A ROBUST AND SELF-RECONFIGURABLE DESIGN OF SPHERICAL MICROPHONE ARRAY FOR MULTI-RESOLUTION BEAMFORMING Zh˙ıyun L˙ı, Ramani Duraiswami ABSTRACT We describe a robust and self-reconfigurable design of a spherical microphone array for beamforming. Our approach achieves a multi-resolution spherical beamformer with per- formance that is either optimal in the approximation of de- sired beampattern or is optimal in the directivity achieved, both robustly. Our implementation converges to the optimal performances quickly while exactly satisfying the specified frequency response and robustness constraint in each iter- ation step without accumulated round-off errors. The ad- vantage of this design lies in its robustness and self-recon- figuration in microphone array reorganization, such as mi- crophone failure, which is highly desirable in online mainte- nance and anti-terrorism. Design examples and simulation results are presented. 1. INTRODUCTION Spherical microphone arrays are recently becoming the sub- ject of some study as they allow omnidirectional sampling of the 3D soundfield, and may find applications in multi- resolution soundfield capture and recreation [4]. In [5], a modal beamformer design in orthogonal beam-space was presented. In [3], we proposed a preliminary extension to allow relatively flexible microphone placements with min- imal performance compromise. Our main contributions in this paper are: 1) we balance the trade-off between accuracy and robustness to allow even more flexible layouts with op- timal performances; 2) we design a self-reconfigurable im- plementation to make the beamformer robust to microphone reorganization; 3) it seamlessly achieves multi-resolution beampatterns, either regular beampatterns or optimal direc- tivity, both robustly. The rest of this paper is organized into four sections. In section 2, we present the basic principle of spherical beam- former. In section 3, we formulate the beamformer for dis- crete array into a finite linear system for specified beam- forming direction. The solution optimally approximates the desired beampattern in least mean square (LMS) sense. In section 4, we optimize the accuracy of approximation under This work was partially supported by NSF Award 0205271. PIRL, UMIACS, Univ. of MD at College Park. Email: zli@cs.umd.edu; ramani@umiacs.umd.edu robustness constraint. To allow efficient implementation, we rewrite this constrained optimization problem into an ellipsoidal form under a linear and a spherical constraints. This naturally leads to a self-reconfigurable design in the form similar to [1] and [2], but with different inputs and optimization goals. Obviously, our implementation inherits their advantages, such as absence of round-off error accu- mulation, exact satisfaction of constraints, etc. Design ex- amples and simulations will be presented in section 5. 2. BACKGROUND The basic idea of the spherical beamformer is to make use of the orthonormality of spherical harmonics to decompose the soundfield arriving at a spherical array. Then the orthog- onal components of the soundfield are linearly combined to approximate a desired beampattern [5]. For a unit magnitude plane wave with wavenumber k, incident from direction ( k , k ), the complex pressure field on the surface ( s , s ,r s = a) of the rigid sphere is [6]: p t ( k , k , s , s ) =4 X n=0 i n b n (ka) n X m= n Y m n ( k , k )Y m n ( s , s ), (1) b n (ka)= j n (ka) j 0 n (ka) h 0 n (ka) h n (ka), (2) where a is the radius of the sphere, j n is the spherical Bessel function of order n, Y m n is the spherical harmonics of order n and degree m. denotes the complex conjugation. h n is the spherical Hankel function of the first kind. If we assume that the pressure recorded at each point ( s , s ,a) on the surface of the sphere s , is weighted by W m 0 n 0 ( s , s , ka)= Y m 0 n 0 ( s , s ) 4 i n 0 b n 0 (ka) . (3) Then making use of orthonormality of spherical harmonics: Z s Y m n ( s , s )Y m 0 n 0 ( s , s )d s = nn 0 mm 0 , (4) the total output from a pressure-sensitive spherical surface is: IV - 1137 0-7803-8874-7/05/$20.00 ©2005 IEEE ICASSP 2005