Helsinki University of Technology Signal Processing LaboratoryTeknillinen korkeakoulu Signaalinkasittelytekniikan laboratorio

Espoo 2007 Report 60

Signal processing for arbitrary sensor array configurations:

theory and algorithms

Fabio Belloni

Dissertation for the degree of Doctor of Science in Technology to be pre-sented with due permission of the Department of Electrical and Communi-cations Engineering for public examination and debate in Auditorium S4 atHelsinki University of Technology (Espoo, Finland) on the 12th of October,2007, at 12 o’clock noon.

Helsinki University of TechnologyDepartment of Electrical and Communications EngineeringSignal Processing Laboratory

Teknillinen korkeakouluSahko- ja tietoliikennetekniikan osastoSignaalinkasittelytekniikan laboratorio

Distribution:Helsinki University of TechnologySignal Processing LaboratoryP.O. Box 3000FIN-02015 TKKFinlandTel. +358-9-451 3211Fax. +358-9-452 3614E-mail: Mirja.Lemetyinen@tkk.fiWebpage: http://wooster.tkk.fi

c© Fabio Belloni

ISBN 978-951-22-8969-1 (Printed)ISBN 978-951-22-8970-7 (Electronic)ISSN 1458-6401

MultiprintEspoo 2007

ABABSTRACT OF DOCTORAL DISSERTATION HELSINKI UNIVERSITY OF TECHNOLOGY

P. O. BOX 1000, FI-02015 TKK

http://www.tkk.fi

Author Fabio Belloni

Name of the dissertation

Manuscript submitted 25 June, 2007 Manuscript revised 31 August, 2007

Date of the defence 12 October, 2007

Article dissertation (summary + original articles)Monograph

Department

Laboratory

Field of research

Opponent(s)

Supervisor

Instructor

Abstract

Keywords Direction of arrival (DoA) estimation, array transform techniques, manifold separation, error analysis

ISBN (printed) 978-951-22-8969-1

ISBN (pdf) 978-951-22-8970-7

Language English

ISSN (printed) 1458-6401

ISSN (pdf)

Number of pages 169

Publisher Helsinki University of Technology, Signal Processing Laboratory

Print distribution Helsinki University of Technology, Signal Processing Laboratory

The dissertation can be read at http://lib.tkk.fi/Diss/2007/isbn9789512289707

Signal Processing for Arbitrary Sensor Array Configurations: Theory and Algorithm

X

Electrical and Communications Engineering

Signal Processing Laboratory

Sensor Array Signal Processing

Prof. Michael D. Zoltowski (Purdue Univerity) and Dr. KimmoKalliola (Nokia Research Center)

Prof. Visa Koivunen

Dr. Andreas Richter

X

Sensor array systems are employed in many application areassuch as multi-antenna wireless communications, radarand biomedicine. Among the research areas of sensor array signal processing, the problem of finding the direction-of-arrival (DoA) at which a propagating wavefield impinges on a sensor array is a popular research field. Applicationsexplicitly needing directional information include beamforming, localization, surveillance, and channel sounding.

Most high-resolution and computationally efficient sensorarray processing algorithms have been developed for idealsensor arrays with regular geometry and known sensor response. In practice, the geometry of the array can not be cho-sen freely and the array response is always an unknown quantity which can be estimated only through noisy calibrationmeasurements. Consequently, the above algorithms are not applicable on real-world arrays with arbitrary configuration.

This thesis focuses on deriving and analyzing novel algorithms providing high-resolution, optimal or close to optimalstatistical performance, and low computational complexity, despite the antenna array geometry and imperfections. Inparticular, the problem of reformulating the array signal processing model so that computationally efficient high-reso-lution DoA estimation algorithms can be used with sensor arrays of arbitrary configuration is addressed. The contribu-tions in this thesis are in the areas of array transform techniques, antenna modelling, and signal processing algorithmsusing sensor arrays of arbitrary configurations.

In this thesis, the key ideas and performance of the most common array transform techniques are investigated. Thetransformation errors and their impact on the DoA estimatesare analyzed. Novel algorithms developed for reducingthe bias and mitigating the excess variance in the DoA estimates are introduced. Furthermore, an alternative approachto the above techniques known as manifold separation technique (MST) is analyzed. The introduced MST exploits theeffective aperture distribution function (EADF) and it is amethod for modelling the azimuthal response of sensorarrays with arbitrary configurations by using Vandermonde structured models. A novel MST-based polynomial rootingDoA algorithm is proposed and the effect of noisy calibration data on its statistical performance is also studied.Implementation issues and the use of the developed techniques in real-world arrays are discussed as well.

ABVÄITÖSKIRJAN TIIVISTELMÄ TEKNILLINEN KORKEAKOULU

PL 1000, 02015 TKK

http://www.tkk.fi

Tekijä Fabio Belloni

Väitöskirjan nimi

Käsikirjoituksen päivämäärä 25 June, 2007 Korjatun käsikirjoituksen päivämäärä 31 August, 2007

Väitöstilaisuuden ajankohta 12 October, 2007

Yhdistelmäväitöskirja (yhteenveto + erillisartikkelit)Monografia

Osasto

Laboratorio

Tutkimusala

Vastaväittäjä(t)

Työn valvoja

Työn ohjaaja

Tiivistelmä

Asiasanat Suunnan estimointi, muunnostekniikat, manifold separation, virheanalyysi, antennien kalibrointi

ISBN (painettu) 978-951-22-8969-1

ISBN (pdf) 978-951-22-8970-7

Kieli Englanti

ISSN (painettu) 1458-6401

ISSN (pdf)

Sivumäärä 169

Julkaisija Helsinki University of Technology, Signal Processing Laboratory

Painetun väitöskirjan jakelu Helsinki University of Technology, Signal Processing Laboratory

Luettavissa verkossa osoitteessa http://lib.tkk.fi/Diss/2007/isbn9789512289707

Sensoriryhmien signaalinkäsittelyn teoriaa ja algoritmeja geometrialtaan vapaavalintaisille ryhmille

X

Electrical and Communications Engineering

Signal Processing Laboratory

Sensor Array Signal Processing

Prof. Michael D. Zoltowski (Purdue Univerity) and Dr. KimmoKalliola (Nokia Research Center)

Prof. Visa Koivunen

Dr. Andreas Richter

X

Sensoriryhmiin perustuvia järjestelmiä hyödynnetään monilla eri sovellusalueilla, kuten radiotietoliikenteen älyanten-nijärjestelmissä, tutkatekniikassa sekä lääketieteellisissä mittauksissa. Sensoriryhmien signaalinkäsittelynosa-alueistasignaalien tulosuunnan (DoA) estimointi on keskeisimpiä tutkimusaiheita. Sovelluksia, jotka tarvitsevat yksikäsitteistäsuuntatietoa, ovat esimerkiksi keilanmuodostus, paikantaminen, valvontasovellukset, navigointi ja kanavaluotaus.

Useimmat laskennallisesti tehokkaista sensoriryhmien korkearesoluutioalgoritmeista on kehitetty ideaalisille sensori-ryhmille, joiden geometria on säännöllinen ja sensorien vaste on tunnettu. Käytännössä geometriaa ei kuitenkaan voidavalita vapaasti ja vasteet sisältävät aina suureita, jotkaovat ainoastaan estimoitavissa kalibrointimittauksista. Tästäjohtuen edellä mainitut algoritmit eivät tyypillisesti ole suoraan sovellettavissa reaalimaailman sensoriryhmiin.

Tässä väitöskirjassa johdetaan ja analysoidaan uusia algoritmeja, joiden tavoitteina ovat korkea resoluutio, tilastollises-ti optimaalinen tai lähes optimaalinen suorituskyky, sekälaskennallinen tehokkuus sensoriryhmien geometriasta jaepäideaalisuuksista riippumatta. Erityistä huomiota kiinnitetään sensoriryhmien signaalinkäsittelymallien muokkaami-seen siten, että voidaan hyödyntää laskennallisesti tehokkaita suunnan estimointialgoritmeja mielivaltaisiin sensori-ryhmien kokoonpanoihin. Työn päätuloksia ovat uudet laskennalliset muunnostekniikat, antenniryhmämallit jasignaalinkäsittelyalgoritmit mielivaltaisille sensoriryhmän kokoonpanoille.

Tässä työssä käydään läpi yleisimpien antenniryhmien signaalinkäsittelyssä käytettävien laskennallisten muunnostek-niikoiden periaatteet ja suorituskyky. Muunnosten aiheuttamia virheitä ja niiden vaikutusta suunnan estimoinnin suori-tuskykyyn analysoidaan. Työssä kehitetään uusia algoritmeja estimointiharhan ja suuntaestimaattien varianssin pienen-tämiseksi. Lisäksi kehitetään vaihtoehtoinen MST-periaateeseen (Manifold Separation Technique) perustuva menetel-mä. MST-tekniikka hyödyntää efektiivistä apertuurin jakaumafunktiota (EADF) ja mallintaa sensoriryhmien kulma-vastetta mielivaltaisilla kokoonpanoilla Vandermonde-rakenteisen mallin avulla. Työssä johdetaan uusi MST-pohjainenpolynomijuuriin perustuva tulosuunnan estimointialgoritmi ja tutkitaan kohinaisen kalibrointitiedon vaikutustasen ti-lastolliseen suorituskykyyn. Myös tekniikoiden soveltuvuutta ja toteutettavuutta todellisilla antenniryhmillä arvioidaan.

Acknowledgements

The research work for this doctoral thesis was carried out in the SignalProcessing Laboratory, Helsinki University of Technology (TKK), during theyears 2003–2007. The Statistical Signal Processing group, led by Prof. VisaKoivunen, is a part of SMARAD (Smart and Novel Radio Research Unit)Center of Excellence in research nominated by the Academy of Finland.

First, I wish to express my sincere gratitude to my thesis advisor Prof.Visa Koivunen for his continuous encouragement, guidance, and constantsupport during the course of this work. I also want to express my sinceregratitude to Dr. Andreas Richter for this guidance and friendship. It hasbeen an honor to work with such deeply dedicated scientists. The discussionsI had with Prof. Koivunen and Dr. Richter, their comments, suggestions,and constructive criticism have greatly contributed to the technical qualityof the thesis.

I would like to thank the thesis pre-examiners, Prof. Sergiy Vorobyovand Prof. Magnus Jansson, for their constructive comments that helped toimprove the final version of the manuscript. Furthermore, Prof. Iiro Har-timo, the former director of the Graduate School in Electronics, Telecom-munications and Automation (GETA) and the GETA coordinator MarjaLeppaharju deserve many thanks. The work of the laboratory secretaries,Anne Jaaskelainen and Mirja Lemetyinen, in assisting and helping me withall practical issues and arrangements is also highly acknowledged.

I would like to thank all the colleagues at the Signal Processing Labora-tory for having created such a nice working atmosphere. In particular, I amgrateful to M.Sc. Traian Abrudan, Dr. Jan Eriksson, Dr. Maarit Melvasalo,Dr. Timo Roman, M.Sc. Jussi Salmi, and M.Sc. Eduardo Zacarias for allthe constructive and relaxing discussions.

This research work was funded by the Academy of Finland (TULE pro-gram), the Finnish Defence Forces Technical Research Center (PvTT), andthe GETA graduate school. Also, the financial support of Tekniikan edista-missaatio (TES), Nokia and Wihuri foundations, is gratefully acknowledged.

I wish to express my sincere gratitude to my parents, Luciano and Ilva,and my sister Veronica for their love, constant support, and encouragementduring all this time. I also would like to thank my Finnish family, in partic-ular Lila, Eino and Kalle, for welcoming me into their lives and for caring

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about me. In addition, I would like to thank my friends Daniel, Anna, Jean-Luc, and Hanna for their genuine friendship. Also all my friends in Italydeserve a special thank for having always made me feel close to them.

Finally, I haven’t got enough words for thanking my dear wife Kaisa forher love, close support and encouragement and for having recently given mea beautiful daughter, Lilja, who brought so much more love and happinessto my life.

Espoo, September 2007

Fabio Belloni

vi

To Kaisa and Lilja

vii

viii

Contents

Acknowledgements v

List of original publications xi

List of abbreviations xiii

List of symbols xv

1 Introduction 1

1.1 Motivation of the thesis . . . . . . . . . . . . . . . . . . . . . 11.2 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . 51.5 Summary of the publications . . . . . . . . . . . . . . . . . . 6

2 Array Signal Processing Models 9

2.1 Ideal Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . 102.2 Real-World Arrays . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Array Calibration Data Model . . . . . . . . . . . . . . . . . 142.4 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 18

3 Data-Driven Transform Techniques 19

3.1 Array Interpolation Techniques . . . . . . . . . . . . . . . . . 203.2 Beamspace Processing Techniques . . . . . . . . . . . . . . . 24

3.2.1 Phase-Mode Excitation Principle . . . . . . . . . . . . 253.2.2 Davies and Beamspace Transforms for UCA . . . . . . 27

3.3 Beamspace Mapping Errors . . . . . . . . . . . . . . . . . . . 293.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Model-Driven Transform Techniques 37

4.1 Wavefield Modelling and Manifold Separation . . . . . . . . . 384.2 Effective Aperture Distribution Function . . . . . . . . . . . . 404.3 Element-Space root-MUSIC . . . . . . . . . . . . . . . . . . . 42

ix

4.4 Modelling Errors due to Calibration Noise . . . . . . . . . . . 444.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Comparison of Transform Techniques 53

5.1 Vandermonde Approximation Errors . . . . . . . . . . . . . . 545.1.1 Theoretical Scenario . . . . . . . . . . . . . . . . . . . 555.1.2 Realistic Scenario . . . . . . . . . . . . . . . . . . . . . 57

5.2 Approximate CRLB for Real-World Arrays . . . . . . . . . . 595.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Summary 63

Appendix I 67

Bibliography 69

x

List of original publications

(I) F. Belloni, and V. Koivunen, ”Unitary root-MUSIC technique for Uni-form Circular Array,” in Proceedings of the IEEE International Sym-posium on Signal Processing and Information Technology (ISSPIT),Darmstadt, Germany, December 14-17, 2003.

(II) F. Belloni, and V. Koivunen, ”Reducing Bias in Beamspace Methodsfor Uniform Circular Array,” in Proceedings of the IEEE InternationalConference on Acoustics, Speech, and Signal Processing (ICASSP),Philadelphia, PA, USA, March 19-23, 2005.

(III) F. Belloni, A. Richter, and V. Koivunen, ”Reducing Excess Variancein Beamspace Methods for Uniform Circular Array,” in Proceedings ofthe IEEE Workshop on Statistical Signal Processing (SSP), Bordeaux,France, July 17-20, 2005.

(IV) F. Belloni, A. Richter, and V. Koivunen, ”Avoiding Bias in CircularArrays Using Optimal Beampattern Shaping and EADF,” in Proceed-ings of the 39th Annual Asilomar Conference on Signals, Systems andComputers, Pacific Grove, California, USA, Oct. 30 - Nov. 2, 2005.

(V) F. Belloni, A. Richter, and V. Koivunen, ”Extension of root-MUSICto non-ULA Array Configurations,” in Proceedings of the IEEE In-ternational Conference on Acoustics, Speech, and Signal Processing(ICASSP), Toulouse, France, May 14-19, 2006.

(VI) F. Belloni, and V. Koivunen, ”Beamspace Transform for UCA: ErrorAnalysis and Bias Reduction,” IEEE Transactions on Signal Process-ing, vol. 54 no. 8, pp. 3078-3089, August 2006.

(VII) F. Belloni, A. Richter, and V. Koivunen, ”Performance of root-MUSIC algorithm using Real-World Arrays,” in Proceedings of the14th European Signal Processing Conference (EUSIPCO), Florence,Italy, September 4-8, 2006.

(VIII) F. Belloni, A. Richter, and V. Koivunen, ”DoA Estimation via Man-ifold Separation for Arbitrary Array Structures,” IEEE Transactionson Signal Processing, vol. 55 no. 10, pp. 4800-4810, October 2007.

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xii

List of Abbreviations

[P. n] nth original publication1-D one-dimensional2-D two-dimensional3-D three-dimensionalAIT Array Interpolation TechniqueBT Beamspace TransformCRLB Cramer-Rao Lower BoundCUBA Circular Uniform Beam ArrayDoA Direction of ArrivalDFT Discrete Fourier TransformDT Davies TransformEADF Effective Aperture Distribution FunctionEEG ElectroEncephaloGramENR EADF to calibration Noise RatioES Element-SpaceESPAR Electronically Steerable Parasitic Antenna RadiatorESPRIT Estimation of Signal Parameter via Rotation Invariance TechniqueEVD EigenValue DecompositionIDFT Inverse Discrete Fourier Transformi.i.d. independent and identically distributedLS Least SquaresMBT Modified Beamspace TransformMEG MagnetoEncephaloGraphyMIMO Multiple-Input Multiple-OutputMODE Method Of Direction EstimationMSE Mean Square ErrorMST Manifold Separation TechniqueMUSIC MUltiple SIgnal ClassificationMVDR Minimum Variance Distorsionless ResponcePSSPA Polarimetric Semi-Spherical Patch ArrayPUCPA Polarimetric Uniform Circular Patch ArrayPULPA Uniform Linear Patch ArrayRARE RAnk Reduction Estimator

xiii

RHS Right Hand SideRMSE Root Mean Square ErrorSDMA Space-Division Multiple AccessSNR Signal-to-Noise RatioSPUCPA Stacked Polarimetric Uniform Circular Patch ArraySVD Singular Value DecompositionTLS Total Least SquaresUCA Uniform Circular ArrayULA Uniform Linear ArrayURA Uniform Rectangular ArrayURPA Uniform Rectangular Patch ArrayWLS Weighted Least SquaresWSF Weighted Subspace Fitting

xiv

List of Symbols

a estimate of parameter a[a]n nth element of the vector a

diag{a} diagonal matrix of the argument vector a

[A]n,m (n,m)th element of the matrix A

A−1 inverse of matrix A

rank{A} rank of matrix A

E{·} expectation operatorℜ{a} real part of complex scalar a‖ · ‖F Frobenius norm (matrix norm)‖ · ‖ L2-norm (vector norm)(·)H Hermitian or conjugate-transpose(·)T transpose(·)∗ conjugate⊙ Hadamard-Schur (i.e. element-wise) product∠(z) phase angle of the complex variable z (polar coordinates)an(φp) response of the nth sensor in the direction φp

a ∈ CN×1 array steering vector

a ∈ CN×1 a perturbed by calibration noise

aI ∈ CN×1 ideal array steering vector

ac ∈ CN×1 steering vector of a calibrated array

as ∈ CN×1 steering vector of a directional synthetic UCA

auca ∈ CN×1 ideal UCA array steering vector

aula ∈ CN×1 ideal ULA array steering vector

A ∈ CN×P array steering vector matrix

Ac ∈ CN×Q calibration matrix

Alc ∈ C

N×Ql lth sub-block of Ac

Ac ∈ CN×Q measurements noise perturbed calibration matrix

A array manifoldbb ∈ C

Mv×1 virtual array steering vector after BTbd ∈ C

Mv×1 virtual array steering vector after DTBl

v ∈ CNv×Ql lth virtual array steering vector matrix

c wavefield propagation speedcm mth Fourier series coefficientC complex domain

xv

d interelement spacingd ∈ C

M×1 truncated Vandermonde structured vector used in the MSTd′ ∈ C

M0×1 first order derivative of d

db ∈ CMv×1 Vandermonde structured vector used in the BT and DT

de ∈ CQ×1 Vandermonde structured vector used in the MST

ds ∈ C∞×1 Vandermonde structured vector used in wavefield modelling

D ∈ CM×P matrix formed by P Vandermonde structured vector d

E ∈ CN×N eigenvectors matrix

Es ∈ CN×P eigenvectors matrix defining the signal subspace

Eη ∈ CN×(N−P ) eigenvectors matrix defining the noise subspace

Eη ∈ CN×(N−P ) perturbed noise subspace

f frequencyfc center frequencyf c

m mth far-field pattern for the ring aperturefs

m mth far-field pattern for the sampled circular array (UCA)F pseudo-spectrum of the ES-MUSIC algorithmF ∈ C

