Real-Time Passive Source Localization: A Practical Linear-Correction Least-Squares Approach

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1 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER Real-Time Passive Source Localization: A Practical Linear-Correction Least-Squares Approach Yiteng Huang, Jacob Benesty, Member, IEEE, Gary W Elko, and Russell M Mersereau, Fellow, IEEE Abstract A linear-correction least-squares estimation procedure is proposed for the source localization problem under an additive measurement error model The method, which can be easily implemented in a real-time system with moderate computational complexity, yields an efficient source location estimator without assuming a priori knowledge of noise distribution Alternative existing estimators, including likelihood-based, spherical intersection, spherical interpolation, and quadratic-correction least-squares estimators, are reviewed and comparisons of their complexity, estimation consistency and efficiency against the Cramér Rao lower bound are made Numerical studies demonstrate that the proposed estimator performs better under many practical situations Index Terms Estimation theory, Lagrange multiplier, least squares, measurement error, real-time implementation, source localization I INTRODUCTION LOCATING radiative point sources using passive, stationary sensor arrays is of considerable interest and has been a repeated theme of research in radar and underwater sonar A common method is to base the estimate on the time delay of arrival (TDOA) measurements between distinct sensor pairs Recently, we have been motivated to study acoustic source localization techniques because of emerging applications of camera pointing in video-conferencing environments [1] [4], and beamformer steering for robust speech recognition systems [5] [7] In these systems, measurement errors and real-time implementation are the two major challenges To estimate the location of a single sound source using estimated TDOAs, in general one needs to select a data model The additive measurement error model assumes that (possibly mutually dependent) additive errors are independent of the measurements Such a model locates the source at the intersection of a set of hyperboloids Finding this intersection is a highly nonlinear problem Although the model does not easily lend itself to modification in the presence of measurement errors, it is able to describe the principal constraints imposed by the TDOA Manuscript received May 23, 2000; revised August 17, 2001 The associate editor coordinating the review of this manuscript and approving it for publication was Dr Michael S Brandstein Y Huang and J Benesty are with Bell Laboratories, Lucent Technologies, Murray Hill, NJ USA ( arden@researchbell-labscom; jbenesty@bell-labscom) G W Elko is with Agere Systems, Murray Hill, NJ USA ( gwe@agerecom) R M Mersereau is with the Center for Signal and Image Processing (CSIP), School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA USA ( rmm@ecegatechedu) Publisher Item Identifier S (01) data and thus is widely used in studying the source localization problem There is a rich literature of source localization techniques that use the additive measurement error model Important distinctions between these methods include likelihood-based versus least-squares and linear approximation versus direct numerical optimization (maximization or minimization), as well as iterative versus closed-form algorithms Several procedures [8], [9] have been based on the maximum likelihood (ML) principle because of the proven asymptotic consistency and efficiency of the ML estimator (MLE) Unfortunately, the number of microphones in an array for camera pointing or beamformer steering is always limited, which makes this a finite-sample rather than a large-sample problem Moreover, ML estimators require additional assumptions about the distributions of the measurement errors One approach is to invoke the central limit theorem and assumes a Gaussian approximation, which makes the likelihood function easy to formulate Although a Gaussian error was justified by Hahn and Tretter [8] for continuous-time processing, it can be difficult to verify and the MLE is no longer optimal when sampling introduces additional errors in discrete-time processing To find the solution to the MLE, a linear approximation and iterative numerical techniques have to be used because of the nonlinearity of the hyperbolic equations The Newton Raphson iterative method [10], the Gauss Newton method [11], and the least mean squares (LMS) algorithm are among possible choices However, for these iterative approaches, selecting a good initial guess to avoid a local minimum is difficult and convergence to the optimal solution cannot be guaranteed Therefore, it is our opinion that an ML-based estimator is not suitable for the real-time implementation of a source localization system For real-time applications, closed-form estimators are desired and appropriately, have also gained wider attention Of the closed-form estimators, triangulation is the most straightforward [2] However, with triangulation it is difficult to take advantage of extra sensors and the TDOA redundancy Currently, most closed-form algorithms exploit a least-squares principle, which makes no additional assumption about the distribution of measurement errors To construct a least-squares estimator, one needs to define an error function based on the measured TDOAs Different error functions will result in different estimators with different complexity and performance Schmidt [12] showed that the TDOAs to three sensors whose positions are known provide a straight line of possible source locations in two dimensions and a plane in three dimensions By intersecting the lines/planes specified by different sensor triplets, he obtained an estimator called plane intersection /01$ IEEE

