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Alternatives to Spherical Microphone arrays: Hybrid Geometries Aastha Gupta & Prof. Thushara Abhayapala Applied Signal Processing CECS To be presented at ICASSP, 20-24 April 2009, Taipei, Taiwan Outline • Spherical harmonic analysis of wavefields • Spherical microphone arrays and limitations • Theory of Non-spherical (Hybrid) arrays • Combination of Circular Arrays • Conclusions Spherical Coordinates • : Elevation • : Azimuth • r: Radial Distance Spherical Harmonics Wave Propagation • Wavefields/ soundfields are governed by the wave equation. • Homogeneous fields. They could be due to scattering, diffraction, and refraction. • Basic solution can act as a set of building blocks Modal Analysis Arbitrary Soundfield – General Solution Spherical Microphone arrays • Spherical microphone arrays capture soundfield on a surface of a sphere. • Natural choice for harmonic decomposition. • Open Sphere [Abhayapala & Ward ICASSP 02] • Rigid Sphere [Meyer & Elko, ICASSP 02]. • Bessel zeros are a problem in open spheres. • Rigid spheres are less practical for low frequencies. • Strict orthogonality condition on sensor locations. Problem • Spherical harmonic decomposition of wavefields/soundfield is a great way to solve difficult array signal processing problems. • How can we estimate spherical harmonics from an array of sensors? • What are the alternatives to spherical arrays? Circular Microphone Arrays Let be the soundfield on a circle at Spherical Harmonics Hybrid Arrays We multiply by and integrate with respect to over to get where Circular harmonic Decomposition Left hand side of this equation is a weighted sum of soundfield coefficients for a given . It can be evaluated for where the truncation number is dependent on the radius of the circle. We show how to extract from a number of carefully placed circular arrays Sampling of Circles In practice, we can not obtain soundfield at every point on these circles. Thus, needs sampling According to Shannon's sampling theorem for periodic functions, can be reconstructed by its samples over with at least samples . We approximate the integral in by a summation: are the number of sampling points on the circle . Least Squares Suppose our goal is to design a Nth order microphone array to estimate (N + 1)^2 spherical harmonic coefficients. By placing Q ≥ (N + 1) circles of microphones on planes given by (rq, θq), q =1, . . . ,Q, for a specific m, we have where The harmonic coefficients can be calculated by solving the simultaneous system of equations or evaluating a valid Moore- Penrose inverse of the matrix Legendre Properties Bessel Properties Infinite summation can be truncated by using properties of Bessel functions: Number of Coefficients Combination of Circles • Consider two circles placed at and where 0 ≤ ≤ . • That is one circle above the x-y plane and the second circle below the x-y plane but equal distance rq from the origin • The circular harmonics of the soundfield on the circle on or above the x-y plane are given by Above xy plane Below xy plane Circular Harmonic combination • Right hand side is a weighted sum of coefficients for a specific • For l=0 the sum only consists of a weighted sum of with n is even. • For l=1 the sum only consists of a weighted sum of with n is odd. Findings so far.. •Thus, we can separate odd and even spherical harmonics from the measurement of soundfield on two circles placed on equal distance above and below the x-y plane. •This is a powerful result, which we can use to extract spherical harmonics from soundfield measurements on carefully placed pairs of circles. Odd Coefficients Guidelines to choose systematically such that is always non singular: There are specific patterns of the normalized associated Legendre function when n−|m| =1, 3, 5... There are number of different range of elevation angles we can choose for θq. Note that θq could be same for all q or a group of values. For a Nth order system, there are N(N +1)/2 odd spherical harmonic coefficients from total of (N +1)2 coefficients. We use N (for N odd) or N − 1 (for N even) pairs of of circular microphone arrays. We choose the radii of these circles as Findings •With this choice, the soundfield at frequency k on a circle with rq is order limited to due to the properties of Bessel functions. This property limits the higher order components of the soundfield present at a particular radius rq. Also, the lower order components are guaranteed to be present due to the choice of radii in (14) which avoids the Bessel zeros. • Thus, selecting rq and θq from the legendre and Bessel plots, we can guarantee that is non singular. Normalised Legendre function-odd Even Coefficients Suppose, we have selected Q pairs of such that when is even. We have following guidelines to choose systematically such that is always non singular: As in the case of odd coefficients, we can choose range of values for θq, which plots for even. • Note that on the x-y plane (θ = π/2), all even associate Legendre functions are non zero. Thus, placing circles on the x-y plane seems to be an obvious choice to estimate even coefficients, where we do not need pairs of circles. Normalised Legendre Function-even Findings Depending on our choice, we can design different array configurations, which will be capable of estimating spherical harmonic coefficients. For a Nth order system, we place N/2 (N even) or (N+1)/2 (N odd) circles on the x-y plane. We choose the radii of these circles based on the bessel plots Simulations-5th Order System We first place four circular arrays (two pairs) with 11, 11, 7 and 7 microphones at (4/ko, π/3), (4/ko, π − π/3), (5/ko, π/6), and (4/ko, π − π/6). Then we place a pair of microphones at (5/ko, 0) and (5/ko, π). This sub array consists of 38 microphones are designed to calculate all odd spherical harmonics up to the 5th order (total of 15 coefficients). We place three circular arrays on the x-y plane together with a single microphone at the origin to complete the design. We have 7, 11, and 13 microphones in three arrays on x-y plane at radial distances 2/ko, 4/ko, and 5/ko, respectively. Simulations •Test Octave - 3000Hz to 6000Hz (kℓ = 55.44) •40dB signal to noise ratio (SNR) at each sensor, where the noise is additive white Gaussian (AWGN). •Estimate all 36 spherical harmonic coefficients for a plane wave sweeping over the entire 3D space and for all frequencies within the desired octave. •We plot the real and imaginary parts of against the azimuth and elevation of the sweeping plane Real part of the estimated harmonic coefficient α54 for a plane wave sweeping over entire 3D space: (a) Theoretical pattern wave for lower, mid, (b), (c), (d) are at frequencies 3000, 4500 and 6000Hz, and upper end of the frequency respectively, and all at SNR= 40dB band. Conclusions Spherical harmonic decomposition is a useful tool to analyse 3D soundfields. Spherical arrays have inherent limitations that make them unfeasible for practical implementation. Circular microphone arrays and hybrid arrays need carefully designing based on underlying wave propagation and theory. Combining circular arrays enables us to calculate odd and even harmonics independently, providing cleaner more accurate results. Thanks & Questions/Feedback ?