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A Method for Source-Microphone Range Estimationin Reverberant Environments, Using an Array of

Microphones of Unknown Relative PositionDenis L. McCarthy and Frank Boland

TrzTity College DublinDublin2Ireland

E-mail: demccart@tcd. ie, fbolandftcd. ie

Abstract Estimating the range between an acoustic source anda microphone is a central problem in microphone-array processing.Although many approaches have been proposed in the literature,these tend to require knowledge of the relative microphone positions.We propose a range estimation method, for use in reverberant en-vironments and where the relative positioning of the elements of amicrophone-array is unknown. We demonstrate the effectiveness ofthis method under real, reverberant conditions.

Keywords Range Estimation, Microphone Arrays

I INTRODUCTION

Range estimation, between an acoustic source andmultiple microphones, is a central problem in thefield of microphone-array processing. Such es-timates could inform decisions regarding micro-phone selection, allowing us to select the micro-phone nearest the source or farthest from somelikely interference. Indeed, range estimates are re-quired to calculate optimal filter weights for near-field beamforming. Range estimates could alsohave application in determining appropriate dere-verberation strategies as well as automated camerasteering/focusing. Given knowledge of the rela-tive microphone positions, the source-microphonerange may easily be obtained from estimates of rel-ative source location - an end to which a variety ofsolutions have been proposed.Much work has focused on the use of time-delay

estimates (TDEs) between microphones. In thetwo-dimensional case, source location may be con-sidered a practical application of Apollonius' Prob-lem of tangent circles. The mathematical solu-tion to this problem, as discovered by Viete, maybe easily expanded to the three-dimensional caseand given TDEs between a minimum of four mi-crophones (three in the two-dimensional case), asource location may be found. In [1], TDEs aredetermined for pairs of microphones in a seriesof four-element, square microphone arrays. Fromthese, a series of source bearing-lines are calcu-

lated, with the final source location estimate be-ing calculated as a weighted average of the closestintersections between bearing-line pairs. In [2] theauthors estimate the source location via a leastsquares fitting of the time delay estimates.

Several methods locate the source by a pollingof candidate locations. In [3], the steered re-sponse power (SRP) is determined for a seriesof discrete locations, in the near-field of an ar-ray, with the maximum returned power being as-sumed to correspond to the source location. In [4]the authors formulate a approximate maximum-likelihood location estimator. This ML estimator,which they show to be equivalent to a weightedcross-correlation criterion, is then used to selectthe source location from among the candidate lo-cations.

In non-reverberant acoustic environments, or insituations where the effects of reverberation arenegligible, relative range estimates may be ob-tained from a comparison of received signal power.In [5] the authors combine TDEs and relative sig-nal power measurements to determine the proxim-ity and direction of arrival of a source in the ex-treme near-field of a two-element array. In [6] theauthors present a method for source localisationthat utilizes received signal energy only. Whilstthis technique is reported as returning consistentlyaccurate source-bearing estimates, range estima-tion is seen, in the presence of reverberation, to be

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subject to a significant bias.Whilst many source localization algorithms have

been shown to be effective in the presence of re-verberation, these typically require knowledge ofthe relative microphone positions. For many prac-tical applications, such information is unavailableor unreliable. Indeed, the question of how to es-timate the range between a sound source and amicrophone, in the presence of reverberation andwith the relative positions of the microphones un-known, remains largely unaddressed. We proposea solution to this problem. Our method combinesrelative power measurements with TDEs to ob-tain absolute source-microphone range estimatesfor microphones at unknown locations. In Section2 we derive a generalized function for estimatingthe source-microphone range in adverse acousticconditions. In Section 3 we address the specificcase of source-microphone range estimation in thepresence of reverberation and in Section 4 we de-scribe a series of experiments designed to test ourmethod under simulated and real, reverberant con-ditions. We conclude in Section 5.

II RANGE FINDING

In air, the power of the received direct-path com-ponent of sound is inversely proportional to thesource-microphone range, r, squared.

