Convention Paper 6764


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1 Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or cosideratio by the Review Board. The AES takes o resposibility for the cotets. Additioal papers may be obtaied by sedig request ad remittace to Audio Egieerig Society, 60 East 4 d Street, New York, New York , USA; also see All rights reserved. Reproductio of this paper, or ay portio thereof, is ot permitted without direct permissio from the Joural of the Audio Egieerig Society. Optimisatio of Cocetred Rigid ad Ope Spherical Microphoe Arrays Abhaya Parthy 1, Craig Ji, ad Adré va Schaik 1 School of Iformatio Techology, The Uiversity of Sydey, NSW, 006, Australia School of Electrical ad Iformatio Egieerig, The Uiversity of Sydey, NSW, 006, Australia {craig, ABSTRACT We preset a ovel microphoe array that cosists of a ope spherical array with a smaller rigid spherical array at its cetre. The distributio of microphoes, which results i the array havig the largest frequecy rage, for a give beamformig order, was obtaied by aalysig microphoe errors. For a fixed umber of microphoes, the results for several examples idicate that the maximum frequecy rage is obtaied whe the microphoes are relatively evely distributed betwee the ope ad rigid spheres. 1. INTRODUCTION May spherical microphoe array cofiguratios, such as that preseted by Meyer ad Elko [1] ad Abhayapala ad Ward [], have a limited useable frequecy rage, typically 34 octaves depedig o the umber of microphoes used. This frequecy rage limitatio is due to spatial aliasig at the high frequecies ad microphoe positioig errors at the low frequecies. For a fixed umber of microphoes o a sphere, spatial aliasig ca be reduced by reducig the spacig betwee microphoes ad by reducig the radius of the sphere. The trade off i reducig the radius of the sphere, however, is that the microphoe positioig error icreases due to the smaller size of the sphere. The spatial aliasig error ad the microphoe positioig error for a give arragemet of microphoes o a spherical microphoe array is depedat o the dimesioless parameter kr, where k is the wave umber ad r is the radius of the sphere. Utilisig multiple arrays of differig radii is a techique which allows a larger frequecy rage to be covered. Gover [3] uses two ope spherical microphoe arrays to capture a larger frequecy rage, a smaller spherical microphoe array for capturig high frequecies ad a larger spherical microphoe array for capturig low frequecies. Multiple ope spherical microphoe arrays ca be cetred at the same poit allowig the soud
2 Broadbad Spherical Microphoe Arrays field to be recorded at oe locatio, however, ope spherical microphoe arrays are disadvataged i that their error rises dramatically for certai values of kr. Rigid spherical microphoe arrays do ot have this problem [4]. It is preferable to use rigid spherical microphoe arrays whe recordig a soud field for this reaso. However, multiple rigid spherical microphoe arrays ca ot be cetred at the same poit, ad usig multiple rigid spherical microphoe arrays for recordig a soud field at locatios close to each other is ot practical, as the rigid spheres will scatter soud ad affect the soud field beig recorded by the other rigid spherical microphoe arrays. We preset a spherical microphoe array cofiguratio, which we have ot see reported previously i the literature, with microphoes mouted o both a ope ad rigid sphere with a commo cetre. The smaller rigid spherical microphoe array is used for recordig high frequecies, while the larger ope spherical microphoe array is used for recordig low frequecies. The soud scattered by the cetral rigid sphere ca be calculated aalytically at ay poit surroudig the sphere, ad thus the soud field at the surface of the ope sphere is kow [1]. This cofiguratio allows soud field recordig at oe locatio while retaiig the advatages of the rigid spherical microphoe array. Buildig a spherical microphoe array with this cofiguratio, usig a fixed umber of microphoes, requires that a umber of microphoes be placed o the rigid spherical microphoe array ad the remaiig microphoes be placed o the ope spherical microphoe array. I additio, the frequecy rages that the two spherical microphoe arrays cover must overlap. We preset a optimisatio algorithm that calculates the umber of microphoes that should be placed o the rigid ad ope spherical microphoe arrays to maximise the frequecy rage of the combied arrays for a give spherical harmoic order of the beamformer.. METHODS The optimisatio program was writte usig the MATLAB software eviromet. The useable kr rage is defied as the kr rage for the spherical microphoe array for which the microphoe positioig error ad the spatial aliasig error remai