Design of time-domain modal beamformer for broadband spherical microphone arrays

An approach to real-valued time-domain implementation of modal beamformer for spherical microphone arrays is proposed. The advantage of the time-domain implementation is that we can update the beamformer when each new snapshot arrives. Our technique is based on a modified filter-and-sum spherical harmonics domain (SHD) beamforming structure. The time series received at the microphones are converted into SHD data using spherical Fourier transform. The SHD data input to the steering unit and then feed a bank of finite impulse response (FIR) filters. The filter outputs are summed to produce the beamformer output time series. The FIR filters tap weights are optimally designed by making a compromise among multiple conflicting array performance measures such as directivity, mainlobe spatial response variation (MSRV), sidelobe level, and robustness. The design problem is formulated as a multiply constrained problem which is solved using second-order cone programming (SOCP). Results of simulations show good performance of the proposed time-domain SHD beamformer design approach. INTRODUCTION Spherical Harmonics Domain (SHD) beamforming technology has recently become an important research issue in threedimensional (3D) sound reception, sound field analysis for room acoustics, direction of arrival (DOA) estimation, and so on. A spherical array is more flexible than other array geometries for forming 3D beampattens, and the modal beamforming can be performed using the elegant spherical harmonics framework. Several modal beamforming approaches to spherical arrays have been studied, e.g., regular phase-mode pattern design [1], non-adaptive and adaptive multiple-null steering techniques [2], Dolph-Chebyshev pattern design approach [3], and optimal beamforming methods [4-6], etc. The studies presented above, however, are all based on a signal model in the frequency domain, where complex-valued modal transformation and array processing is employed. In order to achieve a broadband beamformer, which is usually required for speech and audio applications, the broadband array signals are decomposed into narrow frequency bins using the discrete Fourier transform (DFT) and each frequency bin is independently processed using a narrowband beamforming algorithm, and then an inverse DFT is employed to synthesize the broadband output signal. Since the frequency-domain implementation is performed with block processing, it might be unsuitable for time-critical speech and audio applications due to its associated time delay. It is well known that, in classical element space array processing, the broadband beamformer can be implemented in the time domain using the filter-and-sum structure [7]. The key point of the filter-and-sum beamformer design is how to calculate the FIR filters’ tap weights, in order to achieve the desired beamforming performance. The spherical array modal beamforming can also be implemented in the time domain with the real-valued modal transformation and the filter-and-sum beamforming structure. Meyer and Elko recently proposed a novel time-domain implementation structure for a spherical array modal beamformer [8], within the spherical harmonics framework. The real and imaginary parts of the spherical harmonics are employed as the spherical Fourier transform basis to convert the time domain broadband signals to the real-valued spherical harmonics domain, and the look direction of the beamformer can be tactfully decoupled from its beampattern shape. To achieve a frequency independent beampattern, Meyer and Elko proposed to employ inverse filters to decouple the frequency-dependent components in each signal channel, however, such kind of inverse filtering could damage the system robustness [1]. Moreover, since no systematic performance analysis framework has been formulated, all the mutually conflicting broadband beamforming performance measures, such as directivity factor, sidelobe level, and robustness, etc. cannot be effectively controlled. In this paper, an optimal broadband modal beamforming framework implemented in the time domain is presented. Our technique is based on a modified filter-and-sum modal beamforming structure. We derive the expression for the array response, the beamformer output power against both isotropic noise and spatially white noise, and the mainlobe spatial response variation (MSRV) in terms of the FIR filters’ tap weights. With the aim of achieving a suitable trade-off among multiple conflicting performance measures (e.g., directivity index, robustness, sidelobe level, mainlobe response variation, etc.), we formulate the FIR filters’ tap weights design problem to a multiply constrained optimization problem which is computationally tractable. 23-27 August 2010, Sydney, Australia Proceedings of 20th International Congress on Acoustics, ICA 2010 2 ICA 2010 BACKGROUND Spherical Fourier transform The standard Cartesian ) , , ( z y x and spherical ) , , (   r coordinate systems are used. Consider a unit magnitude plane wave impinging on a sphere of radius a from direction ) , ( 0 0 0     . The spherical harmonics domain expression of the sound pressure on the sphere surface at an observation point ) , ( s s s     is given by [9] ) , ( 0  ka pnm * 0 )] ( )[ (   m n n Y ka b . (1) where c k /   is the wavenumber with c being the sound speed, and f   2  being the temporal radian frequency with f being the frequency, the superscript * denotes complex conjugation, ) (ka bn is a function of array configuration, with available analytical expressions [9], and m n Y is the spherical harmonics of order n and degree m given by    im m n m n e P m n m n n Y ) (cos ] )! ( 4 /[ ] )! )( 1 2 [( ) (      , where ) (cos m n P denotes the associated Legendre function and 1   i . The sound pressure is spatially sampled at the microphone positions s  , M s , , 1   , where M is the number of microphones. The microphone positions are required to satisfy the following discrete orthonormality condition: ' ' 1 * ' ' )] ( )[ ( m m n n M s s m n s m n s Y Y           , (2) where ' n n  and ' m m  are the Kronecker delta functions, and s  is a real value depending on sampling scheme. For uniform sampling, which is assumed through this paper, in order that   4 2 1         S M s s d , we have M s / 4   . In order to compute up to N th order spherical harmonics, the number of microphones M should be larger than or equal to 2 ) 1 (  N to avoid spatial aliasing. We assume that the time series received at the sth microphone is ) (t xs and the frequency-domain notation is ) , ( s f x  . Its discrete spherical Fourier transform is given by * 1 )] ( [ ) , ( ) ( s m n M s s s nm Y f x f x      . (3) Using (3), the sound field is transformed from the spatial (element-space) domain into the spherical harmonics domain. Spherical harmonics domain beamforming Using spherical harmonics domain beamforming, the array output, denoted by ) ( f y , can be calculated as [4]: ) ( f y      N