N×h orthonormal transformation matrix with h ≤ NG ∈ C

N×M truncated EADF matrix

G ∈ CN×M G perturbed by calibration noise in W

G ∈ CN×M0 deterministic matrix, i.e. G perturbed by W

Ge ∈ CN×Q EADF matrix

Ge ∈ CN×Q EADF matrix perturbed by calibration noise in We

Ge ∈ CN×Q deterministic matrix, i.e. Ge perturbed by W

Gs ∈ CN×∞ sampling matrix

H Hilbert spaceIN ∈ R

N×N identity matrixj imaginary unitJm Bessel function of the first kind of order mJ ∈ C

Mv×Mv diagonal matrix containing Bessel functionsk discrete time instantK number of observations (snapshots)L number of angular sectorsL ∈ R

Q×Q selection matrixm excitation mode indexM number of selected modes after truncation in MSTMb largest excitation mode index used in the BTM0 optimal number of selected modesMt largest selected modes index after truncation in MSTMv virtual array size used in the BTN number of sensors in the arrayNv number of virtual array elements in the AITn(k) ∈ C

N×1 observation noise vector at time instant kN ∈ C

N×K observation noise matrix

xvi

Nb ∈ CMv×K beamspace noise matrix after BT

Nb,l ∈ CMv×K left beamspace noise matrix after BT

Nb,r ∈ CMv×K right beamspace noise matrix after BT

Nz ∈ CMv×K noise matrix after bias correction

N set of natural numbersP number of sourcesPE EADF powerPη calibration noise powerQ number of calibration directionsQl number of calibration points in the lth angular sectorr UCA radiusrn distance of the nth sensors from the array centroidR radius of the smallest circle enclosing all the sensorsRs ∈ C

P×P signal covariance matrixRx ∈ C

N×N element-space array covariance matrixR real domainsp(k) pth signal at time instant ks ∈ C

1×K signal vector, i.e. first row of S

S ∈ CP×K signal matrix

T set of array interpolation matricesTa ∈ C

N×Mv general beamspace transform matrixTb ∈ C

N×Mv beamspace transform matrix for the BTTb,l ∈ C

N×Mv left correction beamspace after the BTTb,r ∈ C

N×Mv right correction beamspace after the BTTd ∈ C

N×Mv Davies transform matrixTg ∈ C

N×M transformation matrix for data mapping using EADFTl ∈ C

N×Nv array interpolation matrix for the lth angular sectorTo ∈ C

N×Mv non-unitary beamspace transform matrixv ∈ C

Mv×1 Vandermonde structured steering vector used in the MBTw(γ) generic excitation functionwm(γ) mth excitation functionwm ∈ C

N×1 mth beamforming weight vector for UCAsW ∈ C

N×M block of the calibration noise matrix We

W ∈ CN×M0 matrix containing particular realization of W

Wc ∈ CN×Q random matrix containing the measurements noise

We ∈ CN×Q random matrix containing the calibration noise

x(k) ∈ CN×1 array output snapshot at time instant k

X ∈ CN×K array output matrix

Ya ∈ CMv×K beamspace data matrix

Yb ∈ CMv×K beamspace data matrix after BT or DT

Yl ∈ CNv×K virtual array data matrix after AIT

Z ∈ CMv×K bias corrected beamspace data matrix

αn direction of maximum sensor gain

xvii

βn ∈ C complex sensor gainγ angular location on a point on the unit circleγn angular position the nth sensorsδ ∈ C

N×1 derivative of the array steering vector∆ ∈ C

N×P derivative of the array steering vector matrixεm residual term of order m

ε ∈ CMv×1 dominant term of the additive perturbation matrix , ε(1)

ε(1) ∈ CMv×1 dominant term of the additive perturbation matrix

ε(2) ∈ CMv×1 residual term of the additive perturbation matrix

εl ∈ CMv×1 left component of the residual term

εr ∈ CMv×1 right component of the residual term

ǫ ∈ CN×1 modelling error after MST

ζ variable function of θ, i.e. ζ = κr sin θθ co-elevation angleκ wavenumberλ wavelengthλn nth eigenvalueΛ ∈ R

N×N diagonal matrix containing eigenvaluesΛs ∈ R

P×P diagonal matrix containing the signal eigenvalues

Λη ∈ R(N−P )×(N−P ) diagonal matrix containing the noise eigenvalues

Πs ∈ CN×N projection matrix to the signal subspace

Πη ∈ CN×N projection matrix to the noise subspace

σ2W calibration noise varianceσ2

η observation noise variance

τ time delayφ azimuth angleφp DoA of the pth sourceφq qth calibration directionφ ∈ R

1×P set of DoAs along the azimuthal directionφl ∈ R

1×Ql sub-set of φc defining the lth angular sectorφc ∈ R

1×Q set of Q consecutive calibration directionsΦ space of φψo

ai theoretical normalized fitting errors for AITψr

ai realistic normalized fitting errors for AITψo

b theoretical normalized fitting errors for BTψr

b realistic normalized fitting errors for BTψr

me realistic normalized fitting errors for mapping using EADFψo

ms theoretical normalized fitting errors for MSTψr

ms realistic normalized fitting errors for MSTω angular frequency

xviii

Chapter 1

Introduction

1.1 Motivation of the thesis

In the past decades there has been increased interest in spatial techniquesand sensor array signal processing. The demand from the wireless com-munications market has been the driving force for the development ofsmart antenna systems, diversity techniques, multiple-input multiple-output(MIMO) systems, and spatial division multiple access (SDMA). These tech-niques exploit the spatial domain in addition to the time and frequencydomains [12, 118]. The use of adaptive antenna arrays for wireless commu-nications provides benefits such as improved spectral efficiency, higher radio-link robustness (e.g. via diversity), improvement in signal-to-noise (SNR)ratio, interference reduction, and larger coverage area [54, 116, 121]. Theuse of multiple-antenna in multi-user scenarios allows increasing the systemcapacity and designing the frequencies reuse in a more efficient manner.

Array processing is also playing an important role in many diverse ap-plication areas and emerging services [120]. Most modern radar and sonarsystems rely on antenna arrays or hydrophone arrays as an essential com-ponent of the system. Seismic arrays are widely used for oil exploration andunderground analysis. Various medical diagnosis and treatment techniquessuch as EEG and MEG also exploit arrays. Radio astronomy utilizes verylarge antenna arrays to increase resolution. Sensor networks are employedin applications concerning environmental monitoring, military systems, andagriculture. Localization services may exploit sensor arrays in order to esti-mate the position of signal sources.

A sensor array is formed by a collection of sensors, located at distinctspatial locations, which record signals propagating in space. Figure 1.1 de-picts an example of working environment for an array formed by N = 6sensors. The N -elements array records the propagating wavefronts andit produces an N -dimensional observation vector at a given time instant.Hence, by smartly combining the output of each sensor, the array processorallows detecting signals and estimating their parameters. The structure of

1

Figure 1.1: Sketch of a general sensor array processing problem.

the recorded data is influenced by the array response (or steering vector),which in turn depends on the array configuration parameters such as geom-etry, interelement spacing, and sensor type. The harmonic retrieval problem[76], adaptive beamforming [121, 125, 126], interference cancellation [120],MIMO radar [65], channel sounding [12, 44], and high-resolution direction-of-arrival (DoA) estimation [5, 38, 47, 77, 94] are some of the research areasinvolving sensor array processing.

The problem of finding the DoA of a propagating wavefield impinging ona sensor array is a topic which has been studied since the beginning of the20th century [127]. Today, DoA estimation is still a popular research fieldand there exist applications such as beamforming, localization, tracking,surveillance, and navigation where the explicit directional information ofemitters is needed. Originally, array signal processing algorithms have beenmainly derived for arrays with a regular geometry and by assuming thatthe sensor responses are known. For example well known fast, robust, andcomputationally efficient DoA estimators [77, 79, 80, 84, 115] exploit theVandermonde structure and shift-invariance property of the ideal uniformlinear array (ULA) steering vector for estimating the DoAs of signals.

Motivated by the benefits of providing 360◦ uniform coverage in the az-imuthal plane as well as the 3-D information (azimuth and elevation) on thesources location, in the 60’s researchers realized that applying array process-ing algorithms designed for ideal linear (1-D) arrays to ideal uniform circulararrays (UCA) is not straightforward [14, 15]. Since then effort has been putin extending ULA estimation algorithms to arbitrarily structured planar (2-D) and volumetric (3-D) array configurations [50, 60, 89]. In particular, theextension of subspace-based high-resolution DoA estimation algorithms hasbeen of interest. These algorithms can be implemented by using fast and

2

low-complexity procedures [5, 77, 80] while providing a resolution capabilitythat exceeds the traditional Rayleigh resolution limit1 [37, 71].

In practical scenarios sensor arrays with a simple array geometry oftencan not be used. The hardware cost, the shape and size of the mountingplatform, and other mechanical and structural limitations set constraintson the geometry and response of a real-world sensor array [26, 61, 86].Furthermore, the response of a real-world sensor array is always unknownand degraded by imperfections. It can only be estimated through noisycalibration measurements. The structure of the resulting array configurationmay not fit into the ideal model used by well known sensor array processingalgorithms [33, 34, 60], e.g. polynomial rooting DoA estimation algorithms.

In the past twenty years researchers have proposed several pre-processingtechniques for extending the applicability of well known estimation algo-rithms to arbitrary sensor array configurations [7, 20, 26, 46, 63, 69]. Thesetechniques have been derived following various design and implementationstrategies, which offer different advantages depending on the structure of thesensor array and the working environment. Today, sensor array processingtechniques like array interpolation techniques (AIT), beamspace transform(BT), and manifold separation techniques (MST), are employed in differ-ent applications areas and research fields. They allow reformulating thearray signal processing model so that well known estimation algorithms canbe used on real-world sensor arrays with arbitrary configurations. Thesepre-processing techniques often introduce additional errors into the systemwhich, if not properly mitigated and handled by specifically designed algo-rithms, can have a significant impact on the accuracy of the estimates.

1.2 Scope of the thesis

The scope of this thesis is to develop novel spatial signal processing tech-niques that may be applied on real-world sensor arrays of arbitrary geometry.The main application areas of interest are in smart antennas, as well as de-tection, localization and tracking of signal sources. The focus is to analyzeand derive novel algorithms providing high-resolution, optimal or close tooptimal statistical performance, and low computational complexity despitethe antenna array configuration and imperfections. In particular, the prob-lem of estimating the DoA (azimuth-only) of emitting sources located all atthe same elevation angle is considered.

The techniques considered in this thesis are known as AITs, the BT forUCAs, and the MST. They allow applying fast and robust algorithms orig-inally developed for ULA to different array configurations. They are used

1For an ULA of N identical sensors, the Rayleigh criterion for resolution states thattwo incoherent plane waves propagating into two slightly different directions can only beresolved if the distance of their spatial frequencies is at least 2π/N .

3

for transforming the steering vectors of sensor arrays along the azimuthaldirection. The performances of the proposed techniques are validated andverified by using ideal, simulated, and real-world antenna arrays. In orderto estimate the response of real-world sensor arrays and to develop ad-hoctransforms, array calibration data including information on the array im-perfections have been used.

This thesis has two main goals. The first one is to extend the applicabilityof high-resolution DoA algorithms to arbitrarily structured array configura-tions, while avoiding transformation errors and sector-by-sector processing.The array imperfections are handled in the process by using array calibra-tion data and by exploiting the concepts of effective aperture distributionfunction (EADF) and MST. The accuracy of MST-based DoA estimators isaffected by calibration noise and modelling errors. The impact of modellingerrors on the statistical performance of the DoA algorithms is analyzed.

The second goal of this thesis is to study and analyze the behavior of thebeamspace transform for UCA. The accuracy of the estimates of BT-basedalgorithms are impaired by transformation errors. By deriving algorithmswhich estimate and remove the errors introduced by the BT, a significantincrease in the accuracy of the DoA estimates can be achieved.

1.3 Contributions

This dissertation contributes to the field of array transform techniques, an-tenna modelling, and signal processing algorithms for antenna arrays ofarbitrary configurations. The results derived in the original publications areof interest in research fields exploiting the spatial domain, as well as in ap-plications requiring the directional information of source signals. In order tovalidate the original contributions contained in this manuscript, real-worldscenarios have also been investigated.

First, the Unitary root-MUSIC algorithm for UCA is introduced. Byexploiting a unitary transform, the proposed algorithm can provide lowervariance estimates than the Real-Beamspace root-MUSIC algorithm [69]. Arobust extension of this algorithm is also proposed. It is based on nonpara-metric statistics and it gives highly reliable results in the face of heavy-tailednoise and interference. The additional computational cost is negligible. Inorder to work on UCAs the Unitary root-MUSIC algorithm exploits theBT. This pre-processing technique impacts on the asymptotic performanceof DoA algorithms by introducing bias and excess variance in the estimates.

A qualitative and a quantitative analysis of the effects of BT mappingerrors on subspace-based DoA estimation algorithms are carried out. Aclosed-form expression describing the effects of these errors on the DoAestimates is derived. Novel algorithms developed for reducing the bias andmitigating the excess variance in the DoA estimates are introduced. The im-

4

provement in the estimates’ accuracy is shown via simulations. Furthermore,it is shown that BT mapping errors can be avoided by optimally designingthe beampatterns of the UCA sensors. In the derivation of both the excessvariance reduction techniques and the algorithm for optimal beampatternshaping the concept of EADF have been exploited.

The problem of modelling the azimuthal response of sensor arrays witharbitrary configurations by using Vandermonde structured models is ana-lyzed. The array configuration includes linear, planar, and volumetric arraystructures containing non-idealities such as mutual coupling, antenna man-ufacturing errors, and sensor misplacement. By exploiting the idea of wave-field modelling, a pragmatic approach using the EADF for implementing theMST in practical scenarios is developed.

Finally, the problem of applying fast and robust DoA estimators on real-world antenna arrays is investigated. The focus is on estimating the DoAs(azimuthal angle) of sources located at the same fixed elevation angle. Afast polynomial rooting-based DoA estimation algorithm named Element-Space root-MUSIC (ES-root-MUSIC) is developed. It exploits the conceptof EADF and MST in order to provide an accurate model of the used an-tenna array. Simulation results carried out using both ideal and real-worldarrays show that ES-root-MUSIC can provide fast and accurate DoA estima-tion despite the array configuration. The effect of noisy calibration data onsubspace-based DoA algorithms using MST has been analyzed. First, an op-timality criterion for selecting the optimal number of modes which minimizesthe error in the DoA estimates is proposed. Then, two closed-form expres-sions describing the effect of the array model accuracy and calibration noiseare derived. The second expression is of particular interest in real-worldscenarios, because it allows predicting the performance for subspace-basedalgorithms using MST.

1.4 Structure of the thesis

This thesis consists of an introductory part and eight original publications.The publications are listed at page xiii, and are appended at the end ofthe thesis. The introductory part of the thesis is organized as follows. InChapter 2 the signal model is defined. In Chapter 3, the AITs and the BTare described. Interpolation and mapping errors are presented and theirimpact on the DoA estimates is investigated. Novel algorithms for improv-ing the accuracy of the estimates by removing the effects of BT mappingerrors are introduced. In Chapter 4, the focus is on wavefield modelling, theEADF, and the MST. A novel version of root-MUSIC (ES-root-MUSIC) isintroduced. It allows fast and low-complexity DoA estimation while avoidingtransformation errors and sector-by-sector processing. Furthermore, the im-pact of modelling errors on the estimation accuracy is analyzed. Closed-form

5

expressions describing the DoA estimation error are derived. In Chapter 5, acomparison among the AITs, the BT, and the MST is introduced. Practicalissues concerning real-world implementation and array performance evalua-tion are also discussed. Finally, Chapter 6 summarizes the results and thecontributions of the thesis. Future research topics are also discussed.

1.5 Summary of the publications

The aim of this section is to give a brief overview of the author’s originalpublications and to establish a logical link between them. References toother publications by the author of this thesis are also included in the fol-lowing summary, where appropriate. Publications I, II, III, IV, and VI aredealing with the BT for DoAs estimation with UCAs. Whereas PublicationsV, VII, and VIII focus on the MST for arrays with arbitrary configuration.

In publication I [P. I], a unitary version of the root-MUSIC algorithm forUCAs is derived. The proposed algorithm exploits the BT and a real-valued(unitary) formulation of root-MUSIC in order to achieve low computationalcomplexity as well as improved performance. Furthermore, a robust versionof the method based on multivariate extensions of non-parametric statisticsis introduced. This provides reliable estimates in the face of heavy-tailednoise and interference. The additional computational cost is negligible.

Several DoA algorithms, including UCA Unitary root-MUSIC, exploitthe BT because it allows using computationally efficient techniques such aspolynomial rooting and dealing with coherent sources. The performanceof BT-based estimators is degraded whenever the transform does not workunder suitable conditions on the array configuration such as number of ele-ment and interelement spacing. The mapping error leads to bias and excessvariance in the estimates. Hence, the estimates are not statistically opti-mal. Publications II, III, IV, and VI address this phenomenon by derivingan error analysis as well as solutions to the mapping problem.

In publication II [P. II], a novel algorithm for reducing the bias intro-duced by the BT when applied to UCAs is proposed. By deriving beamform-ers able to synthesize the dominant term of the mapping error, a three-stepstechnique for bias removal is developed. Practically bias-free DoA estimatescan be achieved. Observe, the bias depends on the DoAs and it can not beremoved by offline techniques. The computational complexity of the overallalgorithm remains low since all quantities are computed in closed-form.

In publication III [P. III], a procedure for reducing the excess varianceintroduced by BT is proposed. A criterion for selecting the number of vir-tual array elements is formulated. It stems from an analysis of the inverseFourier series of the array impulse response, called the EADF. The proposedcriterion is optimal in the sense that it maximizes the aperture of the virtualarray while satisfying certain constraints on the maximum number of virtual

6

array elements. Simulation results show that the proposed criterion leads toDoA estimates with a lower variance such that they are closer to the CRLB.

Publication IV [P. IV] investigates the problem of reducing the bias in-troduced by the BT under a different point of view. The key idea stemsfrom observing the characteristics of the EADF for real-world arrays. Inthe paper it is shown that by forming a sensor array with optimal shapedelements’ beampattern, the BT will perform a practically error-free map-ping between UCA and ULA. The design can be formulated as a total leastsquares (TLS) problem. Even without using techniques for bias removal,simulation results show that unbiased DoA estimates can be obtained whenthe beampattern of the array sensors are optimally designed.

