pair, l = 1,2, , L, has its center on the y-axis at an arbitrary y = 8¡.
For the /th COLD pair, the dipole parallel to the 2-axis is referred to as the 2-axis
dipole and the loop parallel to the x-y plane as the x y plane loop.
Assume K (with K < L) narrowband plane waves impinge on the array from
angular directions described by 6 and , where 0 and denote the azimuth and
elevation angles, respectively, as shown in Figure 5.1. Furthermore, suppose each
signal is a completely polarized transverse electromagnetic wave with an arbitrary
elliptical electromagnetic polarization [103]. Assume that the electric field of an
incoming signal has transverse components
E E$e$ +
(5.1)


71
where unit vectors eg, e, and er, in that order, form a right-hand coordinate system
for the incoming signals and Eg and E^ are the horizontal and vertical components of
the electric field respectively. In general, as time progresses, Eg and will describe
a polarization ellipse. For a given signal polarization, specified by constants 7 and 7,
the electric field components are given by (aside from a common narrowband phase
factor s0(t) x)
Eg
= E cos 7,
(5.2)
E^
= Esin "fe,
(5.3)
where E denotes the amplitude of the incident signal. The 7 and 7 can be used to
compute a and /?, which are the ellipticity and orientation angles of the polarization
ellipse, respectively. 7 is always in the range 0 < 7 < 7t/2 and 7 is in the range
7T < 7 < 7T. a and (3 can also be used to compute 7 and 7 [31, 40].
Assume that each dipole in the array is a short dipole (i.e., the length of the
dipole is equal to or less than one-tenth of a wavelength) with the same length LS(¡
and each loop is a small loop (i.e., the perimeter of the loop is equal to or less than
three-tenths of a wavelength) with the same area Asi. Thus the output voltages from
each dipole and loop are proportional to the electric field components parallel to
dipole and loop, respectively.
1For a narrowband BPSK (binary phase-shift keyed) signal, for example, s0(t) =
where u>0 is the carrier frequency and ip(t) is the modulating phase.


72
An incoming signal described by arbitrary electric field components Eg and E#
can be written as
E = E
(cos7)e0 + (sin 7eJT?)e,] .
(5.4)
Let us define the spatial phase factor
j sin 9
qi = e A ,
(5.5)
where A0 is the wavelength of the signal. The effective heights of the short dipoles
and small loops are given by [104]
^sd Lsd, Sin (f) ,
(5.6)
and
.27tAsi .
hsi = J:sin (5.7)
respectively. Including the time and space phase factors in (5.4), we find that an
incoming signal characterized by (9, (j), 7,7, E) produces a signal vector in the COLD
pair centered at y = 6¡ as follows:
z i(t) =
x¡(t)
Xd(t)
^COLD^so(^)Qh
(5.8)


73
where
Vcold
j 2^gi sin (j) cos 7
Lsd sin^sin 7eJi?
(5.9)
An advantage of the COLD array is that its antenna elements are not sensitive to
the azimuth angle 6 of the signal because both the loops and dipoles have the same
s'mcj) field pattern, as may be seen from Equation (5.9). Hence the incoming signal
described by (5.4) is independent of 6. We assume that the antennas and the incident
signals are co-planar, i.e., (j) = 90. Thus (5.9) becomes
Vcold
V$ cos 7
sin7e-??)
(5.10)
where
Vo = j
.27tAsi
Ac
V(h L sd
(5.11)
(5.12)
Note that V$ and represent the complex voltages induced at the loop and dipole
outputs by a signal with a unit electric field parallel to the loops and dipoles, respec
tively. Let s(t) = Es0(t)Vg cos 7. The z¡(t) in (5.8) can be rewritten as
z (t) = u s(t)qi,
(5.13)


74
where
u =
tan 7*
:v
(5.14)
Assume that K signals, specified by incident angles 9k, k 1,2, are
incident on the array. In addition, we assume a thermal noise voltage vector n/(f) is
present at each output vector z¡(t). The n¡(t) are assumed to be zero-mean circularly
symmetric complex Gaussian random processes that are statistically independent of
each other and to have covariance matrix cr2I, where I denotes the identity matrix.
Under these assumptions, the total output vector received by the COLD pair
centered at y 6¡ is given by
K
z¡(t) = J2ukSk(t)qik + n,{t), / = 1,2, ,£, (5.15)
k=l
where and qik are given by (6.9) and (5.5), respectively, with subscript k added to
each angular quantity. Further, Sk(t) = EkS0k(t)Vo cos 7^, where EkS0k(t) denotes the
Arth narrowband signal. The incident signals may or may not be correlated (including
completely correlated, i.e., coherent) with each other.


