Downloaded from orbit.dtu.dk on: Sep 11, 2017 Eigenbeamforming array systems for sound source localization Tiana Roig, Elisabet; Jacobsen, Finn; Jeong, Cheol-Ho; Agerkvist, Finn T. Publication date: 2014 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Tiana Roig, E., Jacobsen, F., Jeong, C-H., & Agerkvist, F. T. (2014). Eigenbeamforming array systems for sound source localization. Technical University of Denmark, Department of Electrical Engineering. 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Elisabet Tiana Roig Eigenbeamforming array systems for sound source localization PhD thesis, November 2014 Eigenbeamforming array systems for sound source localization PhD thesis by Elisabet Tiana Roig Technical University of Denmark 2014 This thesis was submitted to the Technical University of Denmark (DTU) as partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (PhD) in Electronics and Communication. The work presented in this thesis was completed between October 1, 2010 and August 1, 2014 at Acoustic Technology, Department of Electrical Engineering, DTU, under the supervision of Associate Professor Finn Jacobsen, until June of 2013, and Associate Professors Cheol-Ho Jeong and Finn T. Agerkvist, from March of 2013 to the end. The project was funded by the Department of Electrical Engineering at DTU. Cover illustration: Circular microphone array mounted on a rigid sphere, by Elisabet Tiana-Roig Department of Electrical Engineering Technical University of Denmark DK-2800 KONGENS LYNGBY, Denmark Printed in Denmark by Rosendahls - Schultz Graﬁsk a/s c 2014 Elisabet Tiana Roig No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the author. In memory of Finn Jacobsen. I al Toni, el meu far, i al Quim, la meva llum. Abstract Microphone array technology has been widely used for the localization of sound sources. In particular, beamforming is a well-established signal processing method that maps the position of acoustic sources by steering the array transducers toward different directions electronically. The present PhD study aims at enhancing the performance of uniform circular arrays, and to a lesser extent, spherical arrays, for two- and three-dimensional localization problems, respectively. These array geometries allow to perform eigenbeamforming, beamforming based on the decomposition of the sound ﬁeld in a series of orthogonal functions. In this work, eigenbeamforming is particularly developed to improve the performance of circular arrays at low frequencies. Compared to conventional delayand-sum beamforming, the proposed technique, named circular harmonics beamforming, provides a better resolution at the expense of being more vulnerable to noise. A simple way to further improve the array performance is to ﬂush-mount the transducers on a rigid scatterer. For a circular array, an ideal solution is a rigid cylindrical scatterer of inﬁnite length. Due to its impracticality, the use of a rigid spherical scatterer is recommended instead. A better visualization in the entire frequency range can be achieved with deconvolution methods, as they allow the recovery of the sound source distribution from a given beamformed map. Three efﬁcient methods based on spectral procedures, originally conceived for planar-sparse arrays, are adapted to circular arrays. They rely on the fact that uniform circular arrays present an azimuthal response that is rather independent on the focusing direction. Finally, a method based on the combination of beamforming and acoustic holography is introduced for both circular and spherical arrays. This new approach, also expressible in terms of eigenbeamforming, extends the frequency range of operation of conventional delay-and-sum beamforming toward the low frequencies. Keywords: uniform circular arrays, spherical arrays, circular harmonics beamforming, deconvolution methods, spherical harmonics beamforming, holographic virtual arrays v Resumé Mikrofon-array-teknologi har været meget anvendt til lokalisering af lydkilder. Navnlig beamforming er en veletableret signalbehandlingsmetode, som kortlægger placeringen af akustiske kilder ved elektronisk at styre array transducere mod forskellige retninger. Dette Ph.d.-projekt stræber efter at forbedre præstationen af ensartede cirkulære array-systemer, og i mindre grad sfæriske arrays, for hhv. to- og tredimensionale lokaliseringsproblemer. Disse array-geometrier giver mulighed for at udføre eigenbeamforming, dvs. beamforming baseret på dekompositionen af lydfeltet i en række ortogonale funktioner. I dette arbejde er eigenbeamforming specielt udviklet for at forbedre præstationen af cirkulære arrays ved lave frekvenser. Sammenlignet med konventionel delay-and-sum beamforming giver den foreslåede teknik, kaldet circular harmonics beamformning, en bedre opløsning på bekostning af at være mere sårbar over for støj. En enkel måde til yderligere at forbedre array-præstationen er at placering mikrofonerne på overﬂaden af en hård scatterer. For et cirkulært array er en ideel løsning en hård cylindrisk scatterer af uendelig længde. På grund af vanskeligheder ved implementeringen anbefales en hård sfærisk scatterer i stedet for. En bedre visualisering i hele frekvensområdet kan opnås med deconvolutionmetoder, da de tillader gendannelse af lydkilders distribution fra et givet beamformed kort. Tre effektive metoder, baseret på spektrale procedurer, der oprindeligt er udtænkt til plane, sparse arrays, er tilpasset cirkulære arrays. De er afhængige af det faktum, at ensartede cirkulære arrays viser et azimut respons, der er temmelig uafhængig af fokuseringen retning. Endelig indføres en metode, der bygger på en kombination af beamforming og akustisk holograﬁ, for både cirkulære og sfæriske arrays. Denne nye fremgangsmåde, som også kan udtrykkes i form af eigenbeamforming, udvider frekvensområdet for brugen af konventionel delay-and-sum beamforming mod de lave frekvenser. Nøgleord: ensartede cirkulære arrays, sfæriske arrays, circular harmonics beamforming, spherical harmonics beamforming, deconvolution-metoder, holograﬁske virtuelle arrays vii Acknowledgments I would like to dedicate these ﬁrst lines to pay my deepest homage to my supervisor, Finn Jacobsen. This work would have not been possible without his teachings, his generous dedication, and his wise advice. I will always admire him for being committed to his work and to his students until the very end of his life. Moving the project forward without him has been extremely tough. However, his legacy has given me strength and has accompanied me every single day until the end of the project. His memory will always live within me. I will never forget the support of the other professors of the group, Cheol-Ho Jeong, Finn T. Agerkvist, and Jonas Brunskog, when Finn Jacobsen was forced to step aside a year and a half ago. In particular, I would like to thank Cheol-Ho for taking over the main supervision, and for doing it with motivation and efﬁciency. I am also extremely grateful to Efrén Fernández-Grande for his great support, guidance, and help, all the way through. All my colleagues, former colleagues, and friends, at the ‘House of Acoustics’ have contributed immensely to my personal and professional time at DTU. In particular, I would like to express my gratitude to Jørgen Rasmussen and Tom A. Petersen for their help with the equipment and the experimental work, and to Nadia J. Larsen for her help with administrative tasks and for her empathy. I am very grateful to Aggeliki Xenaki, Oliver Lylloff, and Marco Ottink for letting me participate in their Master’s theses; I have really learned a lot from you. Many thanks to Salvador Barrera Figueroa for his advice and also for his subtle way of cheering me up. Thanks to Marton Marschall for his Matlab code and for the fruitful discussions regarding spherical arrays. I am also very grateful to Torsten Dau for his experienced advice during a complicated peerreview process of one of the articles. Special thanks to my wonderful ofﬁce mates, Joe Jensen, Gerd Marbjerg, and Alba Granados, for creating a fantastic working atmosphere. Joe, thanks for your permanent good mood, and your (almost perfect) musical taste. Gerd, Alba, I have really enjoyed this short, but intense, time together in our particular fortress. Without your support in the last months, this would have been a lot harder. ix x Acknowledgments I am very grateful to Karim Haddad and Jørgen Hald from Brüel & Kjær for lending me equipment for the experiments, and for very inspiring discussions. Thanks to Daniel Fernández-Comesaña from Microﬂown Technologies for inviting me to participate in one of his articles. I would also like to thank an anonymous reviewer of the Journal of the Acoustical Society of America, whose valuable comments became the basis of one of the appendices of this dissertation. I am indebted to my family and friends for encouraging me countless times, especially when I needed it most. I deeply thank my mom, my brother Eduard, my dad, Piluca, my sister Vicky, my grandma Carme, in short, all my closest family, for their love and support in all my pursuits. This PhD degree is an accomplishment that belongs to all of them. It also belongs to my dear grandpa Guillem, who left us a few months after the beginning of the project, and to my son Quim, who, born in the course of this project, has been a great source of happiness. Finally, my most sincere gratitude to Toni Torras Rosell. Without your precious and enthusiastic contribution to the papers, your guidance on the dissertation, and your faith in me, this would have simply been impossible. Thanks for being always there, for the dedication to our family, and for your love. Contents List of acronyms xiii List of symbols xv Notations and conventions xix 1 Introduction 1 1.1 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 A brief overview on acoustic array systems . . . . . . . . . . . . . . . 3 1.2.1 Beamforming techniques . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Other array techniques . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Array layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 2 Basic beamforming methods 2.1 2.2 3 11 Delay-and-sum beamforming . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Uniform linear array . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Uniform circular array . . . . . . . . . . . . . . . . . . . . . . 21 2.1.3 Spherical array . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Performance indicators . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Eigenbeamforming 3.1 33 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Eigenbeamforming for circular arrays . . . . . . . . . . . . . . 34 xi xii Contents 3.1.2 3.2 4 5 6 Eigenbeamforming for spherical arrays . . . . . . . . . . . . . 38 Papers A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Deconvolution methods 45 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Paper C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Beamforming with holographic virtual arrays 53 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Papers D and E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Conclusions 59 6.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Insight into circular harmonics beamforming 63 B Acoustic holography with uniform circular arrays 67 B.1 Open array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 B.2 Rigid cylindrical scatterer of inﬁnite length . . . . . . . . . . . . . . . 68 B.3 Rigid spherical scatterer . . . . . . . . . . . . . . . . . . . . . . . . . 70 Bibliography 73 Papers A-E 83 List of acronyms CMF DAMAS DI ESPRIT FFT FFT-NNLS FISTA GSC ISCA MACS MEMS MSL MUSIC NAH SHARP SNR SONAH WNG 2D 3D Covariance matrix ﬁtting Deconvolution approach for the mapping of acoustic sources Directivity index Estimation of signal parameters via rotational invariance techniques Fast Fourier transform Fourier-based non-negative least squares Fast iterative shrinkage-thresholding algorithm Generalized sidelobe canceler Iterative sidelobe cleaner algorithm Mapping of acoustic sources Microelectromechanical systems Maximum sidelobe level Multiple signal classiﬁcation Near-ﬁeld acoustic holography Spherical harmonics angularly resolved pressure Signal-to-noise ratio Statistical optimized near-ﬁeld acoustic holography White noise gain Two-dimensional Three-dimensional xiii List of symbols A An b b C Cn C̃n Cmn C̃mn c cmn d dn F F −1 f G H H (1) Hn (1) hn Jn j jn k Amplitude of the acoustic wave Expansion coefﬁcient of order n Beamformer output Beamformer power response expressed as a vector Cross-spectral matrix nth Fourier coefﬁcient obtained with a continuous circular aperture nth Fourier coefﬁcient estimated with a circular array mnth Fourier coefﬁcient obtained with a continuous spherical aperture mnth Fourier coefﬁcient estimated with a spherical array Speed of sound in the medium of propagation mnth element of the cross-spectral matrix C Sensor spacing of a linear array Constant associated to the nth order in eigenbeamforming Direct FFT Inverse FFT Frequency Grid Point-spread function Point-spread function matrix Hankel function of the ﬁrst kind and order n Spherical Hankel function of the ﬁrst kind and order n Bessel function of the ﬁrst kind and order n √ −1 Imaginary number Spherical Bessel function of the ﬁrst kind and order n Wavenumber xv xvi k ki M N Nv Pnm pm p Qn R Rv r rm s s t Un W wm wn Ynm αm δ δnν ϕ ϕl ϕm ϕs κ̂ λ θ θm θs ϑ List of symbols Wavenumber vector Wavenumber vector of an incident wave Number of array sensors Number of orders in eigenbeamforming Number of orders in eigenbeamforming with a holographic virtual array Associated Legendre function of order m and degree n Sound pressure captured at the mth array sensor Sound pressure Bafﬂe condition function Radius of a circular/spherical array of sensors Radius of a holographic virtual array Position vector Position of the mth array sensor Spatial source power distribution Spatial source power distribution vector Time Chebyshev polynomial of the second kind of degree n Array pattern Weighting of the mth array sensor Weighting of the nth harmonic order Spherical harmonic of order n and degree m Integration factor of the mth sensor of a circular/spherical array Delta function Kronecker delta function Azimuth angle, from 0 to 2π Azimuth angle of the array looking direction, from 0 to 2π Azimuth angle of the mth array sensor, from 0 to 2π Azimuth angle of a sound source, from 0 to 2π Array’s steering vector Wavelength Polar angle, from 0 to π Polar angle of the mth array sensor, from 0 to π Polar angle of a sound source, from 0 to π Polar angle, from −π to π xvii ϑs τm ω Polar angle of a sound source, from −π to π Delay applied to the mth array sensor Angular frequency Notations and conventions Conventions • The convention e−jωt is considered in the entire manuscript. Hence, a progressive plane wave is of the form ej(k · r−ωt) . • Vectors are denoted by boldface lowercase characters, e.g., a. • Matrices are denoted by boldface uppercase characters, e.g., A. • Unit vectors are denoted by boldface lowercase characters with a hat, e.g., â. Mathematical operations ( · )∗ |·| · · 2 Complex conjugate of the argument Absolute value of the argument Ceiling function of the argument 2-norm of the argument xix Chapter 1 Introduction A problem of practical importance when dealing with acoustic measurements is to estimate the directions from which sound waves arrive to the measurement point. While a single microphone cannot provide this information, as microphones are only capable of measuring the sound pressure at that speciﬁc point, combination of simultaneous signals from an array of microphones makes it possible to ﬁlter the sound in space and, thus, achieve directionality. With proper signal processing, array systems can focus into a particular direction, to enhance the signals arriving from there, and attenuate those from other directions. This idea was explored for the ﬁrst time in 1976, when Billingsley and Kinns introduced the acoustic telescope, a system that was able to localize the main contributions of jet engines in real-time [1]. This work laid the foundations of beamforming, which soon became popular among the acoustic community, giving rise to numerous studies not only for sound source localization purposes, but also for signal enhancement and spatial ﬁltering. Nowadays beamforming is an essential tool widely used in the industry for all sorts of applications, such as vehicle assessment, computer games and surveillance, among others. Depending on the application, the most adequate processing techniques and array geometries vary. Generally, beamforming is based on measurements in the far ﬁeld of the sources so that the waves have become planar at the array position. However, it should be noticed that near-ﬁeld beamforming is also possible. Readers interested in the history of beamforming are addressed to the concise monograph by Michel, Ref. [2]. An ideal sound source localization system should present a delta function on the focusing direction and nulls elsewhere. However, beamforming presents two inherent limitations; ﬁrstly, an imperfect resolution on the focusing direction, due to a main 1 2 1. Introduction beam instead of a delta function, and secondly, the appearance of sidelobes in directions other than the focusing direction. Moreover, the array response is frequency dependent. The frequency range of operation of an array is determined, at low frequencies, by the dimensions of the array, and by the microphone spacing, at high frequencies. The larger the array, the better the performance at low frequencies, whereas the closer the microphones, the better at high frequencies. However, the dimensions of the array and the number of microphones are usually limited by practical issues, such as the maneuverability of the array and the overall cost of the equipment. Therefore, dealing with broadband sources poses some challenges. In the almost 40 years of development of acoustic array technology, numerous beamforming algorithms, as well as array geometries, have been suggested to improve the overall performance of array systems. 1.1 Scope of the thesis The present thesis deals with circular and spherical arrays of microphones, to a lesser extent, for localization of sound sources in 2-dimensional (2D) and 3-dimensional (3D) sound ﬁelds, respectively. While spherical arrays have been examined widely in the last decade for speech enhancement and sound source localization purposes, less literature has been devoted to circular arrays. This geometry is particularly interesting for scenarios where sources placed in the far ﬁeld are distributed 360◦ around the array. That is, for instance, the case of many outdoor measurements for environmental noise identiﬁcation, in which reﬂections from the ground are sufﬁciently attenuated, and also the case of measurements in rooms where ﬂoors and ceilings are acoustically treated to reduce reﬂections. One of the main applications involving rooms is conferencing, a scenario that requires beamforming in real-time; see, e.g., Ref. [3]. By contrast, for environmental noise purposes, measurements can be often post-processed at a later stage, which allows the application of more sophisticated algorithms that require a high computational load. The primary goal of the present thesis is to suggest and examine alternatives to the traditional methods for enhancing the performance of these array geometries for sound source localization purposes. It should be noticed that, throughout this dissertation, it is assumed that the acoustic sources are static, placed in the far ﬁeld of the array, and not coherent. It is also assumed that all the array transducers have the same characteristics and are omnidirectional. 1.2 A brief overview on acoustic array systems 3 1.2 A brief overview on acoustic array systems 1.2.1 Beamforming techniques Beamforming techniques are generally classiﬁed in two groups: ﬁxed beamforming and adaptive beamforming. Fixed beamforming algorithms are data-independent, that is, all signals are treated in the same manner without taking into account their individual properties. The simplest method is delay-and-sum beamforming, which is addressed in Chapter 2. Another example is ﬁlter-and-sum beamforming, based on linearly ﬁltering the signals prior to applying delay-and-sum; see, e.g., Ref. [4]. Filtering helps removing disturbances, such as out-of-band noise. A new and specially attractive technique for its simplicity is functional beamforming [5]. This method that results from modifying delay-and-sum beamforming in the frequency domain offers a much higher dynamic range than other beamforming techniques. However, it is very sensitive to microphone positioning errors. On the other hand, adaptive beamforming methods are data-dependent, that is, their parameters follow from statistical observations in the captured signals. As a result, their performance exceeds that of ﬁxed beamforming techniques, at the expense of being more complex to implement and more sensitive to sensor calibration errors [4]. Furthermore, in the presence of coherent sources, most methods fail dramatically. Usually, adaptive techniques rely on narrow-band signals. Several methods are based on solving a constrained mean-squared optimization problem. That is, for instance, the case of the generalized sidelobe canceler (GSC) [6], which basically consists of a ﬁxed beamformer, a blocking matrix, and an interference canceler. The ﬁxed beamformer is steered to the desired direction, while the blocking matrix blocks any signal coming from that direction so that only noise signals from undesired directions pass through. By means of an adaptive algorithm the unwanted signals are emphasized and ﬁnally they are subtracted from the ﬁxed beamformer output. Another type of adaptive techniques, the so-called high-resolution spectral estimation techniques, are derived from parameter estimation theory. One such method is the multiple signal classiﬁcation (MUSIC), which, based on eigenanalysis, relies on the orthogonality between the signals subspace and the noise subspace to improve the quality of the signals [7]. Adaptive methods are out of the scope in this dissertation. Readers interested in adaptive beamforming are, for instance, referred to Chapter 7 in Ref. [4] and Chapter 5 in Ref. [8]. Although most beamforming techniques are in essence independent of the array 4 1. Introduction geometry, there is a group of methods conceived for ‘closed’ arrays, such as circular and spherical arrays, known as eigenbeamforming. Eignebeamforming relies on the decomposition of the sound ﬁeld captured with the array in a series of harmonics, which adds more features compared to traditional beamforming. Fixed eigenbeamforming methods are addressed in Chapter 3. In the last decade, a group of inverse methods, generally referred to as deconvolution methods, has become of interest, as they allow to visualize sound sources with more accuracy than beamforming methods, and even determine their levels. However, the main limitation is that they are computationally expensive, as they are based on iterative algorithms. An overview of these methods is given in Chapter 4. It should be noted that beamforming can be applied also to moving sources. Since in the present study only static sources are considered, the reader is addressed to, e.g., Chapter 8 in Ref. [4] for a basic introduction to tracking problems. 1.2.2 Other array techniques Besides beamforming, there are other sound visualization techniques that rely on array measurements. The most relevant one is acoustic holography, a well-established method that aims at reconstructing sound ﬁelds quantitatively. By means of measurements in a 2D surface (the array), the entire sound ﬁeld, sound pressure, particle velocity, and sound intensity, can be reconstructed in a 3D space. Acoustic holography and beamforming are complementary techniques, as acoustic holography is generally preferred for near-ﬁeld measurements, such as in near-ﬁeld acoustic holography (NAH) [9, 10], whereas beamforming is more adequate for the far-ﬁeld case. Moreover, acoustic holography handles better coherent than incoherent sources, which contrasts with the opposite behavior of beamforming. In most applications, acoustic holography serves to describe the radiation characteristics of the source under analysis. Array technology is also used for blind source separation, although an array layout is not strictly necessary. As the name suggests, blind source separation is a sound source identiﬁcation method that intends to simultaneously recover signals from independent sources without requiring any information on their locations. The main limitation of the method is that it fails when there are more sources than sensors. Blind source separation is not addressed in this thesis. Readers interested in a thorough comparison between 1.2 A brief overview on acoustic array systems 5 blind source separation and beamforming in the time-domain (for speech purposes) are referred to Ref. [11]. 1.2.3 Array layouts Traditionally, beamforming has been carried out mostly with planar-sparse arrays. The simplest conﬁguration is the rectangular grid of elements. However, due to the periodical placement of the sensors, severe sampling error, in the form of aliasing, occurs above the frequency where the spatial Nyquist sampling criterion is not fulﬁlled. This causes a sudden increase in level of the sidelobes, which become replicas of the main lobe in unwanted directions in the worst case. This is addressed in Chapter 2. This characteristic prevents this geometry from being generally used for beamforming purposes. Contrarily, rectangular arrays, with one or two parallel layers, are typically used for NAH. Planar irregular arrays are usually preferred for beamforming over regular arrays, because they do not exhibit an abrupt aliasing pattern. The effect of an aperiodical spatial sampling is a smooth increase in the level of the sidelobes, which leads to a wider frequency range of operation toward high frequencies [12]. Typical irregular arrays used for aeroacoustic purposes are based on spirals, such as the equiangular or logarithmic spiral array with one or more arms [13, 14]. Some other irregular layouts result from optimization processes that determine the position of the sensors that ensures the best possible level of the sidelobes for the frequency range of interest [12]. For some applications, such as wind tunnel measurements, it is convenient to ﬂush-mount the array microphones on a wall or a bafﬂe so that the array structure does not alter the aerodynamic environment. References [12] and [15] examine various planar irregular arrays in detail. Spherical arrays are also widely used for beamforming [16], as well as for acoustic holography [17], whereas circular arrays are less common [18], especially for holography. Generally, spherical and circular arrays used for sound source localization are shift-invariant, that is, the output pattern is independent of the focusing direction so that the system is equally fair in all directions. One way to achieve this characteristic is by keeping a constant microphone spacing all over the array. That is, for instance, the case of a uniform circular array [19]. This interesting feature cannot be achieved with linear or planar-sparse arrays. 6 1. Introduction On the other hand, spherical arrays with non-uniform spacing have also been investigated, e.g., in Ref. [20]. That is, for instance, the case of a rigid spherical array with the sensors placed in horizontal rings with a higher density of sensors on the equator of the sphere suggested in Refs. [21–23] to enhance the horizontal spatial resolution over other directions. This array is suitable for 3D recordings that can afterwards be used to create virtual environments for hearing instrument testing and psychoacoustic purposes via a high-order or a mixed-order Ambisonics loudspeaker system [24]. Circular arrays with a non-uniform spacing are rather unusual. An example can though be found in Ref. [25], where the angular position of the array sensors is determined by the golden-ratio. Variations of the circular and spherical geometries are also found in the literature. For example, Refs. [26, 27] suggest a dual-radius spherical array for beamforming and for NAH that consists of an open spherical array with a smaller spherical array mounted on a bafﬂe in its interior, whereas Ref. [28] introduces an open dual-radius spherical array. Similarly, Refs. [29, 30] examine systems that consist of concentric uniform circular arrays of different radius. For beamforming purposes on a half 3D acoustic scenario, Ref. [31] suggests a bafﬂed hemispherical microphone array that makes use of the image source principle. To achieve a rather constant pattern in the entire frequency range of interest, some arrays are conformed by subarrays, each being responsible for a certain frequency band. Usually, this is approached with planar-sparse arrays [8, 32], although other conﬁgurations are applicable to, such as the previous mentioned concentric circular and spherical arrays. A wise solution to reduce the overall cost of the system is to share, when possible, array elements between different subarrays, giving rise to the concept of nested arrays. The idea of using nested arrays, is, in fact, the essence of constant directivity beamforming, a ﬁxed beamforming method based on applying ﬁlter-and-sum to the the different nested arrays [8, 32]. The main drawback of this technique is that it is impractical at low frequencies, as it requires extremely large arrays. In the presence of stationary signals, ‘scanning arrays’ are alternatives to conventional arrays. The procedure only requires two transducers: while one is kept at a position that serves as a reference, the other is moved along a grid [33]. The main advantage is that the equipment required is obviously cheaper than a that of a conventional array system. Besides, the method offers more ﬂexibility in the sense that the scanning area is not limited to a predeﬁned grid of points. The main drawback, though, is that 1.3 Structure of the thesis 7 measurements are more time consuming. Based on this principle, a scanning array consisting of a rotating microphone set-up is suggested to capture the acoustic behavior of auditoriums in Ref. [34]. The data measured with the array, a large set of impulse responses, is later used for creating virtual acoustic scenes with a 2D or 3D Ambisonics loudspeaker system. Most array systems assume that the sensors are completely omnidirectional. However, microphones with well-deﬁned directivity patterns can also be used, provided that all of them are of the same type and are oriented identically. In such a case, the transfer function of the directive microphones must be taken into account for the beamforming procedure [32]. Besides conventional microphones, pressure-velocity transducers, e.g., Microﬂown PU probes [35], are progressively attracting interest. These transducers provide simultaneous measurements of pressure and particle velocity, which make them particularly suitable for acoustic holography [36]. In fact, some NAH methods rely on the combination of the two quantities to achieve an enhanced performance [37– 39]. Recently, a completely new approach for beamforming based on the acousto-optic effect, i.e., the interaction between sound and light, has been introduced in Ref. [40]. Instead of using a discrete number of sensors as in conventional arrays, the proposed acousto-optic beamformer senses the sound ﬁeld with a laser beam in a continuous manner so that spatial aliasing is totally avoided. So far, only an optical linear aperture has been examined [41]. At the moment, the main drawback of this technique is that the beamformer requires manual steering. However, this problem could be overcome by developing an optical array. 1.3 Structure of the thesis The present PhD thesis follows a paper-based format, that is, the main ﬁndings of the PhD project are presented in a collection of articles elaborated in the course of the project. It is important to emphasize that the articles represent the core of the thesis. The dissertation is structured as follows: Chapter 2, Basic beamforming methods, gives the basic concepts of beamforming required to follow the contributing papers. Readers familiar with the topic can skip this chapter. Chapter 3, Eigenbeamforming, Chapter 4, Deconvolution methods, and Chapter 5, Beamforming with holographic virtual arrays, are devoted to the ﬁndings of the contributing articles. These chapters share 8 1. Introduction the same structure: they begin with an introduction that intends to supplement, when possible, the articles, followed by a synopsis of the articles, a survey on related work, and a discussion of the ﬁndings. Unlike Chapter 2, Chapters 3 to 5 are kept deliberately concise to minimize the repetition in content with the contributing papers. Since these chapters are understood as a complement to the papers, the reader is advised to read the papers before proceeding to the ﬁnal chapter, Conclusions, which concludes the work and suggests further investigations for the future. The thesis also includes two appendices that supplement Chapter 3 and 5, respectively. The contributing papers, ﬁve in total, are appended at the end. Three of them are published in the Journal of the Acoustical Society of America, and the rest are published in the proceedings of two relevant congresses. They are listed in the following: Paper A E. Tiana-Roig, F. Jacobsen, and E. Fernandez-Grande, “Beamforming with a circular microphone array for localization of environmental noise sources,” J. Acoust. Soc. Am., vol. 128, no. 6, pp. 3535–3542, 2010.∗ Paper B E. Tiana-Roig, F. Jacobsen, and E. Fernandez-Grande, “Beamforming with a circular array of microphones mounted on a rigid sphere (L),” J. Acoust. Soc. Am., vol. 130, no. 3, pp. 1095–1098, 2011. Paper C E. Tiana-Roig and F. Jacobsen, “Deconvolution for the localization of sound sources using a circular microphone array”, J. Acoust. Soc. Am., vol. 134, no. 3, pp. 2078–2089, 2013. Paper D E. Tiana-Roig, A. Torras-Rosell, E. Fernandez-Grande, C.