Publication V [P. V] presents an algorithm based on the concept ofEADF and the MST for extending root-MUSIC to arbitrarily structuredarrays. The proposed ES-root-MUSIC algorithm processes the array datadirectly in element-space. It does not require any transformation or arrayinterpolation so that mapping errors can be avoided. In addition to [19, 22],two alternative ways to perform MST are shown. The first approach solvesa LS problem, while the second uses the EADF. The paper also presentsa comparison of the AITs, the BT, and the MST in terms of fitting error.The MST does not require any division into angular sectors and provides asignificantly smaller fitting error over the whole 360◦ coverage area.

Publication VI [P. VI] focuses on the BT for UCA presenting an ex-tended and more detailed discussion of the error analysis [6] and bias re-duction algorithm [P. II]. An expression approximating the bias in the DoAestimates that is caused by the BT is derived. A modified beamspace trans-form (MBT) that performs mapping from the element-space to beamspacedomain taking the error caused by the BT into account is introduced. Ananalysis of the difference in the statistical performance of MUSIC and root-MUSIC algorithms for UCA is also given. Moreover, it is shown that thereis a significant difference in the performance of UCA root-MUSIC techniquedepending whether an even or odd number of elements is used. In order toreduce the mapping error, some antenna design guidelines for choosing thenumber of sensors, array radius and interelement spacing are provided.

Publication VII [P. VII] is the companion paper of [P. V]. The per-formance of ES-root-MUSIC algorithm is investigated by using real-worldantenna arrays. The EADF is computed from calibration data containinginformation on array non-idealities such as mutual coupling and antennamanufacturing errors. The derivation of an approximate CRLB is also in-cluded in the paper. An extension of this work for estimating both the DoAsand the sources polarization can be found in [89].

Publication VIII [P. VIII] focuses on an error analysis of subspaces-based DoA estimation algorithms using MST. In real-world applications,the calibration measurements used to determine the sampling matrix (andthe EADF) are corrupted by noise. This impacts on the performance of the

7

estimation algorithms. A link between the optimal number of selected modesand the calibration noise power is established. Expressions describing the er-ror in the DoA estimates due to calibration noise and truncation are derived.A formula that allows predicting the performance limit of subspaces-basedDoA estimation algorithms using MST is introduced. This is of particularinterest in practical applications because it depends only on quantities whichare known or computable in real-world scenarios.

All the simulation software for all the original publications included inthis dissertation were written solely by the author of this thesis.

In Publication I, the derivation of a unitary version of root-MUSIC forUCAs was carried out by the first author. All of the simulations were per-formed by the first author as well. The co-author provided useful guidanceand ideas for the robustness analysis, in the design of the experiments, andhelped in writing the paper.

In Publications II and VI, the derivation of the bias-reduction algorithmand error analysis were carried out by the author of this thesis. All ofthe simulations were performed by the first author as well. The co-authorcollaborated providing a useful guidance during the derivation of the erroranalysis and theoretical results, in the design of the experiments, and helpedin the writing the paper.

The algorithm in Publication V was derived by the author of this the-sis. For Publications III, IV, and VII the author of this thesis is the maincontributor of each paper. The theoretical results of Publication VIII werealso established by the author. The co-authors collaborated in the deriva-tions, provided guidance in the theoretical modelling, in the design of theexperiments, and helped in writing the papers.

8

Chapter 2

Array Signal Processing

Models

A sensor array is a collection of sensors located at distinct spatial locationsused to sample signals in space. A wavefront which propagates across thearray of sensors is recorded by each sensor and the observed multichanneloutput is known as array signal. Depending on the sensors used, e.g. an-tenna, electrode, microphone, or hydrophone, sensor arrays can be employedin different research areas such as radio frequency scenarios, biomedical stud-ies, acoustic or under-water environments.

Generally, the sensors’ outputs contain information of a signal waveformwhich is corrupted by noise and other interferences. The array outputsalso contain information on the sensor array such as its geometry, elementscharacteristic and structural imperfections. Array processing consists ofusing the multichannel observations collected by a sensor array in an optimalmanner in order to detect signals or estimate their parameters. Adaptivebeamforming, interference cancellation, and high-resolution direction findingare some of the important areas of sensors array processing [44, 49, 80].

The development of the signal model is based on a number of simplifyingassumptions [39, 54, 80]. Throughout this work the sources are assumed tobe narrowband, where narrowband means that the signal bandwidth is smallcompared to the inverse of the propagation time of the wavefront acrossthe antenna array aperture [74]. The physical size of an antenna array,measured in wavelength, is known as the array aperture whereas the effectiveaperture is the array aperture seen from a certain direction [71]. The sourcesare assumed to be situated in the far field of the array and considered asconcentrated entities (point emitters). In addition, it is assumed that thepropagation medium is homogeneous, i.e. not dispersive. Consequently, thewaves impinging at the sensor array can be considered to be planar [49, 71].

Under these assumptions the sensors observe time delayed versions ofthe same signal and the location of the emitter may be characterized bythe direction-of-arrival (DoA) of the transmitted wavefront [39, 105, 123].

9

Figure 2.1: Sketch of an ideal uniform linear array (ULA) with N elementsand interelement spacing d. Here, τ and φ denote the propagation delayand the direction-of-arrival (DoA) of a signal wavefront, respectively.

The angular information of the impinging wavefront is contained in the arrayresponse, also known as array steering vector a(φ, θ) ∈ C

N×1. The collectionof the steering vectors over the parameter space of interest is called the arraymanifold. For example, in a 3-D space it can be defined as

A ={a(φ, θ) | φ ∈ [0, 2π) , θ ∈ [0, π]

}, (2.1)

where φ and θ are the azimuth and co-elevation angles, respectively.This thesis focuses on the area of DoA estimation, where the objective is

to obtain accurate and robust estimates of the angular location of sources.In this context resolution refers to the ability to distinguish two closelyspaced signal sources. Typically this is described through some spectral-likemeasure, which will exhibit peaks at the estimated DoAs of the sources [54].Whenever there are two peaks near the true DoAs, the sources are said tobe resolved. To retrieve the angle of arrival of source signals is of interestin applications where the explicit directional information is needed such asbeamforming, localization, tracking, surveillance, and navigation.

This chapter is organized as follows. First, the theoretical models andclosed-form representations of ideal array steering vectors are defined. InSection 2.2, examples of real-world arrays are shown. In Section 2.3, aprocedure for calibrating antenna arrays is described. The calibration datacontain information on the array response including array non-idealities suchas mutual coupling and manufacturing errors. In Section 2.4, the signalmodel used in this thesis is presented. The model is general in the sensethat it can be applied on arbitrary ideal and real-world array configurations.Finally, in Section 2.5 concluding remarks are discussed.

2.1 Ideal Antenna Arrays

An ideal uniform linear array (ULA) consists of N identical sensors uni-formly placed along a line as depicted in Figure 2.1. Observe that for the

10

Figure 2.2: Sketch of an ideal uniform circular array (UCA) with N elementsand radius r. The wavefield impinges to the array from a direction denotedby θ (co-elevation) and φ (azimuth). The array lies on the xy-plane.

scope of the thesis ideal refers to a theoretical array formed by omnidirec-tional sensors with identical gains and placed at known locations.

In order to avoid spatial aliasing1 the interelement spacing is constrainedso that d ≤ λ/2, where λ = c/fc is the signal wavelength at the centerfrequency fc and c is the wavefront propagation speed. A ULA is a 1-Darray2, which may be used for estimating the azimuthal DoA φ ∈ [−π

2 ,π2 ]

of a propagating wavefront. The time delay between the sensors is definedas τ = d

c sinφ. The steering vector of an ideal ULA shows a convenientVandermonde structure of the form

aula(φ) =[1, e−jω d

csin φ, . . . , e−jω(N−1) d

csin φ

]T, (2.2)

where ω = 2πf is the angular frequency.An ideal uniform circular array (UCA) consists of N identical sensors

evenly placed on a circle of radius r in the xy-plane; see Figure 2.2. Theinterelement spacing is given by d = 2r sin (π/N) [60]. The 2-D structureof the array allows the estimation of both azimuth and elevation angles.The co-elevation angle θ ∈ [0, π

2 ] is measured down from the z-axis andφ ∈ [0, 2π) is the azimuth angle measured counterclockwise from the x-axis. The origin of the coordinate system is located in the center of thecircle. Consequently, the position of the nth sensor is given by the vector~rn =

{r cos γn, r sin γn, 0

}, where γn = 2πn

N (for n = 0, . . . , N − 1) is countedcounterclockwise from the x-axis [39, 69]. The steering vector of a UCA canbe expressed as

auca(ζ, φ) =[ejζ cos (φ−γ0), ejζ cos (φ−γ1), . . . , ejζ cos (φ−γ(N−1))

]T, (2.3)

1and respecting the Nyquist sampling theorem [49, 71].21-D refers to the fact that only one angle (e.g. azimuth) can be estimated.

11

0°

30°

60°

90°

120°

150°

180°

210°

240°

270°

300°

330°

OmnidirectionalDirectional, syntheticDirectional, URPA

Figure 2.3: Examples of normalized sensor beampattern: for omnidirec-tional beampattern the gain is constant in all directions, while for directionalbeampatterns the gain is a function of φ.

where ζ = κr sin θ, and κ = ω/c is the wavenumber. For a fixed elevationangle θ the steering vector in (2.3) depends on the azimuth angle φ only, i.e.auca(ζ, φ) becomes auca(φ).

The ULA and UCA array models can be generalized by considering a2-D array of arbitrary geometry. Throughout this thesis the interelementspacing between adjacent sensors is constrained to d ≤ λ/2 despite the arrayconfiguration. For convenience the origin of the coordinate system is locatedat the centroid of the array and the co-elevation is fixed at θ = 90◦, i.e. inthe array plane. Consequently, the nth element of an ideal array steeringvector aI ∈ C

N×1 can be written as [22]

[aI(φ)]n = βn(φ, γn, rn) e−jωτn(φ) = βn(φ, γn, rn) ejκrn cos (γn−φ) , (2.4)

where rn is the distance of the respective element from the centroid of thearray, and τn(φ) = − rn

c cos (γn − φ) is the propagation delay associated witha signal impinging from direction φ at the nth sensor. Here βn(φ, γn, rn) ∈ C

is a scalar defining the sensor gain, which depends on both the azimuthangle φ and the sensor position (given by γn and rn). If the sensors areomnidirectional, the sensors have equal sensitivity in all directions, implyingthat {βn(γn, rn)}N−1

n=0 is independent of φ.Figure 2.3 depicts the normalized beampatterns of three sensors located

at the origin of the coordinate system. One of the sensors is omnidirectionalwith unit gain, one has a directional beampattern generated by βn(φ) =1.2 + cos(φ), and the last one represents a real-world sensor beampatterntaken from the URPA depicted in Figure 2.4-5. The sensor gain is generallyaffected by the mounting platform, mutual-coupling, shielding, etc.

12

Figure 2.4: Antenna arrays which have been used in this research work.

2.2 Real-World Arrays

The mathematical model of the steering vector for real-world antenna arraysis more complex than the ones described in Section 2.1. In literature severalmodels have been proposed. They consider various kinds of non-idealitiessuch as mutual coupling between sensors [28, 66], manufacturing errors re-lated to elements orientation and position [40, 101, 111], and sensor gains[27, 97]. In this thesis all array imperfections are considered jointly. Thejoint information about imperfections is acquired by array calibration.

Depending on the application a sensor array is composed of N arbitrarilyplaced sensors forming a 1-D, 2-D, or 3-D structure. Conformal arrays [50],biomedical electro-arrays, and sensor networks are examples of such arrays.In practice each sensor has its individual directional characteristics and aphase center which may not correspond to its nominal location.

In this thesis, the following antenna arrays have been used. In Figure 2.4-1, a polarimetric3 uniform linear patch array4 (PULPA) is depicted. Thisarray has N = 8 patch sensors with a interelement spacing d = 0.4943λ.Each element has one port for horizontal and one port for vertical polariza-tion. In Figure 2.4-2, the polarimetric semi-spherical patch array5 (PSSPA)with N = 21 elements is depicted. In Figure 2.4-3, we show the N = 24elements polarimetric uniform circular patch array5 (PUCPA) and, in Fig-ure 2.4-4, the N = 96 elements stacked polarimetric uniform circular patch

3An array is said to be polarimetric whenever each antenna element has two feedsdetecting the orthogonal components of the impinging wavefront.

4Courtesy of the Electronic Measurement Research Laboratory, TU-Ilmenau, Germany.5Courtesy of the Radio Laboratory, SMARAD CoE, Department of Electrical and

Communications, Helsinki University of Technology (TKK), Finland.

13

Figure 2.5: On the left, general calibration setting used in an anechoicchamber. Here, an ULA is mounted on a positioner. On the right, a sketchof the calibration procedure is shown. The response of the antenna array ismeasured by rotating the antenna about its centroid in a controlled manner.The response of each sensor is measured at every calibration direction.

array5 (SPUCPA). The SPUCPA is comprised of 4 stacked rings of 24 polari-metric patches. It has 192 output ports in total. In Figure 2.4-5 a uniformrectangular patch array6 (URPA) with N = 16 polarimetric patch elementsis depicted. Finally, in Figure 2.4-6, a uniform circular array5 (UCA) withN = 16 conical elements is shown. The sensor beampatterns for some ofthese antenna arrays are shown in [57, 100] and [P. III].

2.3 Array Calibration Data Model

The response of a real-world antenna array to a far-field source can be mod-elled through a calibration measurement by measuring the directional char-acteristics of the array in an anechoic chamber. For convenience it is as-sumed that the origin of the coordinate system is located at the centroid ofthe array and the antenna is calibrated along the azimuthal direction, e.g.at θ = 90◦. In Figure 2.5 a possible setting for array calibration is depicted.In practice, the centroid is defined by the calibration setup.

The azimuthal response of an antenna array to a far field source canbe measured by moving the source around the array at a fixed co-elevationangle along φ ∈ [0, 2π)6. Alternatively, the same result can be obtained byfixing the source location and rotating the array about its centroid as shownon the right in Figure 2.5. Hence, by measuring the array response alongQ≫ N consecutive calibration directions φc = {φ1, φ2, . . . , φQ}, samples of

6For example the response of an array with N = 8 elements may be measured on auniform grid of Q = 120 calibration points, i.e. every 3◦ in azimuth.

14

a 2π-periodic function in azimuth are recorded.The calibration matrix Ac ∈ C

N×Q is formed by combining the mea-surements of an N elements antenna array into a matrix so that

Ac(φc) = [ac(φ1), . . . ,ac(φQ)] , (2.5)

where ac(φq) ∈ CN×1 is the steering vector of a real-world array measured

in the direction φq, and Ac is assumed to have full row-rank if φk 6= φq fork 6= q. The calibration data in Ac contain the joint information on array non-idealities such as mutual coupling between sensors, antenna manufacturingerrors, sensors orientation and position, at the time of the calibration.

In practice, the calibration data are always corrupted by measurementnoise. The perturbed calibration matrix Ac(φc) ∈ C

N×Q is modelled as

Ac(φc) = Ac(φc) + Wc , (2.6)

where Wc ∈ CN×Q is the noise matrix containing the receiver noise. The

noise is assumed to be zero-mean, white, complex circular Gaussian dis-tributed with variance σ2

W . The measurement noise appears as one of thesources of uncertainty in the signal model. For more details on the impactof the calibration noise on the DoA estimates and antenna modelling seeSection 4.4, [48, 72], and [P. VIII]. Other sources of uncertainty, related toplane wave assumptions [58] and phase drifting [59], which may be containedin the calibration are not considered in this thesis.

2.4 Signal Model

For the scope of this thesis, the focus of the discussion is on estimatingthe DoAs of signals along the azimuthal direction. Hence, unless explicitlystated, the co-elevation is assumed either fixed or known. For examples of2-D (azimuth and elevation) estimation techniques see [19, 26, 39, 69, 70,79, 88, 120, 133, 138].

Suppose there are P (P < N) narrowband signal sources on the xy-plane, impinging an array from directions φ = {φ1, φ2, . . . , φP }. The sensoroutputs are appropriately preprocessed and sampled at time instants k (k =1, . . . ,K). Based on the simplifying assumptions discussed earlier in thischapter, the array output vector x(k) ∈ C

N×1, also called a snapshot, is amultivariate observation made at time instance k and it is modelled as

x(k) = A(φ)s(k) + n(k), (2.7)

where A(φ) = [a(φ1),a(φ2), . . . ,a(φP )] ∈ CN×P , with a(φp) =

[a0(φp), a1(φp), . . . , aN−1(φp)]T ∈ C

N×1, is the array steering vector matrix,s(k) = [s1(k), . . . , sP (k)]T ∈ C

P×1 is the signal vector, and n(k) ∈ CN×1

15

contains the observation noise at time instant k. The signal s(k) is mod-elled as either stochastic or deterministic depending on the application. Theobservation noise is assumed to be independent of the signals, stationary,second-order ergodic, zero-mean, spatially and temporally white, with vari-ance σ2

η. Observe that, depending on the antenna array under consideration,different array steering vectors a are used throughout this thesis, i.e. aI ,aula, auca and ac.

Assuming that K (with K > N) consecutive and independent snapshotshave been recorded, the model in (2.7) can be rewritten as

X = A(φ)S + N, (2.8)

where S = [s(1), . . . , s(K)] ∈ CP×K is the signal matrix, N ∈ C

N×K con-tains observation noise, and X ∈ C

N×K is the element-space data matrix.Assume that the P -dimensional signal vector s(k) is wide sense station-

ary and that the signal covariance matrix Rs = E{s(k)sH(k)} ∈ CP×P is of

rank P , where E{·} stands for the expectation operator. As examples, twoclasses of stochastic signals fulfilling this requirement are incoherent signalsor coherent signals with a delay difference larger than their coherence timetc ≈ 1/Bs, where Bs is the signal bandwidth [36, 80]. It also applies tolinearly independent deterministic signals.

The element-space spatial covariance matrix Rx = E{x(k)xH(k)} ∈C

N×N can be expressed as [39, 54, 60, 80]

Rx = ARsAH + σ2

ηIN , (2.9)

where σ2ηIN = E{n(k)nH(k)} ∈ R

N×N denotes the noise covariance ma-trix. When P < N , the data model is said to be low-rank since the signalpart of the observations is confined to a P -dimensional subspace of the N -dimensional complex data space [73].

Using matrix decomposition techniques such as SVD and EVD, thesample covariance matrix in (2.9) can be rewritten in terms of eigende-composition as Rx = EΛEH , where E ∈ C

N×N is a unitary matrix andΛ ∈ R

N×N is a diagonal matrix having real eigenvalues ordered such thatλ1 ≥ λ2 ≥ . . . ≥ λN > 0. The N − P smallest eigenvalues of Rx are equalto σ2

η and the corresponding eigenvectors are orthogonal to the columns ofA. These eigenvectors span the noise subspace and the eigenvectors corre-sponding to the P largest eigenvalues span the signal subspace.

