Computations of the Directivity of Spherical Microphone Arrays

Transcription

1 paper ID: 132 /p.1 Coputations of the Directivity of Spherical Microphone Arrays C.F. Cardoso, P.A. Nelson Institute of Sound and Vibration Research, University of Southapton, University Road, Highfield, Southapton, SO17 1BJ, ABSTRACT: The ai of this research is to create a icrophone array on a rigid sphere that could be used for teleconferences and for recording of usic with the ai of five-channel loudspeaker surround sound reproduction. Experients were undertaken, within a MATLAB environent, by running coputer siulations that tested the directivity of a rigid sphere icrophone array in both near and far field. The siulations perfored calculated the directivity of three different icrophone arrays with 128 icrophones: placed circularly around a rigid sphere, placed uniforly distributed in a rigid sphere and placed randoly distributed in a rigid sphere. It is shown that the directivity of the array of icrophones placed uniforly in the sphere show a better directivity than the other arrays since it shows a narrower ainlobe and less sidelobes. 1. INTRODUCTION Interaction between parties at reote locations can be ade ore effective and efficient if it akes use of as uch perceptual inforation as possible. In audio, this perceptual inforation is achieved by reproducing the voices of the talkers in a way that they are perceived as eanating fro the positions associated with the visual stiuli [1]. A icrophone array is a network of sensors whose output signals are cobined in order to enhance the sound waves coing fro a desired direction [2]. Early techniques deonstrated that a icrophone array can be focused, erely by the way in which the signals are processed, with no echanical adjustent [3]. It also has the advantage of enabling hands-free acquisition of speech with little constraint on the user. The decreasing cost of electret icrophone eleents and digital signal processors, which allow adaptive processing and therefore high perforance [4], is aking the application of icrophone array technology increasingly feasible. Microphone array research has been undertaken specifically with regard to spherical icrophone arrays [5,6] capable of steering beas in ultiple directions [7]. Using a three diensional configuration eliinates forward-backward grating lobes which occur in two-diensional arrays, also linear and two-diensional arrays have liitations such as a restricted scan field and a narrow operational frequency bandwidth [8]. In addition, by being spherical full scan coverage is achieved, its aperture direction is independent aking the bea-width and gain of the antenna constant for all scan angles [8]. Therefore spherical icrophone arrays enable sound recordings with high spatial detail [5]. Recent work deonstrated that the acoustic response of arrays where the icrophones are ounted on a scattering body, for instance a rigid sphere, leads to a highly desirable array directivity [9,10,11]. This paper exaines the perforance of three different icrophone arrays ounted

2 paper ID: 132 /p.2 on a rigid sphere, by running siulations in a MATLAB environent, in both near and far field. The beaforer used in the study is a focused beaforer that enhances the icrophone directivity. A focused beaforer attepts to ap the distribution of acoustic source strength associated with a given source distribution by changing the beaforing operation in accordance with the assued source position. 2. MICROPHONE ARRAY Figure 1, shows the block diagra that illustrates the siulations perfored. A point onopole source radiates sound to the icrophone array. When the signal arrives at the icrophones it is scattered due to the presence of a rigid sphere. The icrophone array then processes the resulting signal through an array of filters that ensure the tie alignent of the signals at the suing junction. The output is then passed through an inverse filter with the goal of obtaining an output ipulse response that is an accurate replica, but delayed version, of the source signal. q(n)... g(n) 2 g(n) 1 g(n)... g( 2 -n) g( 1 -n) g(-n) Σ f(n) y(n) Figure 1 Syste used to perfor the siulations 2.1 Scattering of Sound by a Rigid Sphere When an acoustic wave has an obstacle in its path, in addition to the undisturbed plane wave there is a scattered wave, spreading out fro the obstacle in all directions, interfering with the plane wave. The scattered wave is the difference between the actual sound wave and the undisturbed wave, which would be present if an obstacle was not there [12]. The scattered field p s is given by, jρ k p s = ) 4π 0 (2 + 1) b ( j( ka) j n ( ka)) P (cos φ = 0 (1)

3 paper ID: 132 /p.3 where ρ 0 is the air density, k=ω/c 0 where c 0 denotes the sound speed and ω is the angular frequency, j and n are the th order spherical Bessel functions of the first and second kind respectively, a is the distance fro the sphere centre to the receiver, P is the th order Legendre polynoial, φ is the angle subtended between the receiver point, centre of sphere and the source. The constant b needs to be chosen in order to satisfy the boundary condition, which states that the total pressure gradient ust be zero on the sphere surface. Thus, b j = 1 ( kρ ) jn( kρ ) ' ' j n ( ka) / j ( ka) ( ) (2) where ρ is the distance fro the centre of the sphere to the source and the prie denotes differentiation with respect to the arguent of the function. 2.2 Signal Processing for Accurate Replication of Source Strength Tie Histories Beaforing ethods can be used to enhance the aplitude of the desired speech signal while reducing the effects of the interfering signals and reverberation. The beaforer used in the syste is, w H 1 * [ g g + ] g * opt = β (3) where g is the su of the free field Green function with the scattered field Green function and β is a regularisation paraeter introduced in Equation 3 to iprove the tie doain response [11,13]. It should also be noted that this solution for the optial beaforer is essentially that used in the focused beaforer where an attept is ade to ap the distribution of acoustic source strength associated with a given source distribution by changing the assued vector g of Green functions in accordance with the assued source position [14]. The filter f in the syste, illustrated in Figure 1, only represents the ter [g H g] -1, since the filters g * are used to operate on the icrophone outputs. Adapting properties in [15] a tie reversal version of the Green function can be used to calculate g *. Thus G * (e jω )= G(e -jω )=FT{g(-n)} (4) The coplex conjugate of the Green function equals the Fourier transfor of the tie reversed Green function in the tie doain. Therefore the value of g * is calculated by taking the filter g ipulse responses, reversing the and applying a delay, to avoid probles caused by non-causal systes. 3. SIMULATIONS In array theory the coon rule of thub to deterine the distance at which the far field approxiations begin to be valid is R=2L 2 /λ, where L is the sphere diaeter, which was

