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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, YuXian 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 CRAMERRAO BOUND FOR PARAMETER ESTIMATES ..... 119 B PROOF OF EQUATION (5.27) ........................ 121 REFERENCES ................... ............... 123 BIOGRAPHICAL SKETCH .......................... .. 132 . . . . . . . . KEY TO ACRONYMS AOA: AngleOfArrival AR: Autoregressive ARMA: Autoregressive Moving Average AP: Alternating Projection ANPA: Alternating NotchPeriodogram Algorithm CCD: Cocentered Crossed Dipoles COLD: Cocentered Orthogonal Loop and Dipole CRB: CramerRao Bound CSM: Coherent Signalsubspace Method DFT: Discrete Fourier Transform DSCDMA: DirectSequence CodeDivision Multiple Access EM: ElectroMagnetic ESPRIT: Estimation of Signal Parameters via Rotational Invariance Techniques FFT: Fast Fourier Transform IQML: Iterative Quadratic Maximum Likelihood LSESPRIT: LeastSquares based ESPRIT LSML: Large Sample Maximum Likelihood ME: Maximum Entropy ML: Maximum Likelihood MODE: Method Of Direction Estimation MUSIC: Multiple SIgnal Classification NLS: Nonlinear LeastSquares NSF: Noise Subspace Fitting RELAX: RELAXation algorithm for the minimization of the NLS criterion RMSE: RootMeanSquared Error SNR: SignaltoNoise Ratio SSF: Signal Subspace Fitting TLSESPRIT: Total LeastSquares 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 twodimensional 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) CSMESPRIT. 68 5.1 A linear COLD array.............. ........... 90 5.2 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 Rootmeansquared 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 wellknown algorithms including ESPRIT with forward/backward spatial smoothing, MODE/WSF, and AP/ANPA for narrowband signals; and CSMESPRIT 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 Cocentered 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 spacetime 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 spectrumlike 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 wellknown Maximum Entropy (ME) spectral estimation method in geophysics by Burg [8] and the YuleWalker autore gressive estimation method are the early parametric approaches. The introduction of subspacebased 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 nonlinear 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 subspacebased approach results in a resolution that is not limited by the array aperture, provided that the number of data samples or the signaltonoise 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 Fourierlimit. 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 Cocentered 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 anglesofarrival (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 wellknown 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 NotchPeriodogram 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 Signalsubspace Method) based ESPRIT. In Chapter 5 an arbitrary linear array that consists of Cocentered 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 cocentered 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 CramerRao 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 timevarying 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 wellknown plane wave solutions u(t, x) = ej(kTxt), (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 farfield 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 angleofarrival (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)ejkz, (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)ejkz. (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 (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 righthanded and positive for lefthanded po larization. For the case shown in Fig. 2.1a is positive. The parameter 7 is defined as Y =tan1( ), (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: Greatcircle angle = 27, (2.19) Equatortogreatcircle 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 twodimensional 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 twodimensional 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(tkr) (2.23) k=l for purely exponential signals and K y (t) = a(kj(utkr)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 anglesofarrival (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(L1)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, zeromean 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 eigenstructurebased 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 spectrumlike 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 wellknown 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 Rlla(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)Rla(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 modelbased parametric approach. The most important modelbased 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 subspacebased 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 rootMUSIC [60, 61] for a uniform linear array. The subspacebased techniques also lead to the subspacebased 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 subspacebased ML approximation approaches are computationally more attractive. However, the subspacebased 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 leastsquares (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 Kdimensional 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 spectralbased method is a natural choice for an initial estimator provided all sources can be resolved. The alternating projection (AP) [66] and alternating notchperiodogram algorithm (ANPA) [67] are two similar efficient approaches to minimize the nonlinear leastsquares 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 FFTbased 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 zeromean with secondorder 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, zeromean and circularly symmetric Gaussian random vector with covariance matrix R = A(O)R,AH(0) + a21. (2.52) The negative loglikelihood function (ignoring constant terms) of the complete data set y(l), , y(N) is given by 1(0, R,, 02) = loglRI + tr R1R}. (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 nonlinear function of 0 and the minimization 0 cannot be found analytically in general. A Newtontype 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. SubspaceBased 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 subspacebased 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 (signaltonoise 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 MinNorm 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(L1)i ... ej(L1)QK Making use of this special structure results in several computationally and statis tically efficient subspacebased algorithms. This type of algorithm may include ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) [58, 59], and RootMUSIC [60] that will be introduced later. The ESPRIT algorithm exploits a socalled shift structure of the Vandermonde matrix A. let A1 and A2 be the submatrices 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 fullrank 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 = T1 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 leastsquares sense results in LS ESPRIT, while in a TotalLeastSquares sense leads to TLSESPRIT. * RootMUSIC Es2 = Esl, The idea of the RootMUSIC 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 ,L1]T, has K zeroroots 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(z1) to substitute for aH(z), which gives the RootMUSIC 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 RootMUSIC 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 forwardbackward (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 socalled 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(L1)R,(L1)AH + r21, (2.71) where J is an L x L