Efficient approaches to source localization and parameter estimation

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Efficient approaches to source localization and parameter estimation
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Thesis (Ph. D.)--University of Florida, 1996.
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Includes bibliographical references (leaves 123-131).
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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

























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.













TABLE OF CONTENTS




ACKNOWLEDGMENTS .............................

KEY TO ACRONYMS ...........................

LIST OF FIGURES ................................

A BSTRA CT . . . . .

1 BACKGROUND AND MOTIVATION .....................


Background ............
Motivation and Contributions .
Outline of the Dissertation ...


2 WAVE PROPERTIES AND SENSOR ARRAY PROCESSING ......


Introduction ...............
Propagation of EM Waves .......
Polarization of EM Waves .......
Narrowband Signals ...........
2.4.1 Array Data Model and Problem
2.4.2 Nonparametric Methods .
2.4.3 Parametric Methods .
Wideband Signals ............
Polarization Diversity . .
Other Applications . .
Summ ary ................


Formulation


3 NARROWBAND ANGLE AND WAVEFORM ESTIMATION VIA RELAX 41


Introduction .............
Problem Formulation ..........
Angle and Waveform Estimation Using
Results of Simulated and Experimental
Conclusions ...............


4 ANGLE ESTIMATION OF WIDEBAND SIGNALS USING RELAX .. 59


RELAX
Data .









Introduction ..... .. .. .. .. ... .. ..
Problem Formulation ................
Angle Estimation Using RELAX .. .......
Numerical Results ..................
Conclusions .. . . .


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 Sum m ary .. ... .. .. .. .. ... .. .. ...... 114
7.2 Contributions ........... ............ 116
7.3 Future W ork ....... .... ..... .... .. ....... .. 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








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
















LIST OF FIGURES




2.1 Polarization ellipse .. .. .. .. .. .... .. .. .. ... .. 10

2.2 Poincar6 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 01 =

-100 and 02 = 2. (a) Angle estimation. (b) Waveform estimation.. 56








3.3 F3 vs. 01 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 01 = 100

and 02 = 200, 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

01 = -330 and 02 = -360. (a) RELAX. (b) CSM-ESPRIT. 68


5.1 A linear COLD array.............. ........... 90

5.2 Root-mean-squared errors (RMSEs) of estimates versus A0 for the first

of the two signals when 01 = -A0/2, 02 = +A0/2, a = a2 = 45,

/i = 32 = 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








5.3 Root-mean-squared errors (RMSEs) of estimates versus AO for the

second of the two signals when 01 = 50, 02 = 500 + AA, al = 02 = 0,

fI = 02 = 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 01 = 500, 02 = 700, al = 450 Aa and

a2 = 450, /i = 02 = 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 01 = -6, 02 = 6,

al = 02 = 450, l = P2 = 00, 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 01 = 500, 02 = 700, al = a2 = 0, !1 =

02 = 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 01 = 500 and 02 = 700, a = a2 = 450,

/1 = -2 = 00, correlation coefficient = 0.99, and SNR = 10 dB. (a)

Direction estimates. (b) Polarization estimates. . ... 96








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 01 = 500 and 02 = 700a,1 = a = 450, P1 =2 = 0, correlation

coefficient = 0.99, and SNR = 10 dB. (a) Direction estimates. (b)

Polarization estimates. ................. ....... ..97


6.1 A uniform linear COLD array. . .. ... 111

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 01 = 10,

02 = 220, az = 02 = 0, /1 = 00, P2 = 100, 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

01 = 10, 02 = 220, a1 = 02 = 00, 1/ = 00, 12 = 100, correlation co-

efficient = 0.99, and N = 2. (a) Direction estimates. (b) Polarization

estimates. (c) Waveform estimates. . .... 113
















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








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.














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 Capon's methods [3, 4, 5, 6, 7]

are the typical nonparametric methods. Their idea is to form some spectrum-like

function of the parameters) 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








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








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








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.








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.

Chapter 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








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

(TF.M) wave. In a medium with spatially constant permeability pt and permittivity




8



e and with no free charges and currents, the Maxwell's equations are



VB = 0, (2.1)

VE = 0, (2.2)

Vx E = t (2.3)

OE
Vx H = (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:


02 u
V2U pf = 0. (2.5)
at2


The wave equation has the well-known plane wave solutions



u(t, x) = ej(kTx-t), (2.6)



where the temporal frequency w and the magnitude of the wave vector k are related

by

k = Wv/e, (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








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 z 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 = (E1x + E2eJy)e-jkz, (2.8)


where E1 and E2 are real and rj (-180 < q < +1800) is the phase angle between x

and y components. The corresponding magnetic field is



H = (--E2e + El)e-jkz. (2.9)
VIL








The phase and relative amplitudes E1 and E2 determine the state of polarization. In

the most general case of elliptical polarization the polarization ellipse described by E1

and E2e6j, 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


OA
AR- =O (1 OB


(2.10)


Figure 2.1. Polarization ellipse.


We define 3 to be the tilt angle of the ellipse (0 < # < 1800), and a to be the

ellipticity angle, which is given by


a = tan-(AR), (-450 < a < +450).


(2.11)








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

Y =tan-1( ), (0 < 7 90). (2.12)


The geometric relation of a, / and 7 to the polarization ellipse is illustrated in Fig.

