Publication in the conference proceedings of EUSIPCO, Lausanne, Switzerland, 200

ANTONACCI, FABIO

PRANDI, GIORGIO

SARTI, AUGUSTO

TAGLIASACCHI, MARCO

TUBARO, STEFANO

VALENZISE, GIUSEPPE

Antonacci, Fabio

Prandi, Giorgio

Sarti, A.

Tubaro, S.

Valenzise, Giuseppe

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Acoustic Source Localization by Fusing Distributed Microphone Arrays Measurements

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ACOUSTIC SOURCE LOCALIZATION BY FUSING DISTRIBUTED MICROPHONE ARRAYSMEASUREMENTSG. Prandi, G. Valenzise, M. Tagliasacchi, F. Antonacci, A. Sarti, S. TubaroDipartimento di Elettronica e InformazionePolitecnico di MilanoP.zza Leonardo da Vinci, 32 20133 - Milano, Italy{prandi, valenzise, tagliasa, antonacc, sarti, tubaro}@elet.polimi.itABSTRACTThis paper addresses the problem of localizing an acousticsource by optimally fusing the observations provided by dis-tributed microphone arrays. We consider a two-step process.First, each array provides an estimate of the source positionby measuring the TDOAs (time differences of arrival) be-tween each pair of microphones of the array. Then, these esti-mates are optimally fused taking into account the geometry ofthe arrays, in order to provide the final estimate of the sourcelocation. We study the performance of this approach by pro-viding a closed form expression of the Cramer-Rao bound fora few simple geometries.1. INTRODUCTIONThe problem of localizing a source by means of acoustic mea-surements collected by a microphone array has been thor-oughly studied in the literature. A recent survey of the mostrelevant techniques can be found in [1]. A large class of thesemethods divides the problem in two steps: first, the time dif-ferences of arrivals (TDOA) between microphone pairs areestimated; then, the source location is obtained based on thesemeasurements.In some scenarios, especially when the total number ofmicrophone sensors is large, it might be useful to split the lo-calization problem among a set of microphone arrays. Eachsub-array solves the localization problem independently, andthe final estimate of the source location is provided by a cen-tral node that optimally fuses the estimates provided by theindividual sub-arrays. The rationale behind splitting a largearray into smaller sub-arrays lies mainly in the reduced costin terms of computational complexity. It can be shown [2] thatusing all the possible TDOAs in a microphone array leads tobetter result in terms of localization precision. Thus, havingM microphones on the whole, the computational cost due tothe TDOA estimation of the single array strategy grows asO(M2), while the cost of distributing the computation amongS sub-arrays, each with N = M/S microphones, grows asO(M2/S). In addition, when the arrays are deployed as a sen-sor network, the cost of data communication between the mi-crophones and the base station becomes also relevant, and itbecomes more convenient to transmit the local estimate of thesource position from each array instead of the whole acousticsignal from each sensor to the centralized processor.In the literature, the problem of obtaining the source po-1,1( )x t1, ( )Nx t1ˆ , 1p C2,1( )x t2, ( )Nx t2ˆ , 2p C,1( )Sx t,( )S Nx tˆ ,S Sp Cˆ FpFig. 1. Block diagram of the array fusion system.sition by fusing multiple array estimates has been addressed,but mainly in the far-field scenario. In [3], the maximum like-lihood estimation of the direction of arrival is studied, when alarge array is partitioned into sub-arrays. In [4], a distributedprocessing scheme is proposed, which performs direction ofarrival (DOA) estimation at the individual arrays and time-delay estimation between pairs of arrays. Therefore, partialcommunication between sub-arrays is enabled. To the au-thors’ knowledge, the only previous work that addresses