Springer Topics in Signal Processing Boaz Rafaely Fundamentals of Spherical Array Processing Springer Topics in Signal Processing Volume 8 Series editors Jacob Benesty, Montreal, Canada Walter Kellermann, Erlangen, Germany More information about this series at http://www.springer.com/series/8109 Boaz Rafaely Fundamentals of Spherical Array Processing 123 Boaz Rafaely Department of Electricaland Computer Engineering Ben-Gurion Universityof theNegev Beer-Sheva Israel ISSN 1866-2609 ISSN 1866-2617 (electronic) Springer TopicsinSignal Processing ISBN 978-3-662-45663-7 ISBN 978-3-662-45664-4 (eBook) DOI 10.1007/978-3-662-45664-4 LibraryofCongressControlNumber:2014955795 SpringerHeidelbergNewYorkDordrechtLondon ©Springer-VerlagBerlinHeidelberg2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer-VerlagGmbHBerlinHeidelbergispartofSpringerScience+BusinessMedia (www.springer.com) To my parents, Nitzan and Rivka Rafaely Preface Microphonearraysandassociatedarrayprocessingtechniqueshavebeendeveloped for a wide range of applications over the past few decades. These applications include speech communication, music recording, room acoustics analysis, noise control and acoustic holography, defense and security, entertainment, and many more.Inthecasesofspeechinroomsandmusicinconcerthalls,thesoundtendsto travel throughout the entire enclosed space, producing a three-dimensional sound field. Microphone arrays that effectively measure and process three-dimensional sound fields typically require the positioning of microphones within a volume in three-dimensional space. Planar arrays, mounted on an enclosure wall, have been studied for several decades, while more recently, spherical arrays, in which microphonesaremountedaroundarigidsphere,forexample,havebeendeveloped. These offer several advantages over classical linear, rectangular, or circular arrays: (i) Thesphere,havingcompleterotationalsymmetry,facilitatesspatialfiltering, or beamforming, that can be designed to effectively enhance or attenuate sources in any direction. (ii) Array processing and performance analysis can be formulated in the spher- ical harmonics domain, which is the Fourier domain for the sphere. This domain facilitates efficient algorithms and extensive acoustic modeling of both the array and the surrounding sound field. (iii) Beamforming can be efficiently implemented by decoupling beam pattern design from beam pattern steering, therefore providing simplicity and flex- ibility in array realization. Theseadvantageshavemotivatedanincreasing numberofresearchersinrecent years to develop spherical microphone array systems, to study spherical array configurations,todevelopalgorithmsforsphericalarrays,andtoapplythesearrays inawiderangeofapplications.Thisgrowingactivityhasprovidedtheauthorwith the motivation and inspiration to write this book, with the aim of presenting the fundamentals of spherical array processing in a tutorial manner suitable for researchers, graduate students, and engineers interested in this topic. vii viii Preface The first two chapters provide the reader with the necessary mathematical and physical background, including an introduction to the spherical Fourier transform and to the formulation of plane-wave sound fields in the spherical harmonics domain. The third chapter covers the theory of spatial sampling, which becomes useful when selecting the positions of microphones to sample sound pressure functionsinspace.Thenextchapterpresentsvarioussphericalarrayconfigurations, including the popular configuration based on a rigid sphere. The fifth chapter introduces the concept of beamforming and its basic equations, including popular design methods such as delay-and-sum and regular beamforming. The following chapter presents methods for the optimal design of beam patterns, formulated to achieve various objectives, such as maximum robustness, maximum directivity, or minimum side-lobe level. The final chapter develops more advanced array processing algorithms, such as the minimum variance distortionless response (MVDR) algorithm. These algorithms aim to enhance a desired signal while attenuatingundesirednoisecomponentsinthesoundfieldbyexploringtheirunique formulation in the spherical harmonics domain. Myowninterestinspherical array processing beganduringasix-monthvisitto the sensory communication group at MIT in 2002, working with Julie Greenberg andgreatlyenjoyingthestimulatingvibeofBoston.IwouldliketothankJuliefor providing this opportunity, for the hospitality, and for the helpful discussions. During myvisittoBostonIwasexposed totheinspiringpublications onspherical arraysbyJens MeyerandGaryElko.Theirpioneeringworkplantedtheseeds that later flourished to an extensive research effort at my lab, the acoustics laboratory, Ben-Gurion University of the Negev. The research at the acoustics laboratory was pursued by an invaluable cooperation with a great number of research students, postdoctoral researchers, and visitors. The relaxed atmosphere at the lab, the great teamwork, and the endless discussions were the fuel that kept the