Partitioning the eigenvalue/eigenvector pairs into noise eigenvectors (cor-responding to eigenvalues λP+1 = . . . = λN = σ2

η) and signal eigenvectors(corresponding to eigenvalues λ1 ≥ . . . ≥ λP > σ2

η), equation (2.9) may alsobe expressed as a low-rank data model by

Rx = [Es Eη]Λ[Es Eη]H = EsΛsE

Hs + EηΛηE

Hη , (2.10)

16

where Λη = σ2ηI(N−P ) ∈ R

(N−P )×(N−P ) and Λs ∈ RP×P contains the larger

eigenvalues associated with the signal eigenvectors. The projection ma-trix to the signal subspace is Πs = Es(E

Hs Es)

−1EHs = EsE

Hs ∈ C

N×N .In addiction, since the columns of A span also the signal subspace [37],Πs = A(AHA)−1AH . The projection matrix to the noise subspace is givenby Πη = EηE

Hη = I−Πs ∈ C

N×N . Furthermore, under the aforementionedassumptions on the observation noise n(k) and considering an orthonor-mal (unitary) transformation matrix such as FHF = IN , the transformednoise vector FHn(k) is white [69], i.e., FHEηΛηE

Hη F = FHσ2

ηINF = σ2ηIN .

Hence, the statistics of n(k) does not change after unitary transformation.In practical situations only a limited number of snapshots is available.

Hence, only an estimate of (2.9) computed from the observed data is ob-tained. A natural way for estimating the second-order statistics of data isthe sample covariance matrix [10, 44, 54, 60, 123]

Rx =1

K

K∑

k=1

x(k)xH(k) , (2.11)

for which a spectral representation similar to (2.10) can be written as

Rx = EsΛsEHs + EηΛηE

Hη . (2.12)

If x(k) is ergodic stationary with finite second-order statistics, then Rx →Rx as K → ∞ [54, 60].

In array signal processing the standard estimator for the array outputcovariance matrix is the sample covariance matrix of (2.11). Use of thesample covariance matrix may cause the estimation methods to yield un-reliable estimates if the observation noise is non-Gaussian and heavy-taileddistributed. This is due to the fact that the sample covariance matrix is anoptimal estimator for Gaussian data, and it is not designed for robustness.Several solutions to this problem can been found in the literature [16, 53,124, 136]. The idea is to replace the sample covariance matrix by a robustestimator of Rx, which would reliably estimate the eigenvectors or eigenval-ues of Rx also for non-Gaussian data. For example, the sample covariancematrix is very sensitive to outliers (highly deviating observations) and es-timating strategies that put a bound on the influence of outliers have beenproposed; see [123] for more details.

Subspace-based estimation using decomposition techniques is a powerfultool in many signal processing applications where a low-rank signal modelis observed. Examples are sensor array processing, system identification,filtering, image processing, blind channel estimation, timing estimation, andsignal separation in wireless communications [73].

17

2.5 Concluding Remarks

In array signal processing it is often convenient to work with arrays having asteering vector matrix showing Vandermonde structure or some shift invari-ance property. Several robust and computationally efficient DoA estimatorssuch as spatial smoothing techniques [80, 84, 115] and polynomial rooting-based methods [5, 38, 47, 77] exploit these characteristics. However, only alimited group of array geometries exhibit some of these desirable features,e.g. ideal ULA and URA, CUBA [18, 92] and ESPAR [93, 112].

In real-world applications the design constraints and the implementa-tion issues often lead to utilize sensor arrays which do not exhibit the aboveproperties. For example, UCAs have received a lot of attention in surveil-lance applications since they can provide almost uniform resolution in theazimuthal plane over 360◦. However, the mathematical structure of the UCAsteering vector makes several attractive algorithms not directly applicableon such arrays. Another example is given by the incorporation of multi-ple antennas into a handset. The hardware cost, the shape and size of themounting platform, and other mechanical and structural limitations set theconstraints on the geometry and response of the antenna array [26, 61, 86].It can be shown that real-world array elements have individual directionalbeampatterns and that their effective radiation patterns (phase centers) maynot correspond to their nominal locations. Therefore, the structure of theresulting array configuration may not fit into the model used by well knownDoA estimation algorithms [33, 34, 60].

In the past thirty years a lot of effort has been made in extending al-gorithms originally developed for regular (e.g. ideal ULA) to arbitrarilystructured arrays. In this thesis the best known techniques are groupedinto two distinct families: the data-driven and model-driven transform tech-niques. They allow applying fast and robust DoA estimators to real-worldantenna arrays with arbitrary configurations.

The remaining part of the thesis is dedicated to describe and comparethe above techniques both under a theoretical and a practical point of view.In Chapters 3 and 4 the two families of transform techniques are described,their key concepts defined, and the original contributions contained in thismanuscript are fitted into the general framework of mapping and modellingtransform techniques. Finally, Chapter 5 shows a comparison which pointsout the pros and cons of the different transformation approaches.

18

Chapter 3

Data-Driven Transform

Techniques

Most of the sensor array processing algorithms that possess desirable prop-erties in terms of their statistical performance, high-resolution, and low com-putational complexity are derived for regular array geometries such as ULAand UCA. In order to extend these estimation algorithms to antenna ar-rays with arbitrary configuration, several transform techniques have beenproposed in literature. In this thesis, these techniques are divided in data-driven and model-driven transform techniques.

The first group is comprised of array interpolation techniques (AIT) [8,26, 60, 78] and beamspace transform (BT) [69, 70, 139] [P. I]. They aredesigned for linearly transforming the steering vector of a physical arrayinto a steering vector of a virtual array with more desired features. Thesetechniques modify the structure of the observed data to best fit a previouslydefined model. Data-driven transform techniques often lead to bias andexcess variance in the estimates. This results in DoA estimates that are notstatistically efficient [45, 46, 63] [P. II-IV, VI].

The second group includes manifold separation techniques [19-22, 72][P. V, VII]. They aim at creating a model of the physical array whichbest fits to the array data structure. In order to exploit fast polynomial-rooting algorithms on arbitrarily structured arrays, the physical array ismodelled so that the needed Vandermonde structure explicitly appears. Incontrast to data-driven transform techniques, model-driven techniques donot suffer from mapping errors because data mapping is not required intheir implementation. However, in practical scenarios the accuracy of theestimates may be subjected to error caused by array modelling errors andcalibration noise [P. VIII]. Hence, the computed DoA estimates are notstatistically efficient. For more details see Chapter 4.

This chapter focuses on data-driven transform techniques and it is orga-nized as follows. First, the AITs are considered. Some interesting transfor-

19

Figure 3.1: Example of angular sector processing: ” ◦ ” and ” × ” are theelements of an N = 8 UCA and Nv = 7 virtual ULA, respectively. Herethe in-sector is the region φ ∈ [75◦, 105◦], while the rest is the out-of-sectorregion. In AIT the lth in-sector corresponds to transformation matrix Tl.

mation methods are discussed and their key concepts are given. In Section3.2, the idea of beamspace processing techniques is presented. The discus-sion will then concentrate on the BT and the Davies transform (DT) forUCAs. In Section 3.3, the original contribution in this work concerningthe beamspace method, the mapping errors produced, and signal processingtechniques for reducing those errors are presented. This includes a descrip-tion of the modified beamspace transform (MBT), a novel algorithm forbias removal, and a method for reducing excess variance. Finally, Section3.4 summarizes the key features of the discussed transform techniques.

The discussion in this chapter focuses on transforming the steering vec-tors of planar arrays along the azimuthal direction (fixed elevation angle).For 2-D (azimuth and elevation) transform techniques see [69, 133].

3.1 Array Interpolation Techniques

Array interpolation techniques (AIT) [9, 26, 34, 46, 64] are designed forlinearly transforming the array response vector of a planar array into thatof a preliminarily specified virtual array over a given angular sector, calledthe in-sector. Thus, the approach involves sector-by-sector processing tocover the full azimuth; see Figure 3.1. For example, AITs map the arrayresponse of a UCA into that of a ULA for a given in-sector.

The core of the AITs is in the offline process where the transformationmatrices are computed. As described in Section 2.3, the array calibrationyields the calibration matrix Ac(φc) ∈ C

N×Q given in (2.5). In order toachieve a sufficiently low transformation error [26, 60] Ac(φc) is partitionedinto L sub-blocks of size N × Ql so that for the lth in-sector Al

c(φl) =[ac(φ1), . . . ,ac(φQl

)] ⊂ Ac(φc) and Ql = Q/L ≫ N . In practice, this can

20

Figure 3.2: Block diagram describing the key idea of array interpolationtechniques (AIT). Exploiting the transform matrices T = {Tl}L

l=1 (com-puted offline), the output of a physical array can be interpolated onto theoutput of a virtual array, having more desirable properties.

be interpreted as dividing the azimuth range into L angular sectors1 so thatΦ = {φ1, . . . ,φL}, where Φ is the space spanned by φ. The lth in-sectorthen comprises of a set of adjacent calibration directions φl = {φ1, . . . , φQl

}.For the lth angular sector φl, a common way of designing the transform

matrix THl ∈ C

Nv×N is to find a least squares (LS) fit between the calibratedand virtual array responses. The problem can be formulated as [26, 45]

Tl = arg minTl

{∥∥∥THl Al

c(φl) − Blv(φl)

∥∥∥2

F

}, (3.1)

where Alc(φl) ∈ C

N×Ql and Blv(φl) ∈ C

Nv×Ql , with2 Nv ≤ N , are theelement-space and the virtual array steering vector matrices, respectively.As usual Bl

v = [aula(φ1), . . . ,aula(φQl)], that is, Bl

v is assumed to be formedby the ideal ULA steering vector defined in (2.2).

Solving the minimization problem in (3.1) for each angular sector pro-vides the set of interpolation matrices T = {Tl}L

l=1 used in the online pro-cess. Provided that Al

c(φl) ∈ CN×Ql has full row rank, the LS solution to

(3.1) is given by

Tl =(Al

cAlcH

)−1

AlcB

lvH. (3.2)

In the online process the data recorded by the physical array, see (2.8), are

1Sectors of 30◦ are commonly used. However, recently proposed interpolation tech-niques allow using larger angular sectors; for example see [8, 46, 64, 78].

2This condition can be relaxed in case of non-redundant and partially augmentablearrays [119].

21

then linearly transformed so that

Yl = THl X , (3.3)

where Yl ∈ CNv×K is the lth virtual array data matrix. If the virtual array

has a ULA-like steering vector, the data structure of Yl allows well knownDoA estimation algorithms to be applied on planar arrays with arbitrarygeometries [8, 29, 64, 86, 133]. The transformation may also be appliedin different ways. In, e.g., Hyberg et al. [45, 46], the estimated signaleigenvector matrix Es is mapped and not the physical array output data.

Generally, AITs introduce errors in the DoA estimates. By adjusting thefitting error and the condition number of Tl (ratio between its largest andsmallest singular value), the degradation of the estimates may be controlled[9, 26, 34, 45]. Examples of AIT fitting error can be found in Section 5.1.

Figure 3.2 depicts the key concept of AITs. The overall procedure canbe divided into an off- and on-line processes. In the offline process thecalibration measurements of a given antenna array are used to computethe transformation matrices according to the interpolation strategies. Inthe online process the block T combines the data recorded by the physicalarray and the set of interpolation matrices previously computed in the offlineprocess. Finally, the DoA estimation block receives data, which have beentransformed to posses the desired structure.

The concept of array interpolation was originally proposed by Bronez [7].The design process of the transformation matrix was formulated by includingspatial filtering. The transformation matrix was derived by minimizing thetotal response of the physical array under the constraint that the virtualarray steering vector matches the one of a ULA for a grid of angles withinthe in-sector. The proposed interpolation technique can provide a smallapproximation error over the in-sector while spatially filtering the arrayresponse in the out-of-sector region.

Later, Friedlander [26] presented an elegant formulation of the arrayinterpolation problem addressing the root-MUSIC algorithm. For each an-gular sector the transformation matrix is found by solving a LS problem.Extensions of the Friedlander approach to ESPRIT [133] and MODE [132]algorithms as well as to coherent [29, 86, 129] and wide-band signals [30,34] have been proposed. Further studies have also established relationshipsbetween the interpolation errors and the physical [34] and virtual array [9]geometries. The Friedlander formulation obtains a small interpolation errorwithin the in-sector while neglecting the out-of-sector response. As shownin [11, 61] this may cause degradation in the DoA estimation if highly cor-related sources are not all confined to the same in-sector region.

Pesavento et al. [78] have formulated the interpolation approach as aminimization problem with multiple inequality constraints. The proposedAIT minimizes the interpolation error in the in-sector of interest while set-

22

ting multiple stop-band constraints in the out-of-sector region. This pro-vides robustness against source signals arriving from outside the consideredin-sector area. The interpolation matrix design is formulated as the second-order cone programming which can be solved by the SeDuMi MATLABtoolbox [110]. Between the in- and out-of-sector regions exists a roll-offarea where the response is unaccounted for. Similarly to the Friedlanderapproach, whether highly correlated sources are located in this transitionregion, a degradation in the angular performance may be observed.

Lau et al. [61] have extended the above techniques by presenting aninterpolation approach that takes into account the array response over thefull azimuth. The formulation deals explicitly with the entire out-of-sectorregion by setting a target response for this region in addition to the targetresponse for the in-sector. It can be formulated as a WLS problem. Theattempt to control the response of the interpolated array over the entireazimuth (including roll-off area) comes at a cost of relatively high interpo-lation errors. These larger errors in turn increase the DoA estimation bias.In a following paper Lau et al. [64] have proposed an approach to reducethe bias in a data-dependent manner. They designed the adaptive weightingfunction as the conventional beamformer’s power response. A direct conse-quence is that the transformation matrix must be constantly updated foreach of the angular sectors. Due to a smaller approximation error, the sectorsizes can be enlarged and the number of sectors reduced.

Hyberg et al. [45] proposed a transformation matrix design that reducesthe bias in the DoA estimates. They incorporate the performance of theangular estimator into the design of the interpolation matrices by taking intoaccount the orthogonality between the manifold mapping errors and certaingradients of the estimator criterion function. The transformation matrix isfound by solving a LS problem. Similarly to the Friedlander approach, theout-of-sector response is ignored. The proposed design algorithm is basedon a one-emitter analysis. A suboptimal strategy for dealing with multipleemitters is suggested in the paper. In the same spirit, the authors extendedtheir bias reduction approach considering also finite sample effects due tonoise. The technique proposed in [46] allows designing a transformationmatrix which minimizes the MSE of the DoA estimates.

Buhren et al. [8] presented an interpolation procedure with the objectiveof creating a virtual array manifold, which is a shifted version of the phys-ical array manifold. This allows avoiding the selection of the virtual arraystructure (number of sensors, interelement spacing, array orientation, etc.)and reducing the bias in the DoA estimates. The artificial shift-invarianceis then exploited by an ESPRIT-like algorithm called interpolated ESPRIT.The proposed technique does not consider the out-of-sector response andit requires sector-by-sector processing. The design of the two interpolationmatrices involves solving a LS problem and computing a SVD.

23

3.2 Beamspace Processing Techniques

Beamspace processing refers to spatial pre-filtering techniques, which pre-process the data at the output of an antenna array before applying an estima-tion algorithm. Working in beamspace domain offers several advantages overconventional element-space processing. The benefits include reduced com-putational cost, improved performance if the noise is spatially correlated,and enhanced resolution; see [23, 54, 60, 70, 120] and references therein.

On the other hand, the performance of the estimators after preprocessingmay be poorer. For example, reducing the dimension of the data can implya loss of available degrees of freedom to combat interference. The problemof ambiguity in the estimates within the in-sector can also occur [2]. In [42]an approach to beamspace processing with an improved robustness againstout-of-sector interfering sources is developed. Furthermore, a comparativeperformance study of element-space and beamspace estimators can be foundin [109, 134, 135] while a description of an optimal beamspace preprocessoris given in [3, 4, 131].

The key idea of beamspace processing is that the matrix THa ∈ C

Mv×N

linearly transforms the N -dimensional element-space data matrix X ∈C

N×K into the Mv-dimensional beamspace data matrix Ya ∈ CMv×K as

Ya = THa X , (3.4)

where THa Ta = IMv

and Mv ≤ N . Similarly to AITs, a sector-by-sectorprocessing is usually required. In some applications, the initial preprocessingmatrix To may not be unitary. Then, the transform matrix satisfying theunitarity constraint can be constructed as Ta = To(T

Ho To)

−12 [120].

Beamspace processing can be applied to arbitrary array geometries. Fasthigh-resolution DoA estimation algorithms such as root-MUSIC and ES-PRIT can be directly used in the beamspace domain [31, 51, 138, 139] whensuitable array structures are used; e.g. for ULA and URA. Otherwise, AITsand beamspace processing need to be combined [33, 99].

In the special case of UCA, two very attractive beamspace preprocessorscan be used: the Davies transform (DT) [15, 63, 128] or the beamspacetransform (BT) [69, 70, 87]. They aim to map the steering vector of a UCAinto one having the desired Vandermonde structure. Similarly to the AITsdiscussed in Section 3.1, DT and BT allow spatial smoothing and fast DoAestimators to be applied on UCAs [62, 63, 68, 69, 87, 128]. In contrast toAITs and general beamspace preprocessors, DT and BT do not require anydivision into angular sectors. The whole 360◦ coverage area can be mappedusing one predefined transformation matrix.

The DT and BT stem from the pioneering work presented by Davies etal. in the 60’s [14, 15, 67]. They considered a beam lying in the plane ofthe array (φ ∈ [0, 2π) and θ = 90◦) and analyzed the far-field directionalpattern of the array through its Fourier components. The authors found that

24

any aperture distribution of a circular array may be electronically rotatedby simple phasing techniques normally associated with a linear array. Theirtechnique is known as the phase-mode excitation principle [69, 70, 113, 140].

3.2.1 Phase-Mode Excitation Principle

The phase-mode excitation is a synthesis procedure for circular arrays andring apertures [120]. The discussion in this section begins with ring aperturesand extends to more practical circular arrays (UCAs).

The ring aperture, or continuous circular array, is an ideal configura-tion for applying the principle because it leads to an error-free transforma-tion. On a continuous circular aperture any excitation function is periodicin γ with a period of 2π and can hence be expressed in terms of Fourierseries. A generic excitation function w(γ) may be defined using the in-verse Fourier series w(γ) =

∑∞

m=−∞cme

jmγ , where the mth phase modewm(γ) = ejmγ represents a spatial harmonic of the array excitation, cm isthe corresponding Fourier series coefficient and γ represents the angular lo-cation of a point, which runs continuously on the circular aperture. Thearray excitation wm(γ) synthesizes the far-field pattern [15, 69]

f cm(θ, φ) =

1

2π

∫ 2π

0wm(γ)ejζ cos(φ−γ)dγ = jmJm(ζ)ejmφ , (3.5)

where Jm(ζ) is the Bessel function of the first kind of order m with argumentζ = κr sin θ. Here the superscript c stands for continuous, while κ and r arethe wavenumber and the ring radius, respectively.

From the RHS of (3.5) it can be observed that the far-field patternf c

m(θ, φ) is given by the product of two quantities, where Jm(ζ) implicitlydepends on the co-elevation angle θ and ejmφ explicitly depends on theazimuth angle φ. Hence, by fixing the co-elevation (e.g. θ = 90◦) thefar-field pattern depends only on φ. A rule of thumb for determining themaximum mode indexMb that can be excited by the aperture at a reasonablestrength3 is to choose Mb as the smallest integer that is close or equal to κr[15, 70]. Consequently, the modes m ∈ [−Mb,Mb] set the dimension of thebeamspace domain at Mv = 2Mb + 1. For more details see [14, 15, 69, 70].