-H. Jeong, and F. T. Agerkvist, “Towards an enhanced performance of uniform circular arrays at low frequencies,” in Proc. of Inter-Noise 2013, Innsbruck, Austria, 2013. Paper E E. Tiana-Roig, A. Torras-Rosell, E. Fernandez-Grande, C.-H. Jeong, and F. T. Agerkvist, “Enhancing the beamforming map of spherical arrays at low frequencies using acoustic holography,” in Proc. of BeBeC 2014, Berlin, Germany, 2014. ∗ Paper A is based on the Master’s thesis by Elisabet Tiana Roig “Beamforming Techniques for environmental noise”, Technical University of Denmark, 2009. The paper, written in 2010 during the application process of the PhD project, is included as part of this PhD thesis as it led to the research topic of the project. 1.3 Structure of the thesis 9 Besides the aforementioned articles, the following articles were also produced in the course of the PhD project: 1 E. Tiana-Roig and F. Jacobsen, “Acoustical source mapping based on deconvolution approaches for circular microphone arrays,” in Proc. of Inter-Noise 2011, Osaka, Japan, 2011. 2 Fernandez-Comesaña, E. Fernandez-Grande, and E. Tiana-Roig, “A novel deconvolution beamforming algorithm for virtual phased arrays”, in Proc. of Inter-Noise 2013, Innsbruck, Austria, 2013. However, these articles are not explicitly mentioned in the thesis, as the contents of Paper 1 overlap with Paper C and Paper 2 does not directly relate to the work done in eigenbeamforming. 10 1. Introduction Chapter 2 Basic beamforming methods This chapter provides the basic knowledge required to comprehend the main contributions of the PhD project. The chapter begins with an introduction to classical beamforming theory. Two array geometries are examined in detail, the uniform linear array, which serves to explain the basic concepts of beamforming, and the uniform circular array as most contributing papers (Papers A to D) elaborate on this array conﬁguration. In addition, the case of a spherical array is touched upon, as paper E deals with this geometry. The chapter ends with a description of the measures of performance commonly used to evaluate beamforming systems. 2.1 Delay-and-sum beamforming Delay-and-sum beamforming is the oldest and simplest array signal processing algorithm [4]. The principle behind this technique is shown in Fig. 2.1: in the presence of a propagating wave, the signals captured by the microphones are delayed by a proper amount before being added together, to strengthen the resulting signal with respect to noise or waves propagating in other directions. The delays required to reinforce the output signal correspond to the time it takes for the wave to propagate between microphones so that, after applying the delays, the microphone signals are aligned in time. Mathematically, delay-and-sum is formulated as b(t, κ̂) = M −1 wm pm (t − τm (κ̂)), m=0 11 (2.1) 12 2. Basic beamforming methods 0 1 2 M −1 p0 (t) τ0 p1 (t) τ1 p2 (t) τ2 pM −1 (t) τM −1 w0 w0 p0 (t − τ0 ) w1 w2 + wM −1 b(t) wM −1 pM −1 (t − τM −1 ) Figure 2.1: Sketch of a delay-and-sum beamformer. The signals captured by the sensors are delayed (and weighted) before adding them together. κ̂ array rm origin Figure 2.2: Array focused in the direction given by κ̂. where M is the number of microphones, pm is the pressure measured with the mth microphone, wm is its associated amplitude weighting, and τm (κ̂) is the delay applied to the mth microphone required to focus the array in the direction given by κ̂ depicted in Fig. 2.2. The delays are given by τm (κ̂) = κ̂ · rm , c (2.2) where rm is the position vector of the mth microphone and c is the speed of sound in the medium of propagation (approximately 343 m/s in air at 23◦ C). Assuming a plane wave that impinges on the array, the pressure captured by the mth array microphone, expressed in complex notation, is pm (t) = Aej(ki · rm −ωt) , (2.3) 2.1 Delay-and-sum beamforming 13 plane wave ki main lobe sidelobes κ̂ array rm origin Figure 2.3: Plane wave impinging on an array. The beamformer response presents a main beam when steered in the direction of the impinging wave, whereas other directions are partially or totally attenuated. where A is the amplitude, ω represents the angular frequency, related to the frequency f by ω = 2πf , and ki is the wavenumber vector, with magnitude |ki | = k = ω/c. It can be shown that the beamformer output, Eq. (2.1), results in b(t, κ̂) = Ae −jωt M −1 wm ej(ki +kκ̂) · rm . (2.4) m=0 A close inspection of this equation reveals that when the array is steered in the precise direction κ̂ that satisﬁes ki = −k κ̂, the beamformer response presents its maximum value. When focused toward other directions, the response is partially or totally attenuated. If the array scans all possible directions with the appropriate associated delays, the resulting beamformed map will present a main lobe around its maximum and sidelobes elsewhere. This is illustrated in Fig. 2.3. The weightings wm , often referred to as shading, inﬂuence the shape of the main- 14 2. Basic beamforming methods and sidelobes. They act as spatial windows and their effect is analogous to that observed with temporal windows in conventional signal processing. In fact, Eq. (2.4) can be rewritten as b(t, k) = Ae−jωt W (k − ki ), (2.5) where k = −k κ̂, and W (k) is the spatial discrete Fourier Transform of the weightings W (k) = M −1 wm e−jk · rm . (2.6) m=0 This function is usually known as array pattern. When the weightings follow a uniform distribution, the array pattern exhibits the narrowest possible main lobe, whereas tapered distributions, such as the triangular and the Hann windows, yield lower sidelobes at the expense of a wider main lobe [42]. Furthermore, the location of the nulls in the pattern also depends on the weightings. In real case scenarios, the pressure captured by the microphones is contaminated by noise, e.g., background noise and electronic noise. In case of a single plane wave, the pressure is (cf. Eq. (2.3)) pm (t) = Aej(ki · rm −ωt) + nm (t), (2.7) where nm (t) is uncorrelated noise present at the mth microphone. Obviously, the presence of noise inﬂuences the response of beamforming systems. However, compared to a measurement with a single microphone, the combination of many measurements, as in the case of using a microphone array, leads to a better signal-to-noise ratio (SNR). Amongst all techniques, it can be shown that delay-and-sum is the most robust against noise and has, moreover, the ability to suppress uncorrelated noise equally at all frequencies [8]. For broadband signals, it is convenient to implement delay-and-sum beamforming in the frequency domain. The signal is decomposed in a set of monochromatic plane waves (i.e., single-frequency waves), each treated independently in the beamforming 2.1 Delay-and-sum beamforming 15 procedure so that the applied phase shifts correspond to the desired delays. That is b(ω, κ̂) = M −1 wm pm (ω)ejωτm (κ̂) , (2.8) m=0 where pm (ω) is the discrete Fourier Transform of the signal measured with the mth sensor. The time domain version can be simply obtained with the inverse discrete Fourier Transform of b(ω, κ̂). A common practice when dealing with stationary sound ﬁelds is to formulate delayand-sum beamforming in the frequency domain using the averaged cross-spectra of the input signals. From Eq. (2.8), the power output can be written as 2 |b(ω, κ)| = M −1 M −1 wm wn∗ pm (ω)p∗n (ω)ejω(τm (κ̂)−τn (κ̂)) , (2.9) m=0 n=0 where ( · )∗ denotes complex conjugation. For the sake of simplicity, the weightings are set to unity in the following. Let us now consider the averaged cross-spectral matrix∗ ⎡ ⎢ ⎢ ⎢ ⎢ C=⎢ ⎢ ⎢ ⎣ c00 c01 c02 ··· c0(M −1) c11 c11 c12 ··· c1(M −1) c20 .. . c21 .. . c22 .. . ··· .. . c2(M −1) .. . c(M −1)0 c(M −1)1 c(M −1)2 ··· c(M −1)(M −1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦ (2.10) where cmn (ω) = pm (ω)p∗n (ω), (2.11) is the averaged cross-spectrum between the signals captured at the mth and nth sensors, being cmn (ω) = c∗nm (ω). (2.12) By using the cross-spectral matrix, the output power of the beamformer can be rewritten ∗ Note that the frequency dependence in the matrix is omitted. 16 2. Basic beamforming methods as 2 |b(ω, κ̂)| = M −1 m=0 cmm (ω) + −1 M −1 M cmn (ω)ejω(τm (κ̂)−τn (κ̂)) . (2.13) m=0 n=0 n=m As can be seen, the ﬁrst sum of this expression involves the diagonal elements, i.e., the auto-spectral terms, cmm (ω), whereas the second sum accounts for off-diagonal terms, cmn (ω). Note that the diagonal elements contain amplitude information plus self-noise. However, they do not carry phase information, and therefore, do not help in determining the source location. That is not the case for the cross-spectra elements; they contain the relative phase between each pair of sensors, and thus are essential for the beamforming process. Furthermore, self-noise is not present in these terms, as it is uncorrelated across channels. Therefore, it seems reasonable to remove the diagonal elements. This procedure, known as diagonal removal, decreases the level of the sidelobes, resulting in a clearer beamformed map [43]. However, the price to pay for this operation is that the resulting levels are biased. The sound ﬁeld at the array position can be generically described as the superposition of waves created by different sources so that waves arrive from different directions. When waves are incoherent, the beamformer output is equivalent to the superposition of outputs for each wave [12]. If the sources are sufﬁciently far from each other, they can be successfully identiﬁed. However, in the presence of coherent waves most beamformers fail. This problem, which appears, for instance, when dealing with (coherent) reﬂections, is often disregarded in the modern literature [2]. However, some studies have elaborated on this aspect, e.g., Refs.[44, 45]. Until now only sources in the far ﬁeld, and thus, plane waves at the array position, have been assumed. In this situation only the (angular) direction of the sources can be identiﬁed. Their distance to the array cannot be determined, as, in fact, the beamformer is focused toward an inﬁnite distance. In contrast, to localize sources in the near ﬁeld, a ﬁnite focus distance has to be considered, together with spherical wavefronts. This is illustrated in Fig. 2.4, where the array is steered toward a point located at r. Geometrical considerations show that, in order to align the signals at the sensor positions, the delays required for delay-and-sum beamforming are given by 2.1 Delay-and-sum beamforming 17 focus point r |r| − |r − rm | |r − rm | rm origin array Figure 2.4: Beamformer focused toward a point in the near ﬁeld. Spherical waves are expected at the array position. τm = |r| − |r − rm | . c (2.14) Since the amplitude of spherical waves decays with the distance, it is possible to compensate for it by including amplitude corrections in the beamforming algorithm [46]. In what follows, only sources in the far ﬁeld of the array, and thus, planar wavefronts at the array position, are considered. 2.1.1 Uniform linear array A uniform linear array, the simplest array geometry, serves to illustrate the performance of a delay-and-sum beamformer. This array consists of a number of sensors placed in a line with uniform spacing, as shown in Fig. 2.5. z ϑs ki ϑ κ̂ 0 M −1 1 d x d Figure 2.5: Plane wave impinging on a uniform linear array with M sensors. 18 2. Basic beamforming methods From the geometrical considerations given in Fig. 2.5, the mth array sensor is located at ⎤ ⎡ md ⎥ ⎢ m = 0, . . . , M − 1, (2.15) rm = ⎣ 0 ⎦ , 0 where d is the spacing between sensors. Moreover, when the system is steered toward the direction given by the polar angle ϑ, here deﬁned from −180◦ to 180◦ , the steering vector κ̂ becomes ⎡ ⎢ κ̂ = ⎣ sin ϑ 0 ⎤ ⎥ ⎦. (2.16) cos ϑ Let us assume that the array captures a plane wave with amplitude A and wavenumber vector ⎡ ⎤ sin ϑs ⎢ ⎥ (2.17) ki = −k ⎣ 0 ⎦, cos ϑs where ϑs is the angular position of the source. Expressed in these terms, delay-and-sum beamforming, Eq. (2.4), becomes b(t, ϑ) = Ae −jωt M −1 wm e−jk(sin ϑs −sin ϑ)md . (2.18) m=0 Notice that this expression does not depend on the azimuth angle ϕ, which implies that a linear array cannot discriminate between waves arriving from different azimuthal directions. Considering a uniform amplitude weighting wm = 1, the output reduces to b(t, ϑ) = Ae−jωt 1 − e−jk(sin ϑs −sin ϑ)M d . 1 − e−jk(sin ϑs −sin ϑ)d (2.19) After some rearrangement, the corresponding directivity pattern (or beam pattern) results in sin(π(sin ϑs − sin ϑ)M d/λ) , (2.20) |b(ϑ)| = |A| sin(π(sin ϑs − sin ϑ)d/λ) where λ is the wavelength, λ = c/f . As an example, the directivity pattern of a uniform linear array with 10 microphones is shown in Fig. 2.6, when A = 2, d/λ = 0.3 and 2.1 Delay-and-sum beamforming 19 20 |b| 15 10 5 0 −180 −150 −120 −90 −60 −30 0 30 ϑ [◦ ] 60 90 120 150 180 Figure 2.6: Directivity pattern of a uniform linear array with 10 sensors, when d/λ = 0.3 and a plane wave with amplitude 2 impinges on the array from 0◦ . the impinging wave comes from ϑs = 0◦ . As can be seen the curve presents a main lobe around 0◦ as the present wave propagates in this direction. The value of the main beam peak is 20, which corresponds to A × M . However, another main lobe appears around 180◦ . This is a consequence of the line array geometry, as the delays applied at the microphones for both 0◦ and 180◦ are exactly the same. Hence, the uniform linear array presents a front-back ambiguity. This is inherent to all types of linear arrays. From this example, it is worth mentioning that the same directivity pattern can be achieved at all frequencies, as long as d varies according to the frequency to keep the ratio d/λ constant. This implies that at low frequencies, larger inter-spacings, and thus larger array dimensions, are needed in order to keep d/λ constant. In fact, this is a very important property as it suggests that the lowest frequencies that a beamformer can resolve properly are determined by the total size of the array. Besides the two main lobes, the pattern also presents sidelobes with signiﬁcantly lower amplitudes in the other directions. By adding more transducers and keeping d/λ constant, the main lobes become narrower and the number of sidelobes increases, but their overall amplitude decreases. This is illustrated in Fig. 2.7, where the patterns for three different number of sensors are shown in polar plots. This feature has two possible interpretations: 1) considering a ﬁxed array length, having more sensors results in a smaller inter-spacing, which implies that the pattern becomes more directive toward 20 2. Basic beamforming methods -30◦ 0◦ 30◦ -60◦ -30◦ 60◦ -120◦ 120◦ -150◦ 180◦ (a) M = 5 150◦ 30◦ -60◦ -30◦ 60◦ 90◦ -90◦ -30-20 -10 0 -90◦ 0◦ 120◦ -150◦ 180◦ (b) M = 10 150◦ 30◦ -60◦ 60◦ 90◦ -90◦ -30-20 -10 0 -120◦ 0◦ 90◦ -30-20 -10 0 -120◦ 120◦ -150◦ 180◦ (c) M = 20 150◦ Figure 2.7: Inﬂuence of the number of transducers on the beamforming pattern of a uniform linear array. The magnitude is expressed in dB and normalized to A × M . A plane wave approaches the array from 0◦ . In all cases d/λ = 0.3 high frequencies (a smaller wavelength is needed in order to keep d/λ constant.) And 2) assuming a constant space between sensors, increasing the number of sensors is equivalent to extending the array, which implies that for a given frequency, the larger the array, the more directive the pattern. This agrees with the previous discussion regarding array size and low frequencies. Similarly to the effect of increasing the number of sensors, the pattern becomes more directive with increasing the ratio d/λ, as can be seen in Fig. 2.8. However, the case d/λ = 1.2 shows replicas of the main lobe in unexpected directions. These replicas, usually called grating lobes, are caused by the aliasing effect, which is a consequence of undersampling the space with a ﬁnite number of transducers. The aliased replicas occur when d/λ > 0.5, which corresponds to the Nyquist sampling criterion in space. In fact, this criterion determines the highest frequency the array can capture without sampling error. The aliasing effect can be pushed beyond the Nyquist criterion, and thus, toward higher frequencies, by using irregular arrays. With an irregular layout, the level of the sidelobes is kept relatively low for a wider frequency range, and aliasing occurs at those frequencies where the average element spacing is several wavelengths; up to about 4λ according to Ref. [47], which is signiﬁcantly above the Nyquist criterion (λ/2). Ideally, aliasing can only be totally avoided in the hypothetical case of using an array of sensors placed inﬁnitely close to each other, or alternatively, by means of scanning the sound ﬁeld in a continuous manner. Recent studies have shown that this is possible with a laser beam, as in the acousto-optic beamformer [40]. 2.1 Delay-and-sum beamforming -30◦ 0◦ 30◦ -60◦ 21 -30◦ 60◦ 0◦ -60◦ -120◦ 120◦ 150◦ 180◦ (a) d/λ = 0.3 0◦ 120◦ 30◦ -60◦ 60◦ 90◦ -90◦ -30-20 -10 0 -120◦ -150◦ -30◦ 60◦ 90◦ -90◦ -30-20 -10 0 -90◦ 30◦ 90◦ -30-20 -10 0 -120◦ -150◦ 150◦ 180◦ (b) d/λ = 0.5 120◦ -150◦ 150◦ 180◦ (c) d/λ = 1.2 Figure 2.8: Inﬂuence of the ratio d/λ on the beamforming pattern of a uniform linear array with 10 sensors. The magnitude is expressed in dB and normalized to A × M . A plane wave approaches the array from 0◦ . -30◦ 0◦ 30◦ -60◦ -30◦ 60◦ -120◦ 120◦ -150◦ 180◦ (a) ϑs = 0◦ 150◦ 30◦ -60◦ -30◦ 60◦ 90◦ -90◦ -30-20 -10 0 -90◦ 0◦ 120◦ -150◦ 150◦ 180◦ (b) ϑs = 45◦ 30◦ -60◦ 60◦ 90◦ -90◦ -30-20 -10 0 -120◦ 0◦ 90◦ -30-20 -10 0 -120◦ 120◦ -150◦ 150◦ 180◦ (c) ϑs = 90◦ Figure 2.9: Inﬂuence of the direction of an incident plane wave on the beamforming pattern of a uniform linear array with 10 sensors. The magnitude is expressed in dB and normalized to A × M . In all cases d/λ = 0.3 When an incident wave comes from directions other than 0◦ and 180◦ , the resulting main lobe becomes progressively wider toward ±90◦ . This can be seen in Fig. 2.9, for waves arriving from 0◦ , 45◦ , and 90◦ . Due to the front-back ambiguity, a replica of the main lobe always appears at (180◦ − ϑs ) if ϑs ≥ 0, or at (−180◦ − ϑs ) if ϑs < 0. A pattern that depends on the focusing direction is usually referred to as shift-variant. 2.1.2 Uniform circular array Two of the main weaknesses that uniform linear arrays exhibit, namely the front-back ambiguity and the pattern dependency on the steering direction, can be solved if uniform 22 2. Basic beamforming methods y z ki κ̂ ϕ ki θs 1 θ ϕs κ̂ 0 1 0 x M −1 R x M −1 (b) Side view (a) Top view Figure 2.10: Plane wave impinging on a uniform circular array with M sensors. circular arrays are used instead. This array geometry is characterized by having M sensors uniformly distributed in a circle, as illustrated in Fig. 2.10. The position of the mth sensor is in this case given by ⎤ cos (2πm/M ) ⎥ ⎢ = R ⎣ sin (2πm/M ) ⎦ , 0 ⎡ rm m = 0, . . . , M − 1, (2.21) where R is the radius of the circle. According to geometrical model given in Fig. 2.10, the array steering vector, and the wavenumber vector of a wave arriving from (θs , ϕs ), are and ⎤ sin θ cos ϕ ⎥ ⎢ κ̂ = ⎣ sin θ sin ϕ ⎦ , cos θ ⎡ ⎡ sin θs cos ϕs (2.22) ⎤ ⎢ ⎥ ki = −k ⎣ sin θs sin ϕs ⎦ . cos θs (2.23) Using the three previous expressions, the delay-and-sum output can be easily obtained 2.1 Delay-and-sum beamforming 30◦ 0◦ 30◦ 330◦ 60◦ 300◦ 90◦ 23 240◦ 30◦ 330◦ 60◦ 300◦ 270◦ 90◦ -30-20 -10 0 120◦ 0◦ 240◦ 150◦ 210◦ 180◦ (a) ϕs = 0◦ 330◦ 60◦ 300◦ 270◦ 90◦ -30-20 -10 0 120◦ 0◦ 270◦ -30-20 -10 0 120◦ 150◦ 210◦ 180◦ (b) ϕs = 130◦ 240◦ 150◦ 210◦ 180◦ (c) ϕs = 250◦ Figure 2.11: Inﬂuence of the direction of an incident plane wave on the beamforming pattern of a uniform circular array with 10 sensors. The magnitude is expressed in dB and normalized to A × M . In all cases d/λ = 0.3 and θs = θ = 90◦ . using Eq. (2.4). In a compact form, this can be expressed as [48] b(θ, ϕ) = M −1 wm ejkρ cos(ξ− 2πm M ) , (2.24) m=0 where ρ=R (sin θ cos ϕ − sin θs cos ϕs )2 + (sin θ sin ϕ − sin θs sin ϕs )2 , and cos ξ = R sin θ cos ϕ − sin θs cos ϕs . ρ (2.25) (2.26) In order to obtain a shift-invariant pattern, the weightings wm must be uniform. In what follows, they are set to unity. Let us consider a uniform circular array with 10 sensors that captures a plane wave with frequency such that d/λ = 0.3. It is assumed that the wave propagates in the plane of the array and that the beamformer also looks into this plane. Looking at the right panel of Fig. 2.10, this implies that θs = θ = 90◦ . The delay-and-sum response for three different azimuth angles ϕs of the incident wave, 0◦ , 130◦ , and 250◦ , is shown in Fig. 2.11. At ﬁrst sight, it can be clearly seen that the circular geometry does not exhibit the front-back ambiguity and that the directivity pattern has practically the same shape regardless the direction of incidence of the wave. However, for a given d/λ and 24 2. Basic beamforming methods 30◦ 0◦ 30◦ 330◦ 60◦ 300◦ 90◦ 240◦ 150◦ 210◦ 180◦ (a) d/λ = 0.3 30◦ 330◦ 60◦ 300◦ 270◦ 90◦ -30-20 -10 0 120◦ 0◦ 240◦ 150◦ 210◦ 180◦ (b) d/λ = 0.5 330◦ 60◦ 300◦ 270◦ 90◦ -30-20 -10 0 120◦ 0◦ 270◦ -30-20 -10 0 120◦ 240◦ 150◦ 210◦ 180◦ (c) d/λ = 1.2 Figure 2.12: Inﬂuence of the ratio d/λ on the beamforming pattern of a uniform circular array with 10 sensors. The magnitude is expressed in dB and normalized to A × M . In all cases a plane wave arrives from (θs , ϕs ) = (0◦ , 90◦ ) and θ = 90◦ . a ﬁxed number of sensors the main lobe is in general wider to that of a uniform linear array; compare, for instance, panels (a) and (b) in Fig. 2.9 with Fig. 2.11. As can be seen in Fig. 2.12, with increasing d/λ the circular array behaves similarly to a uniform linear array: the main beam becomes more directive and the sidelobes increase in number. Due to the regular geometry of a uniform circular array, aliasing occurs at about d/λ = 0.5. However, aliasing does not take the form of replicas of the main lobe (grating lobes) as in the case of a uniform linear array, yet in a dramatic increase in level of the sidelobes. It is worth mentioning at this point that the distance between consecutive sensors d does not follow the curvature of the circular geometry, but the straight line between consecutive sensors. Geometrical considerations show that d = 2R sin(π/M ). (2.27) Although a uniform circular array is shift-invariant in the azimuthal direction (ϕ), it should be emphasized that this property is not valid with respect to the polar angle (θ). For this reason, uniform circular arrays are normally steered toward directions contained in the plane of the array, by ﬁxing θ = 90◦ , and considering 2D sound ﬁelds, i.e., only waves propagating in that plane. This is a good assumption as long as waves from other directions are sufﬁciently attenuated. If that is not the case, but the beamformer still expects a 2D sound ﬁeld, the resulting map becomes gradually ambiguous to such an extent that the pattern is totally omnidirectional for waves propagating from 2.1 Delay-and-sum beamforming 30◦ 0◦ 25 30◦ 330◦ 60◦ 300◦ 90◦ 0◦ 60◦ 300◦ 270◦ 90◦ -30-20 -10 0 120◦ 240◦ 150◦ 210◦ 180◦ (a) θs = 0◦ and θs = 180◦ 30◦ 0◦ 60◦ 90◦ 270◦ -30-20 -10 0 120◦ 240◦ 150◦ 210◦ 180◦ (b) θs = 30◦ and θs = 150◦ 30◦ 330◦ 300◦ 240◦ 150◦ 210◦ 180◦ (c) θs = 60◦ and θs = 120◦ 0◦ 330◦ 60◦ 300◦ 270◦ 90◦ -30-20 -10 0 120◦ 330◦ 270◦ -30-20 -10 0 120◦ 240◦ 150◦ 210◦ 180◦ (d) θs = 90◦ Figure 2.13: Inﬂuence of the polar angle of an incident plane wave on the beamforming pattern of a uniform circular array with 10 sensors. The magnitude is expressed in dB and normalized to A × M . In all cases d/λ = 0.3, ϕs = 0◦ and θ = 90◦ . θs = 0◦ and θs = 180◦ . This can be seen in Fig. 2.13. In practice, waves with polar angles θs up to ±30◦ off-plane are generally detected successfully. 2.1.3 Spherical array Spherical arrays consist of a number of sensors distributed over the surface of a sphere, which can be open (or transparent) or not. Unlike circular arrays, which have difﬁculties with waves propagating out-of-plane, spherical arrays have the ability to map 3D sound ﬁelds effectively. Furthermore, some layouts can provide a shift-invariant pattern for the entire 3D space. This will be later addressed in Sec. 3.1.2. The geometrical model assumed for a spherical array is shown in Fig. 2.14. In this case, the position of the mth microphone is given by 26 2. Basic beamforming methods θs z θ ki κ̂ y ϕ x ϕs Figure 2.14: Plane wave impinging on a spherical array with M sensors. ⎡ sin θm cos ϕm ⎤ ⎢ ⎥ rm = R ⎣ sin θm sin ϕm ⎦ , cos θm m = 0, . . . , M − 1, (2.28) where R is the radius of the sphere, and θm and ϕm are the polar and the azimuth angles of the mth transducer. The focus direction κ̂ and the wavenumer vector of an incident plane wave are given by Eqs. (2.22) and (2.23), respectively. 2.2 Performance indicators It is apparent from the previous section that several aspects, for instance, geometry, number of transducers, and spacing, inﬂuence the beamformer response. It is therefore necessary to make use of performance indicators to assess and compare beamforming systems. Most measures of performance found in the literature are often adapted from other ﬁelds, such as electromagnetism (antenna theory) and optics. A brief description of the most relevant ones is given in the following. 2.2 Performance indicators 27 Measures of beam pattern Resolution: Deﬁned as the −3 dB width of the main lobe of the directivity pattern and measured in degrees or radians. It is also known as the 3 dB beamwidth, the half-power beamwidth, and the angular resolution. This measure, which is adapted from antenna theory (see, e.g., Ref. [49]) gives the minimum angle at which two incoherent sources can be resolved. Moreover, it is an indicator of directivity. The lower the value, the more directive the beamformer is. Maximum sidelobe level (MSL): Given by the difference in level in dB between the peak of the highest sidelobe in the beam pattern to the peak of the main lobe [12]. This measure is also adapted from antenna theory and in the antenna community it is commonly referred to as SLL (sidelobe level) [50]. The MSL is complementary to the resolution, as it is not about directivity, yet a descriptor of how sensitive the beamformer is toward unwanted directions. Obviously, the larger the level difference between main- and maximum sidelobe, the better. It can be shown that this measure is more sensitive to noise than the resolution. Peak-to-zero distance or Rayleigh resolution limit: Given by the angular difference between the position of the peak of the main lobe of the beam pattern and the position of its closest null. It determines the ability of the array to resolve two incoherent plane waves based on the Rayleigh criterion [51], adapted from optics theory. This criterion states that two plane waves are resolved when the main peak of the beam pattern of one falls on the null closest to the main peak of the beam pattern of the other one. This measure is an alternative to the resolution based on the −3 dB width, as half the beamwidth between the nulls of the main lobe is approximately equal to the −3 dB beamwidth [52]. Figure 2.15 illustrates how the resolution, the MSL, and the peak-to-zero distance can be extracted from a given beam pattern. Other measures Directivity index (DI): Deﬁned as the ratio of the beamformer response in the looking direction to the average response over all directions. Expressed on logarithmic 28 2. Basic beamforming methods 5 Magnitude of b [dB] 0 −3 dB −5 MSL RES −10 −15 −20 −25 −30 −35 PTZ 0 30 60 90 120 150 180 210 240 270 300 330 360 ϕ [◦ ] Figure 2.15: Calculation of the resolution (RES), the MSL, and the peak-to-zero (PTZ) distance from a given beam pattern. scale [52, 53], the DI can be written as DI = 10 log |b(θs , ϕs )| 1 4π 2π π 0 0 2 2 |b(θ, ϕ)| sin θ dθ dϕ . (2.29) This measure can be regarded as the array gain against isotropic noise (noise distributed uniformly over a sphere) [42]. The higher the DI, the better. Array gain: Reﬂects the improvement in SNR achieved by using an array. It is deﬁned as the ratio of the SNR at the array output to the SNR at a single sensor subject to different types of noise [4]. Usually isotropic acoustical noise is considered [8]. White noise gain (WNG): Measured as the array gain, but considering that the SNR at every sensor is due to spatially uncorrelated white noise [8]. It can also be regarded as the ratio of the signal power at the output of the beamformer to the sensor self-noise power assuming a unity variance noise [54]. The WNG is an indicator of the robustness of the array against deviations in the practical implementation, such as sensor self-noise, positioning errors, and amplitude and phase variations. The higher the WNG, the more robust the array is. It can be shown that the optimal WNG equals the number of microphones, for frequencies below 2.2 Performance indicators 29 the Nyquist frequency, and is achieved with delay-and-sum beamforming with a uniform weighting [42]. Usually, the resolution and the MSL are examined together as they provide direct values related to the beam pattern that complement each other to give an idea of its shape. These two indicators can be seen in Fig. 2.16 as a function of frequency for a delay-and-sum beamformer based on a uniform circular array with 12 sensors and 11.9 cm of radius.† . A uniform weighting, a focusing direction of 180◦ , and ideal (noise free) conditions are assumed. As can be seen, the resolution improves with increasing frequency, which means that the main beam becomes narrower. On the other hand, the MSL indicates that at low frequencies the sidelobes are non-existent, but they arise with increasing frequency. Inspection of these two performance indicators together suggest that the beamformer response is omnidirectional at the lowest frequencies, but with increasing frequency the directivity increases, although this is accompanied by an increase in level of sidelobes, which stagnates at a certain frequency. The DI of the array previously examined is showed in Fig. 2.17, together with the WNG. In this case, the DI at the lowest frequencies is 0 dB, which means that the beam pattern is omnidirectional. With increasing frequency, the DI increases progressively, indicating, thus, that the array becomes more and more directive. In contrast to the resolution, the DI provides a value related to the array directivity that considers not only the width of the main lobe, but also the sidelobes. This makes it impossible to predict a beam pattern via the DI, as becomes apparent from the inspection of Fig. 2.17. On the other hand, the WNG is constant across frequency, being equal to 10.8 dB, i.e., 12 in a linear scale, corresponding to the number of array microphones, as expected from the fact that delay-and-sum with uniform weighting provides the optimal WNG. It should be noted that while the resolution, the MSL, and the DI can be extracted from experimental results, this is not possible with the WNG, as this is a theoretical measure. In the microphone array community, the DI and the WNG are typically preferred for the numerical analysis of array designs, see, e.g., Refs. [8, 19, 54, 55]. The resolution and the MSL are less used, but nevertheless they have proven to be a useful evaluation tool, e.g., in Refs. [12, 56–58]. While the resolution can be related to the DI, as both are measures of directivity, the MSL can be used to examine the robustness † An array with these characteristics is used in Paper A. Its Nyquist frequency is about 2.7 kHz. 30 2. Basic beamforming methods 360 Resolution [◦ ] 300 240 180 120 60 0 0 500 1000 1500 Frequency [Hz] 2000 2500 2000 2500 (a) Resolution 0 −10 MSL [dB] −20 −30 −40 −50 −60 −70 0 500 1000 1500 Frequency [Hz] (b) MSL Figure 2.16: Resolution and MSL obtained with a delay-and-sum beamformer based on a uniform circular array with 12 sensors and 11.9 cm of radius. The array is focused toward 180◦ . 2.2 Performance indicators 31 10 8 DI [dB] 6 4 2 0 −2 0 500 1000 1500 Frequency [Hz] 2000 2500 2000 2500 (a) DI 14 12 WNG [dB] 10 8 6 4 2 0 0 500 1000 1500 Frequency [Hz] (b) WNG Figure 2.17: DI and WNG obtained with a delay-and-sum beamformer based on a uniform circular array with 12 sensors and 11.9 cm of radius. The array is focused toward 180◦ . 32 2. Basic beamforming methods of the system, although this is not a direct measure as the WNG is. Since the noise captured by the system affects basically the level of the sidelobes of the beamforming response, this makes the resulting MSL deviate from the MSL that would be obtained in the absence of noise. The higher the deviation, the less robust the system is. In the contributing papers of this thesis, the performance of the suggested methods is examined basically by means of the resolution and the MSL for the following main reasons: 1. For sound source localization purposes the spatial sensitivity of the array is a very important factor. In this sense, the resolution is a very appropriate parameter. In combination with the MSL, these measures provide a better picture of the behavior of the beam pattern than DI and WNG. 2. Both resolution and MSL can be extracted from experimental data and not only from numerical data as in the case of the WNG‡ , which is crucial for the validation of the methods suggested in the papers. 3. For the reader not familiar with microphone arrays, the resolution and the MSL are simpler to interpret than the DI and the WNG. ‡ Note that although the DI can also be extracted from experimental measurements, this is rarely seen in the literature. Chapter 3 Eigenbeamforming 3.1 Introduction Eigenbeamforming, also known as eigenbeam beamforming, is a rather new category of methods that rely on ‘closed’ geometries, such as a sphere or a circle. The sound ﬁeld captured by arrays that fulﬁll this condition can be decomposed into a sum of orthogonal terms that satisfy the wave equation in the coordinate system that best suits the array geometry. Combination of these orthogonal terms, known as harmonics or phase modes, makes it possible to form a detection beam. An eigenbeamforming system consists of two stages, see Fig. 3.1; in the ﬁrst stage, the pressure measured with the array is decomposed in a set of harmonics, and in the second stage, usually referred to as modal beamformer, the coefﬁcients of the harmonics are weighted and added together to provide the ﬁnal beamforming output [32]. The fundamental difference from traditional beamforming lays on the fact that the latter is based on applying the signal processing algorithms directly to the signals captured by the microphones. It should be noted that the concept of using phase modes had been already explored in the past in the ﬁeld of electromagnetism, for antenna design; see, e.g., Refs. [59–68]. Although in principle many array shapes are possible for eigenbeamforming, only those that have a well-deﬁned geometry in the conventional coordinate systems are used in practice. The most popular geometry is the spherical one [69–75], followed by the circular [19, 55, 76–79] and the spheroidal [80]. 