702 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 15, NO. 2, FEBRUARY 2007 Flexible and Optimal Design of Spherical Microphone Arrays for Beamforming Zhiyun Li, Member, IEEE, and Ramani Duraiswami, Member, IEEE Abstract—This paper describes a methodology for designing a ﬂexible and optimal spherical microphone array for beamforming. Using the approach presented, a spherical microphone array can have very ﬂexible layouts of microphones on the spherical sur- face, yet optimally approximate a desired beampattern of higher order within a speciﬁed robustness constraint. Depending on the speciﬁed beampattern order, our approach automatically achieves optimal performances in two cases: when the speciﬁed beam- pattern order is reachable within the robustness constraint we achieve a beamformer with optimal approximation of the desired beampattern; otherwise we achieve a beamformer with maximum directivity, both robustly. For efﬁcient implementation, we also developed an adaptive algorithm for computing the beamformer weights. It converges to the optimal performance quickly while exactly satisfying the speciﬁed frequency response and robustness constraint in each step. One application of the method is to allow the building of a real-world system, where microphones may not be placeable on regions, such as near cable outlets and/or a mounting base, while having a minimal effect on the performance. Simulation results are presented. Index Terms—Beamforming, beampattern, directivity index (DI), optimization, quadrature, spherical microphone array, white noise gain (WNG). LIST OF SYMBOLS Speed of sound. m/s in air. Laplacian operator. Frequency of wave. Angular position. Look direction. Wave incident direction. Observation direction. Observation point. Surface of a unit sphere. Pressure at the spatial location and the time . Plane wave incident from the direction with the wavenumber Complex pressure in frequency domain at the location for plane wave . Radius of the spherical microphone array. Manuscript received June 7, 2005; revised November 30, 2005. This work was supported in part by National Science Foundation Award 0205271. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Futoshi Asano. Z. Li was with the Perceptual Interfaces and Reality Lab, UMIACS, Univer- sity of Maryland, College Park, MD 20742 USA. He is now with Leica, San Ramon, CA 94583 USA (e-mail: zli@cs.umd.edu). Ramani Duraiswami is with the Perceptual Interfaces and Reality Lab, UMIACS, University of Maryland, College Park, MD 20742 USA (e-mail: ramani@umiacs.umd.edu). Digital Object Identiﬁer 10.1109/TASL.2006.876764 Spherical Bessel function of order . th order spherical Hankel function of the ﬁrst kind. Spherical harmonics of order and degree . Complex conjugation. Kronecker delta. Delta function. Quadrature coefﬁcient for at . Band limit of spatial frequency in terms of spherical harmonics orders. Order of beamformer. Maximum order of a robust beamformer. Regular beampattern of order . White noise gain. Directivity index. Actual beampattern looking at . Vector of complex pressure at each microphone position produced by the plane wave of unit magnitude from the desired beamforming direction Vector of complex weights for each microphone. One component of at order and degree Decomposition result of Othonormality errors caused by discreteness. Frequency-dependent scale factor in solving quadrature of orthonormalities. Coefﬁcient matrix of soundﬁeld expansion. Coefﬁcient vector of beampattern. Normalizing coefﬁcient to satisfy the speciﬁed frequency response. Speciﬁed minimum WNG. Parameter in Tikhonov regularization. Step size in adaptive implementation. I. INTRODUCTION S PHERICAL arrays of microphones are recently becoming a subject of interest as they allow three dimensional sam- pling of the soundﬁeld, and may have applications in sound- ﬁeld capture [14]. The paper [15] presented a ﬁrst analysis of such arrays, and showed how sound can be analyzed using them. 1558-7916/$25.00 © 2006 IEEE