writing of this book viable. I would like to express great thanks to the acoustics laboratory researchers: Dr. Jonathan Sheaffer, Dr. Jonathan Rathsam, Dr. Noam Shabtai, Dr. Dror Lederman, Dr. Yotam Peled, Dr. Etan Fisher, Vladimir Tournabin, Hai Morgenstern,DavidAlon,KobyAlhaiany,MickeyJeffet,EladCohen,DimaLvov, Or Nadiri, Shahar Villeval, Tal Szpruch, Nejem Hulihel, Ilan Ben-Hagai, Tomer Peleg, Amir Avni,Morag Agmon,Maor Klieder, Dima Haykin,ItaiPeer,andIlya Balmages. Also, special thanks to Dr. Franz Zotter for the helpful comments on a draftversionofthemanuscriptmadeduringavisittothelab.ThanksalsotoDebbie Kedar for the prompt and professional editing and proofreading of this book. Finally, thanks to my family, Vered, Asaf, Yonathan, and Tal, for providing love therapy that time and again pulled me out of the writing stumbles and falls. Beer-Sheva, December 2014 Boaz Rafaely Contents 1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Functions on the Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Exponential and Legendre Functions . . . . . . . . . . . . . . . . . . . . 12 1.4 Spherical Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Some Useful Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Rotation of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Spherical Convolution and Correlation . . . . . . . . . . . . . . . . . . . 28 2 Acoustical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 The Acoustic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Spherical Bessel and Hankel Functions. . . . . . . . . . . . . . . . . . . 34 2.3 A Single Plane Wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 Plane-Wave Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 Point Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.6 Sound Pressure Around a Rigid Sphere . . . . . . . . . . . . . . . . . . 47 2.7 Translations of Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Sampling the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Sampling Order-Limited Functions. . . . . . . . . . . . . . . . . . . . . . 57 3.2 Equal-Angle Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3 Gaussian Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Uniform and Nearly-Uniform Sampling . . . . . . . . . . . . . . . . . . 65 3.5 Numerical Computation of Sampling Weights. . . . . . . . . . . . . . 68 3.6 The Discrete Spherical Fourier Transform. . . . . . . . . . . . . . . . . 71 3.7 Spatial Aliasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 Spherical Array Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Single Open Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Rigid Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Open Sphere with Cardioid Microphones . . . . . . . . . . . . . . . . . 85 ix x Contents 4.4 Dual-Radius Open Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5 Robustness to Errors and Numerical Array Design. . . . . . . . . . . 90 4.6 Design Examples with Robustness Analysis . . . . . . . . . . . . . . . 93 4.7 Spherical Shell Configuration . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.8 Other Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5 Spherical Array Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Beamforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Axis-Symmetric Beamforming. . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Directivity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4 White Noise Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Simple Axis-Symmetric Beamformers . . . . . . . . . . . . . . . . . . . 114 5.6 Beamforming Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.7 Steering Non Axis-Symmetric Beam Patterns . . . . . . . . . . . . . . 121 6 Optimal Beam Pattern Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.1 Maximum Directivity Beamformer. . . . . . . . . . . . . . . . . . . . . . 125 6.2 Maximum WNG Beamformer . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.3 Example: Directivity Versus WNG. . . . . . . . . . . . . . . . . . . . . . 133 6.4 Mixed Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.5 Maximum Front–Back Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.6 Dolph-Chebyshev Beam Pattern. . . . . . . . . . . . . . . . . . . . . . . . 142 6.7 Multiple-Objective Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7 Beamforming with Noise Minimization . . . . . . . . . . . . . . . . . . . . . 155 7.1 Beamforming Equations Including Noise . . . . . . . . . . . . . . . . . 155 7.2 Minimum Variance Distortionless Response . . . . . . . . . . . . . . . 160 7.3 Example: MVDR with Sensor Noise and Disturbance. . . . . . . . . 164 7.4 Example: MVDR with Correlated Disturbance. . . . . . . . . . . . . . 166 7.5 Linearly Constrained Minimum Variance . . . . . . . . . . . . . . . . . 169 7.6 Example: LCMV with Beam Pattern Amplitude Constraints . . . . 172 7.7 LCMV with Derivative Constraints . . . . . . . . . . . . . . . . . . . . . 176 7.8 Example: Robust LCMV with Derivative Constraints. . . . . . . . . 178 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191