33 34 3. Eigenbeamforming 0 1 2 p0 C̃−N p1 p2 Sound C̃−1 ﬁeld C̃0 Modal C̃1 beamformer decomposition M −1 pM −1 b C̃N Figure 3.1: Eigenbeamforming procedure. The sound ﬁeld captured by the array sensors is decomposed in a series of orthogonal functions, whose coefﬁcients are weighted in the modal beamformer stage to yield the ﬁnal output b. 3.1.1 Eigenbeamforming for circular arrays The concepts behind eigenbeamforming are here brieﬂy described assuming an open uniform circular array, as the one shown in Fig. 2.10 on page 22. Given the circular symmetry, the sound ﬁeld can be decomposed in a Fourier series in the azimuth coordinate, ϕ, so that, at the array radius, the sound pressure can be written in the spatial frequency domain as [10] ∞ p(kR, ϕ) = Cn (kR)ejnϕ , (3.1) n=−∞ where the terms ejnϕ , often referred to as circular harmonics, form a set of orthogonal functions, 2π 1 ejnϕ (ejνϕ )∗ dϕ = δnν , (3.2) 2π 0 where δnν is the Kronecker delta function, which equals unity when n = ν and zero otherwise, and Cn (kR) is the nth order Fourier coefﬁcient, 1 Cn (kR) = 2π 2π 0 p(kR, ϕ)e−jnϕ dϕ. (3.3) 3.1 Introduction 35 If a plane wave with amplitude A, created at ϕs , is present in the sound ﬁeld, it can be shown that the coefﬁcients become [81] Cn (kR) = AQn (kR)e−jnϕs , (3.4) where Qn (kR) is a function that depends on the boundary conditions of the uniform circular array, that is, for example, whether the array is mounted on a bafﬂe or not, as will be seen later. In the case of a uniform circular array suspended in free-space, Qn (kR) = (−j)n Jn (kR), (3.5) where Jn (kR) is a Bessel function of the ﬁrst kind and order n. In theory, the sound pressure is represented by inﬁnitely many Fourier coefﬁcients. In practice, it can be shown that the contribution of those orders higher in magnitude than kR is very small [42]. Therefore, the representation of the sound ﬁeld is often limited, or truncated, to a maximum order N that satisﬁes N ≈ kR. (3.6) It is worth noting that less orders are required for representing the low frequencies compared to the high frequencies. With an array of transducers, the sound pressure is sampled at discrete positions, rather than in a continuous circle. This implies that the coefﬁcients deﬁned in Eq. (3.3) need to be approximated by C̃n (kR) ≈ M −1 αm p(kR, ϕm )e−jnϕm , (3.7) m=0 where the term αm is an integration factor that ensures the discrete orthogonality property of the circular harmonics M −1 ∗ αm ejnϕm ejνϕm ≈ δnν . (3.8) m=0 Given the constant sensor spacing of a uniform circular array, αm = 1/M . An additional consequence of the sampling theorem in space is that the number of array 36 3. Eigenbeamforming sensors required to capture the sound ﬁeld up to order N must be larger than 2N (M > 2N ) [42]. The error due to sampling a continuous circle with a limited number of sensors and the error for considering a ﬁnite number of harmonics for representing the sound ﬁeld are analyzed thoroughly in Refs. [18, 81]. The output of an eigenbeamforming system based on a circular array results from weighting and combining the Fourier coefﬁcients obtained with the decomposition of the sound ﬁeld, that is, b(kR) = N wn (kR)C̃n (kR), (3.9) n=−N where wn (kR) is the weighting associated to the nth order Fourier coefﬁcient Cn (kR). Analogous to the inﬂuence on the weighting observed in Chapter 2 for delay-and-sum beamforming, the output of the eigenbeamformer strongly depends on the these parameters. When the weightings take the form wn (kR) = 1 ejnϕl , Qn (kR) (3.10) where ϕl is the looking direction of the array, the beamforming technique is referred to as circular harmonics beamforming. This method, described and examined in Papers A and B, provides a response rotationally symmetric around the azimuthal looking direction, i.e., the resulting pattern is shift-invariant. When using these weightings it is particularly important to limit the orders in the beamforming algorithm to a value N close to that given in Eq. (3.6) to avoid ampliﬁcation of noise. Usually, the maximum number of orders is chosen as N = kR, where · is the ceiling function [81]. If higher orders are considered, the value Qn (kR) in the denominator tends to zero, boosting, in this way, noise captured in the measurements. Therefore, the truncation of orders can be regarded as a regularization method. The main characteristic of this technique is that the output depends on the number of harmonics taken into account in the algorithm; see Appendix A for further details. Interestingly, delay-and-sum beamforming can also be characterized in terms of eigenbeamforming, making use of the weightings wn (kR) = Q∗n (kR)ejnϕl . (3.11) 3.1 Introduction 37 These weightings also ensure a shift-invariant pattern [16]. Besides circular harmonics beamforming and delay-and-sum beamforming, other methods based on eigenbeam processing with a uniform circular array of microphones can be found in the literature. For instance, Ref. [18] adapts various adaptive methods, such as MUSIC and ESPRIT∗ , to the circular geometry. Eigenbeamforming makes it easier to deal with bafﬂed arrays, arrays whose elements are ﬂushed-mounted on the surface of an object or a scatterer. By simply modifying the function Qn (kR) of Eq. (3.4) according to the bafﬂe type, the scattering effect can be taken into account in the beamforming algorithm. Common bafﬂes suitable for uniform circular arrays are rigid spheres [19, 55] and cylinders [76]. Less popular are rigid bafﬂes with a spheroidal shape [79], which can be oblate or prolate, and bafﬂes with a certain surface impedance [55]. While the scattering effects of spheres, spheroids, and inﬁnitely-long cylinders, have an exact analytical solution, and so does the corresponding Fourier coefﬁcients, that is not the case with the scattering from a ﬁnite cylinder. Arrays with rigid bafﬂes are usually preferred over open arrays for the following two reasons; ﬁrstly, the boundary conditions of a bafﬂe are well deﬁned compared to open arrays, which in practice are far from being transparent (their structure, preampliﬁers, cables, etc., obviously alter the sound ﬁeld [82]). Secondly, bafﬂed arrays provide a better response than open arrays; in particular, this is noticeable with delay-and-sum beamforming toward low frequencies, as due to the presence of the bafﬂe, waves need to travel longer distances before reaching the microphones, which results in an effective larger array aperture [78]. Of all common types of rigid bafﬂes for circular arrays, delay-and-sum beamforming performs best with cylinders. The behavior with oblate and prolate spheroidal bafﬂes lies between those of an open array and a sphere, and a sphere and an inﬁnite cylinder, respectively [79]. With circular harmonics beamforming, the performance with scatterers is very similar to that with open arrays. The only difference is that with open arrays the output presents singularities at those frequencies that coincide with the zero-crossings of the Bessel functions in the denominator of the algorithm; see Eqs. (3.5) and (3.10). With this technique, there is no signiﬁcant difference between different types of bafﬂes. ∗ ESPRIT stands for ‘estimation of signal parameters via rotational invariance techniques’. 38 3.1.2 3. Eigenbeamforming Eigenbeamforming for spherical arrays Spherical arrays are suitable for decomposing a 3D sound ﬁeld into a series of orthogonal terms of the form [10] Ynm (θ, ϕ) = 2n + 1 (n − m)! m P (cos θ)ejmϕ , 4π (n + m)! n (3.12) n where Pm (cos θ) is the associated Legendre function of order m and degree n. These terms are commonly referred to as spherical harmonics. In the case of a spherical array of radius R, the pressure at the sphere is given by the Helmholtz equation in spherical coordinates [16], p(kR, θ, ϕ) = ∞ n Cmn (kR)Ynm (θ, ϕ), (3.13) n=0 m=−n where the terms Cmn (kR) follow Cmn (kR) = 2π 0 π 0 p(kR, θ, ϕ)Ynm (θ, ϕ)∗ sin θdθdϕ. (3.14) Since with an array the pressure is captured at the sensor positions, the coefﬁcients that result from the decomposition of the sound ﬁeld are in practice approximated by C̃mn (kR) = M −1 αi p(kR, θi , ϕi )Ynm (θi , ϕi )∗ , (3.15) i=0 where θi and ϕi are angular coordinates of the ith microphone and αi is an integration factor associated to the ith microphone. This parameter enforces the orthogonality of the spherical harmonics up to order N , such that the resulting coefﬁcients are free of error up to that order [20]. By analogy to the case of eigenbeamforming with a circular sphere, it can be shown that the coefﬁcients C̃mn (kR) can be weighted before adding them together to form a beam in a particular direction. Eigenbeamforming with spherical arrays is well documented in the literature. The reader interested in this topic is addressed to, e.g., Ref. [16, 54]. 3.1 Introduction 39 A note on the design of spherical arrays for eigenbeamforming The position of the microphones in a spherical array is not as trivial as in the case of a uniform circular array. There are several strategies to sample a sphere so that the discrete orthogonality property of the spherical harmonics, that is M −1 αi Ynm (θi , ϕi )Yνμ ∗ (θi , ϕi ) = δnν δmμ , (3.16) i=0 is fulﬁlled. The integration factor of the ith microphone αi , as well as the relationship between the number of transducers M and the maximum order N that can be captured without error, depends on the sampling scheme. An overview of several sampling schemes is given in details in Ref. [20]. By analogy to the uniform circular array, the most intuitive way to satisfy the orthogonality relationship is by sampling the sphere uniformly so that the transducers are equidistant. In such a case, αi reduces to a constant. However, a uniform distribution of sensors is only possible with a limited set of arrangements based on regular polyhedra (also called platonic solids) that allow a sphere to ﬁt in; speciﬁcally, the tetrahedron (4 faces), the cube (6 faces), the octahedron (8 faces), the dodecahedron (12 faces) and the icosahedron (20 faces). By placing the sensors at the center or at the vertices of each face, the resulting distribution of sensors is uniform. An alternative that presents a distribution close to being uniform is the truncated icosahedron† , which has 32 faces [69]. Yet another solution is the combination of a non-equidistant sampling with a non-uniform weighting αi . That is, for instance, the case of the nearly uniform [83], the equiangle [20, 71], and the Gaussian [20, 71, 73, 74, 76] sampling schemes. The equiangle distribution relies on equally spaced samples on θ and ϕ, whereas the Gaussian sampling scheme is similar, but only half of the samples are considered on θ. The orthogonality property of the spherical harmonics ensures that the decomposition is independent of the microphone positions, allowing, thus, a shift-invariant beam pattern due to the spherical symmetry. † An example of a truncated icosahedron is a football. It consists of 12 pentagonal faces and 20 hexagonal faces. 40 3.2 3.2.1 3. Eigenbeamforming Papers A and B Synopsis Paper A suggests an eigenbeamforming technique for a uniform circular array, called circular harmonics beamforming. The technique, conceived for mapping sources distributed from 0 to 360◦ , is particularly suitable for environmental noise problems. Circular harmonics beamforming is compared numerically to delay-and-sum beamforming for both an open array and an array mounted on an inﬁnitely-long rigid cylinder, by means of the resolution and the MSL. The method is also validated experimentally with an open array. Paper B extends the investigation carried out in Paper A to the case of a uniform circular array mounted on the equator of a rigid sphere, and validates it numerically and experimentally. 3.2.2 Related work In 2001, Meyer presented in Ref. [19] a method for beam pattern synthesis based on the decomposition of the sound ﬁeld into a series of (circular) harmonics that relied on a uniform circular array mounted on a rigid sphere. Although the concept was initially developed for uniform circular arrays, it triggered a series of research projects involving spherical arrays of microphones, pioneered by Meyer and Elko, Ref. [69], and Abhayapala and Ward, Ref. [70], in 2002. Their investigations set the foundations of eigenbeamforming for spherical arrays. In the following years, Rafaely published several articles on the matter, e.g., Refs. [16, 20, 71, 75]. In particular, the principles of eigenbeamforming (in that case referred to as phase-mode processing) were described in Ref. [72], following the approach used in Sec. 3 for the circular geometry, also under the plane wave assumption. In particular, Rafaely analyzed the response of an eigenbeamformer whose weightings provided the so-called regular beam pattern, the most directive pattern (for ﬁxed beamforming) [16], and compared it to delay-and-sum expressed in eigenbeamforming terms. The results of that study showed that the directivity achieved with the regular beam pattern exceeds that of delay-and-sum beamforming, specially at low frequencies, at the expense of robustness to noise. That work was later supplemented in Ref. [75] with an overview of various eigenbeamforming methods. Oddly enough, contemporary to Rafaely’s research, Pedersen in Ref. [84] and Song 3.2 Papers A and B 41 in Ref. [85], following an approach different from Rafaely’s, arrived to the expression for eigenbeamforming with a regular beam pattern, considering, though, spherical waves. They called the method spherical harmonics beamforming. A year after Song’s work was published, Haddad and Hald, in Ref. [86], added a scale factor into spherical harmonics beamforming so that in case of having a rigid sphere, the pressure contribution would be determined correctly, i.e., without the inﬂuence of the scattering effect. This version of spherical harmonics beamforming was referred to as spherical harmonics angularly resolved pressure (SHARP). This method has been recently extended in Ref. [57] to provide a smoother response, by means of adding regularization ﬁlters. Similarly to Rafaely’s article on decomposition of sound ﬁelds with spherical arrays, Ref. [71], Teutsch and Kellerman in Ref. [76]‡ presented a theoretical analysis of plane wave decomposition with circular arrays, unbafﬂed, mounted on a rigid inﬁnitelylong cylindrical bafﬂe, and mounted on a rigid cylinder of ﬁnite-length. In addition, they derived eigenbeamforming based on the circular geometry, assuming a continuous aperture instead of the sampled version, i.e., a microphone array. Also, they expressed two adaptive beamforming algorithms ESPRIT and DETECT, in terms of eigenbeamforming and evaluated them with an array mounted on a rigid cylinder, numerically and experimentally. 3.2.3 Discussion Inspired by the literature on eigenbeamforming with spherical arrays, Paper A adapts the theory behind spherical harmonics beamforming to the 2D case with a circular array, assuming, in this case, plane waves impinging on the array. The proposed beamforming technique, referred to as circular harmonics beamforming, was originally conceived in Paper A for localization of environmental noise sources, but it can obviously be applied to other scenarios where sound sources are distributed over the array azimuth. It should be noted that 1) the approach followed in Paper A to derive circular harmonics beamforming is different from the synthesized derivation given in Sec. 3.1.1; and 2) the article does not include the insight into the technique concerning the inﬂuence on the number of orders given in Appendix A. While delay-and-sum is omnidirectional at low frequencies, circular harmonics beamforming presents a certain directivity, namely a resolution of about 112◦ in the ‡ This work is also presented in the PhD Thesis by Teutsch, Ref. [18]. 42 3. Eigenbeamforming worst case. Indeed, the response of circular harmonics beamforming in terms of directivity is better at the lower frequency range than that of delay-and-sum. At high frequencies, both methods perform similarly. The main drawback of circular harmonics beamforming in comparison with delay-and-sum is its vulnerability to noise, which essentially affects the sidelobe levels. Circular harmonics beamforming implemented with an open uniform circular array presents singularities at a few (single) frequencies, which can be resolved when the array is mounted on a rigid inﬁnitely-long cylinder. In general terms, though, the overall output pattern with the two array conﬁgurations is the same. By contrast, the pattern of delay-and-sum with the cylindrical scatterer improves toward low frequencies, as the scatterer makes the array appear larger. Interestingly, with this conﬁguration both the resolution and the MSL at high frequencies is similar for both beamforming techniques. However, at low frequencies, the performance with circular harmonics beamforming still exceeds that of delay-and-sum. The results of the investigation carried out in Paper A showed that the performance with a rigid inﬁnitely-long cylindrical scatterer was better over that of an open array, especially for delay-and-sum beamforming. Since inﬁnitely-long cylinders are not feasible, they are in practice approximated by ﬁnite length cylinders. With regard to that, Teutsch and Kellerman showed in Ref. [76] that a ﬁnite cylinder whose length is 1.4 times its radius is enough to approximate an inﬁnitely long cylinder, as its modal response becomes fairly similar. This result was later ratiﬁed by Granados in Ref. [87]. As an alternative to cylindrical scatterers of ﬁnite-length, Paper B suggests to ﬂushmount the array on the equator of a rigid sphere, and repeat the comparison carried out in Paper A. The main advantage of this conﬁguration is that the scattering produced by this geometry has an exact analytical solution, in contrast to the ﬁnite-length cylinder. With a spherical bafﬂe, circular harmonics beamforming performs in the same manner as in the inﬁnitely-long cylinder case. However, for delay-and-sum the improvement is not as good as with the cylinder, because the effective aperture achieved with the spherical scatterer is smaller. The novelty of circular harmonics beamforming cannot be entirely attributed to the author of this thesis as Zhang et al. in Ref. [88] also derived the same technique under another name and compared it to delay-and-sum for a circular array mounted on a rigid sphere. Their work, thus, resembles the study presented in Paper B, although they assessed the beamforming techniques using DI and WNG, and the analysis was restricted to numerical simulations. In any case, their ﬁndings agree in general terms 3.2 Papers A and B 43 with those of Paper B. It should be emphasized that Ref. [88] is not cited in Papers A and B as the author of this thesis was not aware of the existence of this work at the time of writing the papers. The results in Paper B concerning delay-and-sum also agree with those shown by Daigle et al., in Ref. [55], where the performance of delay-and-sum was analyzed for circular arrays mounted on spherical bafﬂes using the DI. 44 3. Eigenbeamforming Chapter 4 Deconvolution methods 4.1 Introduction Beamforming systems cause unavoidable effects, namely the frequency dependence of the array resolution and the appearance of sidelobes, which result in beamformed maps that appear blurred and often difﬁcult to interpret, particularly when several acoustic sources need to be detected simultaneously. Deconvolution methods intend to deblur them by removing the artifacts introduced by the array system itself and thereby restoring the original data. These methods rely on the fact that the beamformer output is a linear combination of the spatial distribution of acoustic sources and the so-called point-spread function, deﬁned as the beamformer’s response to a point source∗ . Mathematically, the deconvolution problem can be formulated in the frequency domain as follows 2 |b(r)| = s(r ) · H(r|r ), (4.1) r ∈G where s(r ) is the source power distribution at a position r that belongs to the grid of points G, and H(r|r ) is the point-spread function at r due to a source at r . It should be emphasized that the source power distribution is non-negative. In matrix notation, the previous expression can be written as b = Hs, ∗ The (4.2) point-spread function and the beamformed map are sometimes referred to as the ‘dirty beam’ and the ‘dirty map’, respectively. 45 46 4. Deconvolution methods 0.5 y [m] y [m] 0.5 0 dB 2 4 6 8 10 12 14 16 18 20 0 0.5 0 0.5 0.5 0 x [m] 0.5 0.5 0 x [m] 0.5 Figure 4.1: Beamformed map (left) and clean map after deconvolution (right). Measurement of two uncorrelated sources located at 2.7 m from the array. The stars in the beamformed map indicate the position of the sources. The level of the right source is 10 dB higher. With beamforming the left source is masked, whereas it becomes visible after deconvolution. Adapted from Ref. [90]. where b is a vector with the power response of the beamformer, H is a matrix that in each column contains the point-spread functions of each grid point, and s is the unknown source power distribution vector. The deconvolution methods try, thus, to compensate for the ‘blurring’ effect of the point-spread function to recover the original source distribution. Notice that this a discrete inverse problem, and must be treated carefully to avoid an abrupt ampliﬁcation of noise, which can often lead to a meaningless solution [89]. Deconvolution methods approach this problem by means of iterative algorithms. The resulting plot of the estimated source distribution is a ‘clean’ version of the beamformed map: the resolution is improved, and the sidelobes are reduced, or even suppressed. This is illustrated in Fig. 4.1. Deconvolution methods are relevant in many ﬁelds that involve image restoration. This problem was ﬁrst approached for seismology purposes by Robinson [91, 92] back in 1954, inspired by the previous work done by Wiener in that ﬁeld [93]. Since then, deconvolution has been applied to many other research areas, such as radio astronomy [94], optical microscopy [95] and image processing [96]. It was not until the late nineties, that the aeroacoustic community adapted some of the existing deconvolution methods to deal with sound ﬁeld visualization problems. That is the case of CLEAN [97] and Richardson-Lucy [98, 99], both originally developed for astronomy and modiﬁed for acoustical purposes in Refs. [13] and [100], respectively. While CLEAN acts directly on the beamformed map, i.e., on the image itself, Richardson- 4.1 Introduction 47 Lucy solves the inverse problem posed in Eq. (4.2) using Bayes’ theorem on conditional probabilities. Other algorithms, adapted from classical non-negative least squares (NNLS) procedures [101], seek to solve the following optimization problem minimize subject to 1 2 2 Hs − b2 , q ≥ 0. (4.3) (4.4) That is the case of gradient projection methods, such as the fast Fourier transform-nonnegative least squares method (FFT-NNLS) [100], the gradient projection method with Barzilai & Borwein steps [102], and the fast iterative shrinkage-thresholding algorithm (FISTA) [103], examined in Ref. [104] for sound source localization purposes. Alternatively, there are methods speciﬁcally conceived for acoustic purposes. A number of methods are devoted to static incoherent sound ﬁelds. The ﬁrst method developed, called the deconvolution approach for the mapping of acoustic sources (DAMAS) [105, 106], had the main disadvantage that was computationally very heavy. Seeking for efﬁciency, other algorithms, such as DAMAS2 [107], SC-DAMAS [108], CLEANSC [109], the covariance matrix ﬁtting (CMF) [108], and the iterative sidelobe cleaner (ISCA) [110], were implemented based on some assumptions. For example, DAMAS2 relies on a shift-invariant point-spread function, whereas SC-DAMAS, CLEAN-SC and CMF assume source sparsity. Moreover, DAMAS, DAMAS2, CLEAN, and CLEANSC have been extended in Ref. [111] to deal with moving sources. There are far less methods capable to deal with coherent sound ﬁelds. Examples are DAMAS-C [112], CMF-C [108], the mapping of acoustic sources (MACS) [113], and the wavespacebased coherent deconvolution [114]. In this case, CMF-C and MACS rely on sparsity, whereas the wavespace coherent deconvolution algorithm assumes a shift-invariant point-spread function. The main drawback of deconvolution procedures is that they are in general computationally challenging. It is therefore necessary to ﬁnd a compromise between the degree of accuracy, given by the size of the grid and the number of iterations, and the computational run time. Certain techniques, such as DAMAS2, FFT-NNLS, and Richardson-Lucy, rely on a shift-invariant beamformer’s point-spread function in order to use spectral procedures (Fourier-based) to reduce the complexity of the calculations, and thus, improve efﬁciency. Since a shift-invariant point-spread function only depends 48 4. Deconvolution methods on the distance between the source position and the observer position, H(r|r ) = H(r − r ), (4.5) the beamformer output, Eq. (4.1), results in a convolution 2 |b(r)| = s(r ) ∗ H(r − r ). (4.6) This relationship makes it possible to tackle the problem in the frequency domain, by means of expressing the convolution as a multiplication, and, thereby, speed up the process, 2 |b(r)| = F −1 [F[s(r )]F[H(r)]] , (4.7) where F and F −1 are the direct and the inverse FFT. In general, the assumption of a shift-invariant point-spread function is not valid with 2D imaging using planar-sparse arrays, such as spiral and pseudo-random arrays, unless the source region is small compared to the distance between the array and the source. Therefore, Fourier-based deconvolution approaches are restricted to small regions in space. Otherwise, errors occur. This is examined thoroughly in Ref. [104]. To extend these approaches to a larger, and 3D region, Refs. [107, 115, 116] suggest to make use of a coordinate transformation. The comparison of deconvolution methods is a cumbersome task because it can be done as function of many different parameters, such as convergence, resolution, computational load, number of iterations, etc. In addition their performance strongly depends on the case under analysis. Readers interested in the comparison of various methods are addressed to Refs. [100, 104, 108, 117, 118]. 4.2 Paper C 4.2.1 Synopsis Paper C adapts three deconvolution methods conceived for planar-sparse arrays, DAMAS2, FFT-NNLS, and Richardson-Lucy, to the circular geometry. The main characteristic of these methods is that they rely on a shift-invariant point-spread function, which has the advantage that the deconvolution can be approached with spectral pro- 4.2 Paper C 49 cedures to improve computational efﬁciency. The algorithms are examined via simulations and experimental data with a uniform circular array mounted on a rigid sphere. Their performance is analyzed through the beam patterns obtained with both delayand-sum beamforming and circular harmonics beamforming as a starting point for the deconvolution process. 4.2.2 Related work Deconvolution methods have become popular in the recent years as they are capable to provide more accurate maps than beamforming. Initially, they were implemented for planar-sparse arrays, and therefore, most of the existing literature assumes this layout. Although the methods can obviously be applied to eigenbeamforming arrays, they have been rather overlooked. To the author’s best knowledge, there is a lack of literature for the circular geometry, and only three references are available for the spherical one, Refs. [119–121]. Pascal and Li in Ref. [119] explore the beneﬁts of using DAMAS and Richardson-Lucy with a uniform spherical array, whereas Schmitt et al. in Ref. [120] suggest an NNLS algorithm for a spherical array with a pseudo-random distribution of microphones. On the other hand, Legg and Bradley in Ref. [121] analyze the performance of CLEAN-SC, although they do not specify the array conﬁguration. Surprisingly, none of these works consider eigenbeamforming algorithms, such as spherical harmonics beamforming, as a starting point for the deconvolution process; they simply make use of delay-and-sum. Moreover, only one of the mentioned techniques, Richardson-Lucy, makes use of a shift-invariant array pattern to base the computations on spectral procedures, thereby, lowering the computational running time. Precisely, Richardson-Lucy, together with two other methods that rely on shift-invariant pointspread functions, DAMAS2, and FFT-NNLS, are adapted in Paper C to uniform circular arrays. 4.2.3 Discussion Paper C introduces for the ﬁrst time the use of deconvolution methods to circular arrays, and, in addition to it, the use eigenbeamforming as starting point of the deconvolution process. The results of Paper C indicate that the beamformed maps improve signiﬁcantly after the deconvolution process in the entire frequency range of interest. In particular, 50 4. Deconvolution methods the resulting maps present a very ﬁne resolution, and the sidelobes are reduced and in some cases even removed. For a given number of iterations, the maps obtained with the different deconvolution techniques do not present signiﬁcant differences. In all cases, the main beam becomes narrower with increasing frequency. This actually implies that more iterations are needed at low frequencies in order to achieve the same resolution at all frequencies. In this respect, since circular harmonics beamforming presents a better resolution at low frequencies than delay-and-sum, the deconvolved maps also present a better resolution at these frequencies with this technique. At high frequencies the maps are rather independent on the beamforming technique used prior to deconvolution. However, the sidelobes are more noticeable, though much reduced compared to plain beamforming, with circular harmonics beamforming, as this technique is less robust to noise than delay-and-sum. All in all deconvolution methods are particularly useful when there is more than one source present in the sound ﬁeld. Interestingly, if only one source is present in the sound ﬁeld, the aliasing effect above Nyquist frequencies is removed from the map. This can be explained by the fact that the point-spread function used for the deconvolution is contaminated with aliasing, in such a way that during the deconvolution process the point-spread function matches the beamforming response, which is affected in the same way by aliasing. Paper C also shows that, apart from providing a better localization of the sound sources present in the sound ﬁeld, deconvolution methods also give a good estimate of the level of the sources via an integration process. The levels retrieved with the three deconvolution methods under analysis are very similar. However, the levels obtained from the delay-and-sum beamformed map present a better agreement with the average level captured directly with the microphones than the levels estimated with the circular harmonics beamformed map. During the research on deconvolution methods for the circular geometry, it was observed that the performance of the different methods (DAMAS2, FFT-NNLS, and Richardson-Lucy) depends on the case under analysis. In the examples given in Paper C, Richardson-Lucy converged faster than the other two methods, but this was not systematic; it varied depending on several parameters such as the amplitude of the impinging wave, the frequency, and the angle. It was not the goal of Paper C to judge which method was best, but this could certainly be done in a future study where all the parameters that play a role on the methods where analyzed thoroughly. 4.2 Paper C 51 The shortcoming of deconvolution methods is that they are time consuming, especially when the frequency range of analysis is broad, as they can only deal with one frequency at a time. With the current computers, these methods are generally restricted to those situations where measurements can be postprocessed at a later stage. In any case, the results of Paper C indicate a great potential of these methods with other eigenbeamforming systems, e.g., based on spherical arrays. 