A UCA is a discrete version of the continuous circular array where afinite number of sensors is uniformly placed along a circle. Sampling wm(γ)at the array element locations yields the phase-mode excitation beamformingweight vectors for UCAs. Hence, the normalized beamforming weight vectorthat excites the array with phase mode |m| ≤Mb is [15, 69, 87]

wHm =

1

N

[ejmγ0 , ejmγ1 , . . . , ejmγN−1

], (3.6)

3For modes m > Mb, the array gain becomes small over the entire field of view, asJm(ζ) is small when the order m of the Bessel function exceeds its argument ζ.

25

−10 −5 0 5 10−30

−25

−20

−15

−10

−5

0

Mode Index (m)

Mag

nitu

re [d

B]

a)

−10 −5 0 5 10−30

−25

−20

−15

−10

−5

0

Mode Index (m)

b)

Principal TermFirst Right ReplicaFirst Left ReplicaSelection Boundary

Figure 3.3: Excitation modes of the principal and residual terms: the repli-cas (only v = 1 depicted) are due to the spatial sampling by the UCAsensors. The selection boundaries define the area of interest used for select-ing the size of the virtual array. Settings: d = 0.4λ, N = 7 and N = 8 in a)and b) respectively.

where N is the number of UCA sensors and γn is the angular location of thenth element. The array beampattern can then be expressed as

fsm(θ, φ) = wH

mauca(θ, φ) =1

N

N−1∑

n=0

ejmγnejζ cos(φ−γn) (3.7)

= jmJm(ζ)ejmφ +∞∑

v=1

(jgJg(ζ)e

−jgφ + jhJh(ζ)ejhφ)

(3.8)

= jmJm(ζ)ejmφ + εm , (3.9)

where auca(θ, φ) is the UCA element-space steering vector as defined in (2.3),the variable εm represents the entire sum term in (3.8), and the indices gand h are defined as g = Nv −m and h = Nv +m .

Expressions (3.8) and (3.9) are comprised of two terms. The first termis called the principal term and it is identical to the far-field pattern ofthe continuous array in (3.5). The second term εm is called residual term.It arises from the sampling of the continuous aperture by N sensors. Asdiscussed in [69, 70], the principal term dominates if N > 2Mb. This condi-tion is identical to the Nyquist sampling criterion, as Mb defines the largest”spatial frequency” component in the array excitation.

As depicted in Figure 3.3, the sampling of the ring aperture with Nelements in the spatial domain creates replicas of the principal term in themode domain. The vth replica is shifted by Nv from the principal term (atv = 0). The selection boundary is set at the intersection point between the

26

dominant term and the first replica. Within the boundaries, the residualterm (also known as higher-order distortion modes or aliasing terms) causesthe UCA pattern to deviate from that of the ring aperture and they have tobe minimized in order to get close to ideal (continuous) case performance.Observe that even though the residual term εm is given as a sum of infinitenumber of terms, only the first one (for v = 1) is significant; see Table I in[P. VI]. This behavior is independent from the DoA of the sources, and thefirst component of the residual term (for v = 1) always remains dominant.

3.2.2 Davies and Beamspace Transforms for UCA

The Davies transform (DT) was originally proposed by Davies et al. in [14,15] in the context of beamforming and array beampattern design. LaterMathews et al. [69, 70] have proposed a modified version known as thebeamspace transform (BT). Both transforms stem from the phase-modeexcitation principle presented in Section 3.2.1 and they have been exploitedby several authors [35, 62, 63, 68, 87, 113, 128] [P. I, II, III, VI].

The DT and BT are linear transformations that map the N × 1 UCAsteering vector auca(θ, φ) into a Mv ×1 (with N ≥Mv) Vandermonde struc-tured vector having ULA-like properties. Opposite to AIT in Section 3.1they do not require division into angular sectors and they work over thewhole 360◦ coverage area. Furthermore, the virtual array used by both DTand BT does not have a physical interpretation, it only shows Vandermondestructure. There are not free-parameters to be adjusted.

Recalling (3.7)-(3.9) and combining the Mv excited modes |m| < Mb intoa matrix form, the BT for UCA can be written as

bb(θ, φ) = THb auca(θ, φ) ≈

√NJ(ζ)db(φ) , (3.10)

where

J(ζ) = diag{JMb

(ζ), .., J1(ζ), J0(ζ), J1(ζ), .., JMb(ζ)

}(3.11)

db(φ) =[e−jMbφ, . . . , e−jφ, 1, ejφ, . . . , ejMbφ

]T. (3.12)

In the above expressions J(ζ) ∈ CMv×Mv is a diagonal matrix containing

Bessel functions, bb(θ, φ) ∈ CMv×1 is the beamspace array response, and

db(φ) ∈ CMv×1 is a Vandermonde structured vector. The transformation

matrix THb ∈ C

Mv×N is a unitary DFT matrix satisfying THb Tb = IMv

,

which is computed offline. Assuming {Jm(ζ)}Mb

m=0 6= 0 4 and fixing θ = 90◦,the DT can be written as

bd(φ) = THd auca(φ) ≈ db(φ) , (3.13)

where bd(φ) ∈ CMv×1 is the Davies array response, and TH

d =N−1/2J−1TH

b ∈ CMv×N . Comparing (3.10) and (3.13), a link between the

4This means that no resonance phenomenon occurs [21, 63].

27

Figure 3.4: Block diagram representing the beamspace or Davies transformtechnique. The beamforming matrices Tb (or Td) maps the output of anideal UCA onto the output of a virtual array having ULA-like properties.

BT and the DT can be observed. The DT can be interpreted as a particularcase of the BT for UCAs where the co-elevation is fixed at the array planeand the transform yields directly the Vandermonde structured vector db(φ).Generally, the transformation matrix Td is not unitary and a procedure forwhitening the transformed noise may be required. For more details on theconstruction of Tb and Td see, for example, [60, 63, 69, 70, 128].

The transforms in (3.10) and (3.13) are approximative and the mappingerror can be neglected only when certain conditions (number of elements,interelement spacing, SNR, etc.) are fulfilled. For example, the mappingerror caused by an ideal UCA with a fixed radius and d ≤ λ/2 decreases asthe number of sensors N increases. As suggested in [69], the mapping errormay be neglected whenever the residual term corresponding to m = Mb in(3.7) is ”small enough”, i.e. εMb

≤ 0.01. For a quantitative evaluation ofεm as a function of array parameters see [P. VI]. It is important to observethat even when the BT mapping errors are small they may still impact theperformance of estimators. For more details see Section 3.3.

The BT and the DT tacitly assume that all the array elements have thesame omnidirectional response, that they are located at the correct positions,and that there is no mutual coupling between sensors [63]. Clearly, none ofthese assumptions hold in a real-world implementation. The DT and BTVandermonde approximation errors given by using both an ideal and a real-world array are shown in Section 5.1. As expected, for real-world arraysboth these transforms have a significantly larger fitting error.

Figure 3.4 shows the key idea of BT and DT. In the offline process, the

28

transformation matrix or the beamformer Tb (or Td) is computed. Knowl-edge of the array elements’ locations is required. Similarly to (3.4), in theonline process the X ∈ C

N×K element-space data matrix is mapped to thebeamspace domain by Yb = TH

b X ∈ CMv×K , where N ≥ Mv. The DoA

algorithms can then be applied to the virtual array data matrix Yb.Tewfik et al. [114] presented a DT-based approach that allows using

root-MUSIC algorithm on ideal UCA. They investigated the impact of thetransformation on the background noise. An extension to azimuth and ele-vation estimates is also considered. The authors fixed the size of the virtualarray to the size of the physical array, i.e. N = Mv. As discussed in [60]and in Section 3.2.1, this may cause significantly large fitting errors.

Mathews et al. [69, 70] proposed eigenstructure-based algorithms for 2-Destimation with ideal UCAs. They introduced the BT and gave conditionsunder which the transform should be applied. Furthermore, they deriveda real-valued beamspace version of MUSIC and root-MUSIC as well as aclosed-form algorithm (UCA-ESPRIT), which yields automatically pairedsource azimuth and elevation estimates. The statistical performance of theestimators are also presented in these articles.

Wax et al. [128] presented a calibrated version of the Davies transformfor estimating the DoAs of coherent signals using non-ideal UCAs. Bothmutual coupling and general array imperfections were considered. Theyproposed a procedure for computing the robust transformation matrix bysolving a LS problem, which directly uses the array calibration data.

Lau et al. [63] proposed a robust version of the Davies transformfor non-ideal UCAs. They showed that the robust transformation matrixcan be computed offline. The optimization algorithm involves solving aquadratic semi-infinite programming problem by using the dual parametriza-tion method. The basic idea is to trade-off the Vandermonde approximationerror in the transformation for robustness, that is, by designing a transfor-mation matrix of lower norm. A comparison between the proposed transformtechnique and the calibrated DT in [128] was also discussed in the paper.

3.3 Beamspace Mapping Errors

In general, data-driven transform techniques are not error-free. The lineartransforms used for interpolating or mapping the physical array steering vec-tor into a more convenient representation often introduce systematic errorsimpacting the accuracy of estimators. Several strategies have been proposedfor mitigating the effects of these errors; e.g. see [9, 45, 46, 63, 128].

In this section the discussion focuses on the impact of fitting errors dueto the BT on the DoA estimates. The study of these errors also playsa significant role in the original contributions [P. II, III, IV, VI]. First,the effect of the residual terms εm in (3.9) on the virtual array steering

29

vector is described. Then a novel bias removal algorithm and a criterionfor reducing the excess variance are presented. They can be used as analternative approach to the robust DT proposed by Lau [63] or to the AITfor MSE minimization presented by Hyberg [46]. In addition, the proposedalgorithm and criterion can also be used together with the calibrated DTintroduced by Wax [128] and BT presented by Mathews [69, 70].

In order to describe the effects of mapping errors on the BT, equation(3.10) is modified so that also the residual terms of (3.9) are included in it[87]. The modified beamspace transform (MBT) can be written as

bb(θ, φ) = THb auca(θ, φ) = v(φ) + ε(1)(φ) + ε(2)(φ) , (3.14)

where v(φ) =√NJ(ζ)db(φ) is the Vandermonde structured steering vector

of the virtual array. Here ε(1)(φ) ∈ CMv×1 is the first term (v = 1) of

the sum in (3.8) and it represents an additive perturbation [39, 111] ofthe nominal value of v(φ) with angular dependence. The remaining terms(v = 2, 3, . . . ,+∞) of the sum in (3.8) are captured in ε(2)(φ) ∈ C

Mv×1 [P.VI]. The latter includes terms which are significantly smaller than v(φ) andε(1)(φ), and hence ε(2)(φ) can be neglected. The perturbation ε(φ) , ε(1)(φ)of the virtual elements’ locations in v(φ) depends on the actual DoA. Inorder to remove this perturbation, online processing is required.

For the sake of simplicity the proposed bias reduction algorithm is de-rived here for a single source case. Its extension to multiple sources isstraightforward. Combining equations (2.8) and (3.14), the mapping of theUCA element-space data matrix X ∈ C

N×K into the beamspace data matrixYb ∈ C

Mv×K is defined as Yb = THb X. It can be explicitly written as

Yb =(v(φ) + ε(φ)

)s + Nb =

(v(φ) + εl(φ) + εr(φ)

)s + Nb , (3.15)

where s ∈ C1×K is the signal vector, ε(φ) = εl(φ) + εr(φ), and the terms

εl(φ) and εr(φ) are related to the first (v = 1) left l and right r replicas ofthe principal term; see Figure 3.3 and Section 3.2.1. Note that the statisticsof the transformed noise Nb = TH

b N do not change because the beamformermatrix Tb is unitary, i.e. it satisfies TH

b Tb = I.In order to remove the error terms from (3.15), a novel algorithm has

been introduced in [P. II, VI]. The key idea of the proposed bias removalalgorithm is to derive two beamformers (Tb,l and Tb,r), which synthesizethe perturbations εl(φ) and εr(φ) so that

THb,lX = εl(φ)s + Nb,l (3.16)

THb,rX = εr(φ)s + Nb,r , (3.17)

and then to use these beamformers to remove the perturbations in (3.15). Inthe above equations Nb,l = TH

b,lN and Nb,r = THb,rN are the left and right

transformed noise matrices. Note that in order to derive the correction

30

Table 3.1: Algorithm for bias reduction

Step 1: Form the array observation matrix X as in (2.8),

a) compute the initial beamspace data matrix Yb as in (3.15)

b) estimate the initial set of DoAs

Step 2: with the DoAs computed on step 1

c) form the correction beamformers Tb,l and Tb,r,

d) compute the new beamspace data matrix Z as in (3.18),

e) estimate the second set of DoAs.

Step 3: with the DoAs computed on step 2

f) repeat the points (c) and (d),

g) estimate the bias-free DoAs.

beamformers Tb,l and Tb,r estimates of the DoAs are required. The biasremoval is performed by subtracting the correction beamformers Tb,l andTb,r from the original beamformer Tb. This can be written as

Z =(TH

b − THb,l − TH

b,r

)X = v(φ)s + Nz , (3.18)

where Z ∈ CMv×K is the error-corrected beamspace data matrix, v(φ) is

the Vandermonde structured steering vector of the virtual array, and (THb −

THb,l − TH

b,r)(Tb − Tb,l − Tb,r) ≈ I since ||Tb,l||2F and ||Tb,r||2F are ≪ 1. Asummary of this novel algorithm is given in Table 3.1.

Figure 3.5 depicts the decrease of the estimation error at each iterationstep when the bias removal algorithm is used with the UCA Unitary root-MUSIC algorithm [P. I]. The RMSEs of the DoA estimates computed atstep one from the transformed data Yb show error floors due to bias. Afterthe second step the two error floors are lowered significantly. Finally, afterthe third step the bias is practically removed and no error floor occurs. Thecomputational cost of the overall procedure can be evaluated as number ofsteps times the complexity of the angular estimator. The curves appear tobe parallel to the CRLB. However, some excess variance introduced by theBT remains in the estimates and hence the bound is not attained [109, 120].

A criterion for reducing this excess variance has been presented in [P.III]. The proposed criterion is based on the analysis of the excitation modes(Section 3.2.1) and it is optimal in the sense that it maximizes the apertureof the virtual array, while satisfying Mv ≤ N . The criterion selects Mv sothat it equals the number of principal modes having magnitude larger than

31

0 5 10 15 20 25 30 35 40 45 5010

−2

10−1

100

101

102

SNR [dB]

RM

SE

[deg

rees

]

Original algorithm, source 1Original algorithm, source 22nd step, source 12nd step, source 23rd step, source 13rd step, source 2CRLB

1

2

3

Figure 3.5: Cancellation steps for: the original UCA Unitary root-MUSICalgorithm, after the second iteration and after the third iteration. Settings:UCA with N = 8, r = λ/2.6, K = 256 and {φ1, φ2} = {15◦, 25◦}.

the magnitude of the replicas, that is, the number of principal modes withinthe selection boundaries depicted in Figure 3.3. The proposed criterion setsthe size of the virtual array as

{Mv = N − 1 for arrays with an EVEN number of sensorsMv = N for arrays with ODD number of sensors

(3.19)

where N denotes the number of sensors in the UCA.Other selection criteria have been proposed in literature; see for example

[69, 114, 128]. In certain situations, these criteria have been found to besuboptimal or not intuitive [60] [P. III]. In contrast to them, the approachpresented in (3.19) is simple to use and optimal with respect to the size ofthe virtual array. Figure 3.6 shows that the excess variance is significantlyreduced by properly selecting the number of virtual array elements Mv. Therule of thumb in [69] suggests selecting Mv = 5 elements for the consideredarray configuration. Instead, the criterion proposed in (3.19) suggests in-creasing the aperture of the virtual array by selecting Mv = 7 elements. Asa result the variance of the estimates is closer to the CRLB.

In [P. IV] an alternative approach for suppressing the mapping errorintroduced by the BT has been derived. The approach is based on opti-mal sensor beampattern shaping and EADF (effective aperture distributionfunction); see Chapter 4. The key idea is to reduce the aliasing of consecu-tive replicas within the selection boundaries (see Figure 3.3) by compressingthe support of the magnitude of the EADF while preserving the informationcontained in its phase. Consequently, the impact of the aliasing term on theprincipal term is mitigated. Interestingly, this can be achieved by designing

32

0 5 10 15 20 25 3010

−1

100

101

102

RM

SE

[deg

rees

]

SNR [dB]

Algorithm with bias reduction, source 1Algorithm with bias reduction, source 2Algorithm with bias and excess variance reduction, source 1Algorithm with bias and excess variance reduction, source 2CRLB

Figure 3.6: Effect of excess variance reduction using the UCA Unitary root-MUSIC with bias correction. Here, Mv is selected either with the conven-tional rule of thumb (Mv = 5) or with our EADF-based approach (Mv = 7).Settings: N = 7, d = 0.4λ and {φ1, φ2} = {10◦, 20◦}.

array sensors with a certain directional characteristic. Simulation resultsshow that practical unbiased DoA estimates can be obtained without usingtechniques for bias removal when the beampattern of the array sensors areoptimally designed.

3.4 Discussion

This chapter has presented the key ideas and the derivations of the data-driven techniques known as the AITs, the BT and the DT. This sectionnow focuses on summarizing the main features, advantages, and limitationsgiven by the above methods. Some of the main issues related to using thesetransform techniques in real-world scenarios are also addressed.

The main features and benefits of AITs [7, 8, 26, 45, 46, 61, 64, 78]considered in this thesis are summarized in the Table 3.2. The common pointof all these techniques is that they need sector-by-sector processing, theycan be implemented using array calibration data, and they are applicableto sensor array with arbitrary configuration. The feature of robustness forcorrelated signals refers to the ability of controlling both the out-of-sectorregion and the roll-off area. Whereas, the impact on the DoA estimatesrefers to the uncertainty introduced in the system by the transformationitself. This also considers the size of the virtual array with respect to thephysical array. For example using [7] the degrees-of-freedom of the virtualarray are severely limited and a N = 48 elements UCA is mapped into a

33

Table 3.2: Comparison among the presented AITs.AIT

Features [7] [8] [26] [45] [46] [61] [64] [78]

Authors’ proposedIn-Sector Size

15◦ 90◦ 30◦ 60◦ 80◦ 72◦ 120◦ 30◦

Transformation isData-Independent(1)

yes yes yes yes(2) yes(2) yes no yes

Controls the Out-of-Sector Response

yes(3) no no no no yes yes yes(4)

Robust toCorrelated Sources

no no no no no yes yes no

RequireWhitening

yes no yes no no yes yes yes

Transform Errorswithin in-sector(5)

low low low low low med. low low

Impact on theDoA Estimates(6)

high low med. low low med. low low

DesignStrategy

(7) (8) LS LS LS WLS WLS (9)

(1) The transformation matrices are computed offline.(2) Theoretically it is true only for one source case.(3) The out-of-sector response is reduced but not totally suppressed.(4) The roll-off area is uncontrolled.(5) Vandermonde approximation errors.(6) Degradation in estimation accuracy due to bias and excess variance.(7) Minimum norm solution and an ad-hoc selection criterion.(8) Two steps implementation; first LS and then SVD.(9) Cone programming. It is solved by the SeDuMi MATLAB toolbox.

Nv = 4 elements virtual array.The main features and benefits of BTs [63, 69, 70, 114, 128] considered

in this thesis are summarized in the Table 3.3. The common point of allthese techniques is that they do not need sector-by-sector processing, theyare offline procedures, and they allow applying spatial smoothing.

A common disadvantage of the above data-driven transform techniques isthat they often introduce mapping errors leading to bias and excess variancein the DoA estimates [46, 63]. In literature there exist algorithms which havebeen derived for mitigating these errors. For more details see [45, 46, 63]and [P. II, III, IV, VI].