2.1. The trigonometric interrelations of a, /, 7 and i are given by [38, 39]



cos27 = cos 2a cos 23, (2.13)

tan 2a
tan r = in2 (2.14)
sin 2P

tan 2/ = tan27 cos T, (2.15)

sin2a = sin 27 sin ?. (2.16)



The Poincard sphere representation of wave polarization [40] in Fig. 2.2 clearly shows

the relationship among the four angular variables a, /, 7 and rq. 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 = 23, (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









Polarization state


Figure 2.2. Poincard sphere.


the equator and angle between the great circle and the equator as:



Great-circle angle = 27, (2.19)

Equator-to-great-circle angle = ?1. (2.20)



Thus, it is convenient to describe the polarization state by either of the two sets

of angles (a, /) or (', r). The case when a = 0 corresponds to linear polarization.

The case when a = 450 corresponds to circular polarization, with left circular

polarization (a = +450) 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









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.



( Left circular polarization
a = 45"

Left elliptical polarization
cx = 22.50, p =45








Linear polarization
a = 0, P=450





r p n Linear polarization
Linear polarization = 22.5
X=0, =0 a =0 0 = 22.5
a= O, p=o

Figure 2.3. One octant of Poincare sphere with polarization states.



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.








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 rt = (xt, yj) and an impulse response


hi(t, r) = ai(0)(t)6(ri), = 1,..,L.


(2.21)


Source 2
*
/t


Source 1
*
\


Sensor Array


r
ID


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(t, ) from the impinging emitter signals with parameter 8 =


[ 1, 02, ...' ], to the sensor outputs. The Ikth element of H(t, ) takes the








form

Hlk(t, 0)= at(Ok)(t)6(rl), I=1, L,k 1,2, ,K. (2.22)


From a convolution operation, the sensor outputs can be written as


K
yl(t) = > a(Ok)ej(t-kr) (2.23)
k=l


for purely exponential signals and


K
y (t) = a(kj(ut-kr)sk(t) (2.24)
k=1


for narrowband signals, where w is the common center frequency of the signals and

sk(t) represents the complex amplitude waveformm) of the kth signal. The carrier ejwt

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() a(01), a(02), *. a(OK) s(t) + n(t) = As(t) + n(t), (2.25)







where



a(Ok) al(Ok)ek'rl, a2(Ok)ejk2r2 ... aL(Ok)jkrL (2.26)



s(t) st(t), s2(t),'. SK () (2.27)


and n(t) is the noise vector. The vector a(0k) is referred to as the kth array propa-

gation steering vector. It describes via kT rl how a plane wave impinges at the array

(i.e. AOA) and via al(Ok) how the sensors affect the signal amplitude and phase.

For polarization sensitive sensors, the al(Ok) is related to the polarization state of

the incident electromagnetic plane wave. For omnidirectional isotropic sensors, the

al(Ok) 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(0k)

has the form

a(Ok) 1, dk .., ej(L-1)k (2.28)


where

Qk = 6sin(Ok) (2.29)
c


is called the spatial frequency.
















5sin 0


Source


Figure 2.5. A uniform linear array.


We can see from (2.29) that the vector a(0k) is uniquely defined if and only if

Qk is constrained as |Qkl| < 7. The condition is satisfied if


< -
-2


(2.30)


The collection of these steering vector over the parameter space of interest


A = {a(Ok)10k E O}


(2.31)


is often called the array manifold. The parameterization of A is assumed known. Let

us define the set A' as the collection of all distinct array manifold vectors


(2.32)


AK = {AIA = [ a(01) ... a(0K) ], 1 < 02 < -- -< OK}.








Hence, A^ is parameterized by the parameter vector 0 = [ o0, ..., 0. ]'. The

array is assumed to be unambiguous. In other words, any A E AK 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(t) 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(tl)nH(t2) -= Qtl,t2, (2.33)



E{n(ti)nT(t2)} = 0, (2.34)


where 6Stt is the Kronecker delta, (.)H represents the complex conjugate transpose,

and (.)T 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 = a2I.








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)s"(t)}. (2.35)



For the former case,
1N
Rs = (t)s (t). (2.36)
t=1

For the case of spatially white noise, the array covariance matrix has the form



R= E{y(t)yH(t)} = AR,AH + C2I. (2.37)



The eigendecomposition of R results in the representation


L
R = AieieH = E,A,EH + EAEA (2.3s)
i=1


where AX > *. > AK > AK+l = = AL = a2. The matrix E, = [ e, *-, eK

contains the K eigenvectors corresponding to the largest eigenvalues. The range space

of E, is called the .'igiiil subspace. Its orthogonal complement is the noise subspace

and is spanned by the columns of En = [ eK+L, .* eL ]-

The eigendecomposion of the sample covariance matrix R is given by


N
R = y(yH() = E A + EnAnE, (2.39)
t=1








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 parameters) 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


L
yF(t) = z wyi(t) = wHy(t). (2.40)
il=1


Given {yF(t)}ti, the output power is measured by


N N1 H
S F(1t) = N wHy(i)y"(t)w = w Rw. (2.41)
t=l t=1


Different choices of weighting vector w leads to different beamforming approaches.

There are two types of beamformers.








Conventional Beamformer

The weighting vector for conventional beamformer is given by



WBF = 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(O = aH(O)Rfa(0). (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 2r/L for a ULA.

Capon's 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 0. The Capon spatial

filter design problem is posed as


minwllRw subject to wHa(0) = 1.
W


(2.44)







The optimal w is given by


Rl-la(0)
WCP = 1() (, (2.45)



which leads to the following spatial spectrum upon insertion into (2.41)


1
PCP =aH( l (2.46)
aH(0)R-la(0)


Thus, the Capon AOA estimates are obtained by the locations of the largest peaks

of the spatial spectrum given by (2.46).