thelocalization problem in the near field scenario, by fusing mul-tiple array observations, is [5]. Each sub-array computes aspatial likelihood function using a conventional SRP-PHATtechnique. The spatial likelihood functions are blended to-gether by properly weighting the different level of access thata microphone array has to different spatial positions. Nev-ertheless, the weighting is obtained experimentally, by prob-ing the system in an off-line training phase. Also, no theo-retical evidence is given about the performance of the pro-posed method. Recently, the relation between optimal mi-crophone array geometries and the variance of location es-timates has been formalized by Yang and Scheuing [2][6].When the noise affecting TDOA measurements is Gaussian,the Cramer-Rao bound can be written in closed form, and itdepends only on the noise covariance matrix and on the ar-ray geometry. In [2] necessary and sufficient conditions forthe optimum array geometry which minimizes the bound are16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIPprovided, when TDOAs of all microphone pairs are collected.The work is extended in [6], where only a subset of TDOAs,all relative to a reference microphone, is adopted, such asin least squares based methods (e.g. spherical interpolation,spherical intersection, linear corrected least squares).This paper analyzes the performance of a two-step lo-calization process in a 2-dimensional space based on TDOAmeasurements. Each sub-array provides an estimate of thesource location to a central processing node. Unlike [4], thescheme we study in this paper does not involve any data com-munication between sub-arrays. The central processing nodeperforms optimal fusion, based on the incoming estimates andthe knowledge of the array geometry. The Cramer-Rao boundis derived for a few simple array geometries for which theproblem is mathematically tractable.2. FUSION OF LOCALIZATION MEASUREMENTSConsider the system illustrated in Figure 1, which consists ofS distributed microphone arrays, each with N microphones.The source is located in p = [p1, p2]T . We assume that eacharray provides an unbiased estimate of the location of theacoustic source, i.e. pˆs. Possibly, an estimate of the covari-ance matrix Cs = E[(pˆs−p)(pˆs−p)T ] is also given. As weshall see later, Cs depends on the actual estimator adopted,and it is typically related to the array and source geometry andto the measurement noise. The data fusion module receives pˆsand Cs, s = 1, . . . ,S, and it computes the final estimate of thesource, i.e. pˆF .The data model adopted by the fusion module is summa-rized by the following equation:pˆs = p+ws, (1)where ws ∈R2 denotes the position error in the estimate pro-vided by array s, which is directly related to the error in theestimated TDOAs at each microphone. We postulate that noanomalies occur in the TDOA estimation process, i.e. thereare no spurious peaks in the autocorrelation function due, forexample, to reverberation [7]. If we assume that the SNRis high and no reverberation occurs, then the errors on theTDOA estimates are relatively small, and ws can be modeledas a zero mean, Gaussian distributed random process with co-variance matrix E[wswTs ] = Cs (i.e. ws ∼N (0,Cs)). Thus,the likelihood function can be written asL(p; pˆ1, . . . , pˆS) =S∏s=112pi|Cs|e−12 (pˆs−p)T C−1s (pˆs−p). (2)We assume that the data fusion module adopts a maximumlikelihood estimator (MLE) to compute pˆF . For linear datamodels and Gaussian noise as in (1), the MLE is unbiased andefficient, and its variance attains the Cramer-Rao lower bound(CRLB) also for a finite number of measurements S. If thenoise pdf is unknown, the MLE still represents a meaningfulestimator, as it is the best linear unbiased estimator (BLUE)[8].In order to find the MLE, we need to compute the firstorder derivative of the log-likelihood function with respect top, and set it equal to zero, i.e.:∂ logL(p; pˆ1, . . . , pˆS)∂p=∂∂p[− logS∑s=12pi|Cs|− . . .. . .