75
Let z(t), s(t), and n(i) be column vectors containing the received signals, inci
dent signals, and noise, respectively, i.e.,
Zl(<)
Si(t)
ni(0
Z (t) =
Z2(<)
, s(f) =
52(0
, n(i) =
n2(0
zL{t)
5a()
nL(0
The received signal vector has the form
z (t) = As(i) + n(i),
where A is a 2 Lx K matrix
A
ai a2 a k
= [AI]U,
with (g> representing the Kronecker product,
9n 9i2 ' q\K
?21 922 ' 92A'
9l i 9A2 qLK
(5.17)
(5.18)
(5.19)


76
and
Ul
0
U =
(5.20)
0 uA'
Assume that the element signals are sampled at N distinct times tn, n = 1,2, , N.
The random noise vectors n(t) at different sample times are assumed to be in
dependent of each other. The problem of interest herein is to determine the az
imuth arrival angles 0and the states of polarization described by (7*;, r¡k) or (a*, /?*,),
A: = 1,2,---, K, from the measurements z(tn), n 1,2, , N.
5.3 Angle and Polarization Estimation using MODE
The MODE [25, 26] and, in a related form, the WSF [23] algorithms were derived
for angle estimation with uniformly polarized arrays. We present below how to use
the signal subspace-based MODE algorithm with the COLD array for both angle and
polarization estimation.
Let
R = 4 L *(*>"(<), (5.21)
n-1
H A
where () denotes the complex conjugate transpose and R denotes the estimate of
the following array covariance matrix:
R = E[z(t)zH (t)].
(5.22)


77
It has been shown in [25, 23, 76] that an asymptotically (for large N) statistically
T
efficient estimator of the angles 0 = [ $ $2 ... gK ] and the polarization pa-
T
rameters r = [ rj r2 ... rK] can be obtained by minimizing the following
function:
(5.23)
pj
where the symbol PA stands for the orthogonal projector onto the null space of A ,
and the columns in Es are the signal subspace eigenvectors of R that correspond to
the K largest eigenvalues of R, with K min[7V,rank(S)]. Here S is the source
covariance matrix,
S = _E{s(f)s"(f)}.
(5.24)
Assume that K is known. (If K is unknown, it can be estimated from the data as
described, for example, in [43].) Note that if no components of the signal vector s(t)
are fully correlated to one another, then K = K (provided N > K). Further, the A
in (5.23) is a diagonal matrix with diagonal elements Ai > A2 > > A^-, which are
the K largest eigenvalues of R, and
A = A (t2I,
(5.25)
where


78
We show below that we can concentrate out r first and hence reduce the dimen
sion of the parameter space over which we need to search to minimize (5.23).
It has been shown (in Appendix B) that
where
= I PAI + P
Ax + (aWuv
(5.27)
At = (AhA) 1Ah,
(5.28)
and
Vj 0
0 vA'
(5.29)
with
Vfc =
-rl
, k = 1,2, , K.
(5.30)
Thus minimizing /(0,r) in (5.23) is equivalent to minimizing
f(0, r) = Tr
AI
+ P
(AtHI)V
* 2
-ljL
E,A A E
H
(5.31)
Let
W = |vh[(AhA) 1 i] v j \
(5.32)


79
be formed from some consistent estimates of G and r. Since P i s 0(l/y/),
(AI-"(g>I)V
^[(A^A)'1 I]V}~ can be replaced by W without affecting the asymptotics of
the MODE estimator [25, 26]. Then we have
f(0,r) = f1(0) + f2(0,r),
(5.33)
where
fi(0) = Tr
A 2
(A gtl) {(AWA)-1 i} (Ah I)S A ?
(5.34)
and
f2(0,r) = Tr
(A^ I)VWVh(A^ I)S2 *f
(5.35)
The MODE estimates {0,r} are obtained by minimizing /, i.e.,
{,r} = argmin[/i(0) + f2(0,r)]
u, r
(5.36)
Let
Ah odd columns of (A^H I),
(5.37)
and
A = even columns of (A^H I).
(5.38)


80
Let Vh and Vt, be the following K x K diagonal matrices:
and
Then
V/t = diag | r*, , -r^},
V = I.
(AtHI)V = AfcVA + AV.
Thus /2(0,r) in (5.35) can be rewritten as
/2(0,r) = Tr
+Tr
+Tr
+Tr
V?Ar.""1"AfcVfcW
h **h L's
VfAfs2A XfA^W
- 2
V,hA"s A_1f AfcVhW
Vf A^52A *f A.V.W
Since V/! and V are diagonal matrices, Equation (5.42) can be written i
lowing matrix form:
/2(0,r) =
vf eT
Q(*)
V/,
e
(5.39)
(5.40)
(5.41)
(5.42)
the fol-
(5.43)


81
where
Q(0)
(A^s2 'fA^OW7
(A"s2"1fA0WT
(Af^ 'f A) wt
(A^s2_1fA)0WT
Qi(<9) Q2(0)
Q?(0) Q3(0)
(5.44)
with 0 denoting the Hadmard-Schur matrix product (i.e., the elementwise multipli
cation),
Vfc =
(5.45)
and
(5.46)
Note that the polarization parameters are contained only in v^. By setting dfi/dvh =
0, we obtain
v, = -Qr1(0)Q2(0)e. (5.47)
Using (5.47) in (5.43) gives
h(e) = eT
q3(9)-Q?(9)Qr1(e)Q2(e)] e,
(5.48)
which is a concentrated function depending only on 6. The MODE estimates {0,r}
can be obtained by
0 = argrnin [/i(0) +/3(0)],
U
(5.49)