Capon's 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.







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 MIeth1;il

The ML methods include the deterministic ML and stochastic ML.

* Deterministic ML







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]:


N
NLF = E[y(n) A(O)s(n)]H[y(n) A(0)s(n)]. (2.47)
n=l


Minimizing the cost function with respect to s(1), s(2), s(N) gives



s(t)=(AHA)-lAHy(t), = 1,2,.--,N. (2.48)



By substituting (2.48) in (2.47), we get



S N {I- A(AHA)-IAHy I2 (2.49)

= argmaxTr{A(AHA)-lAHR}, (2.50)



where R is the sample covariance matrix.

Note that a nonlinear K-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








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(ti)sH(t2)} = Ra,b,42, E{s(ti)sT(t2)} = 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(O)R,AH(0) + a21. (2.52)



The negative log-likelihood function (ignoring constant terms) of the complete

data set y(l), -, y(N) is given by



1(0, R,, 02) = loglRI + tr R-1R}. (2.53)



Minimization with respect to a2 and R, leads to the following expressions:


Rs,() = Ai(R 2(O)I)AtH


(2.54)








62 ) 1 tr{PXIi}. (2.55)


where

At = (AHA)-IAH, (2.56)


P =I A (AA)1 AH. (2.57)


Substituting (2.54) and (2.55) into (2.53) results in the following compact form

of the criterion:

0 = arg mn {log ARft,()AH + &2(0)I1}. (2.58)


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

0 and the minimization 0 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 70's. These methods assume that the additive noise is either spatially white








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 E, are

orthogonal to A, i.e.,

Ena() = 0, 0 e {01,.OK}. (2.59)


For unique AOA estimates, the array is usually assumed to be unambiguous. In

particular, a ULA is unambiguous if -900 < 0 < 900 and d < A/2. In practice, the

MUSIC "spatial spectrum" is defined as


aH(0)a(0)
PnM(0) -(2.60)
SaH()E Ha() (2.60)


The MUSIC AOA estimates are given by the locations of the largest peaks of

the "spatial spectrum" in (2.60) provided ElEnH is close to EEH.

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








A uniform linear array steering matrix has the structure of a Vandermonde

matrix as follow
1 ... 1


A = (2.61)


ej(L-1)i ... ej(L-1)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 A1 and A2 be the sub-matrices by deleting the first and last rows from

A respectively. Then, A1 and A2 are related by



A2 = Ai4I, (2.62)



where

4 = diag {e J~, eJ"2, ., ej }. (2.63)


Consider the structure of the eigendecomposition of the array covariance matrix R

given in (2.38). If the signal covariance matrix R, has rank K', then the matrix E,

will span a K'-dimensional subspace of A. This observation implies that there exists








a full-rank K x K' matrix such that



E, = AT.



Deleting the first and last rows of (2.64), respectively, gives


E,s = A1T, E,2 = A2T.


(2.64)


(2.65)


Combining (2.62) and (2.65) leads to


(2.66)


where F = T-1 T. Clearly, and 4 have the same eigenvalues. The eigenvalues

are given by ejok, k = 1, 2, ., K, which are related to AOAs. There are two different

ESPRIT algorithms, depending on how to approximate the relation


ES2 = Esli.


(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


Es2 = Esl,







The idea of the Root-MUSIC dates back to Pisarenko's method [69]. If the

signals covariance matrix has full rank, then the polynomial



pl(z) = efa(z), I = K + 1, L, (2.68)


where a(z) = [1, z, z ,L-1]T, has K zero-roots at ejOk, k = 1,2, K. Consider

the zeros of the MUSIC function



ElI Ha(z)II2 = aH(z)tE Ha(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) = zLaT(z-)EnEHa(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








For highly correlated or coherent signals, a rank deficiency occurs in the source

covariance matrix R,. 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 = AP-(L-1)R,-(L-1)AH + r21, (2.71)



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



RFB = I(R + RB) = ARAH + 2, (2.72)



where R, = (R, + ~-(L-1)R,-(L-1))/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.








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 0 and T are unknown, it appears there are

no values of 0 such that E, = AT when only an estimate E, of E, is available. A

natural solution is to find the best weighted least-squares fit of the two subspaces by

the following criterion:



{(,T} = arg min IE, ATI|Iv, (2.73)
0,T


where IIAI1| = tr{AWAH} 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 = At E, into (2.73) leads to the separated criterion function



OSSF = arg mnTr{PAE,WEH}. (2.74)
0








Different choices of W lead to different methods. The choice W = I yields Cadzow's

method [75]. A statistical analysis suggests other weightings, such as


W = (A, -& 2I)2iA,,


(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 PI using a basis for the nullspace

of AH. The projector PA above may be reparameterized in terms of the coefficients

b = [ b b ... bK ]T of a polynomial defined as


K K
bk -k = b0 ( -(
k=O k=1


(2.76)


Let B" be the following (L K) x L matrix


.. bl bo 0


(2.77)


Then PA = PB, where PB denotes the orthogonal projector onto the range space of

B. Thus the reparameterized SSF criterion for MODE is given by



b = arg minTr{B(BHB)- BHEWEH}. (2.78)
b


e.ik) ; bo 0.