below a fixed value. The iputs for the optimisatio algorithm are the total umber of microphoes, the maximum tolerable sigal error, due to microphoe positioig error ad spatial aliasig error, i the spherical microphoe array, expressed as a oisetosigal ratio, the rage for the uiform distributio of the radom microphoe placemet error for the rigid ad ope spherical microphoe arrays, ad the spherical harmoic order of the beamformer. The optimisatio algorithm calculates the umber of microphoes that should be placed o the rigid ad ope spheres, the ratio of the ope sphere radius to the rigid sphere radius, ad the useable kr rage for the ope array ad the useable kr rage for the rigid array. Several assumptios were made i the desig of the optimisatio algorithm ad are detailed i the followig paragraphs. Firstly, we assume that the spherical harmoic order for the beamformer remais costat to esure reasoably costat gai across the spherical microphoe array s useable kr rage. The gai of the microphoe array is related to the directivity idex which is defied as the peaktoaverage ratio of the beam patter expressed i decibels [3]. For spherical microphoe arrays processed usig spherical harmoics, the directivity idex is related to the spherical harmoic order of the beamformer ad icreases as the spherical harmoic order of the beamformer is icreased. The directivity idex remais relatively costat for a large rage of kr values. By oly beamformig at oe order o both arrays, the directivity idex will remai approximately costat across the etire useable frequecy rage. A secod assumptio is that the microphoes will be arraged o the ope ad rigid spheres with a early uiform spatial samplig scheme []. Nearly uiform spatial samplig schemes have bee show to be the most efficiet i terms of the umber of microphoes required [4]. I additio, oly spatial samplig schemes that satisfy the discrete spherical harmoic orthoormality criterio (see [4]) are used: m m α Y ( Ω ) Y ( Ω ) = δ δmm + ε mm, (1) AES 10th Covetio, Paris, Frace, 006 May 0 3 Page of 6
3 Broadbad Spherical Microphoe Arrays where Ω = ( θ, ϕ ) are the sample positios o a uit sphere i spherical coordiates, α are the weights for those sample positios, Y m is the spherical harmoic fuctio of order ad mode m, δ is the Kroecker delta fuctio, deotes complex cougatio, ad ε mm deotes the error i the sum for,, mm,. Oly spatial samplig positio lists which satisfy the spherical harmoic orthoormality criterio with 6 ε mm 3 10 for all,, mm, such that, N, where N is the spherical harmoic order of the beamformer, are used for arragig the microphoes o the spherical microphoe arrays. Several spatial samplig positio lists exist that do ot satisfy the orthoormality criterio (1) at a specified order, although there are other samplig positio lists, with fewer positios, that do satisfy the criterio at the same order. Spatial samplig positio lists that do ot satisfy the criterio are replaced with a spatial samplig positio list, with a lower umber of positios, which does satisfy the criterio. For example, at 4th order, we have foud spatial samplig positio lists with 37, 38, 39 ad 41 positios that do ot satisfy the orthoormality criterio, but a spatial samplig positio list with 36 positios that does satisfy the criterio. Fially, we also assume that measuremet oise is idepedet of the cofiguratio of the spherical microphoe array ad do ot iclude it i our sigal error calculatios..1. Optimisatio Algorithm The optimisatio algorithm begis with all microphoes cosidered to be o the rigid spherical microphoe array. For each iteratio of the algorithm, the umber of microphoes o the rigid spherical microphoe array decreases by oe ad the umber of microphoes o the ope spherical microphoe array icreases by oe. This iteratio cotiues util all microphoes are o the ope spherical microphoe array. For each iteratio, the sum of the microphoe positioig error ad the spatial aliasig error, herei referred to as PA error, for both the ope ad rigid spherical microphoe arrays is computed. The PA error is calculated assumig a sigle farfield, plaewave source ad that the beamformer is steered i the directio of the icomig plaewave. The spatial aliasig error, E a, is due to spatial samplig of the soud field o the surface of the spherical microphoe array. Spatial samplig limits the order to which a soud field ca be decomposed o the surface of the sphere ad the spherical harmoic decompositio of the soud field is trucated. The spatial aliasig error [4] is defied as E a = N = 0 = N+ 1 M = 1 b b 4π 4π α P (cos Θ ) P (cos Θ ) y s, () where N is the beamformig order, M is the umber of microphoes, Θ is the agle betwee the icomig plae wave ad the samplig positio Ω, s y is the 4 sigal power, ( N + 1) (4 π ), ad