2 944 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 Another closed-form estimator, termed spherical intersection (SX), employed a spherical LS criterion [13] The SX procedure is mathematically simple, but requires an a priori solution for the source range, which may not exist or may not be unique in the presence of measurement errors Based on the same criterion, Smith and Abel [14] proposed the spherical interpolation (SI) method, which also solved for the source range, again in the LS sense Although the SI method has less bias, it is not efficient and it has a large standard deviation relative to the Cramér Rao lower bound (CRLB) With the SI estimator, the source range is a byproduct that is assumed to be independent of the location coordinates Chan and Ho [15] improved the SI estimation with a second LS estimator that accommodates the information redundancy from the SI estimates and updates the squares of the coordinates We shall refer to this method as the quadratic-correction least-squares (QCLS) approach In the QCLS estimator, the covariance matrix of measurement errors is used But this information can be difficult to properly assume or accurately estimate, which results in a performance degradation in practice When the SI estimate is analyzed and the quadratic correction is derived in the QCLS estimation procedure, perturbation approaches are employed and, presumptively, the magnitude of measurement errors has to be small It has been indicated in [16] that the QCLS estimator yields an unbiased solution with a small standard deviation that is close to the CRLB when the noise level is moderate However, whether it is still unbiased and whether its variance is still able to approach the CRLB at a practically high noise level are not clear and cannot be justified by our Monte-Carlo simulations In this article we propose a linear-correction least-squares estimation procedure for acoustic source location under an additive measurement error model No assumption on the covariance matrix of measurement errors is made and no linear approximation that holds only in the case of small perturbation is needed Numerical studies demonstrate that the procedure performs well with a small bias and a small standard deviation for practical error magnitudes and source ranges In Section II, we formulate the source localization problem mathematically In Section III, we consider the additive measurement error model and present the CRLBs In Section IV, we present the proposed estimation procedure along with its underlying LS criterion Simulation results are reported in Section V Finally, conclusions can be found in Section VI II SOURCE LOCALIZATION PROBLEM The problem addressed here is the determination of the location of an acoustic source given the array geometry and the relative TDOA measurements among different microphone pairs The problem can be stated mathematically as follows The array is assumed to consist of microphones located at positions in Cartesian coordinates (see Fig 1) The first microphone ( ) is regarded as the reference and is placed at the origin of the (1) coordinate system, ie, The acoustic source is located at The distances from the origin to the th microphone and the source are denoted by and, respectively, where The distance between the source and the th microphone is denoted by (4) The difference in the distances of microphones and from the source is given by This difference is usually termed the range difference It is proportional to the time delay of arrival If the speed of sound is, then The speed of sound (in m/s) can be estimated from the air temperature (in degrees Celsius) according to the following approximate (first-order) formula The localization problem is then to estimate given the set of and Note that there are distinct TDOA estimates, which exclude the case and count the pair only once However, in the absence of noise, the space spanned by these TDOA estimates is -dimensional Any linearly independent TDOAs determine all of the others In a noisy environment, the TDOA redundancy can be used to improve the accuracy of the source localization algorithms, but this would increase their computational complexity For simplicity and also without loss of generality, we choose as the basis for this space in this paper III MEASUREMENT MODEL AND CRAMÉR RAO LOWER BOUND When the source localization problem is examined using estimation theory, the measurements of the range differences are modeled by where (2) (3) (5) (6) (7) (8)