Clearly, the resulting non-iterative procedure includes:

1) Determine b by minimizing (2.78) with (BHB)-1 set to I;

2) Determine an asymptotically efficient estimate b as


H^-1 H.
b = argminTr{B(3HB)-lBHE,WEI}, (2.79)
b


where (B^HB) 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 R, has full rank, then the following

relation holds:

EHA = 0. (2.80)


Given an estimate of En, a natural estimate of 0 can be obtained by minimizing

the following Noise Subspace Fitting (NSF) criterion:



ONSF = argmintr{UAHEEHA}, (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








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 = AtWEfAtH. (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 0.

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








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(fn) = A(0, f)Tr(fn) + N(fn), n = 1,2, .. J, (2.83)



where y(f,) denotes the L x 1 data vector of the n-th frequency bin, A(0, f)) =

[ a(0i, fn) a(OK, f)) ] is a L x K matrix, whose k-th column is the direction








vector for the k-th signal, and F(fn) = [ l(fn) ... FK (fn) ]T, is the K x 1 signal

vector, and N(f,) = [ Ni(fn) .. NL(fS) ]T, is the L x 1 noise vector.

The resulting spatial covariance matrix is



R(fn) = A(0, f,)R,(f.)AH(O, fn) + O(fn)R(f), (2.84)


where Rs(f,) and Rn(f/) 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(fn) are designed

to align the signal-subspace at all frequencies with the signal subspace at the reference

frequency fo,

T(f,)A(0, f)) A(0, fo), n = 1,--,J. (2.85)


Thus, a coherent averaging of the covariance matrices R(fn), n = 1,..., J, may be

performed as below



R, = A T(f,)R(fn)TH(n)

A(O, fo)RAH(O, fo) + R, (2.86)








where
1J
R, = s( f,), (2.87)
n=l1

and
1 J
Rn .= U (fn)T(fn)R(f.)TH(f.). (2.88)
n=l

Hence any narrowband subspace-based algorithms may be applied to Ry after the

coherent averaging is performed.

It is apparent from (2.85) that the transformation matrix depends on the un-

known AOAs 0. 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 Capon's 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, al(0k) in (2.26) is








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








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







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(9) = a(01) a(02) a(0K) (3.2)


T
where 0 = 01 02 ... 8K and a(0k), k = 1,2,. .,K, is the complex M x 1

direction vector for the kth signal arriving from angle Ok relative to the array normal.







(Here (.)T denotes the transpose.) For the ULA, the kth direction vector a(0k) has

the form:



a(0k)= 1 e sin9k ... Cj (M-1)sinOk k= 1,2, K, (3.3)



where b 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)} = QS,n2,, (3.4)


where (.)H denotes the complex conjugate transpose and 6n,,n, is the Kronecker delta.

The signal waveforms s(n) = s(n) s( 2(n) -. SK(r) n = 1,2, 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 01, 02, OK and the

complex waveforms sl(n), s2(n), ., SK(n) from y(n), n = 1,2, .. N.

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.








The angle and waveform estimates {0, s(1), s(2), .. ,(N)} can be obtained via

RELAX by minimizing, with respect to 0 and s(1), s(2), -**, s(N), the following

nonlinear least-squares (NLS) criterion:


N
F1 [0, s(1), s(2), s(N)] = [y(n)- A(O)s(n)]H[y(n)- A(0)s(n)]. (3.5)
n=l


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



y(n) = y(n)- E a(O;)i,(n), (3.6)
i=l,i k


where {O(i,i;(), ,i(2), -. i (Nl)}1, are assumed to have been previously esti-

mated. Then the cost function for estimating the kth signal parameters becomes


N
F2(0k, Sk(l), Sk(2), Sk(N)) = : [yk(n) a(Ok)sk(n)]H [yk(n) a(Ok)sk(n)],
n=1
(3.7)

The minimization of F2 with respect to Ok and sk(l), ', Sk(N) gives


n aH (Ok)yk(n)
Mk() = n =1,2,. ..,N, (3.8)
S6Ok=6k








and


N [ a(Ok)aH(Ok) 2 N 2
k = argminm1 Im- Mn yk() = argmax=1 aH(Ok)yk(n) (3.9)
fn1l I JY n=1


Hence tk is obtained as the location of the dominant peak of the sum of the peri-

odograms aH k)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 yk(n) padded

with zeros. (Note that padding with zeros is necessary to determine 0k with high

accuracy.) Furthermore, Sk(n) is easily computed from the complex height of the

peak of aH(0k)yk(n)/M (for n = 1,2, N).

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 Q1 and 1(n) from yi(n) as in (3.8) and (3.9),

n = 1,2,---,N.


Step (2): Let K = 2. Compute y2(n) with (3.6) by using 0i and il(n) obtained in

Step (1), n = 1,2, N. Obtain 02 and A2(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 A2(n) and redetermine j0 and

1i(n) from yi(n) as in (3.8) and (3.9), n = 1,2, ..-,N.








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 {0j, .i(l), j(2), .. A(N)}|

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 yl(n) with (3.6) by using {0j, Ai(l), A;(2), ... i(N)}~, and re-

determine 01 and .1(n) from yl(n), n = 1,2,- ,N. Then compute y2(n) with

(3.6) by using

{0i, .i(1), A(2),-.. ,.i(N)}i=1,3 and redetermine 02 and .2(n) from y2(n), n =

1,2,. N.