As described in [34, 45, 46, 63], using a transform matrix derived forminimizing the fitting error does not imply that this matrix will also mini-mize the error in the DoA estimates. For example in [45, 46] the proposedtechniques involve writing a minimization of a weighted cost function com-prising of both the fitting and the orthogonality error. In the same spirit, in[63] the authors presented a robust transform technique for non-ideal UCA

34

Table 3.3: Comparison among the presented BT and DT.BT and DT

Features [63] [69] [114] [128]Designed for usingCalibration Data

no no no yes

RequireWhitening

yes no no yes

Applicable onReal-World UCAs

yes no no yes

Robust forCorrelated Sources

yes no no yes

TransformationErrors

med. low low low

Impact on theDoA Estimates

med. low high low

DesignStrategy

(1) (2) (3) LS

(1) Quadratic semi-infinite programming problem.(2) It uses a DFT matrix for the transformation.(3) It uses Bessel function properties.

which allows improving the accuracy of the DoA estimates by sacrificing theVandermonde approximation error. The minimization of the fitting errorbetween physical and virtual steering vector does not generally lead to thebest possible estimation performance.

It is important to observe that computing the transformation matricesdirectly from calibration data leads to Vandermonde approximation errorswhich are always worse than the expected one. This is due to the fact thatthe transformation matrices are designed for optimally fitting the noisy arraycalibration data to the desired virtual array. The transform matrix is neveroptimally designed for mapping from a real-world to a virtual array becausethis would require the knowledge of the true physical array response. In real-world scenarios the true response of an antenna array is always an unknownquantity that can only be estimated from calibration measurements.

A way for reducing the Vandermonde approximation error is to computethe transformation matrices using a model of the antenna array constructedby using the manifold separation technique described in the next chapter.As described in Section 4.4, the formation of the EADF matrix involvesa truncation procedure, which results in an improvement of the SNR usedduring array calibration [P. VIII]. This can also be interpreted as a reductionof the calibration noise contained in the measured array response.

35

36

Chapter 4

Model-Driven Transform

Techniques

In this chapter model-driven transform techniques are considered. The dis-cussion focuses on describing theoretical [19-22] [P. V] and practical [P. VII,VIII] approaches for implementing the manifold separation technique (MST)using the concept of effective aperture distribution function (EADF). TheMST stems from the wavefield modelling formalism for array processing [19-21] and it is a technique used for modelling the steering vector of a givensensor array. In particular, the key idea of MST is to rewrite the steeringvector of an arbitrarily structured array as a product of the characteristicmatrix describing the array itself (sampling matrix ) and a vector with aVandermonde structure containing the unknown angular parameter (coeffi-cient vector). The resulting MST-based array model allows high-resolutionDoA algorithms designed for ideal ULAs such as root-MUSIC [5, 83] andRARE [76, 79] to be applied on non-ideal, possibly polarimetric, arrays witharbitrary geometry [72, 89].

The model-driven transform techniques can offer several advantages overthe data-driven techniques presented in Chapter 3. Opposite to AITs, theMST does not require sector-by-sector processing. In contrast to the DTand the BT, the MST can be applied on non-ideal arrays. Overall, MSTcan provide a significantly smaller fitting error over the whole 360◦ coveragearea. Furthermore, the model-driven techniques do not suffer from mappingerrors. Data mapping is not required and the DoA algorithms can processthe array data directly in element-space. Observe that in practical scenariosthe accuracy of the estimates may be subjected to error caused by arraymodelling errors and calibration noise.

This chapter is organized as follows. First, the concepts of wavefieldmodelling and MST are introduced. A link to the phase-mode excitationprinciple is also established. In Section 4.2, the EADF is presented and thekey ideas of MST-based DoA estimation algorithms are described. In Section4.3, a fast MST-based estimator called the Element-Space root-MUSIC (ES-

37

root-MUSIC) algorithm is presented. In Section 4.4, the impact of modellingerrors due to truncation and calibration noise on the MST-based algorithmsis analyzed. Two closed-form expressions describing the effects of theseerrors on the DoA estimates are also given. Finally, Section 4.5 summarizesthe key features of the MST using the concept of EADF.

Following the scope of this thesis, the discussion is constrained on mod-elling the steering vector of arbitrarily structured sensor arrays along theazimuthal direction. For 2-D (azimuth and elevation) transform techniquessee [17, 19, 55, 81, 82, 88].

4.1 Wavefield Modelling and Manifold Separation

Doron et al. [22] have presented a technique known as array manifold in-terpolation, which can be interpreted as a generalized Davies transform forarrays with arbitrary configurations in wideband signal environments. Later,in a series of three papers [19-21] Doron and Doron presented an interestingtheoretical development of the idea of manifold interpolation in the contextof wavefield modelling. Their works contain fundamental results supportingthe basis of wavefield modelling and the MST presented in this thesis.

The idea of wavefield modelling is to rewrite the antenna array responsein (2.8) as the product of a sampling matrix Gs (independent from the wave-field) and a coefficient vector ds(φ) (independent from the array) [19]. Thisgives the basis for understanding the concepts of EADF and deriving theES-root-MUSIC algorithm presented in Sections 4.2 and 4.3, respectively.

For the sake of clarity, the discussion is carried out under the followingsimplifying assumptions. It is assumed that the wavefield is generated bya far-field point source only (plane wave assumption) and that the beam-pattern of each antenna element is described by a smooth and continuousfunction. Note that the latter conditions are not restrictive since it is satis-fied by real-world antenna arrays of practical interest [19, 89]. The locationof each element is assumed to be known but it may be arbitrary. In addi-tion, for now the derivation is limited to an ideal array with omnidirectionalelements. This assumption does not restrict the generality of the discussion.An extension of wavefield modelling to non-ideal arrays can be found in [19]while examples of MST applied to real-world arrays are presented in [89],[P. VII], and Section 4.4.

Recalling equation (2.4) with {βn(φ, γn, rn)}N−1n=0 = 1, the nth element

of the array steering vector for arrays with arbitrary configuration havingomnidirectional sensors can be written as [22]

[aI(φ)]n = e−jωτn(φ) = ejκrn cos (γn−φ) , (4.1)

where τn(φ) = − rn

c cos (γn − φ) is the propagation delay associated with thenth sensor and a wavefront impinging from direction φ, rn is the sensor’s

38

distance from the centroid of the array and γn is the angular position of thenth array element in polar coordinates.

Using the Jacobi-Anger expansion [1], the rightmost term in (4.1) canbe expressed as [22]

ejκrn cos (γn−φ)=∞∑

m=−∞

jmJm(κrn)ejm(γn−φ)

=1√2π

∞∑

m=−∞

[Gs(rn, γn)]n,m e−jmφ , (4.2)

where Jm(·) is the Bessel function of the first kind of order m, and[Gs(rn, γn)]n,m =

√2πjmJm(κrn)ejmγn defines the (n,m)th element of the

sampling matrix1 Gs. As described in [19], the magnitude of Gs is an evenfunction of m and it decays superexponentially as |m| → ∞. Observe thatGs(rn, γn) depends on the array configuration only and that it is indepen-dent of the wavefield, in particular of φ.

Equation (4.2) represents the exponential Fourier series expansion of the2π-periodic function [aI(φ)]n that leads from the mode domain to the an-gular domain. From the phase-mode excitation principle in Section 3.2.1 itcan be observed that in the case of ideal UCAs (rn = r and γn = 2πn/N forn = 0, . . . , N − 1), the equation (4.2) is actually the Fourier series represen-tation of the 2π-periodic function in (3.5).

Combining (4.1) and (4.2) into matrix form, we can express the keyconcept of the manifold separation technique (MST) as

aI(φ) = Gsds(φ) for φ ∈ [−π, π) . (4.3)

In the case of 1-D (azimuth) estimation, the mth component of ds(φ) is thecomplex exponential defined as

[ds(φ)]m =1√2π

e−jmφ , for m = . . . ,−1, 0, 1, . . . . (4.4)

For 2-D estimation problems, ds(φ, θ) may be expressed using sphericalharmonics [17, 19, 41, 82]. Equation (4.3) suggests that the MST maybe interpreted as a synthesis procedure, which generalizes the phase-modeexcitation principle originally developed for UCAs (see Section 3.2.1) toarbitrarily structured sensor arrays.

Theoretically the MST can perfectly model the array steering vectoraI(φ) only when an infinite number of Fourier coefficients are used; see (4.3)and (4.4). However, as discussed in [19] and [P. V], this condition can be

1In [19] the sampling matrix is introduced as an operator, which operates on functionsbelonging to an infinite dimensional Hilbert space H so that Gs : H → C

N×1. The opera-tor Gs can also be interpreted as a matrix with N rows and an infinite number of columns.

39

relaxed. Due to the superexponential decay of the magnitude of the Fouriercoefficients for |m| → ∞, a sufficiently accurate model of the array steeringvector can be constructed by using a finite number of coefficients. For moredetails on the truncation to a finite number of coefficients see Section 4.2.

When the objective is to minimize the modelling error, a large numberof Fourier coefficients should be selected, e.g. 2|m| + 1 ≫ N . Intuitively, asmaller array error in array manifold reconstruction leads to a smaller errorin the DoA estimates. Ideally, as discussed in [19] and [P. V], the resultingmodelling error can be made arbitrarily small by increasing the maximumexcitation mode index |m|. An example of this can be found in Section 5.1.

In [19-22, 72, 137], the authors proposed truncating the coefficient vectords(φ) at |m| ≈ 2κR, where R is the radius of the smallest circle centered atthe origin of the array that encloses all the array elements. This selectioncriterion is similar to the one used in [15, 69] in the context of BT. Using theaforementioned rule and constraining the interelement spacing to d ≤ λ/2generally gives a number of coefficients smaller than the number of arrayelements, i.e. 2|m| + 1 ≤ N .

In [P. VIII] a different selection criterion is proposed. It is based onpractical limitations and can be used also in real-world scenarios. For agiven set of calibration measurements the criterion suggests a pragmaticapproach for selecting the maximum excitation mode index |m|. The use ofthis criterion results in the minimization of the array modelling error. Thisis studied in more detail in Section 4.4.

The next section presents an alternative way for defining the wavefieldmodelling principle. It allows implementing the MST using arbitrary sensorarray configurations in both simulated and real-world environments.

4.2 Effective Aperture Distribution Function

A method for determining the sampling matrix Gs is presented here. Thesampling matrix can be approximated by using a finite dimensional matrixknown as the effective aperture distribution function (EADF) matrix.

For the sake of simplicity the following discussion is restricted to arraysformed by uncoupled and omnidirectional sensors and it is assumed thatthe array response is known. These assumptions do not limit the gener-ality of the results. In fact, the discussion can be extended to arbitrarilystructured arrays by estimating the array response through noisy calibrationmeasurements [45, 55, 57, 89, 128] and [P. VII]; see also Section 2.2.

A discrete version of (4.1) can be constructed by taking Q ≫ N con-secutive samples of the 2π-periodic function [aI(φ)]n. Due to sampling, theQ-point IDFT inverse discrete Fourier transform (IDFT) of this function

40

−80 −59 −40 −29 −20 −10 0 10 20 29 40 59 80−300

−250

−200

−150

−100

−50

0

Mode Index (m)

EA

DF

Mag

nitu

re [d

B]

Principal TermFirst Right ReplicaFirst Left ReplicaSelection BoundaryCutoff Point

Figure 4.1: Sampling the array manifold in the spatial domain creates repli-cas of the sampling matrix in the mode domain. Here the first right andleft replicas (for u = 1) are depicted. Settings: UCA with N = 8 elements,interelement spacing d = 0.4λ, and Q = 59 sampling directions.

shows replicas of Gs formed according to

1

Q

Q∑

q=1

ejκrn cos (γn−φq)ejmφq = jmJm(κrn)ejmγn

+∞∑

u=1

(jgJg(κrn)ejgγn + jhJh(κrn)ejhγn

), (4.5)

where m = . . . ,−1, 0, 1, . . . with g = Qu−m and h = Qu+m. Expression(4.5) describes a Q-periodic discrete function comprised of two terms. Thefirst term defines the (n,m)th element of the sampling matrix [Gs(rn, γn)]n,m

in (4.2), while the second term describes the set of infinitely many replicasof the first term. The uth replica is shifted by Qu from the principal termfor which u = 0.

In Figure 4.1, the first right and left replicas of the principal term (sam-pling matrix) due to the sampling of the array manifold are depicted. Itcan be observed that when the number of sampling points Q is sufficientlylarge (Q≫ N), the replicas in (4.5) are further apart than the ones shiftedby Nv in (3.8); see also Figure 3.3. Consequently, the aliasing is negligiblewithin the period captured between the selection boundaries.

An approximate version of the sampling matrix can be found by filteringthe region outside the selection boundaries. Considering the Q-points peri-odicity of (4.5), the EADF matrix Ge ∈ C

N×Q is defined as the sub-matrixof Gs so that

[Ge]n,m = [Gs]n,m , (4.6)

41

where n = 0, . . . , N − 1 and m ∈ [− (Q−1)2 , (Q−1)

2 ] are the Q central columnsof the sampling matrix Gs. Observe that if Q→ ∞, then Ge → Gs.

Furthermore, recalling the superexponential decay of the magnitude ofthe modes within a period, the contribution of modes with a sufficiently largeindex |m| to the array manifold reconstruction is negligible. By choosing thecutoff points so that M ≤ Q, a truncated version G ∈ C

N×M of the EADFmatrix Ge is defined as

[G]n,m = [Ge]n,m , (4.7)

where n = 0, . . . , N − 1, and M = 2Mt + 1 for m ∈ [−Mt,Mt]. In theexample depicted in Figure 4.1, Q = 59 uniformly distributed samples alongφ ∈ [−π, π) have been used. The selection boundaries, defining one periodof (4.5), are set at m = {−29, 29}. Furthermore, the cutoff points are set atm = {−10, 10}, resulting in M = 21 selected modes. Here, it can be clearlyseen that the EADF Ge and the truncated EADF G matrices representreduced versions of the sampling matrix Gs.

Finally, combining (4.5)-(4.7), the MST principle in (4.3) and (4.4) canbe rewritten in terms of the truncated EADF matrix as

aI(φ) = Gd(φ) + ǫ(M,Q) for φ ∈ [−π, π) , (4.8)

where the Vandermonde vector d(φ) ∈ CM×1 is of the form

d(φ) =1√M

[ej

M−12

φ, . . . , 1, . . . , e−j M−12

φ]T

, (4.9)

and ǫ(M,Q) ∈ CN×1 contains the modelling error due to aliasing and trun-

cation. For sufficiently large Q, the error due to truncation dominates thealiasing error and hence ǫ(M,Q) ≈ ǫ(M). For more details on the derivationof Ge and G as well as closed-form expressions of the approximation errors;see [19] and [P. V, VIII].

Figure 4.2 summarizes the key ideas of the MST-based DoA estimationalgorithms. In the offline process, the array is calibrated and the measureddata are used to compute the EADF matrix as shown in (4.5). Accordingto the target modelling error, a proper number of excitation modes M isselected. This also defines the size of the Vandermonde structured vectord(φ) in (4.9). The truncated EADF matrix G is a parameter used by theestimator for describing the used antenna array. It is important to observethat a given algorithm can be applied to different antenna arrays simply bychanging the parameter G.

4.3 Element-Space root-MUSIC

An example of a fast MST-based DoA estimation algorithm is the Element-Space root-MUSIC (ES-root-MUSIC) presented in [P. V, VII, VIII]. It

42

Figure 4.2: Block diagram describing the key ideas of the manifold sepa-ration technique (MST). Exploiting the antenna model computed from thecalibration measurement in the offline process, the DoA estimation algo-rithms can process the array data directly in element-space.

allows azimuthal DoA estimation of non-coherent sources at a given elevationangle using arbitrary array configurations. The algorithm does not requirearray data transformation and it operates directly on the element-spacearray data because it exploits the MST.

In real-world radio systems, the maximum SNR is mainly limited by thetransmit power and the receiver noise2. Moreover, it is reasonable to assumethat the calibration SNR used during measurements can be made larger thanthe operational SNR of the system. Hence, if the target modelling error issmaller than the variance of the DoA estimates at the highest achievableSNR, the modelling error ǫ(M) can be neglected and a(φ) ≃ Gd(φ).

Combining (2.8) and (4.8) for a predefined SNR range, the signal modelcan be rewritten as

X = AS + N ≃ GDS + N , (4.10)

where D ∈ CM×P is a matrix formed as D = [d(φ1), . . . ,d(φP )] with full

column rank, φ = {φ1, . . . , φP } are the true DoAs of the P sources, andG ∈ C

N×M is the truncated EADF matrix of (4.7) with full row rank.Recalling (2.11) and (2.12), the element-space covariance matrix may be

expressed according to

Rx = EsΛsEHs + σ2

ηEηEHη ≃ GDRsD

HGH + σ2ηI , (4.11)

2This is the SNR resulting by connecting directly the transmitter and receiver. Oth-erwise, factors such as path-loss and array gains have to the considered as well.

43

where the matrices of eigenvectors Es ∈ CN×P and Eη ∈ C

N×(N−P ) span theelement-space signal and noise subspaces, respectively. The key idea used inthe derivation of ES-root-MUSIC is that Es, A, and GD span approximatelythe same subspace. An analysis of the orthogonality error between GD andEη has been carried out in [P. VIII] and it is also discussed in Section 4.4.

Exploiting MST, we can express the pseudo-spectrum of ES-MUSIC as

F (φ) =(dH(φ)GHEηE

Hη Gd(φ)

)−1

, (4.12)

which allows applying fast polynomial rooting algorithms such as root-MUSIC [5, 83] instead of performing an exhaustive search. By substitutingz = ejφ into the M × 1 Vandermonde vector d(φ), equation (4.12) can berewritten as a polynomial equation of degree 2M − 1. It can be solvedby polynomial rooting algorithm using (2M − 1)(log(2M − 1))2 arithmeticoperations [43, 75]. The azimuth estimates are then computed from theargument of the P roots zp closest to the unit circle, i.e. φp = ∠(zp).

A summary of the ES-root-MUSIC algorithm is given in Table 4.1. Notethat the truncated EADF matrix G is formed offline and it needs to becomputed only once for a given antenna array. For more details on the per-formance of ES-root-MUSIC in both simulated and real-world environmentssee [72, 89] and [P. V, VII, VIII].

Next, the approximation used in (4.10) is analyzed. The impact of themodelling error ǫ(M) on the orthogonality of Eη and GD is investigated.Assuming that the true array manifold is known, the signals of the sourcesare uncorrelated, and neglecting finite sample effects, the ES-root-MUSICis an unbiased estimator [52, 84, 111]. This motivates the use of ES-root-MUSIC for studying the impact of calibration noise on the DoA estimates.

4.4 Modelling Errors due to Calibration Noise

This section describes the effects of calibration noise on the accuracy of MST-based DoA estimation algorithms. First, the results from Section 4.2 areextended by discussing the impact of calibration noise on the EADF matrixand MST implementation. Then, two closed-form expressions of the angularestimation error due to array modelling imperfections are introduced. Thesecond expression is of particular interest in real-world scenarios, because itallows predicting the performance of subspace-based algorithms using MST.For a description of other practical issues such as plane wave assumptionand phase drift during array calibration measurements see [58, 59].