52 4. Deconvolution methods Chapter 5 Beamforming with holographic virtual arrays 5.1 Introduction As seen in Chapter 2, for a given number of sensors, an array with larger dimensions beneﬁts the response at low frequencies compared to a smaller array, at the expense of limiting the upper frequency range of operation of the array system. However, the dimensions of the array are usually given by manufacturers, and users cannot do much about that. Motivated by that, beamforming with virtual arrays emerges as an alternative for eigenbeamforming arrays with the aim to improve their performance at low frequencies. The principle behind it is the following: the pressure captured with an array of microphones is used to predict the pressure at a larger and virtual concentric array of the same type, by means of acoustic holography. The predicted pressure is then used to conduct beamforming. It should be noticed that this method assumes sources in the far ﬁeld of the array, which contrasts with most acoustic holography problems where sources are placed in the near ﬁeld. The details of beamforming combined with acoustic holography are given in Papers D and E, for an open circular array and a spherical array mounted on a rigid bafﬂe, respectively. Additionally, Appendix B gives the expressions for acoustic holography for circular arrays both open and mounted on a rigid cylinder of inﬁte length. In addition to the derivation given in the contributing papers, this method is expressed in eigenbeamforming terms in the following for the case of uniform circular arrays. It is assumed that plane waves traveling perpendicularly to the z-axis impinge 53 54 5. Beamforming with holographic virtual arrays on a (physical) uniform circular array that rests on the xy-plane. The eigenbeamformer output using a holographic virtual circular array with radius Rv is Nv b(kRv ) = wn (kRv )C̃n (kRv ), (5.1) n=−Nv where this expression follows from Eq. (3.9), but with the limits of the summation set to −Nv and Nv , where Nv ≈ kRv . Making use of the acoustic holography expressions given in Appendix B, it can be shown that the weightings for a virtual array when performing delay-and-sum result in 2 |Qn (kRv )| jnϕl e wn (kRv ) = . Qn (kR) (5.2) As can be seen, these weightings differ from those corresponding to the normal expression of delay-and-sum beamforming (for a physical array); see Eq. (3.11). It should be noted that this derivation is valid for open arrays and for arrays mounted on a rigid cylinder, but not for the case of spherical bafﬂes, as the reconstruction with acoustic holography cannot be expressed in these terms; see the discussion given in Sec. B.3 on page 70. When circular harmonics beamforming is performed with the virtual array, the weightings become 1 wn (kRv ) = ejnϕl . (5.3) Qn (kR) Inspection of this expression, independent of Rv , reveals that it totally coincides with the weightings of circular harmonics beamforming performed directly with a physical array; see Eq. (3.10). The only difference in the beamforming algorithm lies in the limits of the summations in Eqs. (5.1) and (3.9), ±Nv and ±N , respectively. As mentioned in Sec. 3, circular harmonics beamforming requires the truncation of orders higher than about kR to avoid regularization error. Accounting for orders up to about Nv ≈ kRv , which is unavoidably higher than kR, will obviously cause a larger error. This shows that computing circular harmonics beamforming with holographic virtual arrays does not present any advantage compared to doing it directly from physical arrays. This also occurs in the case of a spherical array, i.e., the principle works for delay-and-sum, but not for spherical harmonic beamforming. 5.2 Papers D and E 55 5.2 Papers D and E 5.2.1 Synopsis Paper D introduces for the ﬁrst time the concept of beamforming with holographic virtual arrays. An open uniform circular array is chosen for this purpose. The performance of this method is analyzed by means of simulations and experimental results, making use of the resolution and the MSL. Paper E adapts the principles given in Paper D to a rigid spherical array, and goes one step beyond with the investigation of the performance as a function of the radius of the virtual array. 5.2.2 Related work The combination of acoustic holography and beamforming has been examined recently by Fu et al. in Ref. [122] for visualization of sound sources with high temperatures. Their method consists of, at a ﬁrst stage, conducting near-ﬁeld beamforming with a planar array placed at a (known) distance from the source that prevents the system from being damaged due to the high temperatures. At a second stage, the beamformed map serves as input to acoustic holography, to reconstruct the sound ﬁeld closer to the source. Both the procedure and the ﬁnal goal of this work differ fundamentally from beamforming based on holographic virtual arrays for sound source localization. During the preparation of the work presented in Paper D, it was found out that there was a lack on literature about acoustic holography for circular arrays. This method, introduced in Refs. [9, 123, 124] in the beginning of the 1980’s, was conceived for array measurements in the near-ﬁeld of a source to predict the sound ﬁeld closer to it, with the aim to visualize the source radiation characteristics. In the ﬁrst years, planar NAH, based on measurements with planar arrays, was the main focus. However, already in the paper by Maynard et al. from 1985, Ref. [9], the method was expressed in spherical and cylindrical coordinates, in addition to the Cartesian. The possibilities that spherical arrays offered for spherical NAH became soon of interest; see, e.g., Refs. [17, 82, 125, 126]. A peculiarity of spherical NAH that contrasts with planar NAH is that, due to the closed surface of spherical arrays, the sound ﬁeld can be reconstructed in the entire 3D space without restrictions, as long as the reconstruction ﬁeld is free of sources [127]. The principles of spherical NAH can easily be adapted to the circular geometry. 56 5. Beamforming with holographic virtual arrays Since this geometry has one less dimension than the spherical one, circular arrays can predict an entire 2D sound ﬁeld from measurements in a closed curve (a ring). Although acoustic holography with circular arrays can be regarded as a particular case of cylindrical acoustic holography, the circular geometry has not been much examined. In particular, Cho et al. in Ref. [128] and Lee and Bolton in Ref. [129] made use of an open circular array for statistical optimized near-ﬁeld acoustic holography (SONAH) and for patch near-ﬁeld acoustic holography, respectively. The goal of both studies was to measure sources placed in the interior of the array. However, beamforming with virtual arrays, as described in Paper D, assumes sources outside the array. Moreover, with this technique both the measurement and the reconstruction with acoustic holography are carried out in the far ﬁeld of the sources. 5.2.3 Discussion The combination of acoustic holography and beamforming, by means of using virtual arrays for the beamforming procedure, has been examined in this PhD project for the ﬁrst time. It should be noticed that due to a lack of literature in acoustic holography for interior problems with circular arrays, the expressions for this reconstruction technique had to be derived explicitly for Paper D. The results of Papers D and E have shown that when the method is implemented with delay-and-sum, the performance at low frequencies exceeds that obtained with conventional delay-and-sum beamforming, to the detriment of the high frequencies, as the spacing between the ‘sensors’ of the virtual array is larger. It is thus recommended to perform beamforming directly from the physical array at high frequencies, while taking advantage of holographic virtual arrays at low frequencies. The success of the method depends on the dimensions of the virtual array, as the reconstruction process outside the physical array is an ill-posed problem that leads to an error in the estimated sound ﬁeld that increases with increasing the distance from the array to the reconstruction point [126]. This implies that with increasing radius of the virtual array, noise is ampliﬁed progressively. As a consequence, the value of the radius of the virtual array is crucial for the success of the proposed beamforming technique. This is analyzed in Paper E for a virtual spherical array. In contrast to the positive effect of the suggested technique on delay-and-sum, this new method does not exhibit any improvement when circular harmonics and spherical 5.2 Papers D and E 57 harmonic beamforming are used for its implementation, as seen in Sec. 5.1 for circular arrays. This indicates that the method is not general, and therefore, it should be tested with other beamforming algorithms. On the other hand, delay-and-sum beamforming with holographic virtual arrays should be compared to circular/spherical harmonics beamforming in a future follow up study. Some preliminary results not shown in this dissemination suggest that this technique can be more robust and present a better resolution and MSL than circular/spherical harmonics beamforming at certain frequencies. In a future investigation it could be helpful to address this question using other wellknown performance indicators, such as the DI and the WNG, besides the resolution and the MSL. As shown in Sec. 5.1, beamforming with holographic virtual arrays can be expressed in eigenbeamforming terms. However, in the case of circular arrays, this is only possible when they are either open or mounted on an inﬁnitely-long cylinder, due to some limitations imposed by the implementation of acoustic holography for this geometry; see Appendix B. Despite the intrinsic limitations of acoustic holography, and the fact that this technique does not beneﬁt circular harmonics beamforming, the implementation of beamforming using holographic virtual arrays is a new concept that at the moment has showed positive results for delay-and-sum. The ﬁndings during the PhD project aim at setting the ground for future research. 58 5. Beamforming with holographic virtual arrays Chapter 6 Conclusions 6.1 Summary and conclusions This dissertation has examined the use of uniform circular arrays for sound source localization purposes using beamforming. Uniform circular arrays are suitable for 2D sound ﬁelds in which waves propagate along the array plane, as they provide a 360◦ azimuthal coverage. That is often the need in many environmental noise problems where sound sources are placed in the far ﬁeld. A fundamental characteristic of beamforming based on uniform circular arrays is that the output pattern is rotationally symmetric in the azimuthal direction, and thus, the system can be equally fair in all looking directions from 0 to 360◦ . Traditionally, sound source localization problems have been approached mainly with planar-sparse arrays, and to some extent, with spherical arrays. Surprisingly, the use of circular arrays for this purpose has not been explored much in the literature. The work carried out during the present PhD project contributes to ﬁll this gap. Taking delay-and-sum beamforming as a reference, the present study has suggested several options to improve the performance of uniform circular arrays and extend their operative frequency range in order to cope with broadband sources. In all cases, it has been assumed that the acoustic sources where incoherent, static, and located in the far ﬁeld. The goal of Papers A to D is to improve the performance of uniform circular arrays in two different ways: by means of designing new processing techniques (Papers A, C, and D), and by means of changing the physical characteristics of the array (Paper B). The progress and ﬁndings of the PhD project can be seen as a journey toward the improvement of uniform circular arrays. Curiously, the existing literature on spherical 59 60 6. Conclusions arrays inspired the initial work on uniform circular arrays reﬂected in Papers A and B, while the outcome of Papers C and D showed to have potential with spherical arrays. In fact, Paper E closes this circle, by adapting the results of Paper D to the spherical geometry. The ﬁrst contribution of this thesis, presented in Paper A and complemented by Sec. 3 and Appendix A, is the derivation and examination of an eigenbeamforming method, circular harmonics beamforming, that results from adapting spherical harmonics beamforming, a method for spherical arrays, to the circular geometry. The outcome of this study shows that a slightly better performance can be achieved when the array is mounted on a rigid cylinder of inﬁnite length compared to the case where the microphones are simply suspended in the free space. Inspired by the properties of a theoretical rigid cylindrical scatterer of inﬁnite length, and motivated by the difﬁculty of its practical implementation, Paper B suggests the use of a rigid spherical bafﬂe, as its scattering behavior is, unlike the case of a rigid ﬁnite cylinder, well described analytically. On the other hand, Paper C suggests the use of deconvolution methods to improve the visualization of the beamformed maps and recover the levels of the impinging waves with accuracy. From a given map, these methods make use of iterative procedures to estimate the sound sources present in the sound ﬁeld. Since uniform circular arrays are shift-invariant, they can beneﬁt from those deconvolution methods that rely on a shiftinvariant point-spread function, thereby handling the inverse problem in the (spatial) Fourier domain to achieve a lower computational load. Paper C adapts, for the ﬁrst time, three methods originally conceived for planar-sparse arrays, namely, DAMAS2, FFT-NNLS, and Richardson-Lucy, to the circular geometry, and shows the potential of using eigenbeamforming, such as circular harmonics beamforming, as input to the deconvolution process. Finally, Papers D and E contribute to the current literature by suggesting a new method adequate for circular (Paper D) and spherical arrays (Paper E), based on the combination of beamforming with acoustic holography. Its principle relies on applying beamforming to a holographic virtual array with larger dimensions than the physical array to improve the performance at low frequencies. 6.2 Future work 61 The main contributions of this thesis to the existing literature are highlighted in the following: • Development, examination, and validation of circular harmonics beamforming, an eigenbeamforming technique for uniform circular arrays. • Extension of circular harmonics beamforming to uniform circular arrays mounted on a rigid spherical scatterer. • Adaptation of three deconvolution methods, namely DAMAS2, FFT-NNLS, and Richardson-Lucy, to uniform circular arrays, and examination and validation using both delay-and-sum and circular harmonics beamforming prior to the deconvolution process. • Development, examination, and validation of delay-and-sum beamforming with holographic virtual arrays for the improvement of the performance at low frequencies, for both circular and spherical arrays. • Derivation of the equations governing acoustic holography for circular arrays for interior domain problems. 6.2 Future work The ﬁndings of the PhD project have given rise to some questions and challenges that should be addressed in the near future. Deconvolution methods A natural continuation of the work done in the project on deconvolution methods is the extension of Fourier-based algorithms to the 3D case, using spherical arrays (with shift-invariant point-spread functions). The main challenge is to implement the deconvolution problem as a function of both the azimuth and the polar angle. Beamforming with holographic virtual arrays The study on beamforming using holographic virtual arrays has revealed that delayand-sum beamforming beneﬁts from using this method, but that is not the case of 62 6. Conclusions circular harmonics and spherical harmonics beamforming. Therefore, the method should be implemented with other beamforming techniques and examined to prove its generalization. Some preliminary studies not shown in this dissertation suggest that delay-and-sum implemented with holographic virtual arrays is more robust to noise than circular harmonics and spherical harmonics beamforming. Furthermore, it seems that at some frequencies, both the resolution and the MSL are better. It is therefore necessary to compare the methods thoroughly, making use of the usual performance indicators. New technology The ﬁndings of the thesis have been focused toward the enhancement of beamforming at low frequencies. However, if the aim was to improve the performance at high frequencies, this could be achieved by adding more transducers, as this would lower the spacing between transducers, and hence, increase the Nyquist frequency. However, with conventional microphones, this solution is usually not viable, as systems become way too expensive. One alternative would be to use microelectromechanical (MEMS) microphones, as they are small, and, more importantly, cheap. Despite the fact that, at the moment, MEMS microphones are far from being as stable as conventional microphones, this technology is still developing and has very good prospects. Advances in MEMS technology will, for sure, lead to very attractive array systems. A completely different approach could make use of acoustic ﬁbers∗ , as these allow to measure the sound ﬁeld in all the points of the ﬁber. This technology is based on sending an optical pulse into the ﬁber, and awaiting the reﬂections scattered back from the ﬁber glass walls. By measuring the time lag between the signal sent and the reﬂections received, the acoustic signal is extracted. Although at the moment acoustic ﬁbers are only used for measuring sound pressures, they also seem adequate for beamforming purposes. Since ﬁbers allow to scan a sound ﬁeld in a continuous manner, beamformers that made use of this technology would be able to provide maps free of aliasing. In this sense, a single ﬁber shaped in the form of a ring would be enough to build a continuous circular beamformer. ∗ Such Silixa. as the iDASTM (‘intelligent distributed acoustic sensor’), manufactured by the British company Appendix A Insight into circular harmonics beamforming According to Paper A, the starting point to develop circular harmonics beamforming is that, in the presence of a single source in the far ﬁeld, the ideal beamforming output should be a delta function located at the angular position of the source ϕs bideal (ϕ, ϕs ) = Bδ(ϕ − ϕs ), (A.1) where B is a scale factor. Due to the circular geometry, the beamforemer output is expansible in a set of circular harmonics, ∞ bideal (ϕ, ϕs ) = In ejnϕ , (A.2) n=−∞ where In is the nth Fourier coefﬁcient obtained with an ideal beamformer due to a source located at ϕs . The Fourier coefﬁcients are given by the inverse Fourier series 1 In = 2π 2π 0 bideal (ϕ, ϕs )e−jnϕ dϕ = Be−jnϕs . (A.3) Insertion of Eq. (A.3) into Eq. (A.2) yields ∞ bideal (ϕ, ϕs ) = B n=−∞ 63 ejn(ϕ−ϕs ) . (A.4) 64 Appendix A: Insight into circular harmonics beamforming In order to implement this expression, the number of modes of the Fourier series needs to be truncated at N , N ejn(ϕ−ϕs ) . (A.5) b(ϕ, ϕs ) = B n=−N By making use of trigonometric identities, this equation can be rewritten as b(ϕ, ϕs ) = B sin ((N + 1/2)(ϕ − ϕs )) . sin ((ϕ − ϕs )/2) (A.6) Inspection of this expression reveals that when N → ∞ the output becomes a delta function, as [130] 1 sin((N + 1/2)x) = δ(x), (A.7) lim N →∞ 2π sin(x/2) which, in effect, agrees with the starting point of this derivation. Alternatively, the expression given in Eq. (A.6) can be expressed using the Chebyshev polynomial of second kind, Un (cos θ) = sin ((n + 1)θ) . sin θ (A.8) Therefore, with n = 2N and θ = (ϕ − ϕs )/2, b(ϕ, ϕs ) = BU2N (cos((ϕ − ϕs )/2)). (A.9) One of the main characteristics of circular harmonics beamforming is that the output depends on the number of harmonics taken into account in the calculation. In addition, for a given number of harmonics, the output is rather independent of whether the circular array is mounted or not into a rigid bafﬂe. Note that the number of harmonics used in the algorithm depends on the frequency and the radius of the array, R, as well as the number of microphones, M , as N = kR up to a maximum order equal to M/2 − 1 [77, 78]. Figure A.1 shows the normalized beamforming output for different values of N . In this case, a source is simulated at 180◦ . As can be seen, the main lobe becomes narrower with increasing N , which agrees with the fact that, when N tends to inﬁnity, the output approaches a delta function centered at the angular position of the source. On the other hand, the resolution and the MSL decrease with increasing the number of orders, as shown in Figs. A.2 and A.3 as a function of the number of orders. 65 0 Maginitude of b [dB] −10 −20 −30 N N N N N −40 −50 0 30 60 90 =1 =2 =3 =4 =5 120 150 180 210 240 270 300 330 360 ϕ [◦ ] Figure A.1: Normalized output of a circular harmonics beamformer, for different values of N . A straight black line indicates −3 dB. 120 Resolution [◦ ] 100 80 60 40 20 1 2 3 N 4 5 Figure A.2: Resolution as a function of the number of orders taken into account in the calculation of circular harmonics beamforming. 66 Appendix A: Insight into circular harmonics beamforming −9.5 −10 MSL [dB] −10.5 −11 −11.5 −12 −12.5 −13 −13.5 1 2 3 N 4 5 Figure A.3: MSL as a function of the number of orders taken into account in the calculation of circular harmonics beamforming. Appendix B Acoustic holography with uniform circular arrays B.1 Open array Let us consider a uniform circular array of microphones placed at the xy−plane (z = 0) that captures a plane wave that travels perpendicularly to the z−axis, i.e., the wavefronts are parallel to to the z−axis. In what follows the time dependency e−jωt is omitted. After solving the Helmholtz equation in cylindrical coordinates and applying the boundary conditions (basically that the sound ﬁeld at the origin must be ﬁnite), the sound pressure results in p(kr, ϕ) = ∞ An Jn (kr)ejnϕ , (B.1) n=−∞ where An is an expansion coefﬁcient of order n. This expression can be used to determine the sound pressure at an arbitrary point of the sound ﬁeld by means of acoustic holography. For this purpose the values of the coefﬁcients An are needed. Since the pressure at the uniform circular array (at r = R) is known, p(kR, ϕ) = ∞ An Jn (kR)ejnϕ . (B.2) n=−∞ The coefﬁcients can be computed making use of the continuous orthogonality property of the circular harmonics given in Eq. (3.2) on page 34. After some rearranging, they 67 68 Appendix B: Acoustic holography with uniform circular arrays result in 2π An = 0 p(kR, ϕ)e−jnϕ dϕ . 2πJn (kR) (B.3) This expression implies a continuous integral of the sound pressure. However, the pressure is known at a number of discrete positions, as the sound ﬁeld is sampled with M microphones. Using the discrete orthogonality relationship of the circular harmonics given in Eq. (3.8) on page 35 the coefﬁcients An result in An = B.2 1 M M −1 i=0 p(kR, ϕi )e−jnϕi . Jn (kR) (B.4) Rigid cylindrical scatterer of inﬁnite length Let us now consider that the circular array is mounted on a rigid cylinder of inﬁnite length. The total pressure will present the contributions of the incident pressure and the scattered pressure: pt = pi + ps [131]. The incident sound pressure corresponds to that that would occur if the cylinder was not present, i.e., the pressure given in Eq. (B.1). Its associated radial velocity follows ∞ dJn (kr) jnϕ dpi (kr, ϕ) = e . An vkr,i (r, ϕ) ∝ dr dr n=−∞ (B.5) On the other hand, the scattered pressure follows from solving the Helmholtz equation in cylindrical coordinates considering an exterior boundary problem (applying the Sommerfeld radiation condition), ps (kr, ϕ) = ∞ Bn Hn(1) (kr)ejnϕ , (B.6) n=−∞ (1) where Bn is an expansion coefﬁcient and Hn (kr) is a Hankel function of the ﬁrst kind and order n. The associated radial velocity follows ∞ (1) dps (kr, ϕ) dHn (kr) jnϕ = e . Bn vkr,s (r, ϕ) ∝ dr dr n=−∞ (B.7) B.2 Rigid cylindrical scatterer of inﬁnite length 69 Imposing the boundary condition at the surface of the cylinder, that is, the total radial velocity is zero at r = R provides the relationship between An and Bn , ∞ n=−∞ (1) dJn (kr) dHn (kr) + Bn An dr r=R dr ejnϕ = 0. (B.8) r=R From this expression it follows that Bn = −An Jn (kR) (1) Hn (kR) , (B.9) (1) where Jn (kR) and Hn (kR) are the derivatives of the Bessel function and the Bessel function evaluated at r = R, respectively. Insertion of this relationship into the expression of the scattered pressure, Eq. (B.6), yields ∞ ps (kr, ϕ) = − An n=−∞ Jn (kR) (1) Hn (kR) Hn(1) (kr)ejnϕ . (B.10) Finally the total pressure is p(kr, ϕ) = ∞ An Jn (kr) − n=−∞ Jn (kR) (1) Hn (kR) Hn(1) (kr) ejnϕ . (B.11) Following the same procedure carried out in the previous section to determine the coefﬁcients An we obtain the following relationship: An = 1 M M −1 i=0 Jn (kR) − p(kR, ϕi )e−jnϕi (kR) Jn (1) Hn (kR) (1) Hn (kR) . (B.12) Inserting this expression into the total pressure provides the sound pressure at any other position. It should be emphasized that this solution is only valid if the plane waves propagate perpendicularly to the cylinder so that the pressure along the z-axis is constant and the wavefronts match perfectly the symmetry of the scatterer. 70 Appendix B: Acoustic holography with uniform circular arrays B.3 Rigid spherical scatterer In this section we consider a uniform circular array mounted on a rigid sphere. It can be shown that the pressure due to the incident waves and the scattered ones can be written in spherical coordinates as follows [82] ∞ n p(kr, θ, ϕ) = jn (kR) (1) hn (kr) (1) hn (kR) jn (kr) − Amn n=0 m=−n Ynm (θ, ϕ), (B.13) (1) where Amn is an expansion coefﬁcient of order mnth, jn (kr) and hn (kr) are the spherical Bessel and the spherical Hankel function of the ﬁrst kind and order n, and (1) jn (kR) and hn (kR) their derivatives with respect to r, evaluated at r = R. The total sound pressure on the surface of the sphere, i.e., at r = R, is p(kR, θ, ϕ) = ∞ n jn (kR) − Amn n=0 m=−n jn (kR) h(1) n (kR) (1) hn (kR) Ynm (θ, ϕ). (B.14) The coefﬁcients Amn can be found making use of the continuous orthogonality property of the spherical harmonics, 2π 0 π 0 Ynm (θ, ϕ)Yνμ∗ (θ, ϕ) sin θdθdϕ = δnν δmμ . (B.15) Following a similar procedure as in the case of the array mounted on an inﬁnitely-long bafﬂe, they result in 2π π Amn = 0 0 p(kR, θ, ϕ)Ynm∗ (θ, ϕ) sin θdθdϕ jn (kR) − (kR) jn (1) hn (kR) (1) hn (kR) . (B.16) Inspection of this equation reveals that the pressure in the entire sphere is needed for the computation of the coefﬁcients. This implies that the microphones are required to be distributed over the entire sphere, and not only on the equator, as in the case of the circular array. 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Humphreys, “A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays,” in Proc. of the 10th AIAA/CEAS Aeroacoustics Conference, Manchester, UK, 2004. [106] T. F. Brooks and W. M. Humphreys, “A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays,” J. Sound Vib., vol. 294, pp. 856–879, 2006. [107] R. P. Dougherty, “Extension of DAMAS and beneﬁts and limitations of deconvolution in beamforming,” in Proc. of the 11th AIAA/CEAS Aeroacoustics Conference (26th AIAA Aeroacoustics Conference), Monterey, California, 2005. [108] T. Yardibi, J. Li, P. Stoica, and L. N. Cattafesta III, “Sparsity constrained deconvolution approaches for acoustic source mapping,” J. Acoust. Soc. Am., vol. 123, no. 5, pp. 2631–2642, 2008. [109] P. Sijtsma, “CLEAN based on spatial source coherence,” Int. J. Aeroacoust., vol. 6, no. 4, pp. 357–374, 2007. [110] D. Fernandez-Comesana, E. Fernandez-Grande, E. Tiana-Roig, and K. R. Holland, “A novel deconvolution beamforming algorithm for virtual phased arrays,” in Proc. of Inter-Noise 2013, Innsbruck, Austria, 2013. Bibliography 81 [111] V. Fleury and J. Bult, “Extension of deconvolution algoritms for the mapping of moving acoustic sources,” J. Acoust. Soc. Am., vol. 129, no. 3, pp. 1417–1428, 2011. [112] T. F. Brooks and W. M. Humphreys, “Extension of DAMAS phased array processing for spatial coherence determination (DAMAS-C),” in Proc. of the 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, USA, no. AAIA–20062654, 2006. [113] T. Yardibi, J. Li, P. Stoica, N. S. Zawodny, and L. N. Cattafesta III, “A covariance ﬁtting approach for correlated acoustic source mapping,” J. Acoust. Soc. Am., vol. 127, no. 5, pp. 2920–2931, 2010. [114] C. Bahr and L. Cattafesta, “Wavespace-based coherent deconvolution,” in 18th AIAA/CEAS Aeroacoustics Conference (33rd AIAA Aeroacoustics Conference), Colorado Springs, USA, no. AAIA 2012-2227, 2012. [115] A. Xenaki, F. Jacobsen, E. Tiana-Roig, and E. Fernandez-Grande, “Improving the resolution of beamforming measurements on wind turbines,” in Proc. of ICA 2010, Sidney, Australia, 2010. [116] A. Xenaki, F. Jacobsen, and E. Fernandez-Grande, “Improving the resolution of three-dimensional acoustic imaging with planar phased arrays,” J. Sound Vib., vol. 331, no. 8, pp. 1939–1950, 2012. [117] T. Yardibi, N. S. Zawodny, C. Bahr, F. Liu, L. N.Cattafesta III, and J. Li, “Comparison of microphone array processing techniques for aeroacoustic measurements,” Int. J. Aeroacoust., vol. 9, no. 6, pp. 733–762, 2010. [118] Z. Chu and Y. Yang, “Comparison of deconvolution methods for the visualization of acoustic sources based on cross-spectral imaging function beamforming,” Mech. Syst. Signal Process., 2014. [119] J.-C. Pascal and J.-F. Li, “Resolution improvement of beamformers using spherical microphone array,” in Proc. of Acoustics’08 Paris, Paris, France, 2008. [120] A. Schmitt, L. Lamotte, and F. Deblauwe, “Source identiﬁcation inside cabin using inverse methods,” in Proc. of BeBec 2010, Berlin, Germany, 2010. [121] M. Legg and S. Bradley, “Comparison of CLEAN-SC for 2D and 3D scanning surfaces,” in Proc. of BeBec 2012, Berlin, Germany, 2012. [122] Q. Fu, M. Li, L. Wei, and D. Yang, “An improved method combining beamforming and acoustical holography for the reconstruction of the sound pressure on structure surface,” Acta Acust. united Ac., vol. 100, no. 1, pp. 166–183, 2014. 82 Bibliography [123] E. G. Williams and J. D. Maynard, “Holographic imaging without the wavelength resolution limit,” Phys. Rev. Lett., vol. 45, no. 7, pp. 554–557, 1980. [124] E. G. Williams, J. D. Maynard, and E. Skudrzyk, “Sound source reconstructions using a microphone array,” J. Acoust. Soc. Am., vol. 68, no. 1, pp. 340–344, 1980. [125] E. G. Williams, N. Valdivia, and P. C. Herdic, “Volumetric acoustic vector intensity imager,” J. Acoust. Soc. 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A. Gumerov, “Plane-wave decomposition of acoustical scenes via spherical and cylindrical microphone arrays,” IEEE Trans. Audio, Speech, Language Process., vol. 18, no. 1, pp. 2–16, 2010. [133] E. Fernandez-Grande and T. Walton, “Reconstruction of sound ﬁelds with a spherical microphone array,” in Proc. of Inter-Noise 2014, Melbourne, Australia, 2014. Papers A-E 83 85 Errata list The following typos have been detected in the contributing articles after publication: Paper A The index q in the second sum of Eq. (14) should read h. Paper B The function bn (kR) in Eq. (7) should read bq (kR). Paper D The parameter wi in Eq. (8) should read wm . Paper E The function p(kR, θi , ϕi ) in Eq. (17) should read p̃(kR, θi , ϕi ). Paper A Beamforming with a circular microphone array for localization of environmental noise sourcesa) Elisabet Tiana-Roig,b) Finn Jacobsen, and Efrén Fernández Grande Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark, Ørsteds Plads 352, 2800 Kongens Lyngby, Denmark (Received 11 May 2010; revised 7 September 2010; accepted 13 September 2010) It is often enough to localize environmental sources of noise from different directions in a plane. This can be accomplished with a circular microphone array, which can be designed to have practically the same resolution over 360 . The microphones can be suspended in free space or they can be mounted on a solid cylinder. This investigation examines and compares two techniques based on such arrays, the classical delay-and-sum beamforming and an alternative method called circular harmonics beamforming. The latter is based on decomposing the sound ﬁeld into a series of circular harmonics. The performance of the two signal processing techniques is examined using computer C 2010 Acoustical Society of America. simulations, and the results are validated experimentally. V [DOI: 10.1121/1.3500669] PACS number(s): 43.60.Fg, 43.50.Rq [EJS] Acoustical beamforming is a signal processing technique used to localize sound sources using microphone arrays. Unlike other array techniques such as statistically optimized near-ﬁeld acoustical holography (SONAH), which are based on near-ﬁeld measurements,1,2 beamforming is based on farﬁeld measurements, i.e., the array must be placed relatively far from the sources in order to determine their “position” by processing the signals captured by the microphones.3 The goal of the present work is the design of beamformers for localization of environmental noise sources. In outdoors measurements, the sound ﬁeld is basically generated by sources placed far from the measurement point, in the far ﬁeld. At the measurement point, the direction of propagation of the waves can be considered essentially parallel to the ground, which implies that the sound ﬁeld can be assumed to be two-dimensional. For such purposes, it is suitable to use circular arrays as these are able to map the sound ﬁeld over 360 . The techniques developed for the circular geometry are delay-and-sum beamforming (DSB) and circular harmonics beamforming (CHB). The ﬁrst technique is the classical beamforming technique, which is widely used since it is very robust in the presence of background noise.4 By contrast, CHB is a novel technique that belongs to a more recent category called eigenbeamforming. All techniques in this group are based on decomposing the sound ﬁeld into a summation of harmonics.5–8 CHB has been developed by adapting the theory of spherical harmonics beamforming to the two-dimensional case using circular harmonics (CH). a) Portions of this work were presented in “Beamforming with a circular microphone array for localization of environmental sources of noise,” Proceedings of Inter-Noise 2010, Lisbon, Portugal, June 2010. b) Author to whom correspondence should be addressed. Electronic mail: [email protected] J. Acoust. Soc. Am. 128 (6), December 2010 II. DECOMPOSITION OF THE SOUND FIELD USING CH A. Circular apertures Consider a circular aperture of radius R in the xy-plane and a plane wave with amplitude P0 that impinges on the aperture in a direction perpendicular to the z-axis in free space. The incident pressure at any point of the aperture can be written in polar coordinates, pðkR; uÞ ¼ P0 ejki r r¼R ¼ P0 ejkR cosðuui Þ ; (1) where ki and ui are the wave number vector and the angle of the incident wave. The temporal term ejxt has been suppressed. This expression can be expanded in series of circular waves,9 ejkR cosðuui Þ ¼ J0 ðkRÞ þ 1 X 2jn cosðnðu ui ÞÞ Jn ðkRÞ; n¼1 (2) where Jn is a Bessel function of order n. Developing this expression further, the pressure of the incident plane wave becomes pðkR; uÞ ¼ P0 1 X jn Jn ðkRÞejnðuui Þ : (3) n¼1 The pressure can now be represented by an inﬁnite number of CH ejnu (or modes) using the principle of a Fourier series. The pressure on the (unbafﬂed) aperture can be expressed as a function of the angle of the source us using the relationship ui ¼ us þ p, pðkR; uÞ ¼ P0 0001-4966/2010/128(6)/3535/8/$25.00 1 X ðjÞn Jn ðkRÞejnðuus Þ : (4) n¼1 C 2010 Acoustical Society of America V 3535 Author's complimentary copy I. INTRODUCTION Pages: 3535–3542 When the same aperture