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.,


IF[,q+1, q+1(1), Sq+l(N)] Fi [q,, (1), -,q(N)]I ,
< ),...,(3.10)
F1 [Op, 9q 1), 9q (N)]








where 0q, q(1), q(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, 01= 35.50, 02 = 38.50, si(1) = 1, s2(1) = 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(Ok), k = 1,2, where f, and f2 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 AOA separation 30 is less than the FFT resolution 3.60. 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) = -alem_l(n) + w,(n), (3.11)



where a -= -0.85 and 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 og10 (1- a2 [dB]. (3.12)








We assume that there are K = 2 correlated incident signals, with the correlation

coefficient equal to 0.99, arriving from angles 01 = -10 and 02 = 20. 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. (NoIte 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

9(n) = [AH(b)A(b)]-lAH(6)y(n). (3.13)


The root-mean-squared error (RMSE) of the kth waveform estimate is defined as

VE=1 lsk(n) k(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

Cram6r-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








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:
N
F3(0) = |PA(0)Y()II2, (3.14)
n=l

which can be obtained by first minimizing the cost function in (3.5) with respect to

s(1),s(2),.. ,s(N) [66, 67], where PA = A (AHA)- AH. Although using (3.13)

for waveform estimation assumes that the angle estimates are not nearly identical,

maximizing F3(0) does not guarantee this assumption. Figure 3.3 shows F'(0) as a

function of 01 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 0 (only) and using the estimated

6 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 .1 = 32 horizontally polarized horn antennas. The








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 0, be the estimated angle vector obtained from the sth snapshot by

using one of the algorithms. Then our cross-validatory criterion is:



C N( 1) A(IY(n)12, (3.15)
s=1 n= l,n--6s








where N is the number of those snapshots that do not give almost identical angle

estimates, and P I = I-A (AHA)- AH. (Note that almost identical angle estimates

cause the matrix AHA in P' to be ill conditioned.) What the criterion does here
A(0,)
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(0,)A(0)]-1AH(0)y(n),(n = 1, N, and n = s);

3) obtain the residue by subtracting the signal part from the data with the data model

for each snapshot, i.e, residue(n) = y(n) A(0bs)9(n) = P' y(n),(n = 1, ,N,

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








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:
1N
S( P-:"1 Y(n)y"'"(n)P', (3.16)
(N 1)N =ln A() A(,)

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.








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.








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 3.. 9% 5, 6%

LS- Criterion C 1.5353e+04 3.3716e+04 11.0460e+04 1.s302)-+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





























22 04 026 028 03 0

2 0.22 0.24 0i 028 03 0.32


Frequency (Hz)

(a)


108

14


0.4
012







3403603 04 0


0.34 0.36 0,38 0.4


2 0.22 0.24 026 028 0.3 0.32 0.34 0.36 0.38 0.
Frequency (Hz)

(d)


2 0.22 024 0.26


02 0303


0.28 0 3 0.32
Frequency (Hz)

(b)


0.34 0.36 0.38 0.4


2 0. .2024 026


8
6

4

2
-n02 3 02 04 06 00 0


0.28 0.3 0.32 0.34 0136 038 0.4
Frequency (Hz)

(e)


28 0.3 032 0.34 0.36 0.38 0.4 6.2 022 0.24 026 0.28 0,3 0.32
Frequency (Hz) Frequency (Hz)

(c) (f)


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), K1 = 2 (2nd iteration). (d) Modulus of FFT of Y2 in step (2),

K = 2 (3rd iteration). (e) Modulus of FFT of yl in step (2), K = 2 (4th iteration).

(f) Modulus of FFT of y2 in step (2), K = 2 (5th iteration).


2t


2'













































102
RELAX
o MODE
x ANPA
+ LS-ESPRIT
CRB
S101
w
E




10
o 10'






-5 0 5 10 15 20
SNR (dB)


(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 01 = -10 and 02 = 20. (a) Angle
estimation. (b) Waveform estimation.



































(deg) 010 (de)

-20 -20










(dgeg))
10

5






-10

-15



Q(deg)

(b)

Figure 3.3. F3 vs. 01 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.









































(a) (b)


007


03804
03 03
0o2 -R 2 'E2
L- S-ESPRIT
01 01

0 5 10 15 20 25 30 35 00 5 10 15 20 25 30 35
Sensor Number Sensor Number

(c) (d)

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 C'S [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 fo (Hz). The kth bandpass signal sk(t) observed at a

reference point can be written as



Sk(t) = yk(t)j fot, (4.1)








where 7k(t) denotes the complex envelope. Let the signal be observed over a duration

[to, to + To]. The complex envelope can be written as


L
7'k(t) = E k(f )dJ2~t, to < t < to + To, (4.2)
/=1


where
1 to+To
rk(f) = k(t)e-j2,JXdt, (4.3)
T0 to


with fj = 1-(L+1)/2 1 = 1,2,..-,L. L is the number of frequency components

symmetrically placed around 0 Hz with f+l ft = B Thus the bandpass
-L-l--To"*

signal at the reference point (say the first sensor) can be written as


L
Sk() = k(fe)2(+). (4.4)
l=1


The kth signal at the mth sensor has a propagation time delay Tk,m so that


L
sk(t + Tk,m) k F(f)eJ2(fo+ft)(t+rkm) = Xk(t, Tk,m)ej2f (4.5)
l=i


where

(-r1,m sin Ok, k = 1,2,.., K, m = 1,2,..,. 31, (4.6)


with 6 being the spacing between sensors, C the propagation speed, and Ok the angle

of arrival of the kth incident signal relative to the array normal.