−12S∑s=1(pˆs−p)T C−1s (pˆs−p)]=S∑i=1C−1s (pˆs−p) = 0. (3)Therefore, the MLE of the source position is given bypˆF =(S∑s=1C−1s)−1·(S∑s=1C−1s pˆs). (4)The value pˆF is a stationary point of the likelihood function(2). It is a point of maximum, since the second order deriva-tive is negative definite. In fact:∂∂p[∂ logL(p; pˆ1, . . . , pˆS)∂p]=−S∑s=1C−1s . (5)The CRLB is found as the element (i, i) of the inverse ofthe Fisher information matrix, i.e.var(pˆi)≥ [I−1F (p)](i,i), (6)whereIF(p) =−E{∂∂p[∂ logL(p)∂p]}=S∑s=1C−1s . (7)We notice that, if the MLE is used to perform data fusion,the CRLB of the final estimate pˆF depends on the covariancematrices Cs, s = 1, . . . ,S of the individual arrays. In the fol-lowing section, we introduce an expression for Cs, when theacoustic source localization is performed by means of TDOAmeasurements.3. SOURCE LOCALIZATION FROM TDOAMEASUREMENTSEach microphone array consists of N sensors at locations qn,n = 1, . . . ,N. If we consider the TDOA measurement noiseto be zero-mean, Gaussian and i.i.d. with variance σ2, theFisher information matrix (FIM) is given by [2]:I(p) =GGT(σν)2(8)where ν denotes the speed of sound andG = [. . . ,gi j, . . .](i, j)∈{(i, j)|1≤ j<i≤N} (9)gi j = gi−g j (10)16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIPgi =p−qi‖p−qi‖, i = 1, . . . ,N (11)i.e. gi is the unit norm vector pointing from the ith micro-phone to the source.The Fisher information matrix can be rewritten as [2]I(p) =NggT − (g1)(g1)T(σν)2(12)where g = [g1, . . . ,gN ] and 1 = [1,1, . . . ,1]T .4. A SIMPLE DISTRIBUTED ARRAY GEOMETRYIn order to make the following discussion mathematicallytractable, we consider a simple geometry with S arrays. Eacharray consists of N microphones located along a circle cen-tered in the source position, i.e. p = 0. For the first array, i.e.s = 1, the microphone positions are given byqn = R[cos(αn),sin(αn)]T , n = 1, . . . ,N, (13)where R is the radius of the circle, andαn =2pinK−pi(N−1K). (14)The value of K ≥ N determines the angular aperture of thearray (see Figures 2 and 3). Large values of K correspond tosmall angular distances between microphones. When K = N,microphones are uniformly spaced over the whole circle,while K ≫ N means that the array is located on a small circu-lar sector.Also, we postulate that each array adopts an unbiasedand efficient estimator, which attains the CRLB. Actually,such an estimator does not exist [2], since the MLE estima-tor for source location [1] is only asymptotically efficient, i.e.Cs > I−1s . Nevertheless, as shown in [9], the variance of theMLE is very close to the theoretical CRLB when just a smallnumber of microphones (e.g. N ≥ 6 sensors) is used for eacharray. Therefore, in the following we consider the sensor fu-sion performance in terms of the CRLB, which can be con-sidered a tight bound on the estimator variance, provided thatthe number of microphones for each array is not too small, orequivalently, that the number of arrays S is not too large (seeFigures 2-3).For the problem at hand, gn =−qn/R, and the Fisher in-formation matrix for the first array is given byI1 =G1G1T(σ1ν)2=1(σ1ν)2(1/λ1 00 1/λ2)(15)where1λ1= NN∑n=1cos2(αn)−[N∑n=1cos(αn)]2,1λ2= NN∑n=1sin2(αn) (16)−0.4 −0.2 0 0.2 0.4−0.4−0.200.20.4p 1p2(a) Multi-array geometry.1 2 3 4 5 6 7 8100102104106108    S  CRLB − Single arrayCRLB − Array fusion (MONO)CRLB − Array fusion (MULTI)CRLB − Array fusion (INTER)Simulation − Single ArraySimulation − Array fusion (MONO)Simulation − Array fusion (MULTI)Simulation − Array fusion (INTER)(b) Cramer-Rao BoundFig. 