3 HUANG et al: REAL-TIME PASSIVE SOURCE LOCALIZATION 945 and the s are the measurement errors In a vector form, such an additive measurement error model becomes (9) where In the case of a Gaussian measurement error, the Fisher information matrix is given by [15] where is the Jacobian matrix defined as (12) Further, we postulate that the additive measurement errors have mean zero and are independent of the range difference observation, as well as the source location For a continuous-time estimator, the corrupting noise, as indicated in [8], is jointly Gaussian distributed The probability density function (PDF) of conditioned on is subsequently given by (10) where is the covariance matrix of and denotes the determinant Note that is independent of by assumption Since digital equipment is used to sample the microphone waveforms and estimate the TDOAs, the error introduced by discrete-time processing also has to be taken into account When this is done, the measurement error is no longer Gaussian and is more properly modeled as a mixture of a Gaussian noise and a noise that is uniformly distributed over, where is the sampling period As an example, for a digital source location estimator with an 8 KHz sampling rate operating at room temperature (25 C, ie, m/s), the maximum error in range difference estimates due to sampling is about 2164 cm, which leads to considerable errors in the location estimate, especially when the source is far from the microphone array Under the measurement model (9), we are now faced with the parameter estimation problem of extracting the source location information from the mismeasured range differences or the equivalent TDOAs For an unbiased estimator, a Cramér Rao lower bound (CRLB) can be placed on the variance of each estimated coordinate of the source location However, since the range difference function in the measurement model is highly nonlinear in the parameters under estimation, it is very difficult (or even impossible) to find an unbiased estimator that is mathematically simple and attains the CRLB The CRLB is usually used as a benchmark against which the statistical efficiency of any unbiased estimators can be compared In general, without any assumptions made about the PDF of the measurement error, the CRLB of the th (i 1, 2, 3) parameter variance is found as the element of the inverse of the Fisher information matrix defined by [17] (11) and (13) (14) is the normalized vector of unit length pointing from the th microphone to the sound source IV PROPOSED LEAST-SQUARES ESTIMATOR WITH LINEAR-CORRECTION Two limitations of the MLE are that probabilistic assumptions have to be made about the measured range differences and that the likelihood cost function is difficult to minimize due to its nonlinearity An alternative method is the well-known least squares estimator (LSE) The LSE makes no probabilistic assumptions about the data and hence can be applied to the source localization problem in which a precise statistical characterization of the data is hard to achieve Furthermore, an LSE usually produces a closed-form estimate that is desirable in real-time applications In this section, we will begin by investigating the least squares (LS) error criteria and present a new LSE with linear correction for the source localization problem A LS Error Criteria In the LS approach, we attempt to minimize the power of an error function that is zero in the absence of noise and model inaccuracies Different error functions can be defined for closeness from the assumed (noiseless) signal based on hypothesized parameters to the observed data When these are applied, different LSEs will be generated For the source localization problem two LS error criteria can be constructed and will be presented in the following two subsections 1) Hyperbolic LS Error Function: The first LS error function is defined as the difference between the observed range difference and that generated by a signal model depending upon the unknown parameters Such an error function is routinely used in many LS estimators (15)

4 946 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 Fig 1 Spatial diagram illustrating variables defined in the source localization problem When the statistical characteristics of the corrupting noise are unknown, it is more reasonable to assume that it is uncorrelated white Gaussian noise In this case, it is not surprising that the hyperbolic LSE and the MLE minimize (maximize) similar criteria 2) Spherical LS Error Function: The second LS criterion is based upon the distance error from a hypothesized source location to the microphones The correct source location should be at the intersection of a group of spheres centered at the microphones Therefore, the best estimate of the source location will be the point that yields the shortest distance to those spheres defined by the range differences and the hypothesized source range Consider the distance from the th microphone to the source From the definition of the range difference (5) and the fact that,wehave (17) where denotes an observation based on the measured range difference From the inner product, we can derive the true value for, the square of the noise-free distance generated by a spherical signal model (18) The spherical LS error function is then defined as the difference between the measured and true values (19) Fig 2 Three-dimensional microphone array for passive source localization and the corresponding LS criterion is given by Putting the gives where errors together and writing them in a vector form (20) (16) In the source localization problem, an observed range difference defines a hyperboloid in a three-dimensional (3-D) space All points lying on such a hyperboloid are potential source locations and all have the same range difference to the two microphones and 0 Therefore, a sound source located by minimizing the hyperbolic LS criterion (16) is the one that has the shortest distance to all hyperboloids associated with different microphone pairs and specified by the estimated range differences In (15), the signal model consists of a set of hyperbolic functions Since they are highly nonlinear, minimizing (16) leads to a mathematically intractable solution as gets large Moreover, the hyperbolic function is very sensitive to noise, especially for far-field sources Therefore, it is rarely used in practice and indicates that and are stacked side-by-side The corresponding LS criterion is then given by (21) In contrast to the hyperbolic error function, the spherical error function (20) is linear in and The computational complexity to find a solution will not dramatically increase as gets large

5 HUANG et al: REAL-TIME PASSIVE SOURCE LOCALIZATION 947 TABLE I SIMULATION SUMMARY STATISTICS OF THE SOURCE LOCATION ESTIMATORS FOR A NEAR-FIELD (R =100cm) SOURCE AT AZIMUTH ANGLE OF =45 WITH INDEPENDENT NOISE IN THE MEASURED RANGE DIFFERENCES: SPHERICAL INTERPOLATION (SI), QUADRATIC-CORRECTION LEAST-SQUARES WITHOUT ITERATIONS (QC-i) AND WITH ITERATIONS (QC-ii) IN THE SECOND CORRECTION STAGE, AND LINEAR-CORRECTION LEAST-SQUARES (LC) B Spherical LSE Solution With Linear Correction Finding the LS solution based on the spherical error criterion (21) is a linear minimization problem subject to a quadratic constraint (22) (23) where is a diagonal and orthonormal matrix The technique of Lagrange multipliers is used and the source location is determined by minimizing the Lagrangian where is the Lagrange multiplier Expanding this expression, we have (24) Necessary conditions for minimizing (24) can be obtained by taking the gradient of with respect to and equating the result to zero This produces Solving for yields the constrained least squares estimate (25) (26) where is yet to be determined In order to find,wecan impose the quadratic constraint directly by substituting (26) into (23), so that By using an eigenvalue factorization, the matrix diagonalized as (27) can be (28) where and is the eigenvalue of the matrix Substituting (28) into (27),

6 948 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 Fig 3 Empirical bias and standard deviation of source location estimators versus the source elevation angle Source azimuth angle =45 and range R =300cm are fixed The errors in the mismeasured range differences are IID normal and have mean 0 and standard deviation =0:5 cm (a) Estimators of x, (b) estimators of y, and (c) estimators of z we may rewrite the constraint as Therefore, the function of the Lagrange multiplier (29) where (30) is a polynomial of degree six Due to its complexity, numerical methods have to be used for root searching Since the root of

7 HUANG et al: REAL-TIME PASSIVE SOURCE LOCALIZATION 949 Fig 4 Empirical bias and standard deviation of source location estimators versus the source elevation angle Source azimuth angle =45 and range R =300cm are fixed The errors in the mismeasured range differences are IID normal and have mean 0 and standard deviation =1cm (a) Estimators of x, (b) estimators of y, and (c) estimators of z (30) for is not unique, a two-step procedure will be followed such that the desired source location can be determined In the first step, we assume that, and are mutually independent Then the LS solution minimizing (21) for (the source location as well as its range) is given by (31) where is the pseudo-inverse of the matrix The estimate is an unconstrained global least-squares minimizer of the spherical error criterion In the presence of mea-

8 950 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 TABLE II SIMULATION SUMMARY STATISTICS OF THE SOURCE LOCATION ESTIMATORS FOR A FAR-FIELD (R =300cm) SOURCE AT AZIMUTH ANGLE OF =45 WITH INDEPENDENT NOISE IN THE MEASURED RANGE DIFFERENCES: SPHERICAL INTERPOLATION (SI), QUADRATIC-CORRECTION LEAST-SQUARES WITHOUT ITERATIONS (QC-i) AND WITH ITERATIONS (QC-ii) IN THE SECOND CORRECTION STAGE, AND LINEAR-CORRECTION LEAST-SQUARES (LC) surement errors in the range differences, it deviates from its true value and can be expressed as (32) For such an LS estimator, the bias and the covariance matrix can be approximated by using the perturbation approach when the measurement errors are small, as developed in [15] where the values with superscript 0 are the true values and (33) (34) Since the measurement error in the range differences has zero mean, is an unbiased estimate of when the small error assumption holds (35) In the first unconstrained LS estimate (31), the range information is redundant because of the independence assumption As illustrated in [18], if that information is simply discarded, the source location estimate is the same as the spherical interpolation (SI) estimate [14] but with less computational complexity Although such an estimate can be accurate, as indicated in [14] among others, the redundant information of source range is not exploited and the estimation efficiency is low with a large variance relative to the CRLB Therefore, in the second step, we want to correct to generate a better estimate of This estimate is in the neighborhood of and obeys the constraint (23) We expect that the corrected estimate is still unbiased and has a smaller variance To begin, we substitute (32) into (25) and expand the expression to find Combined with (31), (36) becomes (36) (37)