is mounted on a rigid, inﬁnite cylinder, the incident wave is scattered by the cylinder. The pressure on the bafﬂed aperture is the superposition of the incident pressure and the scattered pressure, p ¼ pi þ ps. The scattered pressure at positions on the aperture becomes10 ps ðkR; uÞ ¼ 1 X An cosðnuÞðJn ðkRÞ þ jYn ðkRÞÞ; (5) n¼0 where the terms An are a set of coefﬁcients and Yn is a Neumann function of order n. Making use of the Hankel functions of ﬁrst kind, Hn() ¼ Jn() þ jYn(), the previous expression can be rewritten as ps ðkR; uÞ ¼ 1 X Bn Hn ðkRÞejnu : (6) n¼1 The terms Bn are obtained by imposing that the total velocity in the radial direction vanishes on the surface of the rigid cylinder, ui,r þ us,r ¼ 0, Bn ¼ P0 ðjÞn Jn0 ðkRÞ jnus e ; Hn0 ðkRÞ (7) where Jn0 and Hn0 are the derivatives of the Bessel and Hankel functions with respect to the radial dimension. Using the expressions given in Eqs. (4) and (6) for the incident and the scattered wave, together with the coefﬁcients obtained in Eq. (7), the total pressure at the surface of the rigid cylinder becomes pðkR; uÞ ¼ P0 1 X ðjÞn n¼1 J 0 ðkRÞHn ðkRÞ jnðuus Þ Jn ðkRÞ n 0 : e Hn ðkRÞ (8) Comparing Eqs. (3) and (8) with a Fourier series in the exponential form11 shows that the pressure on the bafﬂed or the unbafﬂed apertures can be represented as pðkR; uÞ ¼ 1 X Cn ejnu ; FIG. 1. (Color online) Normalized modulus of the four lowest Fourier coefﬁcients of the pressure on an unbafﬂed circular aperture (top) and on a circular aperture mounted on a rigid cylindrical bafﬂe of inﬁnite length (bottom). offset by 6 dB compared with the unbafﬂed case. With increasing values of kR, more and more harmonics gain strength. However, for the unbafﬂed aperture the response exhibits some dips that imply that signals that have components around these dips cannot be totally resolved. This problem disappears when the cylindrical bafﬂe is used. Since the curves of the Fourier coefﬁcients are functions of kR, variation of R implies that the curves are scaled in frequency (or wave number), and vice-versa. For instance, when R is increased, the response is shifted toward low frequencies, whereas a decrease of R results in a shift toward high frequencies. (9) n¼1 B. Implementation using microphone arrays where the Fourier coefﬁcients Cn for the two cases are Cn ðkR; us Þ ¼ P0 Qn ðkRÞejnus ; (10) In principle, inﬁnitely many Fourier terms are needed to represent the sound pressure. However, in practice the number of harmonics must be truncated to a maximum order, N. As a rule of thumb, with n The modulus of the ﬁrst four coefﬁcients Cn is shown in Fig. 1, for bafﬂed and unbafﬂed apertures. At low values of kR, the zero order mode is constant and equals 0 dB in both cases, whereas all the other modes have a slope of 10 n dB per decade. When the aperture is bafﬂed, the response is 3536 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 N kR (12) is usually chosen as a ﬁrst approximation.8,12,13 The reason for this is that the amplitude of the Bessel functions in the Fourier coefﬁcients [see Eqs. (10) and (11)] is small when the order of the Bessel functions n exceeds its argument (kR). Thus, the overall contribution of modes jnj > kR is very small. Besides, microphone arrays rather than “ideal” continuous apertures are used in real-life applications, which implies Tiana-Roig et al.: Beamforming with a circular microphone array Author's complimentary copy 8 < ðjÞn Jn ðkRÞ unbaffled; (11) Qn ðkRÞ ¼ 0 n ðkRÞ : ðjÞn Jn ðkRÞ Jn ðkRÞH baffled: H 0 ðkRÞ that apertures are sampled at discrete points. Assuming that an aperture is sampled with M omnidirectional microphones placed equidistantly, the Fourier coefﬁcients become In ¼ 1 2p ð 2p bideal ðuÞejnu du ¼ Aejnus : (19) 0 It follows that M 1X p~ðkR; um Þejnum ; C~n ¼ M m¼1 (13) bideal ðuÞ ¼ A where p~ is the measured pressure at the mth microphone placed at an angle um. The sampling procedure introduces an error in the Fourier coefﬁcients. For example, it can be shown that in the case of an unbafﬂed circular array, the Fourier coefﬁcients resulting after the sampling are, theoretically,12–14 C~n ðkRÞ ¼ P0 ðjÞn Jn ðkRÞejnus 1 X þ P0 ðjÞg Jg ðkRÞejgus 1 X ejnus ejnu : (20) n¼1 Using Eq. (10), the output of the ideal beamformer becomes bideal ðkR; uÞ ¼ A 1 X Cn ðkR; us Þ jnu e : P Q ðkRÞ n¼1 0 n (21) In real implementations, the number of modes must be truncated at a reasonable value, N, and the aperture must be sampled by a number of microphones, M. Thus q¼1 ðjÞh Jh ðkRÞejhus ; (14) q¼1 where g ¼ Mq n and h ¼ Mq þ n. Note that the ﬁrst term is identical to the Fourier coefﬁcient of the continuous aperture; see Eq. (10), whereas the remaining terms are residuals caused by the sampling. Further examination of Eq. (14) reveals that the ﬁrst term is the dominant one when M > 2jnj. Since the highest mode excited is N, M > 2N: (15) In fact, inserting the approximation for N given in Eq. (12) into Eq. (15) yields the Nyquist sampling criterion: M > 2kR ) M>2 2p R k ) k > d; 2 (16) where k is the wavelength and d is the distance between two consecutive microphones. Hence, by fulﬁlling the relationship between M and N given in Eq. (15), the Nyquist criterion is satisﬁed.14 III. BEAMFORMING TECHNIQUES A. CHB The beamformer response is the output of the beamformer as a function of the steering angle, i.e., the angle at which the main beam of the beamformer is pointing. Ideally, the beamformer response should assume a maximum when the beamformer is steered toward the source at us, and should be zero in all other directions; that is, bideal ðuÞ ¼ Adðu us Þ; (17) where A is a scale factor. This can be described in terms of a Fourier series, bideal ðuÞ ¼ 1 X In ejnu ; n¼1 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 (18) bN;CH ðkR; uÞ ¼ A N X C~n ðkR; us Þ jnu e : P Q ðkRÞ n¼N 0 n (22) Comparing with Eq. (10), Eq. (22) be rewritten as bN;CH ðkR; uÞ ¼ A N X C~n ðkR; us Þ : C ðkR; uÞ n¼N n (23) When the beamformer is steered toward the position of the source, u equals us, so the quotients approximate unity and the output assumes a maximum. Note that, when using unbafﬂed arrays, Eq. (23) has singularities at the frequencies where the Fourier coefﬁcients have dips; see Fig. 1. At such frequencies, the CH beamformer is not capable of resolving the location of the source properly. Inserting the approximated coefﬁcients given by Eq. (13) into Eq. (22), the CH beamformer output becomes bN;CH ðkR; uÞ ! M N X A X 1 jnðum uÞ e : ¼ p~ðkR; um Þ MP0 m¼1 Q ðkRÞ n¼N n (24) Ideally, this should be zero at all angles different from us. However, since a limited number of microphones are used, the response exhibits a main lobe around us and side lobes at other angles. B. DSB The delay-and-sum (DS) technique aligns the signals from the microphones of the array by introducing appropriate delays and ﬁnally adds them together.3,15,16 The delays are determined by the steering direction of the array. The output assumes its maximum when the focusing direction coincides with the position of the source. In this investigation, the output of a DS beamformer is implemented in the frequency domain using matched ﬁeld Tiana-Roig et al.: Beamforming with a circular microphone array 3537 Author's complimentary copy þ P0 1 X processing. This method uses phase shifts to align the signals in phase. Assuming that the beamformer is steered toward the direction u, the beamformer output is M X wm p~ðkR; um Þ p ðkR; um ; uÞ; (25) m¼1 C. Beamformer performance—Resolution and maximum side lobe level (MSL) where (1) wm is the weighting coefﬁcient of the mth microphone; (2) p~ðkR; um Þ is the pressure measured at the mth microphone position due to a plane wave generated by a source at us; and (3) p (kR,um,u) is the theoretical complex conjugated pressure that would be captured at the mth microphone due to plane wave generated at u. Note that the argument u is used to emphasize that this is the variable that deﬁnes the focusing direction of the beamformer. In general, the source position is unknown, and therefore the beamformer must map over all possible source positions, i.e., 0 u < 2p. The key point is that when the beamformer is focused toward the position of the source us, the second and the third terms of Eq. (25) become equal in magnitude but opposite in phase. In these circumstances, the microphone signals are aligned in time, and therefore the maximum output of the beamformer is reached. In the case of an unbafﬂed array, the theoretical pressure is simply the closed form for a plane wave, so the beamformer output is bDS ðkR; uÞ ¼ A M X wm p~ðkR; um ÞP0 e m¼1 ¼ M AP0 X M jki r r¼R p~ðkR; um ÞejkR cosðum uÞ : (27) This expression is also valid for the unbafﬂed case, although it is not as precise as Eq. (26) because of the truncation. Further analysis of Eq. (27) reveals that the beamformer output can be written, according to Eqs. (10) and (13), as C~n ðkR; us Þ Cn ðkR; uÞ: (28) n¼N As opposed to CHB, where the beamformer output could be expressed as the ratio of the approximated coefﬁcients to 3538 The performance of circular arrays with CHB and DSB has been evaluated by means of simulations. The circular arrays have radii of 10 and 20 cm and 10 and 20 microphones, respectively. The number of microphones and the radius of each array were chosen by setting the same maximum frequency that could be represented without any sampling error (around 2.7 kHz); see Sec. II B. The simulations were carried out under ideal conditions, i.e., without background noise. The source was placed at 180 , but the source position has a very limited inﬂuence on the results. The amplitude of the waves impinging on the array was the same at all frequencies. A. Simulations with CHB bN;DS ðkR; uÞ ! M N X AP0 X jnðum uÞ ¼ Qn ðkRÞe p~ðkR; um Þ : M m¼1 n¼N N X IV. SIMULATIONS (26) m¼1 Since all microphones have equal “importance,” the weights wm have been set to 1/M. For the bafﬂed array, the output of the beamformer is obtained by introducing Eq. (9) into Eq. (25), and taking into account that the number of modes used for the processing is truncated at a number N, bN;DS ðkR; uÞ ¼ A The resolution of a beamformer is deﬁned as the 3 dB width of the main lobe of the beampattern. This parameter is of interest because it gives an approximation to the minimum angular difference between two incoherent sources that is necessary in order to distinguish them from each other. The beamformer output will usually exhibit side lobes. This is an unwanted effect as the beamformer seems to be sensitive not only in the focusing direction but also in the direction of the side lobes. Therefore, it is convenient to evaluate the beamformer response by means of the MSL. This parameter is the difference in level between the peak of the highest side lobe and the peak of the main lobe. J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 The resolution and the MSL obtained with CHB and an unbafﬂed circular array are shown in Fig. 2, for the range from 50 Hz to 3.5 kHz. The number of orders used in the CHB algorithm given in Eq. (24) followed N ¼ dkRe, where de is the ceiling function. The maximum value of N was M/2 1 in order to fulﬁll Eq. (15). As can be seen, the resolution and the MSL are constant for a certain interval. This depends on the number of orders N used for the processing. The fact that the main lobe becomes narrower from interval to interval indicates an improvement in the resolution. More intervals, i.e., more orders, result in a better resolution, which is the case of the array of largest radius. The MSL follows the same behavior as the resolution, improving when the number of orders is increased. The staircase pattern in these two measures is also obtained with spherical harmonics beamforming. At some frequencies an “unexpected” response occurs, e.g., around 2.1 kHz for the array of 10 cm and 2.8 kHz for the array of 20 cm. This phenomenon is due to the dips in the Fourier coefﬁcients obtained with unbafﬂed arrays. The frequencies where this phenomenon occurs cannot be resolved as precisely as the neighboring frequencies. This Tiana-Roig et al.: Beamforming with a circular microphone array Author's complimentary copy bDS ðkR; uÞ ¼ A the theoretical ones [see Eq. (23)], the output of the DS beamformer is a multiplication of these terms. Therefore, in the case of unbafﬂed arrays, the singularities that can be present in CHB because of the dips of the Fourier coefﬁcients are totally resolved with a DS beamformer. FIG. 2. Resolution and MSL using CHB and unbafﬂed arrays of radii 10 and 20 cm and 10 and 20 microphones, respectively. The source is placed at 180 . FIG. 3. Resolution and MSL using DSB and circular arrays of radius 10 and 20 cm and 10 and 20 microphones, respectively. Solid lines: Unbafﬂed arrays; dashed lines: Bafﬂed arrays. The source is placed at 180 . effect is avoided when the array is mounted on a rigid cylindrical bafﬂe. The overall behavior of the CH beamformers when bafﬂed arrays are used is very similar to the unbafﬂed case but without the problem of unresolved frequencies. The arrays can be used up to a maximum frequency without any sampling error. For the arrays under analysis, this occurs at about 2.7 kHz. Above this frequency, the effect of the sampling error can be seen especially in the MSL (the magnitude of the side lobes is higher than in the previous interval of frequencies). Furthermore, it can be seen that the bafﬂed array of 10 cm of radius has resolution and MSL similar to the unbafﬂed array of radius 20 cm. Thus it can be concluded that mounting an array on an inﬁnite bafﬂe makes it seem to be “larger” than in the unbafﬂed case. Similar characteristics are found when DSB is applied to spherical arrays. In general the resolution obtained with DSB is much worse than the resolution obtained with CHB. At high frequencies, the MSL with DSB is worse than with CHB for unbafﬂed arrays; but the opposite is the case for bafﬂed arrays. B. Simulations with DSB J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 V. EXPERIMENTAL RESULTS A prototype array with a radius of 11.9 cm has been tested in an anechoic room with a volume of about 1000 m3. The array was constructed by mounting twelve 1/4 in. microphones, Brüel & Kjær (B&K) Type 4935, on a circular frame, corresponding to a microphone for every 30 . The implemented prototype is shown in Fig. 4. The array and the source, a loudspeaker, were controlled by a B&K PULSE Analyzer. In all the measurements the loudspeaker was driven by a signal from the generator, pseudorandom noise of 1 s of period, 3.2 kHz of bandwidth, and 1 Hz of resolution. The microphones signals were recorded with the analyzer and postprocessed with the beamforming algorithms DSB and CHB. The normalized outputs obtained with both CHB and DSB are shown on top of Fig. 5, whereas the simulated outputs are provided in the bottom. To account for the background noise introduced in the measurements, the simulations were carried out with a signal-to-noise ratio (SNR) of 30 dB at the input of each microphone due to uniformly distributed noise. Tiana-Roig et al.: Beamforming with a circular microphone array 3539 Author's complimentary copy The resolution and the MSL obtained under ideal conditions using DSB are shown in Fig. 3 for both bafﬂed and unbafﬂed arrays. In the case of bafﬂed arrays, the number of orders used in the DSB algorithm, stated in Eq. (27), was N ¼ dkRe þ 1, up to a maximum N ¼ M/2 1 according to Eq. (15). It is apparent that the resolution is 360 at low frequencies in all cases, and the MSL is non-existent, meaning that the beamformer is omnidirectional. From a certain frequency depending on the radius of the array, the resolution improves continuously until high frequencies. The curves decay in a similar way for both kinds of arrays, but in the bafﬂed case they exhibit small smooth ﬂuctuations. The MSL curves begin at a certain frequency and grow progressively until a maximum level is reached. In the case of unbafﬂed arrays, this level remains constant, whereas for bafﬂed arrays the MSL exhibits ripples while it increases toward high frequencies. Nevertheless, the MSL is better for bafﬂed arrays than for unbafﬂed. In both cases, the performance improves with increasing radius of the array and is better in the case of bafﬂed arrays. The results agree very well with the theoretical ones for both techniques. A few differences deserve to be mentioned in the case of CHB. The side lobes are somewhat deformed and blurred compared with the simulations. The output is not only distorted at the frequencies that coincide with the dips FIG. 5. Normalized output using CHB [(a), (b)] and DSB [(c), (d)] and an unbafﬂed array with a radius of 11.9 cm with 12 microphones. The source is placed at 180 . The top panels (a) and (c) show the measurements performed with a prototype, whereas the corresponding simulation is presented in the bottom panels (b) and (d). For the simulation, an SNR of 30 dB in each microphone is considered. 3540 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 Tiana-Roig et al.: Beamforming with a circular microphone array Author's complimentary copy FIG. 4. (Color online) Circular array with radius of 11.9 cm and 12 microphones. Prototype by Brüel & Kjær. in the Fourier coefﬁcients, but also at frequencies in their vicinity. This phenomenon is particularly pronounced around 1.7 kHz. These differences are suspected to be caused by the CH beamformer algorithm itself, because of the fact that the approximated Fourier coefﬁcients are compared with the theoretical ones in a ratio. When the approximate coefﬁcients match the theoretical ones, the beamformer output is similar to the pattern expected under ideal conditions. The agreement between measurements and simulations can be further examined by studying the resolution and the MSL. These quantities are shown in Fig. 6. The resolution using CHB is very similar to the one obtained with the simulation. The response follows the simulation curve rather accurately even at the frequencies where singularities occur. Some small deviations can be observed at the lowest frequencies, which are attributed to the inﬂuence of background noise. In contrast to the resolution, the MSL deviates somewhat from the simulation. In general, this measure is slightly higher than the expected one and worsens near singularities. At frequencies below 100 Hz a signiﬁcant inﬂuence of background noise in the measurement becomes apparent. For DSB, there are only a few differences compared with the simulations. The ﬁrst is that the resolution equals 360 up to a frequency 20 Hz higher than expected. The second difference is that the ﬁrst side lobe appears at 634 Hz instead of 556 Hz as obtained in the simulation. Yet another difference is that MSL is better than expected in the range from 1950 Hz to about 2300 Hz. These differences are again mainly attributed to the differences between the measured pressure and the theoretical one. However, the beampattern as well as the resolution and the MSL are not as much affected by these differences as in the case of CHB. This characteristic FIG. 6. Resolution and MSL using CHB (left) and DSB (right) and an unbafﬂed array with a radius of 11.9 cm with 12 microphones. The source is placed at 180 . The theoretical case obtained by simulation is also shown. The SNR in each microphone is set to 30 dB for the simulation. VI. CONCLUSIONS Two different beamforming techniques based on circular arrays have been examined theoretically and experimentally, CHB and the well-known DSB. CHB is an adaptation of the spherical harmonics beamforming technique to a circular geometry. The prototype used for the experimental investigation gave very satisfactory results: The beampatterns, the resolution, and the MSL were found to be in extremely good agreement with simulations for both CHB and DSB. For a given array, CHB has better resolution and lower MSL in a wider frequency range than DSB has. Regardless of the technique, these quantities improve with increasing frequency. The frequency range is limited at low frequencies by the inﬂuence of background noise in the case of CHB and by the fact that the output becomes omnidirectional for DSB. At high frequencies, the limitation is in both cases given by the increase of the sampling error. Keeping the number of microphones constant, the beamformer response is scaled in frequency when the radius of the array is modiﬁed. However, when the radius of the array is kept constant but the number of microphones is increased, the response improves toward higher frequencies since the spacing between the microphones becomes smaller. In fact, by increasing the number of microphones, the array behaves more similarly to a continuous aperture. A given ratio between the number of microphones and the radius of the array determines the upper frequency above which a sampling error occurs. In such case, the overall performance improves considerably toward lower frequencies J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 when increasing the radius. However, the number of microphones should be increased accordingly, otherwise the upper limit frequency would be reduced. In the presence of background noise, DSB is more robust than CHB, and CH beamformers exhibit singularities, i.e., frequencies that cannot properly be resolved, when unbafﬂed circular arrays are used. This problem would be solved if it were feasible to mount the arrays on rigid cylindrical bafﬂes of inﬁnite length. The performance of DS beamformers would also improve substantially by mounting the arrays on rigid cylindrical bafﬂes of inﬁnite length, but this is not realistic. CHB can be used in the entire frequency range except at the frequencies that cannot be properly resolved due to the nature of this technique. At such frequencies, it is convenient to use DSB instead. In addition to this, DSB should not be underestimated in environments with a poor SNR because of its robustness. ACKNOWLEDGMENTS The authors would like to thank Karim Haddad and Jørgen Hald, Brüel & Kjær, for lending us a circular microphone array and other equipment for the beamforming measurements. We would also like to thank Julien Jourdan and Marton Marschall for their notes about spherical harmonics beamforming, which became very helpful for the present work. 1 J. Maynard, E. Williams, and Y. Lee, “Nearﬁeld acoustic holography: I. Theory of generalized holography and the development of NAH,” J. Acoust. Soc. Am. 78(4), 1397–1413 (1985). 2 J. Hald, “Basic theory and properties of statistically optimized near-ﬁeld acoustical holography,” J. Acoust. Soc. Am. 125(4), 2105–2120 (2009). 3 J. Hald, “Beamforming and wavenumber processing,” in Handbook of Signal Processing in Acoustics, edited by D. Havelock, S. Kuwano, and M. Vorländer (Springer, New York, 2008), Chap. 9, pp. 131–144. Tiana-Roig et al.: Beamforming with a circular microphone array 3541 Author's complimentary copy and the fact that the inﬂuence of background noise is lower than CHB demonstrate that DSB is a more robust algorithm. J. Bitzer and K. Uwe Simmer, “Superdirective microphone arrays,” in Microphone Arrays. Signal Processing Techniques and Applications, edited by M. Brandstein and D. Ward (Springer, Berlin, 2001), Chap. 2, p. 26. 5 B. Rafaely, “Plane-wave decomposition of the sound ﬁeld on a sphere by spherical convolution,” J. Acoust. Soc. Am. 116(4), 2149–2157 (2004). 6 W. Song, W. Ellermeier, and J. Hald, “Using beamforming and binaural synthesis for the psychoacoustical evaluation of target sources in noise,” J. Acoust. Soc. Am. 123(2), 910–924 (2008). 7 H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on waveﬁeld decomposition using circular microphone arrays,” J. Acoust. Soc. Am. 120(5), 2724–2736 (2006). 8 H. Teutsch, Modal Array Signal Processing: Principles and Applications of Acoustic Waveﬁeld Decomposition (Springer, Berlin, 2007), pp. 150– 188. 9 P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, p. 828. 3542 J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010 10 P. Morse, Vibration and Sound, 2nd ed. (McGraw-Hill, New York, 1948), pp. 347–348. E. Williams, Fourier Acoustics: Sound Radiation and Nearﬁeld Acoustic Holography (Academic Press, London, 1999), pp. 4–5. 12 C. Mathews and M. Zoltowski, “Eigenstructure techniques for 2-D angle estimation with uniform circular arrays,” IEEE Trans. Signal Process. 42(9), 2395–2407 (1994). 13 D. E. N. Davies, “Circular arrays,” in The Handbook of Antenna Design, edited by A. W. Rudge, K. Milne, A. D. Olver, and P. Knight (Peter Peregrinus Ltd., London, 1983), Vol. II, Chap. 12, pp. 298–310. 14 H. Van Trees, Optimum Array Processing. Part IV of Detection, Estimation, and Modulation Theory (Wiley, New York, 2002), pp. 280–284. 15 D. Johnson and D. Dudgeon, Array Signal Processing Concepts and Techniques (Prentice Hall, Englewood Cliffs, NJ, 1993), pp. 112–119. 16 G. Elko and J. Meyer, “Microphone arrays,” in Springer Handbook of Speech Processing, edited by J. Benesty, M. Sondhi, and Y. Huang (Springer-Verlag, Berlin, 2008), Chap. 50, pp. 1021–1042. 11 Tiana-Roig et al.: Beamforming with a circular microphone array Author's complimentary copy 4 Paper B Beamforming with a circular array of microphones mounted on a rigid sphere (L) Elisabet Tiana-Roig,a) Finn Jacobsen, and Efren Fernandez-Grande Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark, Ørsteds Plads 352, 2800 Kongens Lyngby, Denmark (Received 4 November 2010; revised 6 July 2011; accepted 7 July 2011) Beamforming with uniform circular microphone arrays can be used for localizing sound sources over 360 . Typically, the array microphones are suspended in free space or they are mounted on a solid cylinder. However, the cylinder is often considered to be inﬁnitely long because the scattering problem has no exact solution for a ﬁnite cylinder. Alternatively one can use a solid sphere. This investigation compares the performance of a circular array mounded on a rigid sphere with that of such an array in free space and mounted on an inﬁnite cylinder, using computer simulations. The examined techniques are delay-and-sum and circular harmonics beamforming, and the results are C 2011 Acoustical Society of America. [DOI: 10.1121/1.3621294] validated experimentally. V PACS number(s): 43.60.Fg [EJS] Pages: 1095–1098 I. INTRODUCTION II. PLANE WAVE DECOMPOSITION Consider a plane wave, ejki r , generated by a source placed in the far ﬁeld, at a polar angle hs and azimuth angle us , that impinges on a rigid sphere with radius R. The pressure on the surface of the sphere, at a point with spherical coordinates ½R; h; u, can be written as5,6 pðkR; h; uÞ ¼ 4p 1 X q¼0 bq ðkRÞ q X bq ðkRÞ ¼ ðjÞ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2q þ 1Þ ðq nÞ! n Yqn ðh; uÞ P ðcos hÞejnu : 4p ðq þ nÞ! q 1 Cn ¼ 2p ð 2p J. Acoust. Soc. Am. 130 (3), September 2011 pðkR; p=2; uÞejnu du: (4) Inserting Eq. (1), it can be shown that the coefﬁcients become 1 X ð2q þ 1Þbn ðkRÞ q¼jnj (1) n¼q Author to whom correspondence should be addressed. Electronic mail: [email protected] dtu.dk (3) 0 where a) (2) In the function bn , which accounts for the effect of the rigid scatterer, jq is a spherical Bessel function of order q, hq is a spherical Hankel function of ﬁrst kind and order q, and j0q and h0q are their derivatives. On the other hand, Yqn is a spherical harmonic, in which Pnq is a Legendre function of degree q and order n. Note that in Eq. (1) the temporal term ejxt is omitted; and the angles of the position of the source ½hs ; us are used instead of the angles of the incident wave ½hi ; ui , these being related by hs ¼ p hi and us ¼ ui þ p because ^i is opposite to the unit the unit vector of the incident wave k ^i ¼ ^rs . vector of the position of the source ^rs , k Now a circular aperture of radius R is mounted at the equator of the rigid sphere, in the xy plane. Because the polar angle at all positions of the aperture is constant, i.e., h ¼ p=2, its pressure can be represented in a Fourier series in the u coordinate.5 The resulting Fourier coefﬁcients are Cn ðkRÞ ¼ Yqn ðh; uÞYqn ðhs ; us Þ ; ! j0q ðkRÞ hq ðkRÞ ; jq ðkRÞ 0 hq ðkRÞ ðq jnjÞ! jnj P ð0ÞPjqnj ðcos hs Þejnus : ðq þ jnjÞ! q (5) Figure 1 shows the magnitude of the ﬁrst four coefﬁcients assuming a source located in the plane of the aperture, i.e., at hs ¼ p=2. The advantage of this conﬁguration is that its behavior resembles the one of a circular aperture mounted 0001-4966/2011/130(3)/1095/4/$30.00 C 2011 Acoustical Society of America V 1095 Author's complimentary copy During the past decade, studies on the performance of circular arrays of microphones for localizing sound sources over 360 have been reported. For example, Meyer1 utilized modal beamforming to generate a desired beampattern for a circular microphone array mounted around a rigid sphere. Daigle et al.2 considered delay-and-sum beamforming with circular arrays mounted on the surface of sound absorbing spheres and cylinders and showed that the achieved beamwidth improved over that of arrays mounted on hard spheres and cylinders. Instead of delay-and-sum beamforming, Teutsch and Kellermann3 analyzed various algorithms based on decomposing the sound ﬁeld into a series of modes for a circular array mounted on a cylinder. Still in the ﬁeld of modal beamforming Tiana-Roig et al.4 adapted the theory of spherical harmonics beamforming to the two-dimensional case using circular harmonics. The resulting circular harmonics beamformer was compared to the classical delayand-sum beamformer using both a circular array suspended in free space and one mounted on a rigid, inﬁnite cylinder. This letter to the editor repeats the comparison for the case of a circular array mounted on a rigid sphere. q The output of the unbafﬂed array can be also written as bDSB ðkR; uÞ ¼ on a rigid cylinder of inﬁnite length,4 for which all frequencies can be resolved by means of beamforming procedures. III. BEAMFORMING ALGORITHMS Circular harmonics beamforming (CHB) is a technique implemented speciﬁcally for circular arrays of microphones based on the decomposition of the sound ﬁeld using the principles of a Fourier series. The beamformer output is4 bN;CHB ðkR; uÞ ¼ M N X AX 1 ejnðum uÞ ; p~ðkR; um Þ ðkRÞ M m¼1 Q n n¼N (6) where A is a scale factor, M is the number of microphones, p~ and um are the measured pressure and the azimuth angle of the mth microphone, and N is the maximum order taken into account. Qn ðkRÞ, which is related to the Fourier coefﬁcients as Qn ðkRÞ ¼ Cn ðkRÞ=ejnus , depends on the conﬁguration of the array: 8 ðjÞn Jn ðkRÞ Free space > > > 0 > > J ðkRÞH ðkRÞ n n > n > Rigid cylinder > > ðjÞ Jn ðkRÞ H 0 ðkRÞ > n > < of infinite length Qn ðkRÞ¼ 1 > X > ðq n j jÞ! > > Equator of a ð2qþ1Þbn ðkRÞ > > > ðqþ jnjÞ! > q¼jnj > > > : jnj j nj rigid sphere Pq ð0ÞPq ð0Þ (7) Whereas the expressions for the array suspended in free ﬁeld and mounted on an inﬁnitely long cylinder are taken from Ref. 4, the value for the array on the sphere follows from Eq. (5). Note that in all cases the sound sources are assumed to be in the plane of the array, i.e., hs ¼ p=2. Another technique that can be implemented for the circular geometry is delay-and-sum beamforming (DSB). This technique aligns the signals of the microphones by introducing appropriate delays and ﬁnally adds the signals together.7 Implemented in the frequency domain using matched ﬁeld processing, the beamformer output is4 bN;DSB ðkR; uÞ ¼ M N X AX p~ðkR; um Þ Qn ðkRÞejnðum uÞ : M m¼1 n¼N (8) 1096 J. Acoust. Soc. Am., Vol. 130, No. 3, September 2011 (9) which is more precise than Eq. (8) because it does not imply a truncation at an order N. Ideally, the beamformers output should be zero at all angles u different from the angle of the sound source, us . However, because a limited number of microphones is used, rather than a continuous aperture, the response exhibits a main lobe around us and side lobes at other angles. Therefore, it is convenient to evaluate the performance of the beamformer in terms of resolution, deﬁned as the 3 dB width of the main lobe, and maximum side lobe level (MSL), which is given by the difference in level between the peaks of the highest side lobe and the main lobe. IV. SIMULATION STUDY The performance of CHB and DSB using a circular array mounted on the equator of a rigid sphere has been compared to other conﬁgurations such as a circular array suspended in free space or a circular array mounted on a rigid cylinder of inﬁnite length by means of computer simulations. The impinging plane waves were perpendicular to the plane of the array and were created by a source placed at hs ¼ 90 and us ¼ 180 . (Note, however, that the azimuth angle has a very limited inﬂuence on the results.) The maximum order of the algorithms followed N kR for CHB and N kR þ 1 for DSB,4 up to a maximum N ¼ M=2 1 to satisfy the Nyquist criterion: k=2 > d, where d is the distance between the microphones, or equivalently, M > 2kR.8,9 An array with 10 cm of radius and 10 microphones was considered. The left panels of Fig. 2 show the performances of the unbafﬂed array and the array mounted on a sphere using CHB and considering ideal conditions, i.e., without background noise. The behavior of the array mounted on a rigid cylinder is not depicted because the curves coincide with the ones of the array on the sphere in the frequency range of interest. As can be seen, by mounting the array on the sphere, the performance is very similar to the one of the unbafﬂed array but improves particularly at those frequencies where the unbafﬂed array presents peaks. Note that from 2.7 kHz on the MSL worsens dramatically because in this range the Nyquist criterion is no longer fulﬁlled and consequently aliasing occurs. The resolution and the MSL with DSB can be seen in the right column of Fig. 2. In this case, the response of the array mounted on a cylinder of inﬁnite length is also shown. With this technique, the performance of the array mounted on the sphere is better than the one with an unbafﬂed array, especially toward low frequencies. However, it is not as good as in the case of the array mounted on an inﬁnitely long cylinder. Actually, by mounting the array on a cylinder or on a sphere, the apparent distance between microphones increases, and this improves the performance of DSB at low frequencies. This is in agreement with the observations of Daigle et al., although they claimed that the array has an effectively larger aperture when mounted on a physical Tiana-Roig et al.: Letters to the Editor Author's complimentary copy FIG. 1. Magnitude of the Fourier coefﬁcients of the pressure on a circular aperture mounted on the equator of a rigid sphere. M AX p~ðkR; um ÞejkR cosðum uÞ ; M m¼1 FIG. 2. Resolution and MSL using CHB (left) and DSB (right) and a circular array of radius 10 cm and 10 microphones when the microphones are suspended in free space, mounted on the equator of a rigid sphere, and mounted on a rigid cylinder of inﬁnite length. For ease of comparison, the case of the cylinder with CHB is not plotted because it coincides with the case of the sphere. In all cases, the source is in the far ﬁeld at ½hs ; us ¼ ½90 ; 180 . V. EXPERIMENTAL RESULTS A circular array mounted on a rigid sphere has been tested in an anechoic room. The prototype array consisted of 16 1/4 in. microphones, Brüel & Kjær (B&K) type 4958, mounted on the equator of a rigid sphere with a radius of 9.75 cm, corresponding to a microphone for every 22:5 . With this conﬁguration, the array can operate free of aliasing up to about 4.5 kHz. The array and the source, a loudspeaker, were controlled by a B&K PULSE analyzer. The loudspeaker was driven by a signal from the generator, pseudorandom noise of 1 s of period, 6.4 kHz of bandwidth, and 1 Hz of resolution. The microphone signals were recorded with the analyzer and postprocessed with the beamforming algorithms CHB and DSB. Figure 3 shows the output of the array using CHB and DSB when a source is located 4 m away but at the very same height and at an azimuth angle us ¼ 180 . It can be seen that with the two techniques, the array is capable of localizing the sound source in the frequency range of interest with exception of DSB at the frequencies below about 300 Hz due to the fact that this technique behaves omnidirectionally at such values. The performance of the array is also illustrated in Fig. 4, where the resolution and the MSL for both CHB and DSB are shown. The predictions made with computer simulations are also depicted. To account for the background noise introduced in the measurements, the simulations were carried out with a signal-to-noise ratio (SNR) of 30 dB at the input of each microphone due to uniformly distributed noise. FIG. 3. Normalized output with CHB (left) and DSB (right) using a circular array with 16 microphones mounted on the equator of a rigid sphere with radius of 9.75 cm. The source is in the far ﬁeld at ½hs ; us ¼ ½90 ; 180 . J. Acoust. Soc. Am., Vol. 130, No. 3, September 2011 Tiana-Roig et al.: Letters to the Editor 1097 Author's complimentary copy structure independently of the beamforming technique,2 whereas the present study has revealed that this is not the case with CHB. FIG. 4. Resolution and MSL with CHB (left) and DSB (right) using a circular array of radius 9.75 cm and 16 microphones mounted on the equator of a rigid sphere. The source is in the far ﬁeld at ½hs ; us ¼ ½90 ; 180 . For the simulation, the SNR in each microphone is 30 dB. VI. CONCLUSIONS A beamformer consisting of a uniform circular array of microphones mounted on the equator of a rigid sphere has been examined using CHB and DSB. A simulation study has revealed that this conﬁguration is an improved version of a circular array suspended in free space. Particularly, with 1098 J. Acoust. Soc. Am., Vol. 130, No. 3, September 2011 CHB, the array mounted on the sphere behaves identically to the unrealistic case of an array with the same dimensions mounted on a rigid cylinder of inﬁnite length. Therefore, the array on the sphere is a simple solution of special interest as alternative to beamformers based on cylinders of ﬁnite length because these are often approximated by inﬁnitely long cylinders to overcome the problem that an exact analytical expression for such cylinders does not exist. Various experiments using a prototype array have proved the validity of the model. 