4 paper ID: 132 /p.4 chosen to be 18c, and λ is the acoustical wavelength [16]; it was decided to plot the directivity up to 20kHz. Therefore 0.5 was chosen to be the arbitrary distance between the source and the centre of the arrays in the near field, and in the far field, 5 was the distance used. All the directivities were calculated with increents of 1º in the source, using 4096 saples at a sapling frequency of 41kHz. All directivities are shown for ka where k=ω/c 0 where c 0 denotes the sound speed and ω is the angular frequency, a is the distance fro the centre of the sphere to the receivers, therefore the sphere radius. Three siulations were run where it was tested the y-z plane directivity of arrays of icrophones in a rigid sphere: when the icrophones were placed in a circular array around the x-y plane, and when the icrophones were placed uniforly and randoly distributed in the sphere. Figure 2 Microphone array placed in the x-y plane and sources shown around the y-z plane Figure 3 y-z plane directivity for the source at 0.5 fro an array of 128 icrophones placed in the x-y plane

5 paper ID: 132 /p.5 Figure 4 y-z plane directivity for the source at 5 fro an array of 128 icrophones placed in the x-y plane Figure 5 y-z plane directivity for the source at 0.5 fro an array of 128 icrophones placed uniforly in a 9c radius rigid sphere

6 paper ID: 132 /p.6 Figure 6 y-z plane directivity for the source at 5 fro an array of 128 icrophones placed uniforly in a 9c radius rigid sphere The positioning of sensors in a rando fashion helps to avoid the periodicity inherent in nonrando arrangeents, thus possibly flattening the co-array values [17]. Figure 7 y-z plane directivity for the source at 0.5 fro an array of 128 icrophones placed randoly in a 9c radius rigid sphere

7 paper ID: 132 /p.7 Figure 8 y-z plane directivity for the source at5 fro an array of 128 icrophones placed randoly in a 9c radius rigid sphere 4. CONCLUSIONS Nuerical experients were undertaken, within a MATLAB environent, which copared the directivity perforance of different icrophone arrays. It was shown that when the icrophones are placed uniforly around a sphere the directivity is better, since the ainlobe is narrower and there are less sidelobes than when the icrophones are placed randoly or circularly around the sphere, the iproveent goes up to 30dB. REFERENCES [1] M.J. Evans, A.I. Tew, J.A.S. Angus, Spatial Audio Teleconferencing - Which Way is Better?, The Fourth International Conference on Auditory Display, Palo Alto, California Noveber, [2] Y. Mahieux, G. Le Tourneur, A. Saliou, A Microphone Array for Multiedia Workstations, Journal of the Audio Engineering Society 44 (5), May, [3] J. Billingsley, R. Kinns, The Acoustic Telescope, Journal of Sound and Vibration 48 (4): , 1976.

8 paper ID: 132 /p.8 [4] J.L. Flanagan, D.A. Berkley, G.W. Elko, J.E. West, M.M. Sondhi., Autodirective Microphone Systes, Acustica 73 (2): 58-71, February, [5] B. Rafaely, Plane Wave Decoposition of the Sound Field on a Sphere by Spherical Convolution, ISVR Technical Meorandu No. 910, University of Southapton, May, [6] AES Staff Writer, Novel Surround Sound Microphone and Panning Techniques, A digest of selected recent AES convention and conference papers, Journal of the Audio Engineering Society 52 (1/2), January / February, [7] B.N. Gover, J.G. Ryan, M.R. Stinson, Microphone Array Measureent Syste for Analysis of Directional and Spatial Variations of Sound Fields, Journal of the Acoustical Society of Aerica 112 (5), Pt. 1, Noveber, [8] J.M. Rigelsford, A. Tennant, A 64 Eleent Acoustic Voluetric Array, Applied Acoustics 61: , [9] J. Meyer, Beaforing for a Circular Microphone Array Mounted on Spherically Shaped Objects, Journal of the Acoustical Society of Aerica 109 (1): , January, [10] J. Meyer, G.W. Elko, Echoes, The Newsletter of the Acoustical Society of Aerica, Electroacoustic Systes for 3-D Audio A report fro the Pittsburgh eeting 12 (3), [11] C. Cardoso, P. Nelson, Directivity of Spherical Microphone Arrays, In Proceedings of the Institute of Acoustics, Volue 26 (Part 2), Março, [12] P.M. Morse, Vibration and Sound, McGraw-Hill Book Copany, [13] A. Tikhonov, V. Arsenin, Solution of Ill-Posed Probles, Winston, Washington D.C., [14] P.A. Nelson, Source Strength Estiation in the Presence of Noise, unpublished notes, ISVR, University of Southapton, [15] A.V. Oppenhei, R.W. Shaffer Digital Signal Processing, Prentice-Hall, London, [16] C.A. Balanis, Antenna Theory: Analysis and Design, Wiley, New York, [17] H. Kook, P. Davies, J.S. Bolton, Statistical Properties of Rando Sparse Arrays, Journal of Sound and Vibration 255(5): , 2002.