TI
Let Xk(t) Xk(,Tk,1) Xk(, Tk,2) X k t, Tk,M) Then


L
Xk(t) a(Ok, f + f) rk(fl)e2f, (4.7)
1=1


where a(0k, fo + fl) is the M x 1 direction vector of the kth source and has the form

T
a(0k, fo + fi) = eJ27(fo+fz)hr,1 ej2r(fo+f)k,2 ... C -(fo+fi)Tk,M (4.8)



Hence if K wideband signals along with some noise simultaneously impinge upon

the sensor array, the received data vector has the form


L L
x(t) = : [A(, fo + ft)r() + N(1)] eCi2fit = y(l)j27et, (4.9)
-l=1 =1


where

A(O, fo + ft) a(01, fo + f) *.. a(OK, o+ f) (4.10)


and

r(l) [= (fl) ... rK (f/) ,(4.11)


is the K x 1 signal vector, and



N(1)= N (fl) ... NM (f) (4.12)








is the M x 1 noise Fourier coefficient vector, and {y(l)} are by definition the M x 1

Fourier coefficient vectors of x(t):



y() = A(O, fo + f)r() + N(I), = 1,2,- L. (4.13)



We assume that the noise vector N(I) has zero-mean and



E{N(11)NH(12) = Q61,12, (4.14)



where Q is unknown.

The problem of interest herein is to estimate 01, 02, -*,0K from y(l), I =

1,2, L. 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

1.

4.3 Angle Estimation Using RELAX

The estimates {, (1), -, r(L)} can be obtained via RELAX by minimizing

the following nonlinear least-square criterion with respect to 0 and r(1),..., r(L)

(See (3.5)):


L
G = [y() A(0, fo + f))(1)]H [y(l) A(, fo + fi))(1)]. (4.15)
I=1








To estimate the parameters of the kth signal, consider


L
G2 = [yk(l)- a(0k, f + fl))Fk() [Yk(l a(k, f0 + f))k(l)], (4.16)
l=1


where

yk(l) = y()- a(, fo + f1)F(l), (4.17)
i=1,i k

and where I" is as defined before, and {0, Fi(1), r (L)}1,i k are assumed to be

given or estimated. Let


Yk(1) aH(Ok, f + fl)yk(l)
Yk() = (4.18)
M


which can be obtained by using FFTs with zero padding. Then similar to the nar-

rowband RELAX algorithm, 0k can be obtained by


L
S= arg max Yk l)2 (4.19)
S1=1


and

Fk(l) Yk( l)l0k l = 1, 2,--.,L. (4.20)


Hence Ok is obtained from the location of the dominant peak of the sum of the

focused periodograms Yk(1) 2, = 1,2, L. Then fk(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 (fo+fl)/fo since the spacing







between two adjacent FFT samples are different for different I The steps of our

RELAX algorithm for minimizing (6.20) are summarized as follows:

Step (0): Obtain y(l) from x(t) via DFT (discrete Fourier transform).

Step (1): Assume K = 1. Obtain 01 and fi(l) from y(l) as in (4.19) and (4.20),

1 = 1,2,. -,L.

Step (2): Assume K = 2. Compute y2(l) with (4.17) by using 01 and F1(1) obtained

in Step (1), I = 1,2,. -,L. Obtain 02 and F2(1) from y2(1) as in (4.19) and

(4.20), = 1,2,... L.

Next, compute yl(I) with (4.17) by using 02 and 12(1) and redetermine Q1 and

F1(1) from yl(1), 1= 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.








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 fo. The wideband sources have

the same center frequency fo = 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 01 = 100 and 02 = 200. 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, (CSI-ESPRIT provides biased angle estimates even

when the angle separation of the two sources is within the resolution of the spatial

periodogram.








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(fi) is assumed

to be a complex AR process of order 1, i.e, Nm(fi) = -alNm-i(fi) + Wm(fi), where

al = -0.85e-j4 and Wm(fi) is a zero-mean complex white Gaussian random process

with variance equal to c2. The SNR is defined by -101og_012 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 fo. 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.

















102
RELAX: --
CSM: o- -
10,



10



10




10 -5 0 5 10 15 20 25
SNR (dB)


(a)


10,
1 RELAX: N--
CSM: ----
10'



10


10------
10


-10 -5 0 5 10 15
SNR (b)


(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 01 = 100 and 02 = 200,

M = 8, and L = 33. (a) In the presence of white noise. (b) In the presence of

unknown AR noise.


-291 --- --- --- i -------------

.30

-31 N

.32

.33




-35. N N N N


0 10 20 30 40 50 60
Snapshot Number


Snapshot Number

(b)


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 01 = -330 and 02 = -36. (a) RELAX. (b)

C'S l-ESPRIT.


20 25 30














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








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 lth COLD pair, I = 1, 2, *-, L, has its center on the y-axis at an arbitrary y = 61.

For the lth COLD pair, the dipole parallel to the z-axis is referred to as the z-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 0 and 0, where 0 and 0 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 = Eoee + Ee,


(5.1)








where unit vectors ee, eo, and -e,, in that order, form a right-hand coordinate system

for the incoming signals and Eo and E, are the horizontal and vertical components of

the electric field respectively. In general, as time progresses, Eo and E, 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 so(t) 1)



E = E cos 7, (5.2)

E1 = E sin ye, (5.3)



where E denotes the amplitude of the incident signal. The and 17 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 < 7r/2 and yj is in the range

-7r < < 7r. a and f can also be used to compute 7 and y [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 Lsd

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 Ast. 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, s,(t) = ei[wot+(t)],
where wo is the carrier frequency and 0(t) is the modulating phase.