In practical scenarios, the response of a real-world antenna array isknown only through calibration measurements. As described in Section 2.2,calibrating the entire antenna array implies that the measured data con-tain information on array non-idealities, such as mutual coupling, antenna

44

Table 4.1: The ES-root-MUSIC algorithm

Offline Process: Antenna Modelling using MST,

i) perform the calibration matrix Ac(φc) as in (2.5)

ii) compute the EADF matrix Ge as the Q-points IDFT of Ac(φc)

iii) form the truncated EADF matrix G ∈ CN×M by selecting

the M middle columns of Ge such as

M =

⟨{µ ∈ N : N < µ < Q} for ideal cases{µ ∈ N : N < µ ≤M0 ≪ Q, µ = M0} for noisy calibration

Online Process: Perform DoAs estimation,

1. form the array data matrix X as in (2.8)

2. estimate the element-space noise subspace Eη

3. apply the root-MUSIC algorithm on expression (4.12).

manufacturing errors, orientation and position of the sensors. Therefore,also the EADF matrix Ge contains the information about the measured ar-ray, since it is computed from a set of calibration data as shown in (4.5).Consequently, any real-world antenna array can be modelled by exploitingthe EADF and MST concepts presented in (4.8). Examples of real-worldEADFs computed from calibrated arrays can be found in [89] [P. IV, VII].

Calibration data are always corrupted by measurement noise. In contrastto ideal computer simulations, the modelling error in practice can not bemade arbitrarily small by selecting a larger number of modes M within theselection boundaries (selecting a larger number of columns of the EADFmatrix Ge). The noise perturbed EADF Ge ∈ C

N×Q can be modelled as

Ge = Ge + We , (4.13)

where Ge ∈ CN×Q is the error-free EADF matrix, and We ∈ C

N×Q containsthe calibration measurement noise. The noise refers to the receiver noiseand it is assumed to be zero-mean, i.i.d., white, complex circular Gaussiandistributed with variance σ2

W .Combining (4.8) and (4.13), the reconstructed steering vector of a cali-

brated real-world array can then be expressed as

a(φ,W,M) = Gd + ǫ(M) = Gd + Wd + ǫ(M) , (4.14)

where W is an N ×M block of the matrix We, and G = G + W ∈ CN×M

is the perturbed truncated EADF matrix. Equation (4.14) is comprised of

45

three terms. The first term is the ideal MST using a calibration noise-freeEADF matrix, while the quantities Wd and ǫ(M) represent error termsdepending on the calibration noise and the truncation error, respectively.

The effects of the truncation error on the RMSE of the DoA estimatescan be analyzed by fixing the calibration noise power σ2

W and consideringa particular realization of the calibration noise We so that the perturbedEADF matrix is Ge = Ge + We. Observe that this situation is close toreality, particularly when only one set of calibration measurements is avail-able. There exist an optimality criterion for selecting the optimal number ofmodes M0 which minimizes the RMSE of the DoA estimates. For a singlesource case, the optimality criterion is defined as

M0=arg minM

{EWe

{(φp − φp)

2}}

(4.15)

subject to φp=arg minφ

{dH

e (φ)L(M)GHe ΠηGeL(M)de(φ)

}(4.16)

and φ∈[0, 2π) , (4.17)

where EWe{·} stands for expected value with respect to We, Πη is the

true projection to the noise subspace given in (2.10), and φp and φp arethe true and the estimate DoA of the the pth source. In (4.16), the termL(M) ∈ R

Q×Q represents the selection matrix which selects the M centralcolumns and the M central elements out of Ge and de(φ), respectively.The vector de(φ) ∈ C

Q×1 is defined similarly to (4.9) such that d(φ) isthe truncated version of de(φ). For MST, the impact of calibration noisecan be reduced by optimally truncating the EADF matrix Ge [55, 88, 117].The truncation error can cause a modelling error which may dominate overthe (random) error due to observation noise and finite sample effects. Inliterature, the influence of antenna array model mismatch on estimationerrors has been analyzed by several authors; see for example [24, 45, 46, 48,72, 111] and [P. VI].

In Figure 4.3, both an ideal Ge and a perturbed Ge EADF matrix aredepicted along with three possible choices of M . Here, the M = 17 cen-tral modes show expected values larger than the noise variance. For thesimulation, an ideal UCA with N = 8 omnidirectional elements and radiusr = 0.5λ has been used. The array calibration has been performed usingan ENR= 50 dB. Here, the ENR (EADF-to-calibration Noise Ratio) is de-fined as ENR = PE/Pη, where PE and Pη define the true EADF Ge andcalibration noise powers for the nth sensor [P. VIII].

In Figure 4.4, the impact of M on the RMSE of the DoA estimates isshown. For a fixed value of the ENR, each curve has a minimum representingthe minimum achievable error in that scenario and defining the optimalnumber of modes M0, e.g. M = 17 for ENR= 50 dB. The simulations havebeen carried out using two uncorrelated sources at {φ1, φ2} = {25◦, 45◦},observation noise-free data, an UCA with N = 8 elements, radius r = 0.5λ,

46

−11 −8 −5 0 5 8 11−80

−60

−40

−20

0

20

EA

DF

Mag

nitu

de [d

B]

Mode Index (m)

Ideal EADFNoisy EADFCutoff Point

M = 11

M = 17

M = 23

Figure 4.3: Representation of different cutoff points when the EADF matrixis computed from noisy calibration data. When M = 11 (too small) usefulinformation is discarded. IfM = 23 (too large) modes which are significantlycorrupted by calibration noise are selected.

10 15 20 25 30 35 40

10−5

10−4

10−3

10−2

10−1

100

RM

SE

[deg

rees

]

Number of Modes (M)

ENR = 20 dB

ENR = 50 dB

ENR = 80 dB

ENR = 120 dB

Estimated: source 1Estimated: source 2

Figure 4.4: Behavior of the estimated RMSE using ES-root-MUSIC. For afixed value of ENR, as M increases first the error decreases to a minimumvalue. If M continues increasing the RMSE slowly increases due to thecalibration noise in the EADF.

and 1500 independent realizations of W. As discussed in [P. VIII], theRMSE on the right side of the minima depends on the calibration noise,while on the left side depends on the truncation error.

Comparing Figures 4.3 and 4.4, it is simple to interpret the following ruleof thumb. In order to minimize the modelling error resulting from using aperturbed EADF Ge computed from noisy calibration data, the optimal

47

number of modes M0 is given by the number of modes having a magnitudelarger than the calibration noise level.

Truncating the perturbed EADF Ge to G having M = M0 modes hastwo main effects. It reduces the amount of data to be stored [117] and itimproves the accuracy of the array data model by a factor of Q/M such as

ENR eG[dB] ≈ ENR eGe[dB] + 10 log10

(Q

M

). (4.18)

For a fixed number of selected modes M , the larger is the number of cali-bration points Q, the higher is the resulting ENR. This is an advantage ofMST/EADF based techniques over methods which perform array manifoldinterpolation directly using the calibration data; e.g. see [98, 130].

In [P. VIII], an error analysis studying the impact of modelling errorson the accuracy of the angular estimates is presented. Considering a singlecalibration measurement, the perturbed truncated EADF matrix contains aparticular realization of the calibration noise W and G = G+W ∈ C

N×M0 .A closed-form expression of the estimation error can be given as

(φp − φp) ≈ −ℜ{d′H(φp)GHEnE

Hn Gd(φp)}

d′H(φp)GHEnEH

n Gd′(φp), (4.19)

where d′(φ) ,∂d(φ)

∂φ ∈ CM0×1 and Eη is formed according to (4.11). Ob-

serve, from an application point of view equation (4.19) is not very practical.It requires the knowledge of the span of the true noise subspace En.

Considering to perform multiple calibration measurements at a fixedSNR, an approximate expression of the MSE due to calibration noise can bederived. The resulting expression depends only on quantities that are knownor computable in real-world application, such as the perturbed EADF G, thecalibration noise power σ2

W and the number of selected modes M . Observethat in practice the calibration noise level is given in the specifications ofthe hardware equipment used for calibration or it can be computed at thereceiver side by switching off the transmitter [59]. Following the derivationin [P. VIII] the MSE of the DoA estimates is

EW {(φp − φp)2} ≈ σ2

W

d′H(φp)GHEnE

Hn Gd′(φp)

, (4.20)

where EW {·} stands for expected value with respect to W, and En is the per-turbed noise subspace and it is a function of G. Equation (4.20) is of interestin practical applications because it allows to compute the performance limitby using a perturbed EADF (computed from real-world calibration data) asa model for the antenna array. For more details on the derivation of (4.19)and (4.20), see [P. VIII] and Appendix I.

48

−10 0 10 20 30 40 50

10−2

10−1

100

101

RM

SE

[deg

rees

]

SNR [dB]

Estimated by ES−root−MUSICComputed from eq. (4.19)Computed from eq. (4.20)CRLB

Figure 4.5: Comparison between the RMSEs of the DoA estimates usingthe ES-root-MUSIC and the expressions in (4.19) and (4.20). The proposedexpressions well estimate the error floor due to modelling error.

In order to verify the accuracy of the error analysis, the statistical perfor-mance of ES-root-MUSIC has been compared with (4.19) and (4.20). Thesimulations are carried out using an UCA with N = 8 elements and ra-dius r = 0.5λ measured by uniformly taking Q = 360 calibration points inφ = [−π, π) with an ENR= 50 dB. Here, 1000 independent realizations of Wand N have been used and the perturbed EADF G has been constructed bytruncating at M0 = 17 modes. In the simulations, two uncorrelated sourcesat {φ1, φ2} = {25◦, 45◦} and K = 256 snapshots have been used. Note,(4.20) has been evaluated using a single realization G for each SNR value.

Figure 4.5 shows that even if the MST uses a perturbed and truncatedversion of the EADF, for SNRs < 32 dB the finite sample effects dominateover the antenna modelling error. This also means that the modelling er-ror in (4.10) and (4.11) is negligible and the statistical performance of theES-root-MUSIC algorithm is close to the CRLB. On the other hand, forSNRs > 32 dB the array modelling error (calibration noise) dominates overthe observation noise. The statistical performance of the DoA estimates sat-urates. Both of the expressions in (4.19) and (4.20) predict the error floordue to modelling errors well. For convenience, only the results for source 1are depicted here, since similar results can be observed for source 2.

It is important to stress that only expression (4.20) is usable in real-worldscenarios. It requires only knowledge of quantities, which can be determinedand computed from the measured array calibration data.

49

4.5 Discussion

The concept of wavefield modelling has been used in the area of theoreti-cal physics and mathematics since the 40’s [102]. Later in the 90’s, Doronet al. [19-21] introduced the wavefield modelling concept and the manifoldseparation technique in the field of array signal processing. Only in the pastseven years these ideas have become more and more exploited as an engi-neering tool for both theoretical and practical studies. Nowadays, MST isintensively used in research applications related to modelling and evaluat-ing the performance of antenna arrays [55-57, 59, 89] [P. VII, VIII], channelsounding [58, 88, 91], parameter estimation and tracking [90, 95, 96].

Using MST-based algorithms offers several advantages over the AIT, theBT, and algorithms working directly on the calibration data:

• In real-world scenarios, MST is easy to implement and it allows a jointrepresentation of the array non-idealities at the time of the calibration.

• In contrast to AIT, BT- and MST-based algorithms do not requiresector-by-sector processing.

• In contrast to BT, AIT- and MST-based algorithms can be applied onnon-ideal arrays without additional changes in the implementation.

• Similar to the BT and the DT, MST maps the physical array steeringvector into a Vandermonde structured virtual array which does nothave physical interpretation and does not need to be tuned.

• MST allows fast polynomial rooting-based methods such as root-MUSIC and RARE to be applied on arrays of arbitrary geometry;see [19-22, 72, 89] and [P. V, VII, VIII].

• Using the concepts of EADF and MST can improve the performance ofsearch-based algorithms such as beamformer, MVDR Capon, MUSICand WSF [122]. In fact, exploiting these concepts for modelling the re-sponse of simulated and real-world sensor arrays gives an improvementin calibration SNR by the factor Q/M ; see equation (4.18).

• It allows closed-form representation of the derivatives of real-worldarrays steering vectors [P. VII, VIII] instead of using numerical ap-proximations computed from the calibration data [89, 95, 96].

• The modelling errors may be set according to an SNR target. Forexample it has been shown that the number of selected modes controlsthe trade-off between the array model (DoA estimates accuracy) andthe computational complexity of the ES-root-MUSIC algorithm.

50

• For MST-based algorithms there exists a criterion for selecting theoptimal number of modes which minimizes the impact of calibrationnoise on the DoA estimates [P. VIII].

As a disadvantage of MST, implementing MST using M > N as de-scribed in (4.8) and (4.9) does not allow a direct implementation of tech-niques such as spatial-smoothing and root-WSF [54]. This is due to the factthat it is not longer possible to divide the modelled array response into sub-arrays showing a shifting invariant structure. However, this problem canbe overcome by exploiting suboptimal transformation techniques based onMST, where suboptimal refers to the selection of an non-optimal number ofmodes, i.e. M ≤ N . For more details see [22, 60, 72, 128] and Section 5.1.2.Furthermore, the computational complexity of ES-root-MUSIC depends onthe number of selected modes. The larger is M , the larger is the polynomialorder 2M − 1, and the higher is the computational complexity. There existalgorithms which solve for the polynomial roots using order of n(log(n))2

arithmetic operations, where n is the order of the polynomial [43, 75].The next chapter presents a comparison of the AIT, the BT and the

MST in terms of Vandermonde approximation (fitting) error. It will beshown that MST can provide the smallest fitting error over the whole 360◦

coverage area despite the sensor array characteristics.

51

52

Chapter 5

Comparison of Transform

Techniques

The focus in the preceding chapters has been on analyzing and deriving al-gorithms that provide high-resolution, optimal or close to optimal statisticalperformance, and low computational complexity despite the antenna arrayconfiguration and imperfections. Two different classes of methods have beeninvestigated, the so called data- and model-driven transform techniques.The first group comprises of AITs and the BT for UCAs. These methodsare designed to linearly transform the structure of the observed data to bestfit a previously defined model. The second group includes the MST, which isdesigned to model the azimuthal response of a sensor array of arbitrary con-figuration by using Vandermonde structured models. These transformationtechniques allow using fast and high-resolution DoA estimators originallydesigned for ideal ULAs with sensor arrays of arbitrary configuration. As ithas been extensively discussed in this thesis, a common drawback of thesetechniques is that they introduce additional errors into the system in theform of mapping and modelling errors. Whenever these errors are not prop-erly compensated for and handled by specifically designed algorithms, theymay lead to bias and excess variance in the estimates.

In this chapter, the data- and model-driven transform techniques arecompared in terms of their Vandermonde approximation error, also calledthe fitting error. The fitting error represents a way for analyzing the impactof transformation techniques on the statistical performance of the estima-tors. Generally, techniques providing small transformation errors have asmall impact on the quality of the estimates. In [9, 26, 34, 86, 109, 129] and[P. II-IV, VI, VIII] this phenomenon has been analyzed by showing that thesmaller the mapping and modelling errors are, the smaller are the bias andexcess variance in the DoA estimates. For data-driven transform techniques,the fitting error describes the difference between the steering vectors of thetransformed physical array and the Vandermonde structured virtual array;see Sections 3.1 and 3.3. For model-driven transform techniques, the fitting

53

error describes the difference between the modelled and the physical arraysteering vectors; see Section 4.4. In this case the modelled steering vector iscomputed as a product of the EADF matrix and a Vandermonde structuredvector; see Section 4.2.

It is important to notice that due to the very non-linear relationshipbetween the MSE of the estimates and the fitting error, using a techniquewith a low fitting error does not necessarily imply that the impact on theestimates will be minimized too. Several authors have shown that thereexists a trade-off between the Vandermonde approximation error and theMSE of the estimates. In [63] the proposed DT-based transformation ap-proach sacrifices a lower fitting error for improved robustness and resolutionin the estimates. In [45, 46] the authors presented a strategy for optimallydesigning the interpolation matrices, which minimize the MSEs of the DoAestimates. In the case of AITs, simulation results showing the MSE withrespect to the angular separation of sources and SNR are discussed in [8, 45,46, 64]. For the case of the BT for UCAs, the effects of the mapping errorson the DoA estimated have been analyzed in [63, 70] and [P. II-IV, VI]. Forthe case of MST, an analysis on the effects of modelling errors on the MSEof subspace-based DoA estimation algorithms can be found in [P. VIII]. Inthe paper it is shown that the MSE of the estimates can be minimized byoptimally selecting the number of excitation modes.

This chapter is organized as follows. First, the transform techniques arecompared in terms their Vandermonde approximation errors. Discussionson the trade-off between fitting errors and MSE in the estimates are alsocarried out. In order to prove the applicability of AIT, BT, and MST torealistic scenarios, two different array structures are considered. In Section5.2, an approximate CRLB for real-world arrays is introduced. The deriva-tion exploits the concept of MST to provide a sufficiently accurate modelof the array under consideration. Finally, Section 5.3 summarizes the keyfeatures of the transform techniques discussed in this thesis.

5.1 Vandermonde Approximation Errors

This sub-section is divided into two parts, that is, into a theoretical and arealistic case study. The transform techniques have been tested using twoarray configurations, one formed by omnidirectional (theoretical case) andthe other formed by directional sensors (realistic case). In both scenarios, theantenna arrays consist of sensors having known and identical beampatterns.The locations of the sensors are also assumed to be known.

The reason for considering a synthetic directional array instead of areal-world one is that for real-world antenna arrays it is not possible toprovide a fair comparison of the transform techniques in terms of fittingerror. This is due to the fact that the true response of a real-world antenna

54

array is never perfectly known. In practice the array response is known onlythrough noisy calibration measurements, or it can be estimated using MST.For example, using AIT on real-world arrays yields transformation matricescomputed for optimally mapping the calibration data and not the true arrayresponse. Hence, in real-world scenarios, the idea of fitting error is no longermeaningful.

5.1.1 Theoretical Scenario

Consider an ideal UCA of radius r = 0.5λ formed by N = 8 omnidirec-tional sensors. The physical array response is described by auca(φ) in (2.3).The omnidirectional array elements’ beampatterns are all equal to the onedepicted in Figure 2.3.

For a given array configuration, the fitting errors have been comparedby normalizing them to the respective objective functions. The following fit-ting errors represent the mapping and modelling errors using Vandermondestructured models. The errors are defined as

ψoai(φ)=

∥∥THl auca(φ) − al

ula(φ)∥∥

∥∥alula(φ)

∥∥ (5.1)

ψob (φ)=

∥∥THd auca(φ) − db(φ)

∥∥∥∥db(φ)

∥∥ (5.2)

ψoms(φ)=

∥∥Gd(φ) − auca(φ)∥∥

∥∥auca(φ)∥∥ , (5.3)

where φ ∈ [0, 2π) and ψoai(φ), ψo

bt(φ), and ψoms(φ) stand for the fitting errors

caused by the AIT, the BT, and the MST, respectively. The superscript ”o”stands for the theoretical case study. Figure 5.1 depicts the Vandermondeapproximation errors calculated according to (5.1)-(5.3).

