2017 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics October 15-18, 2017, New Paltz, NY ASYMMETRIC BEAMPATTERNS WITH CIRCULAR DIFFERENTIAL MICROPHONE ARRAYS Yaakov Buchris and Israel Cohen Technion, Israel Institute of Technology Technion City, Haifa 32000, Israel {bucris@campus, icohen@ee}.technion.ac.il Jacob Benesty INRS-EMT, University of Quebec 800 de la Gauchetiere Ouest, Suite 6900 Montreal, QC H5A 1K6, Canada benesty@emt.inrs.ca ABSTRACT Circular differential microphone arrays (CDMAs) facilitate com- pact superdirective beamformers whose beampatterns are nearly frequency invariant, and allow perfect steering for all azimuthal di- rections. Herein, we eliminate the inherent limitation of symmetric beampatterns associated with a linear geometry, and introduce an analytical asymmetric model for N th-order CDMAs. We derive the theoretical asymmetric beampattern, and develop the asymmetric supercardioid. In addition, an N th-order CDMAs design is pre- sented based on the mean-squared-error (MSE) criterion. Experi- mental results show that the proposed model yields optimal perfor- mance in terms of white noise gain, directivity factor, and front-to- back ratio, as well as more flexible nulls design for the interfering signals. Index Terms— Circular differential microphone arrays, asym- metric beampatterns, broadband beamforming, supercardioid. 1. INTRODUCTION Differential microphone arrays (DMAs) beamforming constitute a promising solution to some real-world applications involving speech signals, e.g., hands-free telecommunication [1]. DMAs re- fer to arrays that combine closely spaced sensors to respond to the spatial derivatives of the acoustic pressure field. These small-size arrays yield nearly frequency-invariant beampatterns, and include the superdirective beamformer [2, 3] as a particular case. The modern concept of DMAs employs pressure microphones, and digital signal processing techniques are used to obtain desired directional response [4–8]. Most of the work on DMAs deals with a linear array geometry, which is preferable in some applications involving small devices. Yet, linear arrays may not have the same response at different directions, and are less suitable for applica- tions like 3D sound recording where signals may come from any direction. In such cases, circular arrays are advantageous [9–13]. Previous works on DMAs, both for linear and circular geometries (e.g., [14, 15]), have considered only the case of symmetric beam- patterns, which is an inherent limitation of the linear geometry. Yet, in different array geometries like the circular geometry, asymmetric design may lead to substantial performance improvement. In this paper, we derive an analytical model for asymmetric cir- cular differential microphone arrays (CDMAs) which includes also the traditional symmetric model as a particular case. It is shown This research was supported by the Israel Science Foundation (grant no. 576/16), Qualcomm Research Fund and MAFAAT-Israel Ministry of Defense. that an asymmetric model achieves higher performances in terms of white noise gain (WNG), directivity factor (DF), and front-to-back- ratio (FBR) due to a more flexible design, which can better take into account the constraints regarding the null directions. We first derive the analytical asymmetric beampattern and then derive an asymmet- ric version for the supercardioid which is designed to maximize the FBR [4]. Additionally, a mean-squared-error (MSE) solution for an N th-order CDMA is developed, which enables perfect steering to every azimuthal direction. In the simulations section, we present a third-order asymmetric design and demonstrate its benefits with respect to the symmetric one. 2. SIGNAL MODEL We consider an acoustic source signal, X(ω), that propagates in an anechoic acoustic environment at the speed of sound, i.e., c ≈ 340 m/s, and impinges on a uniform circular array (UCA) of ra- dius r, consisting of M omnidirectional microphones, where the distance between two successive sensors is equal to δ =2r sin  π M  ≈ 2πr M . (1) The direction of X(ω) to the array is denoted by the azimuth an- gle θs , measured anti-clockwise from the x axis, i.e., at θ =0 ◦ . Assuming far-field propagation, the time delay between the mth microphone and the center of the array is τm(θs )= r c cos(θs − ψm),m =1, 2,...,M, where ψm = 2π(m−1) M is the angular posi- tion of the mth array element. The mth microphone signal is Ym(ω)= e j cos(θs−ψm) X(ω)+ Vm(ω),m =1, 2,...,M, (2) where  = ωr c , j = √ −1, ω =2πf is the angular frequency, f> 0 is the temporal frequency, and Vm(ω) is the additive noise at the mth microphone. In a vector form, (2) becomes y(ω)=[Y1(ω) ··· YM(ω)] T = d (ω,θs ) X(ω)+ v(ω), (3) where d (ω,θs ) is the steering vector at θ = θs , i.e., d (ω,θs )=  e j cos(θs−ψ 1 ) ··· e j cos(θs−ψ M )  T , (4) the superscript T is the transpose operator, the vector v(ω) is de- fined similarly to y(ω), and the acoustic wavelength is λ = c/f . It is assumed that the element spacing, δ, is much smaller than the wavelength of the incoming signal, i.e, δ ≪ λ, in order to approxi- mate the differential of the pressure signal.