An incoming signal described by arbitrary electric field components Eo and E,

can be written as

E = E [(cos )eo + (sin yeJ")eo] (5.4)


Let us define the spatial phase factor


j 21- sinO (5)
qi = e oi (5.5)



where A0 is the wavelength of the signal. The effective heights of the short dipoles

and small loops are given by [104]



hsd = Ld sin (5.6)



and
27rAst
h = -i -A sin 0, (5.7)
Ao

respectively. Including the time and space phase factors in (5.4), we find that an

incoming signal characterized by (0, q, 7, 7, E) produces a signal vector in the COLD

pair centered at y = 61 as follows:


z(t) *= [ xi(t) (5.8)
zi(t) I I PCoLDESo(t)ql, (5.8)
[ Xd(t)








where
j 2^- sin 0 cos 7
PCOLD = (5.9)
-Lsd sin 0 sin y7e7j

An advantage of the COLD array is that its antenna elements are not sensitive to

the azimuth angle 0 of the signal because both the loops and dipoles have the same

sin 0 field pattern, as may be seen from Equation (5.9). Hence the incoming signal

described by (5.4) is independent of 0. We assume that the antennas and the incident

signals are co-planar, i.e., q = 900. Thus (5.9) becomes


Vo cos 7
PICOLD = V (5.10)
V, sin 7e7 J


where


2rAsi
V = 32A, (5.11)
Ao

V = -Lsd. (5.12)



Note that Vo and V4 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) = Eso(t)Vo cos 7. The zi(t) in (5.8) can be rewritten as


z(t) = us(t)qi,


(5.13)








where
1 1
u e= -- (5.14)
v_ tan -yej r

Assume that K signals, specified by incident angles 0k, k = 1,2, --, K, are

incident on the array. In addition, we assume a thermal noise voltage vector ni(t) is

present at each output vector z1(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 a21, where I denotes the identity matrix.

Under these assumptions, the total output vector received by the COLD pair

centered at y = bI is given by


K
zi(t) uksk(t)qlk +n1(t) 1, = 1,2, ,L, (5.15)
k=1


where Uk and qik are given by (6.9) and (5.5), respectively, with subscript k added to

each angular quantity. Further, Sk(t) = Eksok(t)V cos 7k, where Eksok(t) denotes the

kth narrowband signal. The incident signals may or may not be correlated (including

completely correlated, i.e., coherent) with each other.








Let z(t), s(t), and n(t) be column vectors containing the received signals, inci-

dent signals, and noise, respectively, i.e.,


z(t) =


zl(t)

Z2(t)




ZL(t)


s1(t)

s2(t)




SK(t)


nl(t)

n2(t)




-L(t)


(5.16)


The received signal vector has the form


where A is a 2L x K matri:


z(t) = As(t) + n(t),



x





S[A I]U,
[A I]U,


with : representing the Kronecker product,


. qiK

S. q2K




SqLK


(5.17)










(5.18)


(5.19)








and
U1 0

U = *.. (5.20)

0 UK

Assume that the element signals are sampled at N distinct times tn, n = 1, 2, N.

The random noise vectors n(tn) 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 Ok and the states of polarization described by (Yk, r/k) or (ak, 3k),

k = 1,2, .* K, from the measurements z(t,), 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

S- z(t)ZH("), (5.21)


where (-)H denotes the complex conjugate transpose and R denotes the estimate of

the following array covariance matrix:


R= E[z(t)z"l(t)].


(5.22)








It has been shown in [25, 23, 76] that an asymptotically (for large N) statistically
T
efficient estimator of the angles 0 = [ 01, 82, ".., K ] and the polarization pa-
T
rameters r = [ri, r2, ... rK ] can be obtained by minimizing the following

function:

f(0,r) = Tr P sH (5.23)

-H
where the symbol PL stands for the orthogonal projector onto the null space of A

and the columns in E, are the signal subspace eigenvectors of R that correspond to

the 1k largest eigenvalues of R, with /K = min[N, rank(S)]. Here S is the source

covariance matrix,

S = E{s(t,)sH(tn)}. (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 A1 > A2 > A -, which are

the K largest eigenvalues of R, and



A = A &21, (5.25)



where
S2L1
2 -A =2 tr(R) (5.26)
2L =K+ 2L K I =








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


(5.27)


P= I PAI + Pi
A P(A HI)V'


where


At = (AHA)-IAH,


(5.28)


0 VK


Thus minimizing f(0, r)


(5.30)


--rk
Vk= k= 1,2,...,K.
1

in (5.23) is equivalent to minimizing


f(0, r) = Tr (-PAI + P (A A I)V A EH
(At HOI)V)


W= {V([(AHA) 0 -I]V


and







with


(5.29)


(5.31)


(5.32)







be formed from some consistent estimates of 0 and r. Since P (AtHI)VEs = O(1/N),

{VH[(AHA)- I]V can be replaced by W without affecting the asymptotics of

the MODE estimator [25, 26]. Then we have


f(0, r) = f(0) + f2(0, r),


(5.33)


fl(0) = -Tr (A 0 I) {(AHA)-1 I} (AH I)E,2A-1H ,


(5.34)


f2(, r) = Tr [(AtH 0 I)VWVH(At 0 I)E8A'E .