2. In the top part of the figure, multi-array geometry withM = 24 microphones, S = 4 arrays, and K = SN = M, i.e. eacharray is distributed over 1/4 of the circle. In the bottom part,the trace of the inverse FIM for different kinds of array ge-ometries, when the available number of microphones M = 24is distributed across S arrays and K = M (K = N for inter-leaved configuration). The values of tr{I−1F } are normalizedby the trace of the inverse FIM of a single array with S = 1.ThereforeC1 > σ2ν2(λ1 00 λ2)(17)We assume that the microphone locations for the otherarrays (i.e. s = 2, . . . ,S) are obtained by rotating the first arrayby an angle equal to αs. By defining different values for αs,we can obtain the following distributed geometries for a setof S microphone arrays:16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP• mono-sector array: αs = 0,s = 2, . . . ,S• multi-sector array:– S = 2, α2 = pi/21– S > 2, αs =2pi(s−1)S,s = 2, . . . ,S• interleaved array: K = N, αs = anyIf we assume that the source is isotropic and the gains of themicrophones are all equal, σ1 = σ2 = . . . = σS = σ .In the following, we compare the three aforementionedgeometries in terms of localization performance. To this end,as in [2], the metrics adopted is the trace of the inverse of theFIM, i.e. tr{I−1F }.4.1. Mono-sector arrayIn this scenario, we have a superposition of S co-located ar-rays. Therefore, Cs = C1, s = 2, . . . ,S andtr{I−1F }MONO =tr{C1}S>tr{I−11 }S=σ2ν2(λ1 +λ2)S. (18)An expression for tr{I−11 } was found in [6], and it is equal totr{I−11 }=4σ2ν2N21−ρ22(1−ρ1)(1+ρ1−2ρ22 ), (19)withρ1 =sin(2piN/K)N sin(2pi/K), ρ2 =sin(piN/K)N sin(pi/K), (20)A more compact approximation of the previous expressioncan be obtained as follows. From [2] we have:tr{I−11 }= tr{(GGT )−1} ≥4σ2ν2tr{GGT}(21)Although the equality holds only for arrays uniformly dis-tributed along the circle, a good approximation is given by:tr{I−11 } ∼4σ2ν2tr{GGT}S3/2 =4σ2ν2S3/2N2−∣∣∣ sin(piN/K)sin(pi/K)∣∣∣2 (22)Therefore, we obtaintr{I−1F }MONO >4σ2ν2S1/2N2−∣∣∣ sin(piN/K)sin(pi/K)∣∣∣2 (23)4.2. Multi-sector arrayIn this scenario each array spans a sector, and there are S dis-tinct sectors uniformly spaced along the circle. Therefore, wecan writeCs = TsC1TTs (24)1It can be shown that α2 = pi is not optimal, while α2 = pi/2 is.−0.4 −0.2 0 0.2 0.4−0.4−0.200.20.4p 1p2(a) Multi-array geometry.1 2 3 4 5 6 7 8100102104106108    S  CRLB − Single arrayCRLB − Array fusion (MONO)CRLB − Array fusion (MULTI)Simulation − Single ArraySimulation − Array fusion (MONO)Simulation − Array fusion (MULTI)(b) Cramer-Rao BoundFig. 3. In the top part of the figure, multi-array geometrywith M = 24 microphones, S = 4 arrays, and K = 4SN = 4M,i.e. each array is distributed over 1/16 of the circle. In thebottom part, the Cramer-Rao bound for different kinds of ar-ray geometries, when the available number of microphonesM = 24 is distributed across S arrays and K = 4M. The val-ues of tr{I−1F } are normalized by the same value of Figure 2for comparison.where Ts is an orthonormal rotation matrix given byTs =[cos(αs) −sin(αs)sin(αs) cos(αs)](25)16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIPSettinga =S∑s=1cos2(αs), b =S∑s=1sin2(αs),c =S∑s=1sin(αs)cos(αs) (26)and combining (7) and (24), we gettr{I−1F }MULTI =(S∑s=1C−1s)−1=(S∑s=1TsC−11 TTs)−1> σ2ν2[aλ1+ bλ2cλ1+ cλ2cλ1+ cλ2bλ1+ aλ2]−1=2σ2ν2S[λ1λ2λ1+λ200λ1λ2λ1+λ2]=4σ2ν2tr{GGT}·S−1=4σ2ν2S−1N2−∣∣∣ sin(piN/K)sin(pi/K)∣∣∣2 (27)4.3. Interleaved arrayIn this scenario, each array has uniformly spaced micro-phones along the circle. The covariance matrix C1 is givenbyC1 = σ2ν2[2/N2 00 2/N2](28)Since Cs = C1, s = 2, . . . ,S,tr{I−1F }INTER >tr{C1}S=4σ2ν2SN2(29)5. SIMULATION RESULTSIt is interesting to compare the three geometries described inSection 4 in terms of localization accuracy, for the same totalnumber of microphones M. Figure 2(b) shows the Cramer-Rao bound when M = 24, N = M/S and K = N, normalizedby 4σ2ν2/M2, i.e. the optimal accuracy attained when theM microphones are