IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 20, NO. 4, MAY 2012 1189 Transformation Between Uniform Linear and Spherical Microphone Arrays With Symmetric Responses Ines Hafizovic, Student Member, IEEE, Carl-Inge Colombo Nilsen, Member, IEEE, and Sverre Holm, Senior Member, IEEE Abstract—Spherical microphone arrays are suited for phase mode processing, which complements spatio-temporal processing and often simplifies both the understanding and development of different beamforming techniques. Since a spherical geom- etry cannot benefit directly from methods specially developed for uniform linear arrays, the weight and algorithm design for spherical arrays is mostly optimized and found numerically. One of the exceptions is a recent study that develops the well-known Dolph–Chebyshev weights in the phase mode framework. We show that for the case of a symmetric response, the spherical and linear array geometries are one-to-one, related through a herein developed linear transformation. With this transforma- tion, uniform linear array specific processing techniques become readily available in closed form for spherical array processing. Any uniform linear array weight design technique can be directly applied to spherical arrays. We also show how this transformation can be used to generate virtual uniform linear array data from spherical array data, enabling us to apply uniform linear array specific adaptive algorithms to spherical arrays. Index Terms—Phase mode processing, spherical arrays, weight design. I. INTRODUCTION L INEAR and spherical arrays are frequently applied in audio and speech acquisition, but they are traditionally treated as two very different geometries. For uniform linear arrays, there exists a vast range of array processing algorithms and weight design techniques with useful properties. These properties might be equally desirable for spherical arrays, but the techniques are generally not directly applicable. Therefore, spherical arrays are often processed using either delay-and-sum or phase-mode weights, which are two closed-form weighting functions that were compared and discussed by Rafaely in [1]. Alternatively, a constrained optimization method can be used to numerically generate weights for a desired response [2], [3]. The work of Koretz and Rafaely [4], stands out as an Manuscript received April 15, 2011; revised August 03, 2011; accepted September 30, 2011. Date of publication October 21, 2011; date of current version February 17, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Michael Lewis Seltzer. I. Hafizovic is with Squarehead Technology AS, 0484 Oslo, Norway (e-mail: ines@sqhead.com). C.-I. C. Nilsen and S. Holm are with the Department of Informatics, Univer- sity of Oslo, 0373 Oslo, Norway (e-mail: carlingn@ifi.uio.no; sverre@ifi.uio. no). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASL.2011.2173194 analytical design method for spherical arrays, showing how the Dolph–Chebyshev weights [5] can be formulated in the spher- ical harmonics domain. This is done by relating the Chebyshev polynomials in the weight calculation to the Legendre polyno- mials in the spherical array response. Finding similar explicit relations for other weights that are not directly based on Cheby- shev polynomials may not be trivial, and adapting of the entire body of uniform linear array theory to spherical arrays is not a straightforward task due to its sheer size. In this paper, we show that spherical arrays do not require redesign of uniform linear array methods, as long as the de- sired response is rotationally symmetric. We develop a linear transformation that can be used to transform any symmetric weighting window from the uniform linear array domain to the spherical array domain. This makes all existing closed-form weight expressions directly applicable to spherical arrays, such as the Dolph–Chebyshev, Taylor, or Kaiser weights. The transformation can also be performed on weights designed by uniform linear array optimization algorithms, such as Remez [6]. The transformation yields weights in the phase mode domain, which means that they can be applied to any well-sam- pled spherical array, regardless of the placement of its elements. We also demonstrate that the hermitian of the transformation makes it possible to transform spherical array data to the form of a uniform linear array, enabling the use of certain adaptive algorithms that assume uniform linear array geometry. In Section II, we present background theory on spherical ar- rays and phase mode beamforming. In Section III, we present the transformation between uniform linear and spherical array processing techniques. In Section IV, we present results and simulations, demonstrating the performance of the transforma- tion. Finally, we present our conclusions in Section V. II. BACKGROUND A spherical array is an array with its elements arranged on the surface of a sphere. Spherical microphone arrays were sug- gested by Meyer and Elko in [7], and by Abhayapala and Ward in [8]. In this paper, we will use spherical coordinates on the standard form with being the radial distance from the origin, being the zenith angle, and being the azimuth angle. Processing in the spatio-temporal domain is done, as for other geometries, by weighting and summing signals from different elements. A properly sampled spherical array can alternatively be processed in the so-called phase mode domain. In this do- main, the wavefield from a plane wave arriving at the points on a sphere of radius from a direction 1558-7916/$31.00 © 2011 IEEE