The MODE estimates {0, r} are obtained by minimizing f, i.e.,


{0, r} = arg min [fi(0) + f2(0, r)].
9,r




Ah = odd columns of (AtH 0 I),


(5.35)


(5.36)




(5.37)


A, = even columns of (AtH 0 I).


where


(5.38)








Let Vh and V, be the following K x K diagonal matrices:



Vh =diag r, ..., -r (5.39)



and

V, = I. (5.40)


Then

(AtH 0 I)V = AhVh + AV'. (5.41)


Thus f2(0, r) in (5.35) can be rewritten as



f2(0,r) = Tr [vHAHEi A Ei"AhVhW
2 -1
+Tr [VHAHEEA A EifAVVW

+Tr VHAHE, 2A-lEfAhVhW
Tr [vH ... a. vw]
+Tr VHHAv'EsA A EAvVvW (5.42)



Since Vh and V, are diagonal matrices, Equation (5.42) can be written in the fol-

lowing matrix form:



hVh
f2, r)= V T Q(0) (5.43)
e







where


(AHESA A EEAh) WT (A EA A -1EA,V) 0 WT
Q(W) =2
(AHE,A A EHAh) WT (AHE A EfA,) 0 W

Ql(e9) Q)2() (5.44)

QH(0) Q3(0)


with 0 denoting the Hadmard-Schur matrix product (i.e., the elementwise multipli-

cation),

Vh = -r -r* (5.45)


and

e= ... (5.46)


Note that the polarization parameters are contained only in vh. By setting 9f2/Ovh

0, we obtain

Vh = -Q,1()Q2(0)e. (5.47)


Using (5.47) in (5.43) gives


f3(0) = eT [Q3() QH(0)QI()()] e, (5.48)


which is a concentrated function depending only on 0. The MODE estimates {0, f}

can be obtained by

0 = argmin[fi(0) + f3(0)], (5.49)
6f








and using 0 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 0 and r (see the discussions below).

Step 2: Determine 0 by minimizing fi(O) + f3(O) 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 0 obtained in Step 2 in (5.47).

Step 4: Determine the j and ij from i with



ik = tan-' ) (5.50)

ik = arg -V k=1,2,'.,K. (5.51)



For signals that are not highly correlated or coherent with each other, the initial

estimates of 0 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 0 may be determined by setting W = I and minimizing

fi () + f3() as shown in Equation (5.49). The initial estimate of r can be calculated

by using the initial estimate of 0 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].








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 Poincar6 sphere that represent the actual state of

polarization (7, 7) and the estimated state of polarization (7, ), respectively. Let (

be the angular distance between M and M. Then [31]



cos ( = cos 27 cos 2j + sin 27 sin 2j cos(r 7). (5.52)



where ( is always in the range 0 < ( < 7r.

Applying the first-order approximation to the left side of 5.52 yields



S= 4(k k)2 + sin2(27k)()k 7k)2. (5.53)



The asymptotic variances of the polarization estimates are obtained with (5.53) and

the accuracy results on j and ii given below.

Let

r = ^YT 1T ]T (5.54)








It follows from [25, 76] that the asymptotic (for large N) statistical distribution of 4

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-1 = NRe [tr {A PAiSA HR-AS}], (5.55)



where A, = OA/dir with 7r denoting the ith 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 IVoEkl = IV,.FE, = 1, k = 1,2. Hence, the signal-to-

noise ratio (SNR) used in the simulations is -101ogo102 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 z-axes dipoles. The








counterpart of Equation (5.9) for the CCD array can be written as


S Lsd cos 7 cos 0 + Lsd sin ye7j sin 0 cos (
I1CCD 1 (5.56)
-Led sin 4 sin 'yej


In the following examples, the antennas and the incident signals are assumed to be

co-planar, i.e., 6 = 900, for both the COLD arira and CCD array. For = 900,

(5.56) becomes
S Lsd cos y cos 0
A CCD = (5.57)
-Ld sin ye~J

(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 (01 = 2 =

450). Figure 5.2 shows the root-mean-squared errors (RMSEs) of the estimates of

the first signal as a function of angle separation AO when two correlated signals

with correlation coefficient 0.99 arrive at the array from angles 01 = -A0/2 and

02 = AO/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








broadside of the arrays, the CRBs for the COLD and CCD arrays are similar. This

case corresponds to small incident angles, which make IACCD similar to PCOLD as

may be seen by comparing Equations (6.5) and (5.57).

In Figure 5.3, we consider the case where the signals with identical horizontal

polarizations (al = a2 = 00, /P = /2 = 00) arrive from angles away from the broadside

of the array. In this case, the CCD array is outperformed by the COLD array.

This result occurs because the signal outputs at the y-axis dipoles are attenuated

by a factor of cos 0 for the CCD array (see Equation (5.56)). For the COLD array,

however, the signal outputs at both the dipoles and the loops are independent of the

incident angle 0 (see (6.5)). Note also that the RMSEs of the angle estimates first

decrease and then increase even as the angle separation increases. This result occurs

because the incident angle of the second signal approaches 900 for very large AO and
1
the RMSEs of angle estimates are approximately proportional to I for large A0
Cos2 0

[71]. We note again that MODE gives better performance than MUSIC and NSF and

achieves the CRB as A0 increases. We have also found that the CRBs for COLD

array are much lower than those for CCD array, especially when 0 approaches 900.

Second, we consider how the polarization separation affects the estimator per-

formance. Consider the case where two incident signals with a correlation coefficient

0.99 arrive from angles 01 = 500 and 02 = 700. We assume that the corresponding

ellipticity angles are al = 450 Aa and a2 = 450 and the orientation angles are

01 = 02 = 00. The polarization separation between the two polarization states is

2Aa. Figure 5.4 shows the RMSEs of the direction and polarization estimates as a




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