uniformly distributed around the sourceto form a single array. This normalized value is referred toin the literature [6] as geometric dilution of precision, sinceit accounts for the loss induced by the array geometry. Theplot shows the Cramer-Rao bound for the mono-sector, multi-sector and interleaved geometries, for a different number ofsub-arrays. The plot also shows as a reference the Cramer-Rao bound for a single array of N = M/S microphones.As an example, Figure 2(a) shows the multi-sector scenariowhen S = 4. We notice that the mono-sector geometry per-forms worse. By comparing (23) and (27), we observe thattr{I−1F }MONO/tr{I−1F }MULTI ∼ S3/2, i.e. the gain of adopt-ing a distributed array geometry increases with the number ofsub-arrays. This is due to the fact that the geometric dilutionof precision of the individual arrays is partially compensatedby the fusion process only in the multi-sector scenario. Theinterleaved scenario achieves the best performance, since theindividual arrays do not suffer from any geometric dilution ofprecision.Figure 3(b) shows similar results, but when K = 4N, i.e.the angular aperture of the individual sub-arrays is 4 timessmaller than before. Figure 3(a) shows an example when S =4. Comparing Figure 3(b) with Figure 2(b), we notice that,for the same number of sub-arrays S, the localization accuracydecreases, due to the higher geometric dilution of precision ofthe individual arrays.6. CONCLUSIONSIn this paper we investigated the Cramer-Rao bound of thelocalization accuracy, when the estimates provided by dis-tributed sub-arrays are fused together in a central node. Fu-ture investigation will address the accuracy-resources trade-offs that arise when computational and/or communicationpower is constrained.7. REFERENCES[1] Y. Huang, J. Benesty, and G. W. Elko, Audio Signal Pro-cessing for the Next-Generation Multimedia Communi-cation Systems, chapter 9, Source Localization, KluwerAcademic Publishers, 2004.[2] B. Yang and J. Scheuing, “Cramer-rao bound and opti-mum sensor array for source localization from time dif-ferences of arrival,” in Proceedings of the InternationalConference on Acoustics, Speech, and Signal Processing,Philadelphia, PA, March 2005.[3] P. Stoica, A. Nehorai, and T. Soderstrom, “Decentralizedarray processing using the MODE algorithm,” Circuits,Systems and Signal Processing, vol. 14, pp. 17–38, 1995.[4] R. J. Kosick and B. M. Sadler, “Source localization withdistributed sensor arrays and partial spatial coherence,”IEEE Trans. Signal Process., vol. 52, pp. 601–616, 2004.[5] P. Aarabi, “The fusion of distributed microphone arraysfor sound localization,” EURASIP Journal on AppliedSignal Processing, vol. 2003, pp. 338–347, 2003.[6] B. Yang and J. Scheuing, “A theoretical analysis of 2dsensor arrays for TDOA based localization,” in Pro-ceedings of the International Conference on Acoustics,Speech, and Signal Processing, Toulouse, France, May2006.[7] J. Ianniello, “Time delay estimation via cross-correlationin the presence of large estimation errors,” IEEE Trans-actions on Acoustics, Speech, and Signal Processing, vol.30, no. 6, pp. 998–1003, 1982.[8] S.M. Kay, Fundamentals of statistical signal process-ing: estimation theory, Prentice-Hall, Inc. Upper SaddleRiver, NJ, USA, 1993.[9] T. Ajdler, I. Kozintsev, R. Lienhart, and M. Vetterli,“Acoustic source localization in distributed sensor net-works,” Conference Record of the 38th Asilomar Con-ference on Signals, Systems and Computers, vol. 2, 2004.16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP

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Acoustic Source Localization by Fusing Distributed Microphone Arrays Measurements

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NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Archivio istituzionale della ricerca - Politecnico di Milano