1 J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects,” J. Acoust. Soc. Am. 109(1), 185–193 (2001). 2 G. A. Daigle, M. R. Stinson, and J. G. Ryan, “Beamforming with aircoupled surface waves around a sphere and circular cylinder (L),” J. Acoust. Soc. Am. 117(6), 3373–3376 (2005). 3 H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on waveﬁeld decomposition using circular microphone arrays,” J. Acoust. Soc. Am. 120(5), 2724–2736 (2006). 4 E. Tiana-Roig, F. Jacobsen, and E. Fernández Grande, “Beamforming with a circular microphone array for localization of environmental noise sources,” J. Acoust. Soc. Am. 128(6), 3535–3542 (2010). 5 E. Williams, Fourier Acoustics: Sound Radiation and Nearﬁeld Acoustic Holography (Academic, London, 1999), pp. 4–5, 224–230. 6 B. Rafaely, “Plane-wave decomposition of the sound ﬁeld on a sphere by spherical convolution,” J. Acoust. Soc. Am. 116(4), 2149–2157 (2004). 7 D. Johnson and D. Dudgeon, Array Signal Processing Concepts and Techniques (Prentice Hall, Englewood Cliffs, NJ, 1993), pp. 112–119. 8 H. Van Trees, Optimum Array Processing. Part IV of Detection, Estimation, and Modulation Theory (Wiley, New York, 2002), pp. 280–284. 9 C. Mathews and M. Zoltowski, “Eigenstructure techniques for 2-D angle estimation with uniform circular arrays,” IEEE Trans. Signal Process. 42(9), 2395–2407 (1994). Tiana-Roig et al.: Letters to the Editor Author's complimentary copy For CHB, the simulations and the measurements agree in terms of resolution in most of the frequency range. Small deviations from the expected values are observed at the lowest frequencies and around 500 Hz. In terms of MSL, the measurements oscillate around the expected values as occurs with circular arrays suspended in free space.4 The differences detected in the resolution at low frequencies and about 500 Hz also appear in the MSL. At low frequencies, the difference is attributed to a presence of background noise higher than expected. At about 500 Hz, the performance is better than expected because the MSL is much lower than the predictions. The good agreement with the simulations is also found in the case of DSB. Just in the range from 400 to 600 Hz, the MSL differs from the expected value. Other measurements with the source placed at different positions revealed that the behavior of the array is practically independent of its azimuth angle. Because the beamforming algorithms given in Sec. III expect a source placed at a polar hs ¼ 90 , i.e., at the plane of the array, the performance is optimal when this happens. However, it can be shown that sources placed in the range hs ¼ 90 645 can still be localized. Paper C Deconvolution for the localization of sound sources using a circular microphone arraya) Elisabet Tiana-Roigb) and Finn Jacobsen Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark, Ørsteds Plads 352, DK-2800 Kongens Lyngby, Denmark (Received 14 September 2012; revised 18 June 2013; accepted 10 July 2013) During the last decade, the aeroacoustic community has examined various methods based on deconvolution to improve the visualization of acoustic ﬁelds scanned with planar sparse arrays of microphones. These methods assume that the beamforming map in an observation plane can be approximated by a convolution of the distribution of the actual sources and the beamformer’s point-spread function, deﬁned as the beamformer’s response to a point source. By deconvolving the resulting map, the resolution is improved, and the side-lobes effect is reduced or even eliminated compared to conventional beamforming. Even though these methods were originally designed for planar sparse arrays, in the present study, they are adapted to uniform circular arrays for mapping the sound over 360 . This geometry has the advantage that the beamforming output is practically independent of the focusing direction, meaning that the beamformer’s point-spread function is shift-invariant. This makes it possible to apply computationally efﬁcient deconvolution algorithms that consist of spectral procedures in the entire region of interest, such as the deconvolution approach for the mapping of the acoustic sources 2, the Fourier-based non-negative least squares, and the Richardson–Lucy. This investigation examines the matter with computer simulations and C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4816545] measurements. V Pages: 2078–2089 I. INTRODUCTION Beamforming with phased arrays of microphones is a well established method for visualization of sound ﬁelds. However, because the sound ﬁeld is mapped with a discrete number of microphones, beamforming techniques present intrinsic limitations, speciﬁcally the frequency dependence of the array resolution and the appearance of side lobes that contaminate the beamforming map with sometimes unexpected results.1 These two factors make it difﬁcult to interpret the map and therefore to visualize the actual sound ﬁeld accurately. The focus of the present investigation is on the improvement of the performance of uniform circular arrays (UCAs) for mapping the sound ﬁeld over 360 around the array to localize sound sources in the far ﬁeld. This array geometry has lately been of interest in various studies about environmental noise localization, conferencing, and measurements in ducts among others; see, for instance, Refs. 2–9. It has been shown that a better performance in terms of resolution and level of the side lobes can be achieved by mounting the array on a scatterer such as a sphere or a cylinder3,4,6 or by designing other techniques than delay-and-sum (DS) beamforming, for example, circular harmonics (CH) beamforming.5 Nevertheless these solutions are not sufﬁcient to result in a clear and unambiguous beamforming map. a) Portions of this work were presented in “Acoustical source mapping based on deconvolution approaches for circular microphone arrays,” Proceedings of Inter-Noise 2011, Osaka, Japan, September 2011. b) Author to whom correspondence should be addressed. Electronic address: [email protected] 2078 J. Acoust. Soc. Am. 134 (3), September 2013 In the recent years, the aeroacoustic community has suggested various methods to improve the beamforming map in two-dimensional (2D) imaging using planar sparse arrays to map the sound ﬁeld in a region parallel to the plane of the array. These methods rely on the fact that the map is a convolution of the acoustic sources and the beamformer’s pointspread function (PSF), which is deﬁned as the response of the beamformer to a point source. By means of deconvolution, the distribution of the sources can be recovered presenting a better resolution and reduced (or even suppressed) side lobes in comparison with direct beamforming. Examples of deconvolution methods can be found in Refs. 10–12 for static uncorrelated noise sources, in Refs. 13–15 for correlated noise sources, and in Ref. 16 for moving sources. The main problem is that these methods require a high computational effort due to the fact that they are based on iterative algorithms. To improve the efﬁciency, certain techniques use spectral (Fourier-based) procedures for the deconvolution, but these can only be applied when the beamformer’s PSF is shift-invariant, that is, when the response of the beamformer to a point source depends only on the distance between the focusing point of the beamformer and the position of the point source. However, for 2D imaging, the assumption that the PSF is shift-invariant is only a good approximation when the source region is small compared with the distance between the array and the source. Therefore the use of such deconvolution approaches is restricted to a small region in space unless it is expanded to a larger (and 3D) region by making use of a coordinate transformation.10,17,18 Interestingly, one could think of adapting the existing deconvolution methods to a UCA to improve its performance. Contrary to the case of planar sparse arrays for which 0001-4966/2013/134(3)/2078/12/$30.00 C 2013 Acoustical Society of America V Author's complimentary copy PACS number(s): 43.60.Fg [BEA] II. CONVOLUTIONAL FORMULATION FOR UNIFORM CIRCULAR ARRAYS A beamformer based on a UCA of microphones is capable of mapping the sound ﬁeld over 360 in the plane of the array to ﬁnd the direction of sound sources located in that plane. By electronically steering the beamformer, the sound ﬁeld is scanned in a grid of azimuth angles u, from 0 to 360 , to detect the propagating acoustic waves that impinge on the array and thereby to identify the direction of the sound sources that emit them. When a single source is present, the beamformer output exhibits a main lobe around the azimuth of the source, whereas other directions are contaminated with side lobes; see Fig. 1. The characteristics of the beampattern, i.e., the shape of the main lobe and the side lobes, are given by the beamformer’s PSF. This function was originally deﬁned as the beamformer response to a point source with unit strength at an arbitrary position of a grid located in a plane parallel to the array plane.10–12 However, this deﬁnition needs to be reformulated for a UCA because the goal is to look into all possible azimuth angles around a UCA beamformer instead of looking to a plane parallel to the array. In the current study, the sources are considered to be sufﬁciently far from the array position, and therefore waves captured by the array can be regarded to be planar. This assumption implies that the direction of the waves can be identiﬁed, although the distance to the sources that emit them cannot be estimated—this would require a near-ﬁeld scenario. Under the plane wave assumption, the PSF can be redeﬁned as the beamformer response to a plane wave of unit amplitude created by a source in the far ﬁeld of the array. Then in the presence of incoherent sources, the beamformer output is related to the PSF as X sðu0 ÞHðuju0 Þ; (1) bðuÞ ¼ u0 2G where sðu0 Þ contains information regarding the direction and the strength (at the array position) of a plane wave created by a source located at an azimuth angle u0 contained on the grid G, whereas Hðu j u0 Þ is the PSF at u due to a source at u0 . From now on s will be referred to as the source distribution. From this expression, it becomes apparent that the information regarding sound sources can be recovered from the measured beamformer map and the beamformer’s PSF. This is done by means of a deconvolution procedure, imposing that the distribution of sound sources must be non-negative [sðu0 Þ 0]. This is an inverse problem, which in matrix notation can be rewritten as b ¼ Hs; (2) where the vectors b and s contain the information about the beamformer output and the source distribution, respectively, and H is a matrix that in each column contains the PSF for one source located at an angle u0 of the grid. Often the matrix H can be singular, which implies that there may be inﬁnitely many solutions for s.11 For a beamformer based on a UCA, the focusing direction can be steered to any position in the plane where the array lies without changing the beampattern signiﬁcantly due to the symmetry of the array.2,19 This implies that the overall shape of the beamformer’s PSF remains practically the same independently of its looking direction as shown in Fig. 2. A PSF that satisﬁes this condition is called to be shift-invariant because it depends only on the difference of the actual focus point u and the azimuth of a source u0 , Hðuju0 Þ ¼ Hðu u0 Þ: Inserting this property into Eq. (1) leads to X bðuÞ ¼ sðu0 ÞHðu u0 Þ; (3) (4) u0 2G FIG. 1. Illustration of beamforming with a UCA for detecting the location of a distant sound source. J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 which corresponds to a discrete circular convolution of sðuÞ and HðuÞ. Making use of the convolution theorem, Eq. (3) can be expressed with the discrete Fourier transform, E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array 2079 Author's complimentary copy the PSF is shift-variant per se, beamformers based on UCAs have a practically shift-invariant PSF along the region of interest,2,19 i.e., 360 , and consequently this scenario seems particularly adequate for the use of Fourier-based deconvolution methods. In the following, the main deconvolution methods that rely on a shift-invariant PSF, namely, the deconvolution approach for the mapping of the acoustic sources 2 (DAMAS2), the Fourier-based non-negative least squares (FFT-NNLS), and the Richardson–Lucy (RL) will be reformulated for the case of plane waves impinging on a UCA. The ﬁrst method, DAMAS2, introduced by Dougherty in Ref. 10, is an extension of DAMAS of Brooks and Humphreys (Ref. 13). The second algorithm, FFT-NNLS, was adapted from the classical NNLS procedures20 by Ehrenfried and Koop in Ref. 11. Finally RL, which was initially developed by Richardson and Lucy (Refs. 21 and 22) for image restoration in astronomy, was also adapted for acoustical purposes in Ref. 11. All these methods will be examined by means of computer simulations and experimental results. Among the the existing methods, DAMAS210 and FFTNNLS,11 appear to be especially attractive for a UCA to map the sound ﬁeld over 360 , because they rely on a shiftinvariant PSF to solve the deconvolution problem as formulated in Eq. (5). Yet another method that can be adapted to a UCA for localizing sound sources is RL, even though this was initially conceived for deconvolution problems in statistical astronomy.21,22 Unlike DAMAS2 and FFT-NNLS, this method solves the inverse problem given by Eq. (5) from a statistical point of view following from Bayes’ theorem on conditional probabilities. In any case, the three methods aim to ﬁnd an estimate of the source distribution s, s~, that convolved with the PSF ~ as similar as gives an estimate of the beamformer output, b, possible to the real beamformer output. An overview and a comparison of the performances of RL, DAMAS2, and FFT-NNLS for planar sparse arrays for localizing sound sources in a small region of a plane parallel to the array plane can be found in Ref. 11. In what follows DAMAS2, FFT-NNLS, and RL are adapted for the case of UCAs when sources in the far ﬁeld of the array are assumed. In this sense, the estimate of the source distribution, s~, will provide information about the direction of the impinging waves and their level at the array position. A. DAMAS2 DAMAS2 addresses the inverse problem formulated in Sec. II directly as stated in Eq. (5), which follows from Eqs. (1) and (2) considering a shift-invariant PSF. The algorithm consists of the following steps: 1. Step 0 s~ð0Þ ðuÞ :¼ 0; and thus one can take advantage of the computational efﬁciency of this operation, bðuÞ ¼ F 1 ½F ½sðuÞ F ½HðuÞ; (5) where the operators F and F 1 stand for the direct and the inverse Fourier transforms, respectively. This relationship is a key issue for deconvolution methods based on spectral approaches. Note that this equation requires that the PSF used for the calculation of the beamformer output, HðuÞ, considers a source placed at an azimuth u0 ¼ 0, according to Eq. (3). 8u 2 G: Compute the value a, which is given by the discrete integral of the PSF, X a¼ Hðu u0 Þ; (7) u2G for a source located at an angle u0 , e.g., u0 ¼ 0 . Calculate the FFT of HðuÞ, FH ¼ F ½HðuÞ: Compute the estimate of the beamformer map b as From the mid 2000s the aeroacoustic array community has suggested various deconvolution methods, such as the DAMAS family of algorithms10–16 or the NNLS algorithms,11 to visualize sound sources with accuracy from a given beamforming map. These methods use iterative procedures to solve the inverse problem expressed by Eqs. (1) and (2). ðnÞ b~ ðuÞ :¼ F 1 ½F ½~ s ðnÞ ðuÞ FH : J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 (8) 2. Step 1 III. DECONVOLUTION METHODS 2080 (6) (9) Note that this expression follows from Eq. (5). In the original algorithm implemented for planar sparse arrays given by Dougherty in Ref. 10, the preceding expression is scaled by a Gaussian ﬁlter to smooth the retrieved sound distribution E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array Author's complimentary copy FIG. 2. PSF of a particular UCA focused to different directions (0 , 110 , and 230 ). Initialize the iteration index, n ¼ 0, and use an estimate of s, s~ð0Þ to start. Typically this value is set to zero for the entire region of interest, i.e., and to minimize the inﬂuence of high wave-number noise, i.e., background noise induced by sources outside the region of interest.11 However, for the case of UCAs, this ﬁlter is not necessary as all sources are placed inside the entire region of interest, between 0 and 360 . 5. Step 4 Compute the auxiliary value g(n) as ðnÞ ðuÞ FH : gðnÞ ðuÞ :¼ F 1 ½F ½w 6. Step 5 3. Step 2 Apply a non-negativity constraint to update the value of s~ ! ~ðnÞ ðuÞ bðuÞ b s~ðnþ1Þ ðuÞ :¼ max s~ðnÞ ðuÞ þ ;0 : a Calculate an optimal step factor k as X gðnÞ ðuÞrðnÞ ðuÞ u k :¼ X ðgðnÞ ðuÞÞ2 : u 4. Step 3 7. Step 6 Increment the iteration index, n ¼ n þ 1. Update the solution s~ðnþ1Þ using the non-negativity constrain as follows 5. Remaining steps Repeat steps 1–3 until the standard deviation of the residual r(n) converges to zero. The residual is deﬁned as the difference between the estimated beamforming map and the actual map rðnÞ ðuÞ :¼ b~ ðuÞ bðuÞ: ðnÞ (10) 8. Step 7 Increment the iteration index, n ¼ n þ 1. 9. Remaining steps Repeat steps 1–7 until the standard deviation of the residual r(n) converges to zero. Unlike DAMAS2, FFT-NNLS tries to minimize the square sum of the residuals, that is minkHs2bk2 : (11) This can be solved by a gradient-type minimization procedure as suggested in Ref. 11. The steps are the following: 1. Step 0 Initialize the iteration index, n ¼ 0, and s~ð0Þ before starting the iterative procedure. As in DAMAS2, s~ð0Þ is usually set to zero; see Eq. (6). Besides, compute FH as in Eq. (8). 2. Step 1 (n) from a given solution rðnÞ ðuÞ :¼ F 1 ½F ½~ s ðnÞ ðuÞ FH bðuÞ: Note that the ﬁrst term of this difference is the estimate of the ðnÞ beamformer output at the nth iteration, b~ , as in Eq. (9). 3. Step 2 Calculate the gradient w(n) as For imaging deblurring purposes, RL assumes a shiftinvariant PSF. One iteration cycle of the algorithm can be written as " # 1 ðnÞ b ðnþ1Þ HðuÞ ; (12) ðuÞ ¼ s~ ðuÞ ðnÞ s~ a s~ ðuÞ HðuÞ where the initial value is 1 s~ð0Þ ðuÞ ¼ bðuÞ: a (13) If the initial value s~ð0Þ is non-negative (so the non-negativity constraint is fulﬁlled), RL guarantees that all generated solutions s~ðnÞ will be non-negative. Note that the scaling factor a given in the two previous expressions was not present in the original formulation. The reason for this is that normalized data was of concern, so that a ¼ 1. However, for the current investigation, the data is not normalized, and this value must represent the discrete integral of the PSF,11 given in Eq. (7). Making use of the Fourier transform to compute the convolutions the method consists of the following steps: 1. Step 0 wðnÞ ðuÞ :¼ F 1 ½F ½r ðnÞ ðuÞ FH : Initialize the iteration index, n ¼ 0, set the initial value of s~, s~ð0Þ , using Eq. (13) and compute FH as in Eq. (8). 4. Step 3 Use the following projection of the gradient 0 if wðnÞ ðuÞ < 0 and s~ðnÞ ðuÞ ¼ 0; ðnÞ ðuÞ :¼ w ðnÞ w ðuÞ otherwise: J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 C. RL 2. Step 1 Compute an estimate of the beamformer output as ðnÞ b~ ðuÞ :¼ F 1 ½F ½~ s ðnÞ ðuÞ FH : E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array 2081 Author's complimentary copy B. FFT-NNLS Compute the residual vector r s~ðnÞ as follows: ðnÞ ðuÞ; 0Þ: s ðnÞ ðuÞ þ kw s~ðnþ1Þ ðuÞ :¼ maxð~ 3. Step 2 Calculate the ratio between the actual beamformer output and the estimated one, xðnÞ ðuÞ :¼ bðuÞ : ðnÞ b~ ðuÞ 4. Step 3 Update the value ~ s according to the following expression 1 s~ðnþ1Þ ðuÞ :¼ s~ðnÞ ðuÞF 1 ½F ½xðnÞ ðuÞ FH : a 5. Step 4 Increment the iteration index, n ¼ n þ 1. 6. Remaining steps Repeat steps 1–4 until the standard deviation of the residual r(n) [deﬁned as in Eq. (10)] converges to zero. IV. SIMULATION AND MEASUREMENT RESULTS A. Beamforming techniques for a UCA DS beamforming and a more recent technique called CH beamforming, which is especially conceived for UCAs,5,6 are the techniques selected to test the deconvolution algorithms. DS beamforming is based on delaying the signals captured at each array microphone by a certain amount and adding them up to focus the system to a speciﬁc direction in space that depends on the applied delay. Instead CH beamforming is based on decomposing the sound ﬁeld into a summation of harmonics (as in a Fourier series) and comparing the resulting coefﬁcients with the ones obtained from decomposing the expected sound ﬁeld in the looking direction of the array. These techniques are implemented to localize the direction of sound sources that lay in the plane of the array, or close to it, but sufﬁciently far so that the generated waves are regarded as planar at the array position; see Fig. 1. Obviously beamforming procedures with such array geometry for mapping the sound ﬁeld over 360 provide information about the azimuth of the source, u0 , but do not account for its polar angle, h0 , which means that those sources with a certain elevation are always projected in the plane of the array. For the present investigation, a UCA with the microphones ﬂush-mounted on a rigid sphere as the one shown in Fig. 3 is assumed. This conﬁguration provides better results than a UCA in which the microphones are suspended in free space in terms of both width of the main beam (resolution) and level of the largest secondary lobe (the so-called maximum side lobe level).6 2082 J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 FIG. 3. (Color online) UCA mounted on a rigid sphere. Assuming a UCA of radius R and M microphones, the output of a CH beamformer focused toward u is given in the Fourier transform domain (spatial frequency domain) by 2 M N X X 1 jnðum uÞ e bCH ðkR; uÞ ¼ A p~m ðkRÞ ; m¼1 Q ðkRÞ n¼N n (14) where k is the wave number of the frequency of interest, A is a scaling factor, p~m is the sound pressure captured by the mth microphone placed at an angle um, and N is the maximum number of harmonics used for the algorithm. This value should follow N ¼ dkRe, where de refers to the ceiling function, up to a maximum equal to M=2 1, to obtain the optimal map (higher orders would amplify the inﬂuence of noise dramatically).5,6 The function Qn(kR) depends on the geometry of the UCA. For a UCA mounted on the equator of a rigid sphere, ! 1 X hq ðkRÞ q 0 ð2q þ 1ÞðjÞ jq ðkRÞ jq ðkRÞ 0 Qn ðkRÞ ¼ hq ðkRÞ q¼jnj ðq jnjÞ! jnj ðP ð0ÞÞ2 ; ðq þ jnjÞ! q (15) where jq and hq are spherical Bessel and spherical Hankel functions of order q, j0q and h0q are their derivatives with jnj respect to the radial direction r evaluated at r ¼ R, and Pq is a Legendre function of degree q and order n. On the other hand, the output of a DS beamformer is expressed by E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array Author's complimentary copy In this section DAMAS2, FFT-NNLS, and RL are examined by means of computer simulations and measurements. The beamforming techniques used prior to the mentioned deconvolution algorithms will be introduced brieﬂy before presenting the results. (16) where B is a scaling factor and the parameter Qn ðkRÞ is again given by Eq. (15). The value of N should be in this case at least N ¼ dkRe þ 1, up to a maximum equal to M/21 as for CH beamforming. The scaling factors of CH and DS beamforming can be chosen such as the maximum value of the beamformer output is equal to one when a plane wave with amplitude unity is detected. To accomplish to be P this A and B need A ¼ 1/(M(2 N þ1)) and B ¼ M m¼1 pm ðkRÞpm ðkRÞ , where pm is the sound pressure of a plane wave of amplitude unity created at, e.g., 0 . Regardless of the beamforming technique, it should be kept in mind that because the sound ﬁeld is sampled at discrete positions with the array microphones, modal aliasing occurs at those frequencies the wavelength of which is less than twice the distance between two consecutive microphones. When aliasing occurs side lobes increase dramatically, becoming replicas of the main lobe in the worst case (the so-called aliased lobes). From the given beamforming techniques, the PSF corresponding to each of them can be obtained assuming that a plane wave of amplitude unity is captured at the array. A detailed description of the calculation of the PSF can be found in the Appendix. B. Test case using computer simulations Let us assume a plane wave with frequency 1.6 kHz and amplitude a ¼ 2 captured by a CH beamformer that consists of a UCA with radius 9.75 cm and 16 microphones mounted on the equator of a rigid sphere (corresponding to a microphone at every 22.5 ). With this conﬁguration, the array is capable of operating up to about 4.5 kHz without aliasing. The wave is generated by a source placed at an azimuth angle of 60 . The PSF of such beamformer and the beamformer output are shown in Fig. 4. As can be seen, the beamforming process successfully detects the wave because a main beam is visualized around 60 . However, the main beam is rather broad, and the map presents side lobes elsewhere, which can lead to confusion. The beamformer map is then postprocessed with DAMAS2, FFT-NNLS, and RL. For these processes, a grid of azimuth angles from 0 to 359 with a resolution of 1 has been used. Note that besides the direction of the sources, the retrieved value s gives the information of the squared amplitude of the plane waves at the array position emitted by them because both the beamforming output and the PSF (given in the Appendix) correspond to magnitude squared functions. The top row of Fig. 5 shows the source distribution recovered with DAMAS2, FFT-NNLS, and RL after 500 iterations. The three algorithms produce a clean map compared to the beamformer response; the direction of the source is pointed out with a narrow main lobe, and the effect of side lobes is practically removed. Theoretically the recovered sources should be represented by a delta function with its maximum being the squared amplitude of the plane wave (a2), 4 in this case. However, none of the methods provides this result after 500 iterations. An estimate of the squared amplitude of each impinging wave, a~2 , can be obtained with an integration method that consists of summing the values s~ðuÞ inside the region of interest, G0 (Ref. 12), X s~ðuÞ: (17) a~2 ¼ u2G0 The resulting level estimates obtained with DAMAS2, FFTNNLS, and RL are 4.13, 4.04, and 4.00, respectively, which agree with the value of the squared amplitude of the plane wave under consideration. The convergence of the algorithms given by the standard deviation of the residual (i.e., the difference between the estimated beamformer output and the actual one) as a function of the number of iterations n is shown in Fig. 6. For all the algorithms, the standard deviation of the residuals converges to a value close to zero when the number of iterations increases as expected. For this particular example, it can be seen that from about 5000 iterations, the standard deviation of the residuals is practically zero. Therefore ideally one FIG. 4. PSF (left) and output (right) of a CH beamformer that consists of a UCA with 16 microphones mounted on the equator of a rigid sphere with radius of 9.75 cm. A source in the far ﬁeld at u0 ¼ 60 is assumed for the simulations. J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array 2083 Author's complimentary copy 2 M N X X jnðum uÞ bDS ðkR; uÞ ¼ B p~m ðkRÞ Qn ðkRÞe ; m¼1 n¼N FIG. 5. Maps retrieved with DAMAS2 (left column), FFT-NNLS (middle column), and RL (right column) after 500 iterations (top row), and after 1 106 iterations (bottom row). A source in the far ﬁeld at u0 ¼ 60 is considered. CH beamforming is used prior to the deconvolution algorithms. FIG. 6. Standard deviation of the residual as a function of the number of iterations n, for DAMAS2, FFT-NNLS, and RL. 2084 J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 Let us consider that besides the source at 60 responsible for the previous plane wave with frequency 1.6 kHz, there is another source at 90 that creates a plane wave with unity amplitude at the same frequency. The two sources are incoherent. The beamformer map obtained with CH beamforming is shown in Fig. 7. As can be seen, the beamforming map reveals only a source located at about 60 (the maximum is actually at 63 ). Although the main beam presents an asymmetric shape that can indicate that there is another source present, it is not possible to state that this is placed at 90 as assumed. After applying deconvolution the maps shown in Fig. 8 are achieved with 5000 iterations. The three methods reveal FIG. 7. Map obtained with CH beamforming. Two plane waves are present in the sound ﬁeld, one with amplitude 2 created at 60 and another one with amplitude 1 created at 90 . E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array Author's complimentary copy would expect a recovered source distribution represented by a delta function with a level of 4 at 60 . However, even after 1 106 iterations, the retrieved source distribution with DAMAS2 and FFT-NNLS differs from a delta function as can be seen in the bottom row of Fig. 5. In the case of RL, the result is much closer to the ideal response, but a close-up of the ﬁgure reveals that there is still some spreading (the peak is not exactly 4 and the neighboring values are not zero). This means that more iterations are needed to get a response as close as possible to the ideal response. The deconvolution processes are particularly useful when more than one source are present in the sound ﬁeld because they make it possible to locate the different sources even when they are not visible in the beamformer output. two sources, the strongest being located at 60 as expected. However, the other source is not exactly located at 90 but close to it: at 94 with DAMAS2, at 92 with FFT-NNLS, and at 91 with RL. The estimates of the squared amplitude of the sources are very close to the expected level of the two waves (22 and 12); 4.2 and 0.8 with DAMAS2, 4.1 and 0.9 with FFT-NNLS, and 4.0 and 0.9 with RL. As in the case of having only one source, it can be shown that with the increase of the number of iterations the maps become clearer, and moreover the source with less energy is located at an azimuth angle that tends to the actual value. Although not shown similar results are obtained with DS beamforming. C. Simulated and experimental data results The deconvolution methods DAMAS2, FFT-NNLS, and RL have been tested experimentally and compared to computer simulations. For this purpose, measurements with the UCA shown in Fig. 3 were carried out in an anechoic room with a volume of about 1000 m3. The array consisted of 16 1/4 in. microphones, Br€ uel and Kjær (B&K) type 4958, mounted on the equator of a rigid sphere with a radius of 9.75 cm. The array and the source, a loudspeaker, were controlled by a B&K PULSE analyzer. The loudspeaker was placed at 4 m from the array center but at the very same height at an azimuth angle of 180 . It was driven by a signal from the generator, pseudorandom noise of 1 s of period, 6.4 kHz of bandwidth, and 1 Hz of resolution. The signal had a duration of 5 s. The microphone signals were recorded with the analyzer, and after averaging for each channel, they were postprocessed with CH beamforming and DS beamforming in the frequency range from 50 Hz to 5.5 kHz. These procedures scanned directions from 0 to 359 with a resolution of 1 . Subsequently the obtained beamforming maps were processed with DAMAS2, FFT-NNLS, and RL, using 500 iterations. 