9 HUANG et al: REAL-TIME PASSIVE SOURCE LOCALIZATION 951 Fig 5 Empirical bias and standard deviation of source location estimators versus the source elevation angle Source azimuth angle =45 and range R = 300cm are fixed The errors in the mismeasured range differences are colored normal and have mean 0 and standard deviation = 0:7 cm (a) Estimators of x, (b) estimators of y, and (c) estimators of z and hence final output of the proposed two-step estimation procedure Substituting (38) into (32) yields (38) (39) (40) Equation (40) suggests how the second-step processing updates the source location estimate based on the first LSE results, ie, the SI estimate If the regularity condition [19] Solving for produces the corrected estimate and also the (41)

10 952 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 Fig 6 Empirical bias and standard deviation of source location estimators versus the source elevation angle Source azimuth angle =45 and range R = 300cm are fixed The errors in the mismeasured range differences are colored normal and have mean 0 and standard deviation = 1:0 cm (a) Estimators of x, (b) estimators of y, and (c) estimators of z is satisfied, then the estimate series can be expanded in a Neumann (42) implies that in order to avoid divergence, the Lagrange multiplier should be small In addition, needs to be determined carefully such that obeys the quadratic constraint (23) Because the function is smooth near (corresponding to the neighborhood of ), as suggested by numerical experiments, the secant method [20] can be used to determine its desired root Two reasonable initial points can be chosen as where the second term is the linear correction Equation (41) (43)

11 HUANG et al: REAL-TIME PASSIVE SOURCE LOCALIZATION 953 TABLE III SIMULATION SUMMARY STATISTICS OF THE SOURCE LOCATION ESTIMATORS FOR A FAR-FIELD (R =300cm) SOURCE AT AZIMUTH ANGLE OF =45 WITH MUTUALLY CORRELATED, JOINT GAUSSIAN NOISE IN THE MEASURED RANGE DIFFERENCES: SPHERICAL INTERPOLATION (SI), QUADRATIC-CORRECTION LEAST-SQUARES WITHOUT ITERATIONS (QC-i) AND WITH ITERATIONS (QC-ii) IN THE SECOND CORRECTION STAGE, AND LINEAR-CORRECTION LEAST-SQUARES (LC) where the small number is dependent on the array geometry Five iterations should be sufficient to give an accurate approximation to the root In summary, the steps of the linear-correction least-squares estimation procedure are as follows: Introduce one supplemental variable, the source range, in addition to the source location coordinates, and construct a linear error function given by (20) With (31), the first LSE estimates independent and that minimize the spherical LS criteria defined by (21) In the second step, we try to take advantage of the information redundancy in the first LS estimate to improve the estimation efficiency and update its estimate with the linear correction (40) The Lagrange multiplier is found by searching the root of the function (30) around zero Via, the quadratic constraint (23) is imposed on the final solution C On the Correction Techniques ChanandHo[15]firstsuggestedexploitingtherelationbetween thesourcerangeanditslocationcoordinatestoimprovetheestimation efficiency of the SI estimator with quadratic correction Accordingly, they constructed a quadratic data model for where denotes the Schur (element-by-element) product (44) is a constant matrix, and is the corrupting noise In contrast to our proposed linear correction technique based on the Lagrange multiplier, the quadratic counterpart needs to know the covariance matrix of measurement errors in the range differences a priori In a real-time digital source localization system, a poorly estimated will lead to performance degradation In addition, the QCLS estimation procedure uses the perturbation approaches to linearly approximate and in (32) and (44), respectively Therefore, the approximations of their corresponding covariance matrices and can be good only when the noise level is low When noise is at a practically high level, the QCLS estimate has a large bias and a high variance Furthermore, since the true value of the source location which is necessary for calculating and cannot be known theoretically, the estimated source location has to be utilized for approximation It was suggested in [15] that several iterations in the second correction stage would improve estimation accuracy However, while the bias is suppressed after iterations, the estimate is closer to the SI solution and the variance is boosted, as will be demonstrated in our simulations Finally, the direct solutions of the quadratic-correction LSE are the squares of the source location coordinates In 3-D space, these correspond to 8 positions, which introduce decision ambiguities Other physical criteria, such as the domain of interest, were suggested but these are hard to define in practical situations, particularly when one of the source coordinates is close to zero By comparison, our proposed linear-correction method updates the source location estimate of the first LSE without making any assumption about the error covariance matrix and without resort to a linear approximation Even though we need to find a small root of function (30) for the Lagrange multiplier that satisfies the regularity condition (41), the function is smooth around zero and the solution can be easily determined using the secant method We claim that the proposed linear-correction method achieves a good balance between computational complexity and estimation accuracy