(1)Pdp (r) OcI

FRom this follows

d-2Pdp (er) 0( r2 (2)

Pdpo is the direct-path power received by some

microphone, mo, at a source-microphone range, ro.From this, the direct-path power received by anymicrophone may be given as a function of r and ro

Pdp (r) =Pdpo [exp -2Inr )] (3)

microphones and so, by substituting (4) into (3)and performing algebraic manipulation, we obtaina simple and well known estimator of ro.

Pdp r

1 Pdp(rx)Pdpo(5)

However, in non-ideal acoustic environments,the ratio of received powers between pairs of micro-phones is distorted by the presence of interferingnoise and reverberation. Equation (6) describesthe power received by a microphone (direct soundand interference), as a function of r, normalizedrelative to the power received at m,.

Px (ry) {Pdp [exp (2 In(rx)] }

(6)

where PIY is the received power due to inter-fering reverberation and noise at my and PT. =

Pdp: + P1x. In the general case, PIY is unknownbut the ratio pIY = Ox v is assumed known. Wenow derive an estimator of ro for non-ideal acousticenvironments.

9x (rx, ry) = /3xypx (rx) -Px (ry) (7)

gx(rxIry) = Pdp, xy-exp (-21n (r(8)

From this

- exp (-2ln (r))

: exp (-2 In ( r )

gx (rxIry)9x (rx I rz)l

(9)

Substituting in (4)

From any one of a variety of time-delay estima-tion techniques (see [7] and references therein), wemay obtain estimates of the absolute difference inthe distances to the source between microphones.

rx-ro = CTx (4)

where c is the speed of sound in air and Tx is thetime-delay estimate between mx and mo. In noise-less, anechoic environments the direct-path soundaccounts for all acoustic energy received by the

gx (rx,ry) - 3fAx exp (-21n (1 + CY "CxT)) '9x (rx, rz ) x,z- exp (-2ln (I + crz-crx)) f

(10)

From (10) we see that gx(r ,ry) is, in fact, a func-gx(rx,rz)Ition of the known values Tr, TY' Tz, Ox v and ox zand a single unknown, ro. Solving (10), which weshall refer to as the Range-Finder algorithm, givesus an estimate of ro.

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ISSC 2006, Dublin Institute of Technology, June 28-30

2 4 6SR (dB)

8 10 12

1.5I-~~~~~~~~~~~05

Q ----I0 - It)C }

0 100 200 300 400 51100.Range Estimates (cm)

8

6 -

2 -

0 K~ ~ 50

0 100 200 300 400Moving Average of Range Estimates (cm)

Fig. 1: A comparison of the performance of theRange-Finder and simple range estimator algorithms w.r.t

SIR

To illustrate the performance of the Range-Finder under adverse acoustic conditions, a seriesof simple simulations was performed. Three micro-phones were simulated as being in an anechoic en-

vironment, at ranges lm, 2m and 4m, respectively,from a source producing a "maximum-length se-

quence" (MLS) of 8191 samples in length. The mi-crophone recordings were corrupted with uncorre-

lated white Gaussian interference signals. The per-

formance of the Range-Finder algorithm was com-

pared to that of the simple range estimator givenin (5), with respect to the signal-to-interferenceratio (SIR). The SIR taken is that at the micro-phone closest to the source and, in order to achievethe best possible performance, the simple range

estimator uses the two microphones closest to thesource only. For simplicity, 1, Vx, y. Fromthe results, shown in Figure(1), it is clear that thesimple range estimator suffers from a significantbias whilst the range-finder algorithm obtains ac-

curate range estimates even at low SIR's.

III RANGE FINDING IN REVERBERANT ROOMS

Typically, in reverberant environments, both thereceived reverberant power, PI, and the ratiosof these powers between microphones, 3, are un-

known. We are forced, then, to use assumed valuesand so we take v = 1, Vx, y. In other words we

assume that received reverberant energy levels are

the same at each microphone. This assumption isa reasonable one in many cases, particularly whenwe are discussing long-term reverberation. How-ever, short-term reverberation levels are heavilydependant on factors such as the proximity of a

microphone to reflecting surfaces and the absorp-tion/diffusion characteristics of these surfaces. Wecan expect greater levels of short-term reverbera-tion in a microphone positioned next to a polished

Fig. 2: The moving average, (b), is calculated byconvloving the range-estimate pulse train, (a),with a