permutation matrix, whose components are zero excepts for ones on the antidiagonal. Then the FB array covariance matrix is obtained by RFB = I(R + RB) = ARAH + 2, (2.72) where R, = (R, + ~(L1)R,(L1))/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. SubspaceBased ML Approximations Subspacebased 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 subspacebased 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 leastsquares fit of the two subspaces by the following criterion: {(,T} = arg min IE, ATIIv, (2.73) 0,T where IIAI1 = tr{AWAH} and W is a K' x K' positive definite weighting matrix. This is a separable nonlinear leastsquares 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 noniterative 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. Twodimensional 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 semidefinite 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 twostep 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 nth frequency bin, A(0, f)) = [ a(0i, fn) a(OK, f)) ] is a L x K matrix, whose kth column is the direction vector for the kth 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 signalsubspace 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 signalsubspace 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 subspacebased 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 DSCDMA (DirectSequence 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 DSCDMA signals. These algorithms have been shown to be robust against the multiuser nearfar problem of DSCDMA 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 wellknown 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 Mutiparameter Adaptive Radar System (MARS) developed at McMaster University. More importantly, we explain why RELAX outperforms other wellknown algorithms including ESPRIT, MODE/WSF, and AP/ANPA. For the experimental data, since we do not know the true incident angles, we introduce a crossvalidatory 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 (M1)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 zeromean 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 leastsquares (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 leastsquares 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 = 103, 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 MonteCarlo 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 zeromean 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 LSESPRIT (least squares based ESPRIT) [12] are obtained by using 9(n) = [AH(b)A(b)]lAH(6)y(n). (3.13) The rootmeansquared error (RMSE) of the kth waveform estimate is defined as VE=1 lsk(n) k(n)2. Note that LSESPRIT 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 Cram6rRao 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 Multiparameter 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 12bit 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 crossvalidatory 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 crossvalidatory criterion is: C N( 1) A(IY(n)12, (3.15) s=1 n= l,n6s where N is the number of those snapshots that do not give almost identical angle estimates, and P I = IA (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 crossvalidatory criterion C obtained by using RELAX, ANPA, MODE, LSESPRIT, and TLSESPRIT (total least squares based ESPRIT) [12] at frequencies 8.62, 9.76, 9.79, and 12.34 GHz. Note that LSESPRIT and TLSESPRIT 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 LSESPRIT and TLSESPRIT have similar performances for this data set, but TLSESPRIT has a much higher failure rate, which indicates that when waveform estimation is desired, LSESPRIT 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 zeropaddings. 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 zeropaddings 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 zeropaddings 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 multisnapshot 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 wellknown 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: Crossvalidatory 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 + LSESPRIT 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 SESPRIT 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 wellknown C'S [29] based LSESPRIT (CSMESPRIT). We also explain why wideband RELAX can outperform the CSMESPRIT. 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)ej2,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 LlTo"* 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 zeromean 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 leastsquare 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 LSESPRIT (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 zeromean 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 signaltonoise 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 MonteCarlo simulations. Figure 4.1(a) shows the rootmeansquared 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 CSMESPRIT stops decreasing when SNR reaches a certain level. This is because, due to focusing, (CSIESPRIT 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 CSMESPRIT 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) = alNmi(fi) + Wm(fi), where al = 0.85ej4 and Wm(fi) is a zeromean 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 CSMESPRIT. Finally, we apply both RELAX and CSMESPRIT 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 lESPRIT. 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 Cocentered 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 cocentered 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 subspacebased MODE algorithm [25, 26] for both angle and polarization estimation. Since the MODE algo rithm is a signal subspacebased 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 noncoherent. 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 2Lelement linear array consisting of L Cocentered 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 yaxis at an arbitrary y = 61. For the lth COLD pair, the dipole parallel to the zaxis is referred to as the zaxis dipole and the loop parallel to the xy 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 righthand 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 onetenth 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 threetenths 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 phaseshift 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 coplanar, 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 zeromean 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 subspacebased 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,2A1H , (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, 2AlEfAhVhW 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 HadmardSchur 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 onedimensional 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 noncoherent 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 firstorder 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 CramerRao bound (CRB), CRB. The ijth element of CRB1 is given by CRB1 = NRe [tr {A PAiSA HRAS}], (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 MonteCarlo simulation results. We compare MODE with MUSIC and NSF for both angle and polarization estimation. The simulation results were obtained by using 50 MonteCarlo 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 signalto 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 zaxes 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 coplanar, 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 coplanar, we will need twodimensional 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 rootmeansquared 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 yaxis 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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