12 954 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 V SIMULATIONS We have proposed a linear-correction LS estimator for the passive source localization problem In this section, we investigate its performance at practical measurement error levels using Monte Carlo simulations For comparison, three existing estimators suitable for real-time applications are also studied: spherical interpolation (SI), and quadratic-correction LS estimators without iterations (QCLS-i) and with iterations (QCLS-ii) in the second correction stage Because they involve quadratic functions, the QCLS estimators are not guaranteed to have a unique estimate To resolve this ambiguity, the QCLS methods choose the location whose coordinates have the same sign as that of the first-step estimate, which is the SI solution Such selection criteria produce better performance for the QCLS estimators from our simulations In our numerical studies, we employed a microphone array of six sensors, as shown in Fig 2, which was applied in a passive acoustic tracking system [18] recently developed at Bell Labs The sensors are located at (distances in centimeters) (45) For this array geometry, the value of in (43) was empirically determined as For all source location estimators, the estimation accuracy in terms of the bias and standard deviation is a function of the source location, since the array is fixed More specifically, the accuracy is, in general, inversely proportional to the source range This dependency was studied in our simulations by positioning a single source at various locations, near-field versus far-field, with different elevation angles At each location, the empirical bias and standard deviation of each estimator were obtained by averaging the results of 2000-trial Monte-Carlo runs The measurement errors were set to have mean zero and prespecified standard deviation We considered independent noise as well as mutually dependent noise since the latter is more common in practice For mutually independent measurement errors, both Gaussian and uniform distributions were investigated For mutually dependent measurement errors, we present a study with a joint Gaussian distribution only The covariance matrix is given by [15] (46) where is the variance of the measurement errors We first considered a near-field source with range cm Different source azimuth angles produce different estimation accuracy for and For ease of presentation, we define which results in the same level of bias and standard deviation magnitude for the estimators of and Table I reports the summary statistics of the studied estimators under IID Gaussian and uniform error distributions, and standard deviations 05 and 1 As seen, these estimators have reasonably small biases and small standard deviations under all situations Notice that the bias and standard deviation diminish as the measurement error is reduced Remarkably, the performance of these least-squares estimators with Gaussian errors is similar to that with uniform errors This appealing feature is a benefit of having made no assumption about the error distribution in the least-squares approaches, in contrast to the maximum likelihood methods A second set of numerical experiments involved far-field sources We moved the single source to positions 300 cm away from the array This range is more than six times the array size and is of more interest in camera steering and beamforming applications The summary statistics are presented in Table II Those with IID Gaussian errors of standard deviations 05 and 1 are plotted in Figs 3 and 4, respectively In the graphs of standard deviation, the CRLBs are also presented for comparison As clearly shown in these figures, the QCLS-i estimator has the largest bias Even worse, the bias grows as the measurement errors and the source range increase Although the QCLS-ii estimator has a very small bias, its variance is relatively quite large Since the source range and azimuth angle are fixed in this numerical experiment, as the source is moved away from the horizontal ( ) plane, the standard deviations of the estimators generally grow while those of the and estimators remain almost the same In terms of standard deviation, all correction estimators perform better than the SI estimator (without correction) Among these four studied LS estimators, the QCLS-ii and the LCLS achieve the lowest standard deviation and their values approach the CRLB at most source locations The last set of numerical experiments studied how these source location estimators performed when the measurement errors were mutually correlated The source is again 300 cm from the array with a 45 azimuth angle Because the covariance matrix of the measurement errors is impossible or difficult to estimate in practice, that information is not provided to the estimators and no emphasis on particular range differences is made in our simulations The results are presented in Table III and are plotted in Figs 