Hamming window

table compared to one positioned next to, say, cur-

tains. This may negatively impact the accuracy ofour estimates. We note that reverberation may

also effect the accuracy of the time-delay estima-tion algorithm used, however, a discussion of theseeffects and the means by which they may be mit-igated against is beyond the scope of this paper

and so is omitted here.Considering this, we must take into account the

possibility (indeed likelihood) that one or more mi-crophones will receive levels of acoustic energy thatdo not correspond to the model assumed in (6).The inclusion of range estimates calculated usingsuch microphones could lead to large errors. Inpractical situations, we would likely require more

than three microphones, to ensure that we obtainat least one accurate range estimate. However,we are then presented with the problem of distin-guishing between accurate and faulty results. Werequire an approach that may effectively discrim-inate between correct results and a comparablenumber of false estimates.

Possible solutions to the problem of erroneous

estimates could involve averaging and/or the re-

moval of outliers. Unfortunately, outliers may notbe easily identifiable and the mean of the range es-

timates cannot always be assumed to approximatethe true range, as the mean may be significantlyaffected by false results. Our first step therefore,is to identify and remove as many false results as

possible.Given any three microphones, there exists six

possible permutations of [m,, my, m,]. All six ofthese yield identical estimates of ro, therefore, us-

ing M microphones, we may obtain up to MC3independent range estimates. Some erroneous es-

timates may be easily spotted. Negative ranges,

497

En

Range Estmate'Range F nder"

3 True Range

252.

1.5 X

( 5 500

=__ x(m) y(m) z(m)'MO 1.5 3.5 1.75m1 2.5 2.5 1.75m2 1.5 1.5 1.8s 1.5 4.81 1.07

Table 1: Coordinates of the simulated microphonesand source, s

for example, may be ignored and so, by selectingmo such that it is the closest microphone to thesource (this information can be obtained from theTDEs) we increase the likelihood that underesti-mates of the range are negative and, hence, eas-ily identifiable as false. Indeed, given any knownrange boundary we can identify and remove erro-neous estimates. Unfortunately, we are, very often,still left with several false but otherwise plausiblerange estimates.The second step is to find a moving average of

the remaining range estimates. Figure(2a) showsa pulse train corresponding to the estimates ofa source-microphone range. Each sample in thepulse train corresponds to an increment in therange estimate, Ar. We obtain a moving aver-age of these estimates by convolving them with aHamming window, Figure(2b).The moving average returns local maxima cor-

responding to clusters of estimates. The rationalmotivating this approach is that correct results willbe grouped around the true range value while falseresults will demonstrate no such correlation.

IV EXPERIMENTS

A series of experiments were performed to testthe performance of the Range-Finding algorithm.A room was simulated using an acoustic soft-ware package. The dimensions of the room were[5.25m 6.95m 2.44m]. Three microphones and asound source were placed at the locations givenin Table(1). Using a raytracing algorithm, thefirst 20ms of the source-microphone impulse re-sponses were generated and statistical reverber-ant tails added. These impulse responses werethen convolved with a MLS of length 32767 sam-ples to obtain the simulated "recordings". Thesampling rate used was 10kHz. The reverbera-tion time, RT60, of the room was varied by alter-ing the rooms surface absorption coefficients. Therecordings were split into segments of 1024 sam-ples and a range estimate for mo was found foreach segment using both the Range-Finder algo-rithm and the simple range estimator given in (5).The mean of these results, plus and minus the stan-dard deviation and with respect to RT60,is shownin Figure(3).To test the Range-Finder in real reverberant

conditions, five microphones were setup at vary-

E 25

T 2

RTO (s)

Fig. 3: A comparison of the performance of theRange-Finder and simple range estimator algorithms w.r.t

RT6o

ing distances from a loudspeaker in a reverber-ant hall (RT60 - 1.ls) and a medium-sized class-room (RT60 = 0.5s). A maximum-length se-quence (MLS) of approximately 5.6s duration wasproduced by the loudspeaker and recorded us-ing each of the microphones at a sampling rateof 10kHz. The time delay between each of therecorded samples was determined using a phase-alignment transform (PHAT). The speed of soundwas taken as 340ms- . The samples were splitinto unwindowed segments and then processed bythe range finding algorithm. The segment lengthused was 4096 samples. Once again range esti-mates were found using both the Range-Finder al-gorithm and the simple range estimator, for eachsegment.