1. Sound source localization To analyze the performance of the deconvolution methods in terms of localization of sound sources, the beamforming maps prior to the deconvolution algorithms were normalized for simplicity. J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 The left side of Fig. 9 shows the normalized output obtained with CH beamforming and the source distribution maps obtained with deconvolution as a function of frequency. The predictions made with computer simulations are also depicted on the right side of the ﬁgure. To account for the background noise present in the measurements, the simulations were carried out with a signal-to-noise ratio (SNR) of 30 dB at the input of each microphone due to uniformly distributed noise. At ﬁrst sight, it can be seen that measurements and simulations yield very similar results. The beamformer procedure (top row) reveals the direction of the main source at 180 in all the frequency range, but the main lobe is rather broad, specially at low frequencies, and side lobes appear along the map. However, the map is satisfactorily improved after applying DAMAS2 (second row), FFT-NNLS (third row), and RL (bottom row) because the main lobe becomes more directive and side lobes are reduced signiﬁcantly. Interestingly, these procedures can still visualize the direction of the source clearly at those frequencies where aliasing in the beamforming map occurs, this is, above 4.5 kHz approximately. This effect could be important for those applications dealing with broadband sources. However, it has been observed that the retrieved map is free of aliasing just when a single source is present. The results obtained with DS beamforming as well as the recovered maps after deconvolution are shown in Fig. 10. In this case, there is also a very good agreement between measurements and simulations. Similar to the results obtained with CH beamforming, the deconvolution algorithms yield an improved version of the beamforming map. Furthermore, they are capable of unveiling the direction of the source at very low frequencies where the DS beamformer is omnidirectional. For both techniques, it can be seen that in the case under analysis 500 iterations are sufﬁcient to demonstrate a clear improvement of the maps after deconvolution. However, the width of the main beam is not constant after the deconvolution processes; it becomes narrower with increasing frequency. This implies that to obtain better results at the lower frequencies, the deconvolution algorithms should include more iterations. Although a comparison of the three techniques could be done at this point, this is out of the scope of the present study E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array 2085 Author's complimentary copy FIG. 8. Maps retrieved with DAMAS2, FFT-NNLS, and RL after 5000 iterations. Two sources in the far ﬁeld at 60 and 90 are considered. CH beamforming is used prior to the deconvolution algorithms. because it has been observed that their performance depends strongly on the case under analysis. 2. Estimated sound pressure level The sound pressure level of the impinging waves at the array can be estimated after deconvolution by means of the 2086 J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 estimated squared amplitude, see Eq. (17). As mentioned in Sec. II, the level of the sources cannot be estimated because these are assumed to be in the far ﬁeld of the array, and therefore the wave fronts at the array position are practically planar. The estimated sound pressure level obtained after deconvolution is similar to the sound pressure level captured E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array Author's complimentary copy FIG. 9. (Color online) Normalized output obtained with CH beamforming (top row) and resulting maps after applying DAMAS2 (second row), FFT-NNLS (third row), and RL (bottom row). The left column shows experimental results and the right column computer simulations. by the array microphones. To show this, in Fig. 11 the sound pressure level averaged across the array microphone signals is plotted together with the estimated sound pressure level retrieved with DAMAS2 when CH and DS beamforming are used prior to the deconvolution process. Note that for this analysis, the beamforming maps were not normalized, so that the recovered sound pressure level is J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 comparable to the averaged sound pressure level captured by microphones. The agreement between the estimated sound pressure level obtained after deconvolution and the averaged sound pressure level of the microphone signals is particularly good at lower frequencies. In fact, the curves are totally overlapped at these frequencies in the case of DS beamforming E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array 2087 Author's complimentary copy FIG. 10. (Color online) Normalized output obtained with DS beamforming (top row) and resulting maps after applying DAMAS2 (second row), FFT-NNLS (third row), and RL (bottom row). The left column shows experimental results and the right column computer simulations. better resolution and are practically free from effects of side lobes. The deconvolution methods methods initially suggested for planar sparse arrays have been adapted to the circular geometry, which has the advantage that beamforming maps can be deblurred very efﬁciently with those deconvolution methods based on spectral procedures, namely, DAMAS2, FFT-NNLS, and RL. The performance of these methods has been examined for two beamforming techniques, DS and CH beamforming, with computer simulations and experimental results. For the three deconvolution methods, the resulting maps are improved by applying the deconvolution algorithms in comparison with the conventional beamforming maps. APPENDIX: CALCULATION OF THE BEAMFORMER’S PSF Consider a plane wave of amplitude a created by a source 0 in the far ﬁeld of a UCA, aejkr . When the array is mounted on a sphere, the pressure captured at the surface of the sphere, at a point with spherical coordinates ½R; h; u, is23,24 pðkR; h; u; h0 ; u0 Þ q 1 X X bq ðkRÞ Yqv ðh; uÞYqv ðh0 ; u0 Þ ; ¼ 4pa (A1) v¼q q¼0 where prior to the deconvolution process, probably due to the robustness of this technique. With increasing frequency, the retrieved level with deconvolution becomes lower than the mean level of the microphones. The difference between low and high frequencies is given by the fact that at low frequencies, the rigid sphere does not affect the sound ﬁeld, whereas at high frequencies, it acts as a scatterer. The averaged pressure at the microphones is ampliﬁed due to the scattering effect at these frequencies. However, with deconvolution, the effect of the scatterer is removed because this is accounted for in the beamforming algorithms given in Eqs. (14) and (16). Although not shown, very similar results were obtained with FFT-NNLS and RL. V. CONCLUSIONS A UCA can be used for scanning the sound ﬁeld over 360 to detect sound sources located in the array plane. By means of beamforming procedures, the direction of the existing incoherent sound sources can be found, but these procedures give rise to a blurred map. An investigation for improving the visualization of the beamforming map has been carried out by applying deconvolution procedures, which are capable of recovering the location of the actual sources with improved precision. The resulting maps present 2088 J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 bq ðkRÞ ¼ ðjÞ ! j0q ðkRÞ hq ðkRÞ ; jq ðkRÞ 0 hq ðkRÞ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2q þ 1Þ ðq vÞ! v Yqv ðh; uÞ P ðcos hÞejvu : 4p ðq þ vÞ! q (A2) (A3) Note that in Eq. (A1), the temporal term ejxt is omitted; and the angles of the position of the source ½h0 ; u0 are used instead of the angles of the incident wave ½hk ; uk ; these being related by h0 ¼ p hk and u0 ¼ uk þ p because the ^ is opposite to the unit vecunit vector of the incident wave k ^ ¼ ^r 0 . tor of the position of the source ^ r0, k At the position of each microphone, ½R; p=2; um , the pressure is p~m ðkRÞ ¼ pðkR; p=2; um ; h0 ; u0 Þ: (A4) In the present study, the PSF is deﬁned as the beamformer response to a plane wave of unit amplitude created by a source placed in the far ﬁeld of the array, but at its very same plane, i.e., at an inclination angle h0 ¼ p/2 and azimuth angle u0 . The PSF is then obtained when the expression given in Eq. (A4) for a plane wave with amplitude a ¼ 1 is inserted into the beamformers output given in Eqs. (14) and (16) for CH and DS beamforming, respectively. It should be emphasized that the PSF considered in the deconvolution methods presented in Sec. III requires that the source used for its calculation is placed at an angle u0 ¼ 0; see Eq. (5). With these considerations the expressions of the PSF for CH and DS beamforming result in E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array Author's complimentary copy FIG. 11. Averaged sound pressure level captured by the array microphones and estimated sound pressure level obtained with DAMAS2. Maps with CH beamforming (top) and DS beamforming (bottom) are obtained prior to deconvolution. q M X C X 4p HCH ðkR; uÞ ¼ bq ðkRÞ Mð2N þ 1Þ m¼1 q¼0 q X v¼q Yqv ðp=2; um ÞYqv ðp=2; 0Þ 2 1 jnðum uÞ e ; ðkRÞ Q n¼N n N X (A5) and M X C 4p X HDS ðkR; uÞ ¼ bq ðkRÞ M m¼1 q¼0 q X v¼q N X n¼N Yqv ðp=2; um ÞYqv ðp=2; 0Þ 2 Qn ðkRÞejnðum uÞ ; (A6) where the coefﬁcients Qn are given in Eq. (15). Although the upper limit of the second summation operator of each technique, C, should be ideally inﬁnity, it has to be truncated for implementation purposes. In the present study, the value used for C followed dkRe þ 5 because this guaranteed that all the coefﬁcients bq that were left out of the summation had a value very close to zero. 1 D. H. Johnson and D. E. Dudgeon, Array Signal Processing Concepts and Techniques (Prentice Hall, Englewood Cliffs, NJ, 1993), pp. 112–119. 2 J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects,” J. Acoust. Soc. Am. 109(1), 185–193 (2001). 3 G. Daigle, M. Stinson, and J. Ryan, “Beamforming with air-coupled surface waves around a sphere and circular cylinder (L),” J. Acoust. Soc. Am. 117(6), 3373–3376 (2005). 4 H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on waveﬁeld decomposition using circular microphone arrays,” J. Acoust. Soc. Am. 120(5), 2724–2736 (2006). 5 E. Tiana-Roig, F. Jacobsen, and E. Fernandez-Grande, “Beamforming with a circular microphone array for localization of environmental noise sources,” J. Acoust. Soc. Am. 128(6), 3535–3542 (2010). 6 E. Tiana-Roig, F. Jacobsen, and E. Fernandez-Grande, “Beamforming with a circular array of microphones mounted on a rigid sphere (L),” J. Acoust. Soc. Am. 130(3), 1095–1098 (2011). J. Acoust. Soc. Am., Vol. 134, No. 3, September 2013 A. Parthy, N. Epain, A. van Schaik, and A. T. Jin, “Comparison of the measured and theoretical performance of a broadband circular microphone array,” J. Acoust. Soc. Am. 130(6), 3827–3837 (2011). 8 A. M. Torres, M. Cobos, B. Pueo, and J. J. Lopez, “Robust acoustic source localization based on modal beamforming and time-frequency processing using circular microphone arrays,” J. Acoust. Soc. Am. 132(3), 1511–1520 (2012). 9 P. Sijtsma, “Circular Harmonics Beamforming with multiple rings of microphones,” in 18th AIAA/CEAS Aeroacoustics Conference, Colorado Springs, CO (June 4–6, 2012), AIAA Paper 2012–2224. 10 R. P. Dougherty, “Extension of DAMAS and beneﬁts and limitations of deconvolution in beamforming,” in 11th AIAA/CEAS Aeroacoustics Conference, Monterey, CA (May 23–25, 2005), AIAA Paper 2005–2961. 11 K. Ehrenfried and L. Koop, “Comparison of iterative deconvolution algorithms for the mapping of acoustic sources,” AAIA J. 45, 1584–1595 (2007). 12 T. F. Brooks and W. M. Humphreys, “A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays,” J. Sound Vib. 294, 856–879 (2006). 13 T. F. Brooks and W. M. Humphreys, “Extension of DAMAS phased array processing for spatial coherence determination (DAMAS-C),” in 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA (May 8–10, 2006), AIAA Paper 2006–2654. 14 T. Yardibi, J. Li, P. Stoica, and L. Cattafesta III, “Sparsity constrained deconvolution approaches for acoustic source mapping,” J. Acoust. Soc. Am. 123(5), 2631–2642 (2008). 15 T. Yardibi, J. Li, P. Stoica, N. Zawodny, and L. Cattafesta III, “A covariance ﬁtting approach for correlated acoustic source mapping,” J. Acoust. Soc. Am. 127(5), 2920–2930 (2010). 16 V. Fleury and J. Bulte, “Extension of deconvolution algorithms for the mapping of moving acoustic sources,” J. Acoust. Soc. Am. 129(3), 1417–1428 (2011). 17 A. Xenaki, F. Jacobsen, E. Tiana-Roig, and E. Fernandez-Grande, “Improving the resolution of beamforming measurements on wind turbines,” in Proceedings of the 20th International Congress of Acoustics, Sidney, Australia (2010), pp. 1–8. 18 A. Xenaki, F. Jacobsen, and E. Fernandez-Grande, “Improving the resolution of three-dimensional acoustic imaging with planar phased arrays,” J. Sound Vib. 331, 1939–1950 (2012). 19 J. J. Fuchs, “On the application of the global matched ﬁlter to DOA estimation with uniform circular arrays,” IEEE Trans. Signal Process. 49(4), 702–709 (2001). 20 C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, Philadelphia, 1995), 351 pp. 21 W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972). 22 L. B. Lucy, “An iterative technique for the rectiﬁcation of observed distributions,” Astronom. J. 79, 745–754 (1974). 23 E. G. Williams, Fourier Acoustics: Sound Radiation and Nearﬁeld Acoustic Holography (Academic Press, London, 1999), pp. 4–5, 224–230. 24 B. Rafaely, “Plane-wave decomposition of the sound ﬁeld on a sphere by spherical convolution,” J. Acoust. Soc. Am. 116(4), 2149–2157 (2004). E. Tiana-Roig and F. Jacobsen: Deconvolution for a circular microphone array 2089 Author's complimentary copy 7 Paper D Towards an enhanced performance of uniform circular arrays at low frequencies Elisabet Tiana-Roig 1, Antoni Torras-Rosell 2, Efren Fernandez-Grande 3, Cheol-Ho Jeong 4, and Finn T. Agerkvist 5 1, 3, 4, 5Acoustic Technology, Dep. Electrical Engineering, Technical University of Denmark, Ørsteds Plads 352, 2800 Kgs. Lyngby, Denmark 2DFM, Danish National Metrology Institute, Matematiktorvet 307, 2800 Kgs. Lyngby, Denmark ABSTRACT Beamforming using uniform circular arrays of microphones can be used, e.g., for localization of environmental noise sources and for conferencing. The performance depends strongly on the characteristics of the array, for instance the number of transducers, the radius and whether the microphones are mounted on a scatterer such as a rigid cylinder or a sphere. The beamforming output improves with increasing frequency, up to a certain frequency where spatial aliasing occurs. At low frequencies the performance is limited by the radius of the array; in other words, given a certain number of microphones, an array with a larger radius will perform better than a smaller array. The aim of this study is to improve the performance of the array at low frequencies without modifying its physical characteristics. This is done by predicting the sound pressure at a virtual and larger concentric array. The propagation of the acoustic information captured by the microphones to the virtual array is based on acoustic holography. The predicted pressure is then used as input of the beamforming procedure. The combination of holography and beamforming for enhancing the beamforming output at low frequencies is examined with computer simulations and experimental results. Keywords: Uniform circular array, Beamforming, Holography 1. INTRODUCTION Beamforming based on a uniform circular array of microphones (UCA) is a well-known method to localize sound sources around the array from 0 to 360◦ . In the present paper, the main concern is the improvement of the performance at low frequencies. In the recent years, various strategies have been suggested 1 [email protected] 2 [email protected] 3 [email protected] 4 [email protected] 5 [email protected] in this matter, such as the design of beamforming techniques other than delay-and-sum. For instance, circular harmonics beamforming is a clear example of how a beamforming technique can be designed to suit a particular geometry, in this case, the circular geometry. This technique is based on the decomposition of the sound ﬁeld in a series of coefﬁcients by means of a Fourier series. 1 With this technique most of the frequency range is improved. Another possibility is to ﬂush-mount the microphones on a rigid bafﬂe, such as a rigid sphere or a spherical cylinder. The effect of the scatterer has proved to be beneﬁcial compared to the case where the array microphones are suspended in the free space. 2–4 Yet another alternative is the use of deconvolution methods, which clean the beamforming map by means of iterative algorithms to ﬁnally recover the distribution of the sources present in the sound ﬁeld. These methods are very effective, but require high computational effort, in particular at low frequencies. Methods such as the Deconvolution Approach for the Mapping of Acoustic Sources 2 (DAMAS2), the Fourier-based Non-Negative Least Squares (FFT-NNLS) and the Richardson-Lucy (RL) have already been adapted to the circular geometry. 5,6 In all cases the performance improves with increasing frequency, up to a frequency where spatial aliasing occurs. The poor performance at the lowest frequencies is especially of concern with delay-and-sum because this presents an omnidirectional pattern, and therefore sources in this frequency range cannot be localized. Although the use of a scatterer improves its performance, it does not completely eliminate this problem. With deconvolution methods the low frequency problems can be resolved, even with beamforming patterns obtained originally with delay-and-sum. However if the same resolution is to be achieved in the entire frequency range, the lower the frequency the more the iterations needed. Or in other words, the deconvolution methods are less efﬁcient at low frequencies. In this article we suggest a simple method to improve the performance of UCAs at low frequencies, which does not imply the design of new beamforming techniques or a modiﬁcation of the geometry of the array. The basic idea is that for a speciﬁc number of microphones, a UCA with a larger radius will perform better at low frequencies than an array with a smaller radius, because the distance between the microphones will be larger, and the wavelengths corresponding to the low frequencies will be better captured. Inspired by this concept, one could measure the sound ﬁeld with a UCA, and by means of acoustic holography predict the sound pressure at a larger and virtual radius. The estimated pressure could be then used as input to the beamforming algorithm. A sketch of the procedure can be seen in Fig. 1. acoustical holography beamforming physical array virtual array beamforming map Fig. 1 – Sketch of the procedure for the calculation of the beamforming map. The pressure captured by a UCA is used for the prediction of the pressure at a larger and virtual array by means of acoustical holography. The predicted pressure is used as input of the beamforming procedure. The combination of holography and beamforming for the improvement of the performance at low frequencies is the subject of study in the present work, and this is examined by means of computer simulations and experiments. 2. 2.1. BACKGROUND THEORY Acoustic Holography Acoustic holography is a sound visualization technique that makes it possible to reconstruct the sound ﬁeld over a three-dimensional space based on a two dimensional measurement. Often, the measurement is performed close to the source, as in near-ﬁeld acoustic holography (NAH) to capture the evanescent waves for an enhanced spatial resolution. 7,8 However, in the present study we are concerned with the reconstruction of the sound ﬁeld in the far ﬁeld, prior to the beamforming processing. In acoustic holography, the measured sound ﬁeld is typically expanded into a series of basis functions from which the entire acoustic ﬁeld can be reconstructed. In this paper we focus on a circular space, making use of the fact that the sound ﬁeld can be predicted on different radii by means of the Bessel functions that account for the propagation in the radial direction. This approach is in a sense analogous to the one commonly used for NAH in spherical coordinates, 9–11 but in this case the radial functions are conventional Bessel functions, and the angular dependency is reduced to the azimuth only, as it follows from a circular harmonic expansion. It is worth noting that the inverse holographic reconstruction, i.e., when propagating towards the source, is an ill-posed problem that requires regularization. It has been shown that in the case of spherical NAH, truncation is an appropriate regularization procedure. 12 Similarly, truncation is adequate for the circular geometry. Let us consider that a plane wave that travels perpendicularly to the z-axis (i.e., the wavefronts are parallel to the z-axis) is captured by a UCA of radius R placed at the xy-plane, at z = 0. The sound pressure at the array can be represented in terms of solutions of the Helmholtz equation in a cylindrical coordinate system with origin at the center of the UCA. After applying the boundary conditions (basically that the sound ﬁeld at the origin must be ﬁnite), the pressure can be expressed as 7 p(kr, ϕ) = ∞ An Jn (kr)ejnϕ , (1) n=−∞ where k is the wavenumber and An is the coefﬁcient of the n’th term. As can be seen the angular dependency of the pressure is given by the circular harmonics ejnϕ , whereas the radial dependency is given by the Bessel functions Jn . Note that the time dependency e−jωt is omitted. The previous expression can be ideally used to determine a particular sound ﬁeld at any point by means of acoustic holography. For this purpose we need to determine the values of the coefﬁcients An . The pressure at the UCA, at r = R is p(kR, ϕ) = ∞ An Jn (kR)ejnϕ . (2) n=−∞ Making use of the orthogonality of the circular harmonics, 2π 1 ejnϕ e−jmϕ dϕ = δmn , 2π 0 (3) the coefﬁcients An can be retrieved by multiplying each side of Eq. (2) by a complex conjugated circular harmonic and integrating over the entire circle, from 0 to 2π. The resulting expression follows 2π 1 −jnϕ dϕ 2π 0 p(kR, ϕ)e . (4) An = Jn (kR) This expression implies a continuous integral of the sound pressure. However the pressure is known at discrete positions, because the sound ﬁeld is sampled with M microphones. Therefore, the integral must be approximated by means of a ﬁnite summation: 2π M −jnϕ p(kR, ϕ)e dϕ ⇒ αi p(kR, ϕi )e−jnϕi , (5) 0 i=1 where the coefﬁcients αi must equal 2π/M to keep the orthogonality properties of the circular harmonics given in Eq. (3) in discrete notation. Finally the coefﬁcients An are calculated as M 1 −jnϕi i=1 p(kR, ϕi )e M Ân = . (6) Jn (kR) By inserting this expression into Eq. (1), the sound pressure can be, in principle, predicted anywhere. As mentioned earlier, regularization is needed in practice. This is done by truncating the limits of the summation presented in Eq. (2) to certain values −N and N , p(kr, ϕ) = N Ân Jn (kr)ejnϕ . (7) n=−N It can be shown that a reasonable value of N follows N = kr + 1, where · is the ceiling function, up to a maximum value M/2 − 1. 2.2. Beamforming Beamforming is a signal processing technique commonly used in acoustics to localize sound sources. The beamforming technique used in the present study is the classical delay-and-sum beamforming, which is a very simple, but robust, method. It is based on delaying the signals of each array microphone by a certain amount and adding them together, to reinforce the resulting signal. Depending on the delay applied to the different microphones the array is steered to a particular direction. 13 Expressed in the spatial frequency domain the beamforming output follows b(kR, ϕ) = A M wi p̃(kR, ϕm )p∗ (kR, ϕm ), (8) m=1 where wm is the weighting coefﬁcient of the m’th microphone, p̃(kR, ϕm ) is the pressure measured at the m’th microphone, and p∗ (kR, ϕm ) is the theoretical complex conjugated pressure due to a plane wave with origin at ϕ. In the presence of a single source, when the beamformer is focused to the direction of the actual source, the maximum output is achieved. Ideally the beamformer would present a peak at the direction of the source and zeros elsewhere, but this is not the case due to the fact that the sound ﬁeld is captured at discrete positions with the microphones. This implies that the beamforming map presents a main lobe around the direction of the source and side lobes elsewhere. In case of an unbafﬂed UCA, the theoretical pressure is simply the closed form for a plane wave, ejk·r , at the array microphones, so the beamformer output is b(kR, ϕ) = M 1 p̃(kR, ϕm )ejkR cos(ϕm −ϕ) . M m=1 (9) Note that the weights wm have been set to 1 and A = 1/M , in order to have a maximum beamformer output equal to one when a plane wave of amplitude unity is present. Although the focus of the current study is the improvement of the performance at low frequencies, it should be mentioned that the operation of a beamformer is limited at high frequencies when the Nyquist sampling criterion is not fulﬁlled, i.e., at those frequencies whose corresponding wavelengths are less than twice the distance between two adjacent microphones. When aliasing occurs side lobes increase dramatically, becoming replicas of the main lobe in the worst case (the so-called aliased lobes). 2.3. Combining acoustic holography with beamforming The aim of this study is to combine acoustic holography and beamforming to improve the beamforming output at low frequencies. As shown in Fig. 1 the pressure is measured with a UCA of radius R and M microphones placed at ϕi . By means of holography the pressure is predicted at a larger and virtual array of radius Rv . In the present study the number of virtual microphones and their azimuth angles are the same as for the actual array. In fact, by means of simulations it has been observed that the position of the microphones is not that relevant as long as the distance between microphones remains constant. This makes sense since UCAs are practically shift-invariant, i.e., the beamforming pattern is the same regardless the focusing direction. 14 The pressure predicted with acoustic holography, which follows from evaluating Eq. (1) at (Rv , ϕi ), is then used as input of the beamforming procedure. The coefﬁcients Ân given in Eq. (6) are obtained with the pressure measured with the actual array microphones. Then the beamforming algorithm follows from inserting Eqs. (6) and (7) into Eq. (9), b(kRv , ϕ) = M M N 1 Jn (kRv ) j(n(ϕm −ϕi )+kRv cos(ϕm −ϕ)) p̃(kR, ϕi ) , e M 2 m=1 Jn (kR) i=1 (10) n=−N where N = kRv + 1, up to a maximum value M/2 − 1. 3. 3.1. RESULTS AND DISCUSSION Computer simulations The effect of combining beamforming and holography is analyzed in this section by means of computer simulations. A UCA like the one shown in Fig. 2 has been assumed. The array radius is R = 11.9 cm and it has 12 microphones. The array used for the simulations coincides with the array used for the measurements Fig. 2 – Prototype UCA of radius 11.9 cm and 12 microphones used for the measurements. presented in the next section. Following from the Nyquist sampling theorem, this array will present spatial aliasing from ca. 2.8 kHz. A plane wave generated at 180◦ is considered. The frequency range of interest is from 50 Hz to 2 kHz. A signal-to-noise ratio (SNR) of 30 dB at each array microphone due to uniformly distributed noise is assumed for the simulations to account for background noise. Beamforming has been performed in the usual way with the pressure at the array microphones following from Sec. 2.2. Besides this, by means of holography, the simulated pressure has been used to predict the pressure at a larger and virtual radius, twice the size of the actual array radius (2R) at the same azimuth angles. The predicted pressure at the virtual array has been used for the beamforming procedure as indicated in Sec. 2.3. In parallel, beamforming has been performed in ideal conditions (in absence of noise) with a UCA of radius 2R and 12 microphones. Note that for this case, as well as for the case of the virtual array, aliasing is expected from about 1.4 kHz; i.e., the operating frequency range is half the range of the array of radius R. For ease of understanding the resulting normalized beamforming outputs for a single frequency, in this case 400 Hz, are shown in Fig. 3. It can be seen that in all cases a main lobe around 180◦ is present, which Beamforming output [dB] 0 −10 −20 −30 −40 UCA – R virtual UCA – 2R UCA – 2R (w/o noise) −50 −60 0 60 120 180 240 Azimuth [◦ ] 300 360 Fig. 3 – Normalized beamforming outputs at 400 Hz obtained with three UCAs with 12 microphones: a real array of radius R = 11.9 cm, a virtual array of radius 2R and a real array of radius 2R. indicates that there is a source in this direction, as expected. However the main lobe obtained with the array with radius R is very wide, which can lead to confusion, whereas the virtual array and the array with radius 2R present a narrower main lobe, which makes the interpretation of the map clearer. The maps obtained for all frequencies are shown in Fig. 4. Note that Fig. 3 corresponds to a vertical cut of the beamforming maps at 400 Hz. $%& ! !"# Fig. 4 – Normalized beamforming maps obtained with three UCAs with 12 microphones. (Top) map obtained with an array of radius R = 11.9 cm, (middle) map obtained with a virtual array of radius 2R by means of combining holography and beamforming, and (bottom) ideal map obtained with an array of radius 2R. A source at 180◦ is assumed. For the small and the virtual array a SNR of 30 dB is accounted for. In all the cases the maps are omnidirectional at the lowest frequencies. With increasing frequency the patterns become more directive, unveiling a source at 180◦ . For the virtual array and the array of radius 2R aliasing is observed at about 1.4 kHz as expected. The virtual array is more directive at low frequencies compared to the actual array of radius R as expected from the theory. In fact the virtual array is omnidirectional in a narrower frequency range (half the range of the actual array) and from the upper frequency limit of the omnidirectional range it becomes more and more directive. Regarding the level of the side lobes, it can be seen that in both cases the levels are similar. The performance of the virtual array is very similar to the ideal performance of the array of radius 2R, up to the Nyquist sampling frequency where differences are observed. This shows that the virtual array behaves in this range as a real array with the same radius. The most apparent difference is the vertical line at 1103 Hz, which shows that for that frequency the beamforming is rather omnidirectional. This is caused by the fact 360 0 300 −10 240 −20 MSL [dB] Resolution [o ] that the Bessel function in the denominator of Eq. (10) is zero for n = 0 at that frequency. Alternatively to the beamforming maps, the performance of the array can be analyzed by means of two measures: the resolution and the maximum side lobe level (MSL). The resolution is the −3 dB width of the main lobe, whereas the MSL is the difference between the highest secondary lobe and the main lobe. In both cases, the smaller the values, the better. The resolution and the MSL are shown in Fig. 5. 180 120 −30 −40 UCA – R virtual UCA – 2R UCA – 2R (w/o noise) −50 60 −60 0 100 1000 Frequency [Hz] 100 1000 Frequency [Hz] Fig. 5 – Resolution (left) and MSL (right) obtained with UCA of radius R = 11.9 cm and 12 microphones, a virtual UCA with radius 2R, and an ideal UCA of radius 2R. A plane wave created by a source at 180◦ is assumed. For the small array and the virtual array a SNR of 30 dB is considered. These two measures conﬁrm that the virtual array behaves like a real array with the same dimensions, especially in terms of resolution, up to the frequencies where sampling error occurs. However, in terms of MSL the levels are slightly higher for the virtual array from about 800 Hz. The peak at 1103 Hz seen in both the resolution and MSL with the virtual array corresponds to the singularity observed previously in the beamforming map. 3.2. Experimental results Measurements with the prototype array with radius 11.9 cm and 12 microphones shown in Fig. 2 were carried out in an anechoic room of dimensions 12.1 m × 9.7 m × 8.5 m. The array microphones were 1/4 in. microphones Brüel & Kjær (B&K) Type 4935. A picture of the set-up is shown in Fig. 6. The array and the source placed in the far-ﬁeld of the array were controlled by a B&K PULSE analyzer. The loudspeaker was driven with a signal from the generator, pseudorandom noise of 1 s of period, 3.2 kHz of bandwidth, and 1 Hz of resolution. Each microphone signal was recorded with the analyzer, and after Fourier transforming they were postprocessed with beamforming. The resulting map can be seen in the top panel of Fig. 7. The data from the microphones were used to predict the pressure by means of acoustical holography at a virtual UCA with twice the radius of the array used for the measurements. With the predicted pressure used as input of the beamforming algorithm, the normalized map shown in the bottom of Fig. 7 was obtained. As can be seen the beamforming maps are very similar to the maps obtained with simulations in the previous section, although they appear slightly more blurry. The resolution and the MSL for the actual and the virtual arrays can be seen in Fig. 8. These two measures resemble the curves obtained with simulations. In terms of the resolution, the major differences are observed in the peak at 1103 Hz, which is more abrupt, and in the region where aliasing occurs, although this region is not of interest. For the MSL it can be seen that the curves appear slightly shifted towards high frequencies compared with the simulations, and that the MSL of the virtual array is a bit higher than expected in the range between 800 Hz and 1 kHz. In any case, the results prove that at the low frequencies the actual (and small) array beneﬁts from using holography to predict the pressure at a larger and virtual radius and combine it with beamforming. 