5 and 6 for error standard deviations 07 cm and 10 cm, respectively We see that the performance of all estimators deteriorates because the errors are no longer independent When the noise is at a low level, ie, 07 cm standard deviation, the QCLS-ii estimate is still able to achieve a low standard deviation with a moderate bias But when the noise reaches a level as high as that of a 10 cm standard deviation, the linear approximation used by the QCLS estimators is invalid and the estimation procedure fails, as shown in Fig 6 However, the proposed LCLS estimation procedure makes no assumption about the covariance matrix of the measurement errors and does not depend on a linear approximation It always produces an estimate whose bias is small and whose variance grows steadily with the noise level The LCLS estimator outrivals the other three estimators when the noise level is high When the noise covariance matrix is available in some cases, the QCLS estimators perform better but such a comparison is well beyond the scope of this paper

13 HUANG et al: REAL-TIME PASSIVE SOURCE LOCALIZATION 955 VI CONCLUSIONS In this paper, we have proposed the linear-correction least-squares approach to the source localization problem in the presence of measurement errors This method is mathematically simple and can be easily implemented in a real-time system, without any assumptions in addition to the additive measurement error model This feature is appealing because many assumptions made by the likelihood-based estimators about the error distribution, as indicated in Section III, can be difficult to verify in practice, particularly for discrete-time systems Meanwhile, one would be interested in the efficiency of the proposed estimator relative to the Cramér Rao lower bound and other closed-form methods Our simulations reported in Section V suggest that with smaller standard deviation, both the linear-correction and quadratic-correction least-squares estimators are more efficient than the spherical interpolation algorithms and are able to approach the CRLB Between these two correction techniques, the LCLS algorithm is more practical making no assumption about the noise covariance Without iterations in the second stage, the QCLS estimator has a big bias But with iterations, the QCLS estimator yields a large variance The proposed efficient estimation procedure is a good compromise of small bias and low variance ACKNOWLEDGMENT The authors would like to thank J H McClellan and D R Morgan for helpful discussions and acknowledge the comments and suggestions of the anonymous reviewers REFERENCES [1] Y Huang, J Benesty, and G W Elko, Microphone arrays for video camera steering, in Acoustic Signal Processing for Telecommunication, S L Gay and J Benesty, Eds Norwell, MA: Kluwer Academic, 2000 [2] H Wang and P Chu, Voice source localization for automatic camera pointing system in videoconferencing, in Proc IEEE Workshop Appls Signal Processing Audio Acoustics, 1997 [3] D V Rabinkin, R J Ranomeron, J C French, and J L Flanagan, A DSP implementation of source location using microphone arrays, Proc SPIE, vol 2846, pp 88 99, 1996 [4] C Wang and M S Brandstein, A hybrid real-time face tracking system, in Proc IEEE ICASSP, vol 6, 1998, pp [5] H F Silverman, Some analysis of microphone arrays for speech data analysis, IEEE Trans Acoust, Speech, Signal Processing, vol 35, pp , Dec 1987 [6] J L Flanagan, A Surendran, and E Jan, Spatially selective sound capture for speech and audio processing, Speech Commun, vol 13, pp , Jan 1993 [7] D B Ward and G W Elko, Mixed nearfield/farfield beamforming: A new technique for speech acquisition in a reverberant environment, in Proc IEEE Workshop Appls Signal Processing Audio Acoustics, 1997 [8] W R Hahn and S A Tretter, Optimum processing for delay-vector estimation in passive signal arrays, IEEE Trans Inform Theory, vol IT-19, pp , May 1973 [9] M Wax and T Kailath, Optimum localization of multiple sources by passive arrays, IEEE Trans Acoust, Speech, Signal Processing, vol ASSP-31, no 5, pp , Oct 1983 [10] Y Bard, Nonlinear Parameter Estimation New York: Academic, 1974 [11] W H Foy, Position-location solutions by Taylor-series estimation, IEEE Trans Aerosp Electron Syst, vol AES-12, pp , Mar 1976 [12] R O Schmidt, A new approach to geometry of range difference location, IEEE Trans Aerosp Electron, vol AES-8, pp , Nov 1972 [13] H C Schau and A Z Robinson, Passive source localization employing intersecting spherical surfaces from time-of-arrival differences, IEEE Trans Acoust, Speech, Signal Processing, vol ASSP-35, pp , Aug 1987 [14] J O Smith and J S Abel, Closed-form least-squares source location estimation from range-difference measurements, IEEE Trans Acoust, Speech, Signal Processing, vol ASSP-35, pp , Dec 1987 [15] Y T Chan and K C Ho, A simple and efficient estimator for hyperbolic location, IEEE Trans Signal Processing, vol 42, pp , Aug 1994 [16], An efficient closed-form localization solution from time difference of arrival measurements, in Proc IEEE ICASSP, vol II, 1994, pp [17] S M Kay, Fundamentals of Statistical Signal Processing: Estimation Theory Englewood Cliffs, NJ: Prentice-Hall, 1993 [18] Y Huang, J Benesty, and G W Elko, Passive acoustic source localization for video camera steering, in Proc IEEE ICASSP, 2000 [19] C D Meyer, Matrix Analysis and Applied Linear