b is defied as where, ( ka) h' ( ka) 4 πi ( ( kr) h ( kr)), (3) h are the spherical Bessel ad Hakel fuctios respectively,, h are their derivatives, i = 1, ad a r is the radius of the cetral rigid sphere. The spatial aliasig error is depedat o the beamformig directio, thus it is calculated for 6 icomig plaewave directios, distributed aroud the sphere as i [6], ad the averaged. It was foud, empirically that after averagig across 6 icomig wave directios the spatial aliasig error varied isigificatly as more icomig wave directios were added ad averaged. The microphoe positioig error, E Ω, is due to errors i the placemet of the microphoes o the spherical microphoe array. The microphoe placemet error, Δ, is the deviatio from the ideal microphoe positio, Ω, to the positio, Ω, give by θ = θ +Δ ad ϕ = ϕ +Δ siθ. (4) The microphoe placemet error, Δ, is assumed to be uiformly distributed such that Δ 0.00 radias. This rage of microphoe placemet error seemed reasoable give the size of the spherical microphoe array. Microphoe positioig error [4] is defied as AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 3 of 6
4 Broadbad Spherical Microphoe Arrays E Ω where = N = 0 = 0 M = 1 b b 4π 4π α P (cos Θ )[ P (cos Θ ) P (cos Θ )] y s,() Θ is the agle betwee the specified microphoe positio ad the icomig wave, ad Θ is the agle betwee the actual microphoe positio ad the icomig wave. The microphoe positioig error is depedat o the directio of beamformig, so a average error is calculated across the sphere for a umber of beamformig directios. The positioig error for the spherical microphoe array is calculated for 100 realisatios of the radom microphoe placemet error for each of 11 icomig plaewave directios, distributed aroud the sphere as i [6], ad the averaged. It was foud empirically that after averagig 100 realisatios of the microphoe placemet error for each of 11 icomig wave directios the error varied isigificatly as more realisatios of the placemet error ad icomig wave directios were added ad averaged. For each iteratio, the PA error with the specified maximum tolerable oisetosigal ratio is used to calculate: firstly, the useable kr rage for the rigid spherical microphoe array, secodly, the largest ratio of the radius of the ope sphere to the radius of the rigid sphere, for which the PA error is less tha the maximum oisetosigal ratio ad such that the highest kr value for the ope spherical microphoe array is idetical to the lowest kr value for the rigid spherical microphoe array, fially, the useable kr rage for the ope spherical microphoe array. It should be oted that the useable kr rage for the ope spherical microphoe array is computed so that the largest value of kr lies before the first local maximum, i the alias error curve, which is greater tha the specified oisetosigal ratio. As show i Fig. 1, the spatial aliasig error curve for the ope sphere cotais umerous rages of kr for which the error rises dramatically. NoisetoSigal Ratio (db) kr Figure 1: Spatial aliasig error curve is show averaged over 6 icomig wave directios, for ope spherical array with 6 microphoes, havig a radius 7 times greater tha the rigid sphere located at its cetre. 3. RESULTS The optimisatio algorithm was executed to fid the optimal distributio for rigidope spherical microphoe array cofiguratios with 96, 64, 3 ad 4 microphoes. All spherical microphoe array cofiguratios were desiged to have a maximum error oisetosigal The first spherical microphoe array cofiguratio cosists of 96 microphoes operatig at 4th order. A miimum of 36 microphoes are required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. Thus, whe the rigid or ope spherical microphoe array cotais less tha 36 microphoes, that spherical microphoe array caot be used for beamformig. Referrig to Fig., the frequecy (i.e., kr) rage, for this cofiguratio, is at a maximum of.06 octaves whe 46 microphoes are placed o the ope spherical microphoe array ad 0 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is.66. The frequecy rage curve show i Fig. is ot smooth whe plotted agaist the umber of AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 4 of 6