4. FINAL REMARKS AND FUTURE WORK In this article it has been shown that the performance of delay-and-sum beamforming improves at low frequencies by combining acoustic holography with beamforming. The procedure is the following: the pressure captured by a UCA is used to predict the pressure at a virtual array with a larger radius by means of Fig. 6 – Measurement set-up. ! ! Fig. 7 – (Top) Normalized beamforming map obtained with a UCA with 12 microphones and radius 11.9 cm. The pressure captured by the array microphones is used to predict the pressure at a virtual UCA with twice the radius of the original array by means of acoustical holography. The predicted pressure is used to compute the normalized beamfroming map (bottom). 0 300 −10 240 −20 MSL [dB] Resolution [o ] 360 180 120 −30 −40 −50 60 −60 0 100 1000 Frequency [Hz] UCA – R virtual UCA – 2R 100 1000 Frequency [Hz] Fig. 8 – Resolution (left) and MSL (right) obtained with UCA of radius R = 11.9 cm and 12 microphones and a virtual UCA with radius 2R. A plane wave was created by a source at 180◦ . acoustical holography. The predicted pressure is then used as input to the delay-and-sum beamforming algorithm. The beneﬁts of using a virtual array have been proven by means of simulations and experimental results. The combination of holography and beamforming adds new features to UCAs without any additional cost. Speciﬁcally the array gains more ﬂexibility, for example at high frequencies the array measurements could be used directly for the beamforming procedure in the usual way, whereas at low frequencies acoustic holography could be used prior to beamforming to enhance the beamforming map at those frequencies. There are still some questions that need to be examined further, e.g., the effect of using other beamforming techniques different from delay-and-sum beamforming, the applicability of the method in noisier conditions, and how large the virtual array can be. In this sense virtual arrays with radius larger than twice the radius of the actual array have been tested. The results, which are not included in the article, reveal that both resolution and MSL become worse than expected with increasing the radius. However this statement needs additional investigation. The idea presented in the present study can be applied to other UCAs mounted on a scatterer such as a rigid cylinder, or to spherical arrays to map a three dimensional sound ﬁeld. ACKNOWLEDGMENTS The authors are thankful to Finn Jacobsen who was involved in this study at an early stage. REFERENCES [1] E. Tiana-Roig, F. Jacobsen, and E. Fernandez-Grande, “Beamforming with a circular microphone array for localization of environmental noise sources”, J. Acoust. Soc. Am. 128(6), 3535–3542 (2010). [2] G. Daigle, M. Stinson, and J. Ryan, “Beamforming with air-coupled surface waves around a sphere and circular cylinder (L)”, J. Acoust. Soc. Am. 117(6), 3373–3376 (2005). [3] H. Teutsch and W. Kellermann, “Acoustic source detection and localization based on waveﬁeld decomposition using circular microphone arrays”, J. Acoust. Soc. Am. 120(5), 2724–2736 (2006). [4] E. Tiana-Roig, F. Jacobsen, and E. Fernandez-Grande, “Beamforming with a circular array of microphones mounted on a rigid sphere (L)”, J. Acoust. Soc. Am. 130(3), 1095–1098 (2011). [5] E. Tiana-Roig and F. Jacobsen, “Acoustical source mapping based on deconvolution approaches for circular microphone arrays”, in Proceedings of Inter-noise 2011, Osaka, Japan (2011). [6] E. Tiana-Roig and F. Jacobsen, “Deconvolution for the localization of d sources using a circular microphone array,”, J. Acoust. Soc. Am. (To be published). [7] E. G. Williams, Fourier Acoustics: Sound radiation and near ﬁeld acoustic holography (Academic, London) (1999). [8] J. D. Maynard, E. G. Williams, and Y. Lee, “Nearﬁeld acoustic holography : I . Theory of generalized holography and the development of NAH”, J. Acoust. Soc. Am. 78(4), 1395–1413 (1985). [9] E. G. Williams, N. Valdivia, and P. C. Herdic, “Volumetric acoustic vector intensity imager”, J. Acoust. Soc. Am. 120(4), 1887–1897 (2006). [10] E. G. Williams and K. Takashima, “Vector intensity reconstructions in a volume surrounding a rigid spherical microphone array”, J. Acoust. Soc. Am. 127(2), 773–783 (2010). [11] F. Jacobsen, G. M. Pescador, E. Fernandez-Grande, and J. Hald, “Near ﬁeld acoustic holography with microphones on a rigid sphere (L)”, J. Acoust. Soc. Am. 129(6), 3461–3464 (2011). [12] A. Granados, F. Jacobsen, and E. Fernandez-Grande, “Regularized reconstruction of sound ﬁelds with a spherical microphone array”, in ICA 2013 Montreal, Montreal, Canada (2013). [13] D. Johnson and D. Dudgeon, Array Signal Processing Concepts and Techniques (Prentice Hall, Englewood Cliffs, New Jersey) (1993). [14] J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects”, J. Acoust. Soc. Am. 109(1), 185–193 (2001). Paper D BeBeC-2014-03 ENHANCING THE BEAMFORMING MAP OF SPHERICAL ARRAYS AT LOW FREQUENCIES USING ACOUSTIC HOLOGRAPHY Elisabet Tiana-Roig1 , Antoni Torras-Rosell2 , Efren Fernandez-Grande1 , Cheol-Ho Jeong1 and Finn T. Agerkvist1 1 Acoustic Technology, Dep. Electrical Engineering, Technical University of Denmark Ørsteds Plads 352, 2800 Kgs. Lyngby, Denmark 2 DFM, Danish National Metrology Institute Matematiktorvet 307, 2800 Kgs. Lyngby, Denmark ABSTRACT Recent studies have shown that the localization of acoustic sources based on circular arrays can be improved at low frequencies by combining beamforming with acoustic holography. This paper extends this technique to the three dimensional case by making use of spherical arrays. The pressure captured by a rigid spherical array under free-ﬁeld conditions is used to compute the expected pressure on a virtual and larger sphere by means of acoustic holography. Beamforming is then applied with the pressure predicted at the virtual array. Since the virtual array has a larger radius compared to the one of the physical array, the low frequencies (the ones with larger wavelength) are better captured by the virtual array, and therefore, the performance of the resulting beamforming system is expected to improve at these frequencies. The proposed method is examined with simulations based on delay-and-sum beamforming. In addition, the principle is validated with experiments. 1 INTRODUCTION Spherical arrays of microphones have been of interest in the last decade, because of the ability to measure in a three-dimensional sound ﬁeld [1, 2]. Typically, these arrays are suitable for sound source localization using beamforming [3–6] and for sound recording in higher order reproduction systems such as Ambisonics [7–9]. 1 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. Several strategies to improve the performance of beamforming systems have been suggested in the recent years. For example, it has been shown that arrays with ﬂushed-mounted microphones on a rigid sphere perform better compared to open (or transparent) spherical arrays [2, 10, 11]. Besides this, different beamforming techniques have been designed for this geometry [6]. Among them, phase-mode (or spherical harmonics) beamforming is of particular interest, because it exploits the spherical geometry by decomposing the sound ﬁeld in a series of spherical harmonics. Compared to the classical delay-and-sum beamforming, phase-mode beamforming presents a better directivity, at the expense of being more sensitive to noise [5]. In fact, delay-and-sum beamforming is a very robust technique, but it performs poorly at low frequencies, being omnidirectional in the worse case. Inspired by an article on uniform circular arrays presented recently in Ref. [12], the present article examines the possibility of enhancing the localization of noise sources with spherical arrays at low frequencies by combining spherical acoustic holography [13–15] and delay-andsum beamforming. The idea behind this concept is that for a given number of transducers, an array with a larger radius will perform better at low frequencies than a smaller array [2]. However, if one cannot change the geometry of the array, a simple solution to obtain a virtually larger array is illustrated in Fig. 1: the sound pressure is captured with a spherical array (rigid or transparent), and by means of acoustic holography the pressure is predicted at a virtual spherical array with larger radius. Finally the pressure at this virtual array is used for the beamforming process. The theory presented in this work is supplemented with simulations and measurements. acoustical holography beamforming physical array virtual array Figure 1: Procedure to obtain the beamforming map: the pressure captured by a spherical array is used to predict the pressure at a larger and virtual array with acoustic holography, and from this beamforming is carried out. 2 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. 2 ACOUSTIC HOLOGRAPHY AND BEAMFORMING WITH A SPHERICAL ARRAY 2.1 Acoustic holography Acoustic holography with a spherical array of transducers is a sound visualization technique that enables the reconstruction of a sound ﬁeld over the three-dimensional space, based only on the sound pressure or particle velocity captured with the array. Acoustic holography measurements are usually performed very close to the source and the reconstruction lies somewhere between the measurement position and the source, as in near-ﬁeld acoustic holography (NAH). However, in the present study, measurements in the far ﬁeld of the sound source are of concern. Let us consider a rigid spherical array with radius R centered at the origin of the coordinate system. The pressure at a point outside the array is given by the sum of the incident sound pressure and the scattered pressure due to the presence of the sphere, p = pinc + psca . (1) Given the spherical geometry, it makes sense to describe both pressures in terms of solutions of the Helmholtz equation in spherical coordinates (r, θ , ϕ) (θ being the inclination angle with respect to the z−axis and ϕ being the azimuth). The incident pressure, which is the one that would be measured if the scatterer was not present, must be described by means of spherical Bessel functions, because these are ﬁnite (even at the origin) [15, 16], pinc (kr, θ , ϕ) = ∞ n ∑ ∑ Amn jn (kr)Ynm (θ , ϕ), (2) n=0 m=−n where jn is the spherical Bessel function of order n, and the terms Ynm are the so-called spherical harmonics, 2n + 1 (n − m)! m (3) P (cos θ )e jmϕ , Ynm (θ , ϕ) = 4π (n + m)! n in which Pnm is the associated Legendre function. Note that the time dependence e− jωt is omitted. The scattered pressure must be described as outgoing waves, represented in this case by the spherical Hankel functions of the ﬁrst kind [17], psca (kr, θ , ϕ) = ∞ n ∑ ∑ (1) Bmn hn (kr)Ynm (θ , ϕ), (4) n=0 m=−n (1) where hn is the Hankel function of the ﬁrst kind and order n. The relationship between the coefﬁcients Amn and Bmn is given by the fact that the total radial velocity at the surface of the rigid sphere (r = R) is zero. From this condition it follows that Bmn = −Amn 3 jn (kR) (1) hn (kR) , (5) 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. (1) (1) where jn and hn are the radial derivatives of jn and hn . Therefore, the total pressure is ∞ n jn (kR) (1) hn (kr) Ynm (θ , ϕ). p(kr, θ , ϕ) = ∑ ∑ Amn jn (kr) − (1) hn (kR) n=0 m=−n (6) Since the pressure at the array is known, the coefﬁcients Amn can be retrieved by making use of the orthogonality relationship of the spherical harmonics, 2π π 0 0 μ Ynm (θ , ϕ)Yν (θ , ϕ)∗ sin θ dθ dϕ = δnν δmμ , (7) where δnν is the Kronecker delta function. Then, it can be shown that the coefﬁcients Amn are Amn = 2π π 0 0 p(kR, θ , ϕ)Ynm (θ , ϕ)∗ sin θ dθ dϕ jn (kR) − jn (kR) (1) hn (kR) (1) hn (kR) . (8) To implement this equation in practice, the integrals must be substituted by discrete summations, capable of fulﬁlling the discrete orthogonality relationship of the spherical harmonics, which has to be accounted for in the design of the array, M μ ∑ αiYnm(θi, ϕi)Yν (θi, ϕi)∗ = δnν δmμ for ν ≤ Nhol , n ≤ Nhol , (9) i=0 where i represents the ith microphone at position (R, θi , ϕi ), M is the number of sensors, and αi is an associated integration weight factor that guarantees orthogonality up to a certain order Nhol . Using the discrete orthogonality, the expression for the expansion coefﬁcients Amn results in [15] ∑M αi p(kR, θi , ϕi )Ynm (θi , ϕi )∗ . (10) Amn = i=1 jn (kR) (1) jn (kR) − (1) hn (kR) hn (kR) This relationship assumes that the highest order of spherical harmonics included in the sound pressure is lower or equal to Nhol . This is a reasonable assumption as long as the value kR is about Nhol . When this requirement is not met aliasing occurs in the coefﬁcients. The coefﬁcients Amn can be used to compute the incident pressure and the scattered pressure separately (see Eqs. (2) and (4)) and the total pressure (see Eq. (6)) at a point (r, θ , ϕ). 2.2 Beamforming Beamforming is a signal processing technique well used for localization of sound sources. There are several beamforming methods, but in the present study, delay-and-sum beamforming is chosen. Although this method is the oldest one, it is still widely used due to its robustness. It consists of delaying the signals of each array microphone by a certain amount and adding them together, to reinforce the resulting signal. Depending on the delay applied to the different microphones, the array is steered to a particular direction, whereas other directions are totally or partially attenuated [18]. Since in the current study the array is mounted on a rigid sphere, 4 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. it is simpler to express the beamforming output in the spatial frequency domain, because this allows us to compensate for the effect of the scatterer. In this domain the output is M b(kR, θ , ϕ) = B ∑ wi p̃(kR, θi , ϕi )p(kR, θi , ϕi |θ , ϕ)∗ , (11) i=1 where B is a scaling factor, wi is a weighting factor, p̃ is the measured pressure at the ith microphone, while p corresponds to the theoretical pressure at the ith microphone due to a source in the far-ﬁeld at (θ , ϕ). It can be shown that the pressure at (R, θi , ϕi ) due to a plane wave created by a source at (θ , ϕ) is [4] p(R, θi , ϕi ) = ∞ n ∑ ∑ Qn (kR)Ynm (θi , ϕi )Ynm (θ , ϕ)∗ , (12) n=0 m=−n where Qn is Qn (kR) = 4π(− j)n jn (kR) − jn (kR) (1) hn (kR) (1) hn (kR) . Making use of this expression the output of the delay-and-sum beamformer is M N i=1 n=0 b(kR, θ , ϕ) = B ∑ wi p̃(kR, θi , ϕi ) ∑ Qn (kR) n ∑ Ynm (θi , ϕi )Ynm (θ , ϕ)∗ (13) ∗ . (14) m=−n Note that the second summation has to be truncated at N for the real implementation. A reasonable value is N ≈ kR + 1. By making use of the addition theorem [19] that states that n 4π Ynm (θ , ϕ)Ynm (θq , ϕq )∗ , ∑ 2n + 1 m=−n (15) cos(ψq ) = cos θ cos θq + sin θ sin θq cos(ϕ − ϕq ), (16) Pn (cos ψq ) = where the beamformer output can be simpliﬁed: 2n + 1 Qn (kR)∗ Pn (cos ψi ). 4π n=0 M N bN (kR, θ , ϕ) = B ∑ wi p(kR, θi , ϕi ) ∑ i=1 (17) To have an output equal to one when a plane wave with amplitude unity is measured at the array, it is easy to show that the value of B should be B= 1 2 ∑M i=0 wi |p(kR, θi , ϕi |θ0 , ϕ0 )| , (18) where θ0 and ϕ0 can be any angle, because with the spherical array the shape of the beampattern is independent of the steering direction, as it is practically shift-invariant [2]. 5 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. 2.3 Beamforming with a virtual array As mentioned in the introduction, the goal of this study is to combine acoustic holography together with beamforming to improve the beamforming map at the low frequencies. To do this, the concept of virtual array has been presented; see Fig. 1. The pressure captured with a rigid spherical array is used to predict the pressure at a virtual spherical array with larger radius Rv , with virtual sensors placed at (Rv , θi , ϕi ). The number of virtual sensors and their azimuth and inclination is kept the same as in the physical array. At this point we can consider two possibilities: 1) A virtual transparent array or 2) a virtual rigid array. For the virtual transparent array the expression is simply the incident pressure given in Eq. (2), evaluated at r = Rv . For the case of the virtual rigid array we should create a virtual spherical scatterer at Rv . To do that the incident pressure with coefﬁcients Amn (the ones obtained with the physical array) would impinge on the virtual sphere creating a virtual scattered pressure distributed at the surface of the virtual array. In accordance with Eqs. (4) and (5), the scattered pressure at the virtual transducers would be ∞ psca (Rv , θi , ϕi ) = − ∑ n ∑ Amn n=0 m=−n jn (kRv ) (1) hn (kRv )Ynm (θi , ϕi ). (1) hn (kRv ) Then, the total pressure at the virtual rigid array (at r = Rv ) would be Nhol n jn (kRv ) (1) hn (kRv ) Ynm (θi , ϕi ). p(Rv , θi , ϕi ) = ∑ ∑ Amn jn (kRv ) − (1) hn (kRv ) n=0 m=−n (19) (20) Since a rigid array has beneﬁts compared to the transparent array, a virtual rigid spherical array is chosen for the current study. To sum up, the procedure for combining holography and beamforming is the following one: 1. With a rigid spherical array measure the pressure at the microphones, p(R, θi , ϕi ), where i = 1, . . . , M. 2. Insert p(R, θi , ϕi ) into Eq. (10) to retrieve the coefﬁcients Amn to be used for acoustic holography. 3. Insert Amn into Eq. (20) to obtain the predicted pressure at the virtual rigid array, p(Rv , θi , ϕi ). 4. Use p(Rv , θi , ϕi ) as input of the beamforming process, given in Eq. (17), but substituting R by Rv and using N = kRv + 1. In the present study, the chosen weighting factor, wi , equals the integration factor of the acoustic holography process, αi . 3 SIMULATION STUDY The focus of this section is to analyze the outcome of combining acoustic holography and beamforming by means of simulations. A rigid spherical array with radius R = 9.75 cm and 50 ﬂush-mounted microphones has been assumed. The characteristics of the array used for the simulations are the same of that used for the measurements (which will be presented in Sec. 4). 6 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. A picture of the array can be seen in Fig. 2. The location of the microphones and their associated integration weights result from an optimization procedure inspired by Ref. [20]. This procedure guarantees that the discrete orthogonality relation across microphone positions given in Eq (9) is valid up to order Nhol = 5, if kR ≤ Nhol . When this condition is not met, that is, above 2.8 kHz, aliasing occurs. Figure 2: Prototype of spherical array used in the measurements. The simulations assume a plane wave created at coordinates (θ , ϕ) = (90◦ , 90◦ ). However, the origin of the plane wave is not important because the array is practically shift-invariant. The frequency range of analysis contains the low frequencies up to 2 kHz. To account for the background noise, a signal-to-noise ratio (SNR) of 30 dB at each microphone due to uniformly distributed noise is considered. Following the procedure described in the previous section, acoustic holography is performed prior to beamforming, considering a virtual array with a radius 4 times larger than the radius of the physical array used to measure the actual sound ﬁeld. The normalized beamformer output obtained with the physical array using conventional beamforming and the output of the virtual array are shown in Fig. 3 for a frequency of 210 Hz. For ease of reference, the ideal beamformer output that would be obtained in absence of noise with a physical array of the same radius is also shown. As can be seen in the leftmost subﬁgure in Fig. 3, the output for the physical array is rather omnidirectional (the level is quite uniform). However the map is signiﬁcantly improved when using the pressures at the virtual array as the source located at (90◦ , 90◦ ) is successfully identiﬁed. Moreover, the beamformer map resembles the map of the physical array of the same radius under ideal conditions to a high extent. The discrepancies are caused by the noise assumed for the virtual array simulation. The performance is also quantiﬁed by two measures: the resolution and the maximum side lobe level (MSL). The resolution is the −3 dB width of the main lobe, whereas the MSL is the difference between the highest secondary lobe and the main lobe. For both measures, the smaller the values, the better. The resulting resolution for the azimuth and inclination angles, as well as the MSL, can be seen in Fig. 4, along the entire frequency range of interest. This ﬁgure includes the results with the physical array with radius R (black curve) and the ones obtained at four virtual arrays with radii 2R (continuous blue curve), 3R (continuous green curve), 4R (continuous red curve), and 5R (continuous cyan curve). The ideal curves obtained with arrays 7 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. !" Figure 3: Normalized beamforming outputs at 210 Hz obtained with three rigid spherical arrays: one with radius R = 9.75 cm (left), a virtual array with radius 4R (middle) that results from the pressure at the physical array with radius R via acoustic holography, and an array of radius 4R with absence of noise (right). A SNR of 30 dB was assumed at each microphone of the physical array with radius R. with radii 2R (dashed blue curve), 3R (dashed green curve), 4R (dashed red), and 5R (dashed cyan) for a SNR of inﬁnity are also depicted. In all cases it can be seen that both the resolution and the MSL are non-existent at low frequencies, meaning that the beamforming map is omnidirectional. From a particular frequency that depends on the array characteristics, the resolution improves, and sidelobes arise resulting in a certain MSL. The resolution for both azimuth and inclination angles is improved towards the low frequencies with increasing radius of the virtual array, in comparison with the physical array of radius R used to capture the signals. Interestingly the curves of the virtual arrays are very similar to the ones of the arrays with the same radius under ideal conditions, although some deviations that become stronger with increasing virtual radius are observed for the virtual arrays of radii 3R, 4R and 5R. On the other hand, the MSL of the virtual arrays is progressively shifted towards the low frequencies with increasing virtual radius. However, the MSL is more sensitive to noise than the resolution, as this measure worsens towards the high frequencies with increasing virtual radius, and the differences with the ideal MSL obtained with the physical arrays of the same radii in absence of noise (dashed curves) become larger. This is a consequence of the holography process itself, as the noise captured with the physical array is ampliﬁed with increasing distance to the reconstruction points, speciﬁcally for r > R. Therefore the reconstructed pressure deviates 8 5th Berlin Beamforming Conference 2014 360 180 R ph = R Rv = 2R Rv = 3R Rv = 4R Rv = 5R 240 180 150 RES Inclination [◦ ] 300 RES Azimuth [◦ ] Tiana-Roig et al. 120 60 120 90 60 30 0 0 100 1000 100 Frequency [Hz] 0 1000 Frequency [Hz] −5 MSL [dB] −10 −15 −20 −25 −30 −35 100 1000 Frequency [Hz] Figure 4: Resolution along the azimuth angle (top left), along the inclination angle (top right) and MSL (bottom) obtained by means of simulations with a physical array of radius R = 9.75 cm and 50 microphones (black continuous curve), as well as with four virtual arrays with radii 2R, 3R, 4R and 5R (blue, green, red and cyan continuous curves), that result from the pressure at the physical array with radius R via acoustic holography. The colored dashed lines show the results with arrays of the same radii as the virtual arrays, but with a SNR of inﬁnity. A plane wave was created at (θ , ϕ) = (90◦ , 90◦ ), and a SNR of 30 dB was assumed for the physical array with radius R. from the ideal one [15], having a direct impact on the beamforming map, particularly on the sidelobes. Although not shown here, simulations reveal that the ampliﬁcation of noise with an virtual array of radius 6R has dramatic inﬂuence on the beamforming map. In conclusion, the results from the simulations show that one could take advantage of virtual arrays using the appropriate radius for each frequency, determined by the MSL. For example, in the case of study, a virtual array with radius 5R is suitable up to 170 Hz, from this frequency to about 280 Hz, one with radius 4R would be preferable, from 280 Hz to 400 Hz, 3R is more adequate, whereas from 400 Hz to 800 Hz a virtual array with radius 2R seems better. Above 800 Hz the physical array should be used as it is. 4 MEASUREMENT RESULTS Measurements with a Brüel & Kjær (B&K) prototype array were carried out in a large anechoic chamber of about 1000 m3 . The array, which can be seen in Fig. 2, had 50 1/4 in. microphones 9 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. B&K Type 4935 ﬂush-mounted on a rigid sphere and 11 video cameras. Its radius, R, was 9.75 cm. The set-up, shown in Fig. 5, consisted of a loudspeaker placed in the far ﬁeld, at 5.8 m from the array. The loudspeaker height was 1 m and the array height 1.30 m. The array was placed such as the loudspeaker was detected at about (θ , ϕ) = (90◦ , 90◦ ). Figure 5: Measurement set-up. The loudspeaker was fed with white noise. The signal level was adjusted so that the SNR at the array microphones was about 30 dB for most of the frequency range, although the SNR at the low frequencies was lower. The signal at each microphone was recorded with a B&K Pulse analyzer for 10 s. The data was segmented in blocks of 1 s using a Hanning window and a 50% overlapping. For each block, the crosspectra between each microphone and a reference, which was chosen to be microphone number one, was computed. The averaged crosspectra were used as input to conventional delay-and-sum beamforming. Besides, the data were used to predict the pressure at several virtual radii Rv , at 2R, 3R, 4R and 5R, before applying beamforming, following the procedure indicated in Sec. 2.3. The resulting resolution for the azimuth and inclination angles, and the MSL with the physical and virtual arrays are shown in Fig. 6. Both performance indicators follow the same trend observed in the simulations shown in Fig. 4: the resolution improves towards the low frequencies with increasing virtual radius, and the MSL is shifted towards the low frequencies, although its level increases with increasing virtual radius. The reader should keep in mind that the simulations were carried out assuming a SNR of 30 dB, which was not exactly the case for the measurements, especially after postprocessing the data, and therefore, some deviations between simulations and results are expected. In this regard, the MSL curves obtained with the virtual arrays are slightly better than the simulated ones. These results conﬁrm that the concept of virtual array can be used to enhance the performance of the beamforming system at low frequencies, with an appropriate virtual radius depending on the frequency. In this study, this makes it possible to extend the lower frequency of the physical 10 5th Berlin Beamforming Conference 2014 360 180 R ph = R Rv = 2R Rv = 3R Rv = 4R Rv = 5R 240 180 150 RES Inclination [◦ ] 300 RES Azimuth [◦ ] Tiana-Roig et al. 120 60 120 90 60 30 0 0 100 1000 100 Frequency [Hz] 0 1000 Frequency [Hz] −5 MSL [dB] −10 −15 −20 −25 −30 −35 100 1000 Frequency [Hz] Figure 6: Resolution along the azimuth angle (top left), resolution along the inclination angle (top right) and MSL (bottom) obtained by means of measurements with a rigid spherical array of radius R = 9.75 cm and 50 microphones (black continuous curve), as well as the resulting resolution and MSL when considering four virtual spherical arrays with radii 2R, 3R, 4R and 5R (blue, green, red and cyan continuous curves), that result from the pressure at the physical array with radius R via acoustic holography. A plane wave was created at about (θ , ϕ) = (90◦ , 90◦ ). array down to about 55 Hz and 75 Hz in terms of resolution for the azimuth and the inclination angles, respectively, and 110 Hz in terms of MSL, in comparison with the original 250 Hz, 350 Hz and 550 Hz. The advantage of combining acoustic holography and beamforming is further illustrated in Fig. 7, where the beamforming map obtained with the physical array at 210 Hz is shown, together with the maps obtained with virtual arrays with radii 2R, 3R and 4R. The larger the virtual radius, the clearer the map becomes, making it possible to localize better the sound source at its actual position, (90◦ , 90◦ ). 11 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. Figure 7: Normalized beamforming outputs at 210 Hz measured with a rigid spherical array with 50 microphones and radius R = 9.75 cm (top left), and three virtual rigid spherical arrays of radii 2R (top right), 3R (bottom left), and 4R (bottom middle), that result from the pressure at the physical array with radius R via acoustic holography. 12 5th Berlin Beamforming Conference 2014 Tiana-Roig et al. 5 CONCLUSIONS Beamforming with spherical arrays is a powerful tool to localize and identify sound sources in a three-dimensional sound ﬁeld. However, the resulting maps are difﬁcult to interpret at low frequencies because such frequencies imply poor directivity, in particular with delay-and-sum beamforming. Inspired by the fact that the performance of the array would improve at low frequencies if a larger array was used, the present paper has presented a simple method that consists of predicting the pressure at a larger and virtual array by means of acoustic holography, and using it as input to the delay-and-sum beamforming procedure. The performance of this combined approach has been assessed with two performance indicators, namely the resolution and the MSL. Both simulations and experimental results show that the resolution improves with increasing virtual radius, at the cost of the MSL, which is more sensitive to noise. This implies that the maximum virtual radius appropriate for each frequency is mainly determined by the MSL. The use of holography prior to delay-and-sum beamforming offers new possibilities without any additional cost. At low frequencies the concept of virtual array can be used to improve the maps at such frequencies, while conventional beamforming can be applied directly at high frequencies. ACKNOWLEDGMENTS The authors would like to thank Karim Haddad, Brüel & Kjær, for lending us the spherical array used in the measurements. REFERENCES [1] T. Abhayapala and Darren B. Ward. “Theory and design of high order sound ﬁeld microphones using spherical microphone arrays.” IEEE ICASSP, II, 1949–1952, 2002. [2] J. Meyer and G. Elko. “A highly scalable spherical microphone array based on an orthonormal decomposition of the soundﬁeld.” IEEE ICASSP, II, 1781–1784, 2002. [3] M. Park and B. Rafaely. “Sound-ﬁeld analysis by plane-wave decomposition using spherical microphone array.” J. Acoust. Soc. Am., 118(5), 3094–3103, 2005. [4] B. Rafaely. “Plane-wave decomposition of the sound ﬁeld on a sphere by spherical convolution.” J. Acoust. Soc. Am., 116 (4), 2149–2157, 2004. [5] B. 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Rafaely. “Analysis and design of spherical microphone arrays.” IEEE Transactions on Speech and Audio Processing, 13(1), 2005. [11] B. Rafaely and B. Weiss. “Spatial aliasing in spherical microphone arrays.” IEEE Transactions on Signal Processing, 55(3), 2007. [12] E. Tiana-Roig, A. Torras-Rosell, E. Fernandez-Grande, C.-H. Jeong, and F. Agerkvist. “Towards an enhanced performance of uniform circular arrays at low frequencies.” Internoise 2013, Innsbruck, Austria, 2013. [13] E.G. Williams, N. Valdivia, P.C. Herdic and J. Klos. “Volumetric acoustic vector intensity imager.” J. Acoust. Soc. Am., 120(4), 1887–1897, 2006. [14] E.G. Williams and K. Takashima. “Vector intensity reconstructions in a volume surrounding a rigid spherical microphone array.” J. Acoust. Soc. Am., 127(2), 773–783, 2010. [15] F. Jacobsen, G. Moreno-Pescador, E. Fernandez-Grande, and J. Hald. “Near ﬁeld acoustic holography with microphones on a rigid sphere (L).” J. Acoust. Soc. Am., 126(6), 3461– 3464, 2011. [16] F. Jacobsen and P. Juhl. Fundamentals of general linear acoustics. Wiley, 2013. [17] F. Jacobsen, J. Hald, E. Fernandez-Grande, and G. Moreno. “Spherical near ﬁeld acoustic holography with microphones on a rigid sphere.” Acoustics’08, Paris, 2008. [18] D. Johnson and D. Dudgeon. Array Signal Processing Concepts and Techniques. Prentice Hall, 1993. [19] G. Arfken and H. Weber. Mathematical methods for physicists. Elsevier Academic Press, Burlington, MA, 2005. [20] I. Sloan and R. Womersley. “External systems of points and numerical integration on the sphere.” Advances in Computational Mathematics, 21, 107–125, 2004. 14 www.elektro.dtu.dk Department of Electrical Engineering Acoustic Technology Technical University of Denmark Ørsteds Plads Building 348 DK-2800 Kgs. Lyngby Denmark Tel: (+45) 45 25 38 00 Fax: (+45) 45 93 16 34 Email: [email protected]

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