In the case of the reverberant hall, an analy-sis of the initial range estimates revealed that es-timates calculated using one of the microphones,Mic 4, showed significant error. Per segment, 6 ofthe 10 range estimates incorporated Mic 4 in theircalculation. In the case of the classroom, it wasseen that range estimates calculated using mea-surements from two microphones, Mic 1 and Mic4, again showed significant error. This meant that9 of the 10 range estimates for each microphonewere false.From the initial range estimates, a moving aver-

age was found for both estimation methods. Therange increment, /r, was lcm and the Hammingwindow was 50 increments wide. The maximumvalue of this was taken to as ro. The results areshown in Figure(4)

V CONCLUSION

The problem of source-to-microphone range find-ing, in the presence of reverberation and withoutknowledge of the relative microphone locations is

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ISSC 2006, Dublin Institute of Technology, June 28-30

Olassmo3rn

~~Range EstimateRange Finder

I~~ ~ I r

O 0.5 1 1.5 2 2.5 3 3.5 4 4.Range(fm)

Fig. 4: Under real reverberant conditions, theRange-Finder outperforms the simple range estimator.

one that has remained largely unaddressed. In thispaper we have proposed a solution to this problemthat uses multiple microphones. Our method is ro-bust against false results, even where the numberof such results is comparable to, or, indeed, signif-icantly greater than, the number of correct rangeestimates.

Future work will investigate, adapting our ap-proach for cases where the acoustic source is ahuman voice. It is expected that such cases willpresent additional difficulty for the range findingalgorithm. In real rooms, levels of received rever-berant power are equal for all microphones in abroadband sense only. Large portions of humanspeech are spectrally sparse (mainly the harmonicor "voiced" sounds) and at individual, and in par-ticular low frequencies, the frequency responses ofdiffering source-microphone channels do not con-sistently conform to the model assumed herein.Preliminary results have shown range finding per-formance to be strongly effected by acoustic envi-ronment and microphone positioning and have in-dicated the need for more effective discriminationbetween accurate and spurious results.

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[3] Griebel, S.M.; Brandstein, M.S., "Microphonearray source localization using realizable delayvectors ," Applications of Signal Processing toAudio and Acoustics, 2001 IEEE Workshop onthe , vol., no.pp.71-74, 2001

[4] Chen, J.C.; Hudson, R.E.; Kung Yao,"Maximum-likelihood source localization andunknown sensor location estimation for wide-band signals in the near-field," Signal Process-ing, IEEE Transactions on [see also Acoustics,Speech, and Signal Processing, IEEE Transac-tions on] , vol.50, no.8pp.1843-1854, Aug 2002

[5] Teutsch, H.; Elko, G.W., "An adaptive close-talking microphone array," Applications of Sig-nal Processing to Audio and Acoustics, 2001IEEE Workshop on the , vol., no.pp.163-166,2001

[6] Birchfield, S.T.; Gangishetty, R., "Acousticlocalization by interaural level difference,"Acoustics, Speech, and Signal Processing,2005. Proceedings. (ICASSP '05). IEEE Inter-national Conference on , vol.4, no.pp. iv/1109-iv/1112 Vol. 4, 18-23 March 2005

[7] DiBiase, J.H.; Silverman, H.F.; Bransd-stein, M.S., "Robust Localization in Rever-berant Rooms", Microphone Arrays; SignalProcessirng Technriques and Applications, M.Branstein, D. Ward (Eds.), Springer Press

VI ACKNOWLEDGEMENTS

The support of the Informatics Commercialisationinitiative of Enterprise Ireland is gratefully ac-knowledged. Denis McCarthy also acknowledgesthe financial support, from Trinity College, of apostgraduate studentship.

REFERENCES

[1] Brandstein, M.S.; Adcock, J.E.; Silverman,H.F., "A closed-form method for finding sourcelocations from microphone-array time-decayestimates," Acoustics, Speech. and SignalProcessing, 1995. ICASSP-95., 1995 Interna-

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