In the case of AIT the simulation settings for the physical and the vir-tual arrays are shown in Figure 3.1. The ideal UCA is mapped into an idealULA of size Nv = 7, which is located on the diameter of the UCA. Theright- and left-most elements of the virtual array are placed on the circum-ference of the UCA. Hence, the interelement spacing of the virtual ULA isd = 2r/(Nv − 1). The orientation of the virtual ULA is chosen so that itsbroadside direction is in the middle of the in-sector φ ∈ [75◦, 105◦]. Note thatwith this configuration the UCA and the virtual ULA have approximativelythe same aperture for sources located within the in-sector. Following thesettings used in [60], the virtual ULA steering vector matrix has columns de-fined as in (2.2) and it is constructed by taking Ql = 2049 calibration pointswithin the in-sector. As expected, AIT provides a low fitting error withinthe considered in-sector, but a large error elsewhere. By reducing the widthof the in-sector or by decreasing the size of the virtual array, the fitting errorcan be made smaller. A lower fitting error usually leads to a reduction of

55

0 50 100 150 200 250 300 35010

−12

10−9

10−6

10−3

100

Azimuth [degrees]

Nor

mal

ized

Fitt

ing

Err

or

BTAITMST, with M = 21MST, with M = 33

Figure 5.1: Comparison of BT, AIT [26], and MST in terms of fitting errorswhen using an ideal UCA formed by omnidirectional sensors. Choosing asufficiently large M for the MST results in a significant and consistentlysmaller fitting error than using the AIT or the BT.

the bias in the estimates. However, from the DoA estimation point of view,a smaller in-sector leads to an increase in the computational complexity ofthe algorithm (more angular sectors need to be processed), while a smallervirtual array usually results in an increase of the excess variance in the esti-mates. Consequently, depending on the system specifications such as SNRtarget and processing power, a compromise between all the above quantitiesneeds to be made.

In the case of BT (or DT), the ideal UCA is mapped into a virtual arrayof size Mv = 7 according to (3.10)-(3.13). Under ideal assumption BT showsuniform performance throughout the whole coverage area. Observe thatagain the fitting error decreases as the size of the virtual array decreases.Similarly to AITs, a smaller size of the virtual array usually leads to anincrease in the excess variance in the estimates. For more details see [106]and [P. III, VI].

In the case of MST, the ideal UCA has been sampled over a grid of Q =360 calibration points uniformly distributed in the whole 360◦ coverage area.In order to show the effect of truncation on the Vandermonde approximationerror, M = {21, 33} selected modes have been used. Figure 5.1 shows thatMST can provide a fitting error that is several orders of magnitude smallerthan those of AIT and BT over the whole coverage area. Theoretically,when using MST the fitting error and the bias in the estimates can be madearbitrarily small by increasing M . Observe that MST-based algorithms donot suffer from excess variance in the estimates since they do not perform

56

data mapping.

5.1.2 Realistic Scenario

In order to make the simulation more realistic, a UCA formed with directiveelements and measured by recording noisy calibration data is considered.The true array response of the synthetic UCA is defined by

as(φ) = diag{β0, . . . , βN−1}auca(φ) , (5.4)

where βn(φ, αn) = 1.2 + cos(φ − αn) is the nth sensor gain depicted inFigure 2.3 and αn = 2πn/N , for n = 0, . . . , N − 1, is the direction ofmaximum gain. Each sensor is oriented so that αn points outside the array.

The calibration data matrix Ac ∈ CN×Q of the array described in (5.4)

is formed by sampling the true array response as(φ) in Q adjacent directionsalong φ. As described in Section 2.3, the noisy calibration data can then bemodelled as Ac = Ac + Wc, where the matrix Wc contains the calibrationnoise. Fixing the noise power at ENR= 50 dB and computing the EADFmatrix as described in (4.5), the resulting EADF is similar to the noisyEADF depicted in Figure 4.3. In contrast to real-world scenarios, in asimulation environment both the noisy and noise-free array responses areknown. This allows a fair comparison of the fitting errors.

Similarly to Section 5.1.1, the normalized fitting errors are defined as

ψrai(φ)=

∥∥THl as(φ) − al

ula(φ)∥∥

∥∥alula(φ)

∥∥ (5.5)

ψrb (φ)=

∥∥THd as(φ) − db(φ)

∥∥∥∥db(φ)

∥∥ (5.6)

ψrms(φ)=

∥∥Gd(φ) − as(φ)∥∥

∥∥as(φ)∥∥ , (5.7)

where φ ∈ [0, 2π) and the superscript ”r” stands for realistic scenario.In Figure 5.2, the fitting errors of (5.5)-(5.7) have been computed using

the antenna array defined in (5.4) and its calibration data. Unless explicitlystated otherwise, the simulation settings are as described in Section 5.1.1.

In the case of AIT, the interpolation matrix Tl for the in-sector φ ∈[75◦, 105◦] is computed from the noisy calibration measurements Ac as de-scribed in (3.1) and (3.2) while, in (5.5), the true array response is used forevaluating the fitting error. Observe that this situation is close to realityand it gives a realistic interpretation of the Vandermonde approximationerror. As expected, AIT provides a low fitting error within the in-sector.However, since the transformation matrix is not computed from the truearray response, the value of the fitting error within the in-sector can not belowered by changing the number of virtual array elements Nv or the widthof the in-sector.

57

0 50 100 150 200 250 300 350

10−3

10−2

10−1

100

Azimuth [degrees]

Nor

mal

ized

Fitt

ing

Err

or

BT

AIT

MST, with M = 17

Mapping using EADF

Figure 5.2: Comparison of BT, AIT [26], MST, and the mapping usingEADF in terms of fitting errors when using an ideal UCA formed by di-rectional sensors. BT does not work in this case. MST can still provide aconsistently smaller error than the other techniques.

In the case of BT, the fitting error is significantly larger in this scenariothan in the theoretical scenario described in Section 5.1.1. As the plots inFigure 5.2 suggests, and as is pointed out in [63], the BT performs wellonly on ideal omnidirectional UCAs. Consequently, using BT-based DoAalgorithms in real-world scenarios generally leads to very poor estimates.Solutions to this problem have been proposed in [63, 128].

In the case of MST, the EADF matrix has been optimally truncated toM0 = 17 modes. As for AIT, the array model Gd(φ) used in (5.7) has beenconstructed from noisy calibration data and it is compared with the truearray response as(φ) in (5.4). As in the theoretical case, MST shows againa fitting error, which is consistently smaller than those of AIT and BT overthe whole coverage area. Due to calibration noise, the fitting error can notbe made smaller by selecting a larger M . For more details on the modellingerrors due to calibration noise see Section 4.4. As presented in [P. VIII], thebias in the estimates due to modelling errors is minimized by selecting theoptimal number of modes using the proposed selection criterion.

Figure 5.2 also depicts the performance in terms of fitting error of analternative way of exploiting the EADF and MST concept. A transformationmatrix for mapping arbitrarily structured antenna arrays into virtual arrayswith ULA-like properties over the whole 360◦ visible area can be found bysuboptimally truncating the EADF matrix at M ≤ N modes. As suggestedin [21, 22, 72], the transform matrix is defined as TH

g = (GHG)−1GH ∈C

M×N . For the simulation, the perturbed truncated EADF matrix G givenin (4.14) has been constructed using M = 7 modes. The resulting fitting

58

error is then expressed as

ψrme(φ) =

∥∥THg as(φ) − db(φ)

∥∥∥∥db(φ)

∥∥ , (5.8)

where φ ∈ [0, 2π) and ψme stands for mapping using EADF. This alterna-tive mapping technique shows uniform performance in the whole coveragearea. Comparing equations (5.6) and (5.8), the latter can be interpretedas a generalized BT [60], which can operate on real-world sensor arrays ofarbitrary configuration.

5.2 Approximate CRLB for Real-World Arrays

In order to evaluate the potential performance of a parameter estimation al-gorithm operating on a given signal model, it is meaningful to derive a boundfor the estimators performance. The Cramer-Rao lower bound (CRLB) pro-vides a bound on the covariance matrix of any unbiased estimate of theparameter of interest [13, 85, 88, 107, 108]. The CRLB is defined as theinverse of the Fisher information matrix [25].

In array signal processing, several bounds for different signal models havebeen proposed [89, 104-109, 120]. This section focuses on the stochasticCRLB for DoA estimation and it presents an approximate CRLB definingthe performance limit of MST-based DoA algorithm [P. VII].

It is important to note that the following derivation gives the exactCRLB for the signal model of (4.10) used in the implementation of the ES-root-MUSIC algorithm but only an approximation of the CRLB for a givenantenna array. This is due to the modelling error given in (4.14), which canbe made small only up to a threshold that depends on the measurementnoise σ2

W in the calibration data; see Section 4.4. However, if the calibra-tion measurements are properly done, the modelling error can be neglectedand the approximated CRLB still describes the performance bounds (at thehighest possible SNR) of a given real-world array with sufficient accuracy.

Starting from the signal model defined in (4.10) and (4.11), and usingboth the property of MST [21, 88, 89] [P. VIII] and the results in [104], theCRLB for the parameters of interest φ (DoAs) is given by [P. VII]

CRLB(φ) =σ2

η

2K

{ℜ

[(∆HΠη∆) ⊙ (RsD

HGHR−1x GDRs)

T]}−1

, (5.9)

where ∆ = [δ1, . . . , δP ] ∈ CN×P with

δp =∂a(φ)

∂φ

∣∣∣φ=φp

= G∂d(φ)

∂φ

∣∣∣φ=φp

. (5.10)

Here δp ∈ CN×1 contains the derivative of the array steering vector with

respect to φ evaluated at the pth DoA, A(φ) ≈ GD(φ) ∈ CN×P is the

59

Table 5.1: Comparison among the presented transform techniques.Data-Driven Model-Driven

Features AIT BT MST map MST

Sector-by-SectorProcessing

yes no no no

Transformation isData-Independent(1)

yes(2) yes yes yes

Applicable onArbitrary ArrayGeometries

yes no yes yes

Applicable onReal-World Arrays

yes no(3) yes yes

AllowSpatial-Smoothing

yes yes yes no

AllowHigh-Resolution

yes yes yes yes(4)

Bias yes yes yes yesExcess Variance yes yes yes noDesignComplexity

high low low low

ImplementationComplexity

med./high low low med./low

TransformationErrors

low(5) medium medium low

(1) The transformation matrices are computed offline.(2) except [64] and [45, 46] for multiple sources.(3) unless jointly used with techniques providing robustness [63, 128].(4) such as ES-root-MUSIC and RARE [89].(5) within each of the in-sector.

MST-modelled array steering matrix, and Rs ∈ CP×P and Rx ∈ C

N×N arethe signal and array covariance matrices, respectively. In equation (5.9), ⊙denotes the Hadamard-Schur product, i.e., element-wise multiplication.

5.3 Discussion

This chapter shows a comparison of the transform techniques presented inthis thesis in terms of Vandermonde approximation errors. From the simula-tion results depicted in Figures 5.1 and 5.2 it can be seen that MST providesa consistently smaller fitting error than the other techniques in both the the-oretical and realistic case. Hence, the impact of the fitting error on the DoAestimation accuracy and the computational complexity of the system can bereduced by using MST-based estimators.

Table 5.1 summarizes the key features of all the transform techniquesconsidered in this manuscript. The techniques have been separated into

60

data- and model-driven transforms. The block named ”design complexity”in the table considers the number of free-parameters [9] and the implemen-tation procedure required by each of the transform techniques. In this table”implementation complexity” refers to the total (transform technique andestimator) computational cost during the online process. For a given estima-tion algorithm (e.g. root-MUSIC), this is a measure of the overall complexityrequired for estimating DoAs. In the case of AIT, the implementation com-plexity includes running the DoA algorithm for each of the angular sectorand the final procedure for selecting the final DoA estimates. Instead, in thecase of MST, the complexity includes solving for the roots of a polynomialequation of degree 2M + 1, where M > N .

The CRLB presented in (5.9) is the bound for the signal model in (4.10)and (4.11) using GD(φ) as array steering vector matrix. The proposedCRLB can hence be used in simulated environments as a benchmark formodelled antenna arrays [32]. Otherwise, in practical scenarios, expression(5.9) can be exploited for predicting the performance of real-world arrays.

61

62

Chapter 6

Summary

The modern trend of utilizing multi-sensor systems in emerging servicesand applications such as localization, wireless communication, radar, andbiomedicine has recently focused attention to the area of sensor array signalprocessing. The benefits offered by multi-antenna systems include the pos-sibility of exploiting the spatial component in addition to the time and fre-quency domain, increased resolution, higher radio-link robustness, reducedinterference, and improved spectral efficiency. Among the research areas in-volving sensor array signal processing the problem of finding the direction-of-arrival (DoA) of the wavefront impinging on a sensor array is a topicthat has been investigated since the beginning of the 20th century. Today,direction finding is still an important research topic and it is used in manydifferent modern applications such as beamforming, tracking, navigation,surveillance, channel sounding, and antenna modelling and testing.

Most high-resolution and computationally efficient sensor array algo-rithms have been developed for ideal sensor arrays with regular geometryand known element response. In practice, however, the sensor array con-figuration (geometry, interelement spacing, and sensor type) can not bechosen freely. Moreover, the array response is always an unknown quantity,which can be estimated only through noisy calibration measurements. Con-sequently, it is generally not possible to apply the aforementioned algorithmson real-world sensor arrays with arbitrary configuration. In real-world sce-narios the array configuration includes linear, planar, and volumetric arraystructures containing non-idealities such as mutual coupling, antenna man-ufacturing errors, and sensor misplacement.

In the past thirty years a lot of effort has been put in deriving pre-processing methods such as array interpolation techniques (AIT), beamspacetransform (BT), and manifold separation techniques (MST), for extendingthe applicability of algorithms originally developed for ideal array configura-tions to arbitrary ones. These methods have been derived following differentdesign and implementation techniques, which offer different advantages de-

63

pending on the structure of the sensor array and the working environment.As a common feature they often introduce additional uncertainties into thesystem that, if not properly compensated for, can affect the performance ofthe estimation algorithms.

The work reported in this thesis provides new solutions which allow us-ing high-resolution DoA estimation algorithms with arbitrarily structuredsensor arrays. This manuscript analyzes and derives novel algorithms pro-viding high-resolution, optimal or close to optimal statistical performance,and low computational complexity, despite the antenna array geometry andimperfections. In particular, the problem of estimating the DoA of emittingsources located all at the same elevation angle has been investigated. In or-der to validate and verify the performance of the proposed techniques ideal,simulated, and real-world antenna arrays have been used. The proposed re-sults contribute to the research field of array transform techniques, antennamodelling and signal processing algorithms using sensor arrays of arbitrarilyconfigurations.

In this thesis the transform techniques have been divided into two dif-ferent classes: the data- and model-driven transform techniques. The firstgroup comprises of AITs and BT. They are designed for linearly transform-ing the steering vector of a physical array into that of a virtual array withmore desired features. They allow using high-resolution algorithms with ar-bitrarily structured sensor arrays at the cost of introducing transformationerror into the system. In this thesis, the key ideas and performance of thesetechniques have been investigated. An analysis of the effects of BT mappingerrors on subspace-based DoA estimation algorithms has been carried out.A closed-form expression describing the effects of these errors on the DoAestimates has been derived. Novel algorithms developed for reducing theimpact of BT mapping error on the estimates have been introduced. Theproposed algorithms allow significantly improving the accuracy of the DoAestimates by removing the bias and mitigating the excess variance due tothe BT. Hence, practically unbiased estimates which are closer to the CRLBcan be achieved. Furthermore, an alternative approach for suppressing theBT mapping error has also been developed. Simulation results show thatpractical unbiased DoA estimates can be obtained without using techniquesfor bias removal when the beampattern of the array sensors are optimallydesigned.

In this work it has been shown that the model-driven transform tech-niques can offer several advantages over the data-driven techniques. Op-posite to AITs, the MST does not require sector-by-sector processing. Incontrast to the BT, the MST can be applied on non-ideal arrays. The MSTcan provide a significantly smaller fitting error over the whole 360◦ coveragearea when using both ideal and real-word sensor arrays. Furthermore, themodel-driven techniques do not suffer from mapping errors. Data mappingis not required and the MST-based algorithms can process the array data

64

directly in element-space. However, in practical scenarios the accuracy ofthe estimates may be subjected to error caused by array modelling errorsand calibration noise.

A pragmatic approach using the concept of effective aperture distributionfunction (EADF) has been developed for implementing the MST in practi-cal scenarios. A novel version of root-MUSIC (ES-root-MUSIC) based onthe concepts of EADF and MST has also been introduced. Practical issuesconcerning the implementation of MST-based algorithms in real-world sce-narios have also been addressed. Simulation results carried out using bothideal and real-world arrays show that ES-root-MUSIC can provide fast andaccurate DoA estimation independent of the array configuration. The effectof noisy calibration data on subspace-based DoA algorithms using MST hasbeen analyzed. First, an optimality criterion for selecting the optimal num-ber of modes which minimizes the error in the DoA estimates is developed.Then, two closed-form expressions describing the effects of calibration noiseon the array model accuracy have been derived. One of these expressionsis of particular interest in real-world scenarios because it allows predictingthe performance of subspace-based algorithms using MST. An approximateexpression of the CRLB using MST has also been introduced. It can beexploited for predicting the performance of real-world arrays.

Future research continuing the work of this thesis should include DoAalgorithms derived for estimating both elevation and azimuth simultane-ously. The solution to the problem may lead to the study of transformtechniques based on spherical harmonics as a potential basis. This can beof interest in applications such as 3-D antenna modelling, localization, andsurveillance services. Another research direction should involve the study ofself-calibration methods for MST-based algorithms. In case of mass produc-tion of handsets containing multiple-antennas it is realistic to use a referenceEADF matrix for describing the antenna array response. Self-calibratingmethods could be used for improving the accuracy of the estimates by miti-gating the effects of manufacturing errors. Last, but not least, further inves-tigation on the performance of MST-based mapping techniques should becarried out. This is of interest for applications using arbitrarily structuredarrays in environments with highly correlated signals.

65

66

Appendix I

A justification for the approximation ‖EHn Gd′

i‖2 ≈ ‖EHn Gd′

i‖2 used in [P.VIII] can be given as follow. Here the equations number refer to [P. VIII].

The vector d′

i is not given by a linear transformation of di and d′

i ⊥ di

for all φi ∈ [−π, π); see (23). Consequently, as also shown in (27) and (28),it is not orthogonal to EH

n G, i.e. ‖EHn Gd′

i‖ 6= 0, while ‖EHn Gdi‖ ≈ 0.

First, the terms G and G = G + W are considered. If ‖G‖F ≫‖W‖F (for sufficiently large ENR and number of selected modes M), then

EHn Gd′

i ≫ EHn Wd′

i, and we can write

‖EHn Gd′

i‖ = ‖EHn Gd′

i+EHn Wd′

i‖ ≈ ‖EHn Gd′

i‖ . (I.1)

Therefore, the terms En and En are considered. The canonical anglesbetween the span of the ideal En ∈ C

N×(N−P ) and the perturbed En ∈C

N×(N−P ) noise subspaces give a measure of the angle between the canonical

bases of these subspaces [103; pages 260-261]. The canonical angles of EHn En

are computed as cos−1(λi), where λi is the i-th singular value of EHn En. If

‖G‖F ≫ ‖W‖F , cos−1(λi) ≈ 0 (for i = 1, . . . , N), i.e. the two subspaceshave approximatively the same span.

Consequently, the projection of Gd′

i into EnEHn or EnE

Hn is approxima-

tively the same and ‖EHn Gd′

i‖2 ≈ ‖EHn Gd′

i‖2 holds.

67

68

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