5 Broadbad Spherical Microphoe Arrays Frequecy Rage (Octaves) Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure : The frequecy rage is show for a microphoe array cofiguratio with 96 microphoes ad workig to 4th order with maximum oisetosigal microphoes. This is caused by the spatial aliasig error which is highly oliear across kr ad chages i a oliear fashio as the umber of microphoes are icreased or reduced. The frequecy rage curves show i Figs. 3, 4 ad are ot smooth for the same reaso. The secod spherical microphoe array cofiguratio cosists of 64 microphoes operatig at 3rd order. A miimum of 6 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage for this cofiguratio is at a maximum of 6.3 octaves whe 3 microphoes are placed o the ope spherical microphoe array ad 3 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is 7.3. The third spherical microphoe array cofiguratio cosists of 3 microphoes operatig at d order. A miimum of 1 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage, for this cofiguratio, is at a maximum of octaves whe 16 microphoes are placed o the ope spherical microphoe array ad 16 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is Number of Microphoes o Rigid Sphere Figure 3: The frequecy rage is show for a microphoe array cofiguratio with 64 microphoes, ad workig to 3rd order with maximum oisetosigal Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure 4: The frequecy rage is show for a microphoe array cofiguratio with 3 microphoes ad workig to d order with maximum oisetosigal The fourth spherical microphoe array cofiguratio cosists of 4 microphoes operatig at d order. A miimum of 1 microphoes is required to satisfy the spherical harmoic orthoormality criterio for this cofiguratio. The frequecy rage, for this cofiguratio, is at a maximum of 9.90 octaves whe 1 microphoes are placed o the ope spherical microphoe array ad 1 microphoes are placed o the rigid spherical microphoe array. The ratio of the ope sphere radius to the rigid sphere radius, with this distributio of microphoes, is 3.7. AES 10th Covetio, Paris, Frace, 006 May 0 3 Page of 6
6 Broadbad Spherical Microphoe Arrays Frequecy Rage (Octaves) Number of Microphoes o Rigid Sphere Figure : The frequecy rage is show for a microphoe array cofiguratio with 4 microphoes ad workig to d order with maximum oisetosigal 3.1. Example Microphoe Array A example spherical microphoe array is discussed to illustrate the practical cosideratios that are required whe costructig a spherical microphoe array. The example spherical microphoe array cofiguratio has 3 microphoes o the ope sphere ad 3 microphoes o the rigid sphere, ad has the highest frequecy rage of all the cofiguratios with 64 microphoes. The kr rage for the rigid microphoe array is , ad the kr rage for the ope microphoe array is First of all, the radius is related to f the frequecy by krc r = π f, (6) where c is the speed of soud. Thus, if we would like the array to work to a maximum frequecy of 14.0 khz, the radius of the rigid sphere would have to be 1.87 cm ad the radius of the ope sphere would the be 14.1cm. With this radius, the ope spherical microphoe array ca work to a low frequecy limit of 17 Hz. However, whe buildig the rigid microphoe array usig DPA type 4060BM microphoes, which have a diameter of 0.4 mm ad a height 1.7 mm, it is ot be possible to place 3 microphoes o a sphere of radius 1.87 cm ad the radius of the rigid sphere has to be icreased to accommodate 3 microphoes. We fid that a sphere with a radius of at least.04 cm has to be used to accommodate 3 microphoes. With this radius for the rigid sphere, the high frequecy limit of the rigid spherical microphoe array becomes 1.8 khz. The radius of the ope sphere the becomes 1.4 cm, ad the low frequecy limit of the ope spherical microphoe array is 160 Hz. 4. CONCLUSION From the results preseted above, it is evidet that the largest useable frequecy rage for a cocetric rigid/ope spherical microphoe array beamformer that operates to a costat order is achieved whe microphoes are placed both o the rigid sphere ad the ope sphere. The results idicate that a relatively eve distributio of microphoes, betwee the ope ad rigid spheres, produces the highest frequecy rage.. REFERENCES [1] J. Meyer ad G. W. Elko, A highly scalable spherical microphoe array based o a orthoormal decompositio of the soudfield, i Proc. ICASSP, vol. II, 00, pp [] T. D. Abhayapala ad D. B. Ward, Theory ad desig of high order soud field microphoes usig spherical microphoe array, i Proc. ICASSP, vol. II, 00, pp [3] B. N. Gover, J. G. Rya, ad M. R. Stiso, Microphoe array measuremet system for aalysis of directioal ad spatial variatios of soud fields, J. Acoust. Soc. Amer., vol. 11, o., pp , 00. [4] B. Rafaely, Aalysis ad desig of spherical microphoe arrays, IEEE Tras. o Speech ad Audio Processig, vol. 13, o. 1, pp , 00. [] R. H. Hardi ad N. J. A. Sloae, McLare s improved sub cube ad other ew spherical desigs i three dimesios, Discrete Computatioal Geometry, vol. 1, pp , 199. [6] J. Fliege ad U. Maier, A twostage approach for computig cubature formulae for the sphere, Ergebisberichte Agewadte Mathematik, No. 139T. Fachbereich Mathematik, Uiversität Dortmud, 441 Dortmud, Germay. September AES 10th Covetio, Paris, Frace, 006 May 0 3 Page 6 of 6
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