Algebra Philadelphia, PA: SIAM, 2000 [20] W H Press, B P Flannery, S A Teukolsky, and W T Vetterling, Numerical Recipes in C: The Art of Scientific Computing Cambridge, UK: Cambridge Univ Press, 1988 Yiteng (Arden) Huang received the BS degree from the Tsinghua University, China, in 1994, the MS and PhD degrees from the Georgia Institute of Technology (Georgia Tech), Atlanta, in 1998 and 2001, respectively, all in electrical and computer engineering From 1998 to 2001, he was a Research Assistant with the Center of Signal and Image Processing, Georgia Tech, and a Teaching Assistant with the School of Electrical and Computer Engineering, Georgia Tech In the summers from 1998 to 2000, he worked with Bell Laboratories, Murray Hill, NJ and engaged in research on passive acoustic source localization with microphone arrays He joined Bell Laboratories as a Member of Technical Staff in March 2001 His current research interests are in adaptive filtering, multichannel signal processing, source localization, microphone array for hands-free telecommunication, speech enhancement, and statistical signal processing Dr Huang is the recipient of the Colonel Oscar P Cleaver Outstanding Graduate Student Award from the School of Electrical and Computer Engineering, Georgia Tech, and the 2000 Outstanding Research Award from the Center of Signal and Image Processing, Georgia Tech He was awarded the Outstanding Graduate Teaching Assistant Award of the School of Electrical and Computer Engineering, Georgia Tech Jacob Benesty (M 98) was born in Marrakech, Morocco, in 1963 He received the MS degree in microwaves from Pierre and Marie Curie University, France, in 1987, and the PhD degree in control and signal processing from Orsay University, France, in 1991 During his PhD studies (from November 1989 to April 1991), he worked on adaptive filters and fast algorithms at the Centre National d Etudes des Telecommunications (CNET), Paris, France From January 1994 to July 1995, he worked at Telecom Paris on multichannel adaptive filters and acoustic echo cancellation He joined Bell Labs, Lucent Technologies (formerly AT&T), Murray Hill, NJ, in October 1995, first as a Consultant and then as a Member of Technical Staff Since then, he has been working on stereophonic acoustic echo cancellation, adaptive filters, source localization, robust network echo cancellation, and blind deconvolution He coauthored the book Advances in Network and Acoustic Echo Cancellation (Berlin, Germany: Springer-Verlag, 2001) He is co-editor/co-author of the book Acoustic Signal Processing for Telecommunication (Norwell, MA: Kluwer, 2000) Dr Benesty was the co-chair of the 1999 International Workshop on Acoustic Echo and Noise Control

14 956 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 9, NO 8, NOVEMBER 2001 Gary W Elko received the BSEE degree from the Cornell University, Ithaca, NY, in 1977, and the MS and PhD degrees from the Pennsylvania State University, University Park, in 1980 and 1984, respectively, all in electrical and computer engineering He was a Distinguished Member of Technical Staff at AT&T/Lucent Bell Laboratories, Murray Hill, NJ, where he supervised a research group working on signal processing and electroacoustics related to the hands-free acoustic telecommunication problem He is currently a Consulting Member of Technical Staff at Agere Systems, Murray Hill His current interests are in microphone array beamforming, acoustic echo cancellation, 3-D audio, room acoustics, and electroacoustic transducers Dr Elko is a past Associate Editor for the IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING ( ), a member of the IEEE Audio and Electroacoustics Technical Committee (1990 present), and a fellow of the Acoustical Society of America He was chosen as one of the 2001 IEEE Signal Processing Society distinguished lecturers Russell M Mersereau (S 69 M 73 SM 78 F 83) received the SB and SM degrees in 1969 and the ScD degree in 1973 from the Massachusetts Institute of Technology, Cambridge He joined the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, in 1975 His current research interests are in the development of algorithms for the enhancement, modeling, and coding of computerized images, synthesis aperture radar, and computer vision In the past, this research has been directed to problems of distorted signals from partial information of those signals, computer image processing and coding, the effect of image coders on human perception of images, and applications of digital signal processing methods in speech processing, digital communications, and pattern recognition He is the coauthor of the text Multidimensional Digital Signal Processing (Englewood Cliffs, NJ: Prentice-Hall, 1984) Dr Mersereau has served on the Editorial Board of the PROCEEDINGS OF THE IEEE and as Associate Editor for signal processing of the IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING and IEEE SIGNAL PROCESSING LETTERS He is the corecipient of the 1976 Bowder J Thompson Memorial Prize of the IEEE for the best technical paper by an author under the age of 30, a recipient of the 1977 Research Unit Award of the Southeastern Section of the ASEE, and three teaching awards He was awarded the 1990 Society Award of the Signal Processing Society He is currently the Vice President for Awards and Membership of the SP Society