82
and using 9 in (5.47) to obtain r.
To summarize, we have the following MODE algorithm for angle and polariza
tion estimation:
Step 1: Obtain initial estimates of 9 and r (see the discussions below).
Step 2: Determine 9 by minimizing f\(0) + fz(Q) as shown in Equation (5.49)
with W in (5.32) formed from the initial estimates obtained in Step 1.
Step 3: Calculate r by using the 9 obtained in Step 2 in (5.47).
Step 4: Determine the 7 and r) from r with
Ik
Vk
tan 1
k = 1,2, , K.
(5.50)
(5.51)
For signals that are not highly correlated or coherent with each other, the initial
estimates of 9 and r in Step 1 may be obtained by using MUSIC [84], which requires
a one-dimensional search over the parameter space. For highly correlated or coherent
signals, the initial estimate of 9 may be determined by setting W = I and minimizing
fi(9) + /3(0) as shown in Equation (5.49). The initial estimate of r can be calculated
by using the initial estimate of 9 in (5.47). The initial estimates obtained by using
MUSIC for non-coherent signals or MODE with W = I are known to be consistent
[26, 71].


83
5.4 Statistical Performance Analysis
We present below the asymptotic (for large N) statistical performance of MODE
for both direction and polarization estimation with the COLD array.
Before we present the analysis results, however, we first describe the method we
use to describe the accuracy of the polarization estimates. For reasons discussed in
[31], we define the polarization estimation error to be the spherical distance between
the two points M and M on the Poincare sphere that represent the actual state of
polarization (7,77) and the estimated state of polarization (7,7), respectively. Let (
be the angular distance between M and M. Then [31]
cos £ = cos 27 cos 27 + sin 27 sin 27 cos(7 fj). (5.52)
where ( is always in the range 0 < ( < w.
Applying the first-order approximation to the left side of 5.52 yields
Cl = 4(7fc Ik)2 + sin2(27fc)(7* ~ Vkf- (5.53)
The asymptotic variances of the polarization estimates are obtained with (5.53) and
the accuracy results on 7 and 77 given below.
Let
9r 7r VT
T =
T
(5.54)


84
It follows from [25, 76] that the asymptotic (for large N) statistical distribution of f
is Gaussian with mean r and covariance matrix equal to the corresponding stochastic
Cramer-Rao bound (CRB), CRB. The ijth element of CRB 1 is given by
CRB
*7
Re [tr {a^PyASAHR-1 AS}] ,
(5.55)
where A; = <9A/cb with r denoting the z'th element of r.
5.5 Numerical Results
We present below several examples showing the performance of using the MODE
algorithm with the COLD array and comparing the asymptotic statistical perfor
mance analysis results with the Monte-Carlo simulation results. We compare MODE
with MUSIC and NSF for both angle and polarization estimation. The simulation
results were obtained by using 50 Monte-Carlo simulations. In the examples, we as
sume that there are K = 2 incident signals and both signals are assumed to have the
same amplitude Ek, such that \VgEk\ = |ViLk| = 1, k = 1,2. Hence, the signal-to-
noise ratio (SNR) used in the simulations is 101og10<72 dB. The array is assumed
to have L = 8 COLD pairs that are uniformly spaced with the spacing between two
adjacent COLD pairs equal to a half wavelength. We also compare the estimation
performance of using the COLD array with that of using a CCD array with the
same array geometry. The CCD array consists of crossed y- and 2-axes dipoles. The


85
counterpart of Equation (5.9) for the CCD array can be written as
Vccd
Lsd cos 7 cos 6 + Lsd sin 7e:v sin 6 cos cf>
Lsd sin (5.56)
In the following examples, the antennas and the incident signals are assumed to be
co-planar, i.e., = 90,
(5.56) becomes
Pccd ~
Lsd cos 7 cos $
(5.57)
-Lsd sinjejv
(We remark that if the antennas and the incident signals are not co-planar, we will
need two-dimensional CCD or COLD arrays for angle and polarization estimation,
which is the case not considered herein. For this case, however, the COLD array will
not always perform better than the CCD array.)
First, we present two examples that illustrate how the angle separation between
the two incident signals affects both the direction and polarization estimates. We
begin with the case of two signals with identical circular polarizations (au = a2 =
45). Figure 5.2 shows the root-mean-squared errors (RMSEs) of the estimates of
the first signal as a function of angle separation A6 when two correlated signals
with correlation coefficient 0.99 arrive at the array from angles 0\ A0/2 and
02 = A6/2. We note that MODE performs better than MUSIC and NSF. Further,
MODE achieves the best possible unbiased performance, i.e., the corresponding CRB,
as the angle separation increases. Because the signals arrive from angles near the


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