EIGEN-BEAM PROCESSING FOR DIRECTION-OF

EIGEN-BEAM PROCESSING FOR DIRECTION-OF-ARRIVAL ESTIMATION USING SPHERICAL APERTURES Heinz Teutsch and Walter Kellermann Multimedia Communications and Signal Processing University of Erlangen-Nuremberg Cauerstr. 7, 91058 Erlangen, Germany Email: {teutsch,wk}@LNT.de

In [1] a method of decomposing a 3D wave field into spherical harmonics (’eigen-beams’) by using spherical microphone arrays was introduced. It was further shown how the eigen-beams can be used for beamforming applications by means of a modal beamformer structure. This paper introduces a method for the unambiguous localization of multiple wideband acoustic sources in 3D space using the eigen-beams of [1]. It was shown in [1] that the response of a continuous spherical aperture of radius R to a plane wave impinging from (θ, φ), can be expressed as, Fnm (kR, θ, φ) = in bn (kR)Ynm (θ, φ)∗ = in bn (kR)

s

2n + 1 (n − m)! m P (cos θ)e−imφ , 4π (n + m)! n

(1)

where (·)∗ denotes the conjugate complex operation, i2 = −1, k is the wavenumber, Ynm (·) is the spherical harmonic of order n and degree m, Pnm (·) is the associated Legendre function of order n and degree m, and

bn (kR) =

  jn (kR)

open spherical aperture (1)0 . hn (kR) (1)  jn (kR) − hn (kR) rigid spherical aperture 0 jn (kR) (1)

Here, jn (·) denotes the spherical Bessel function and hn (·) denotes the spherical Hankel function of the first kind. (·)0 denotes the derivative with respect to the argument. We now use these eigen-beams for acoustic source localization starting from a recurrence relation for the associated Legendre functions [2], 2m cot θPnm (cos θ) = (m − n − 1)(n + m)Pnm−1 (cos θ) − Pnm+1 (cos θ).

(2)

The idea of using recurrence relations for modal processing applied to circular apertures has been introduced in [3]. The following discussion follows the path shown in [3] and modifies it for its application to spherical apertures. By keeping the order fixed to n = N , an eigen-space array vector is defined as F N (θ, φ) = [FN−N (θ, φ), FN−N +1 (θ, φ), . . . , FN0 (θ, φ), . . . , FNN (θ, φ)]T , (l)

where [·]T denotes transposition. Now, M = 2N − 1 eigen-space subarray vectors are extracted as a N = ∆(l) D0 F N (θ, φ), l = −1, 0, 1, where ∆(−1) , ∆(0) , and ∆(1) extract the first, middle, and last M elements from D 0 F N (θ, φ), and where D 0 = diag{(−1)N , . . . , (−1)0 , 1, . . . , 1N }.

By considering d plane waves impinging on the spherical aperture, the recurrence relation, Eq. (2) can be written as (0)

(−1)

D 1 AN = D 2 A N

(1)

Φ + D 3 A N Φ∗ ,

(3)

where (l)

AN

(l)

(l)

= [aN (θ1 , φ1 ) | . . . | aN (θd , φd )], l = −1, 0, 1

Φ = diag{µ1 , . . . , µd }, µs = tan θs · e−iφs , s = 1(1)d D1

=

D2

=

D3

=

(4)

−(N −1) N −1 1 2 diag{(N − 1)/cN , . . . , 1/c−1 } N , 0, 1/cN , . . . , (N − 1)/cN ν−1 diag{[−(N − 1) − ν] · (N − ν)/cN }, ν = −(N + 1)(1)(N − 1) −(N −2) 1 N diag{1/cN , . . . , 1/c−1 N , −1, 1/cN , . . . , 1/cN }

and cm N denotes the square-root term in Eq. (1). Since the eigen-space array matrix AN is related to the signal subspace matrix S N by a non-singular matrix T , we get (l) S N = ∆(l) S N , l = −1, 0, 1. The signal subspace matrix can be obtained by determining the d principal eigenvalues of the data covariance matrix. This covariance matrix is estimated based on the decomposed output (spherical harmonics) of the spherical aperture. Equation (3) can then be expressed as  T  Ψ (0) D1 S N = E , (5) ΨH where (·)H denotes the Hermitian operation and E Ψ

(−1)

= [D 2 S N = T

−1

(−1)∗

| D3 S N

]

ΦT

By solving Eq. (5) in a least-squares sense, an estimate for Ψ can be obtained. Finally, by realizing that the complex eigenvalues of Ψ are the entries of Φ, the azimuth of the impinging plane waves, φ s , s = 1(1)d, can be readily identified by the phase value of these eigenvalues. Similarly, the direction-of-arrival in elevation, θ s , s = 1(1)d, simply correspond to the inverse tangent of the magnitude of the eigenvalues, see definition of µ s in Eq. (4). Several observations are of interest. Firstly, it can be shown that for the number of spherical harmonics to be extracted from the wave field, N ≥ d + 1 must hold. Secondly, the algorithm derived above can be identified as ESPRIT-like. Thirdly, source localization using eigen-beams obtained by wave field decomposition along the surface of a spherical aperture is inherently frequency-independent. In order to arrive at a realizable system the continuous aperture needs to be sampled by discrete microphones. Unfortunately, the optimum microphone arrangement on the sphere for extracting the eigen-beams with minimum modal aliasing is not obvious. First simulations suggested, however, that best results are obtained by sampling the (θ, φ)-grid by equi-spaced microphones in φ and microphones placed on Gaussian nodes with appropriate weights in θ [4]. In order to decompose the wave field into (N + 1)2 spherical harmonics using this configuration, a minimum of N + 1 and 2N microphones in θ and φ, respectively, are required. This sampling grid can then be written as [5],   q −1 (θp , φq ) = cos (gp ), 2π , 0 ≤ p ≤ N, 0 ≤ q < 2N, (6) 2N where gp denotes the position of the Gaussian nodes. Using this particular arrangement allows for an efficient implementation of the decomposition, known as the fast spherical harmonics transformation [5]. References [1] J. Meyer, G.W. Elko, “A Highly Scalable Spherical Microphone Array Based on a Orthonormal Decomposition of the Soundfield”, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, 2002, vol. II, pp. 1781–1784. [2] G. Arfken, Mathematical Methods for Physicists Academic Press, New York, 3rd ed., 1985. [3] C.P. Mathews, M.D. Zoltowski, “Eigenstructure Techniques for 2-D Angle Estimation with Uniform Circular Arrays”, IEEE Transactions on Signal Processing, vol. 42, No. 9, pp. 2395–2407, September 1994. [4] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, 1972. [5] M.J. Mohlenkamp, ”A Fast Transform for Spherical Harmonics”, Journal of Fourier Analysis and Applications, 5(2/3):159-184, 1999.