REFERENCES. A Barankin-Type Bound on Direction Estimation Using Acoustic Sensor Arrays

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO., ANUARY 0 43 CRB( detection ) where p = 0 represents the so-called translation factor [] which is determined due to the probability of detection P d and the probability of false alarm P fa as follows: P fa = Q (&) and P d = Q ( ) (&) where Q (:) and Q ( ) (:) denote the right tail probability of and 0 ), respectively. This conclude the proof. ( REFERENCES [] D. Donno, A. Nehorai, and U. Spagnolini, Seismic velocity and polarization estimation for wavefield separation, IEEE Trans. Signal Process., vol. 56, no. 0, pp , Oct []. Li and R. Compton, Angle and polarization estimation using esprit with a polarization sensitive array, IEEE Trans. Antennas Propag., vol. 39, no. 9, pp , Sep. 99. [3] I. Ziskind, M. Wax, and. Rafael, Maximum likelihood localization of diversely polarized sources by simulated annealing, IEEE Trans. Antennas Propag., vol. 38, no. 7, pp. 4, ul [4] K. Wong and M. 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Moses, Efficient maximum likelihood DOA estimation for signals with known waveforms in presence of multipath, IEEE Trans. Signal Process., vol. 45, pp , Mar [3] A. Renaux, Weiss Weinstein bound for data aided carrier estimation, IEEE Signal Process. Lett., vol. 4, no. 4, pp , Apr [4] B. Ng and C. See, Sensor-array calibration using a maximum-likelihood approach, IEEE Trans. Antennas Propag., vol. 44, pp , un [5] P. Stoica and R. Moses, Spectral Analysis of Signals. Englewood Cliffs, N: Prentice-all, 005. [6] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, N: Prentice-all, 993, vol.. [7] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Englewood Cliffs, N: Prentice-all, 998, vol.. A Barankin-Type Bound on Direction Estimation Using Acoustic Sensor Arrays Tao Li, oseph Tabrikian, and Arye Nehorai Abstract We derive a Barankin-type bound (BTB) on the mean-square error (MSE) in estimating the directions of arrival (DOAs) of far-field sources using acoustic sensor arrays. We consider narrowband and wideband deterministic source signals, and scalar or vector sensors. Our results provide an approximation to the threshold of the signal-to-noise ratio (SNR) below which the performance of the maximum likelihood estimation (MLE) degrades rapidly. For narrowband DOA estimation using uniform linear vector-sensor arrays, we show that this threshold increases with the distance between the sensors. As a result, for medium SNR values the performance does not necessarily improve with this distance. Index Terms Acoustic sensor array, acoustic vector sensor, Barankin bound, direction of arrival estimation, threshold SNR. I. INTRODUCTION The Barankin bound [] [4] is a useful tool in estimation problems for predicting the threshold region of signal-to-noise ratio (SNR) [5] [8], below which the accuracy of the maximum likelihood estimation (MLE) degrades rapidly. Identification of the threshold region enables to determine the operation conditions, such as observation time and transmission power, to obtain a desired performance. In the recent years, many works have been carried out for identification of the threshold region of the MLE. One approach is based on the method of interval estimation (MIE) [9] in which the performance of the MLE in the threshold region is approximated. owever, Manuscript received October 6, 009; accepted September 07, 00. Date of publication September 0, 00; date of current version December 7, 00. The work of T. Li and A. Nehorai was supported by the ONR Grant N The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Xavier Mestre. T. Li and A. Nehorai are with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO, 6330 USA ( litao@ese.wustl.edu; nehorai@ese.wustl.edu).. Tabrikian is with the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 8450, Israel ( joseph@ee.bgu.ac.il). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TSP X/$ IEEE

2 43 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO., ANUARY 0 it does not provide a performance lower bound. Another well investigated approach is the use of the non-bayesian bounds which can predict the threshold region and provide a performance lower bound. In this correspondence, we derive a Barankin-type bound (BTB) on the directions-of-arrival (DOAs) estimation of acoustic signals based on measurement models with deterministic unknown parameters. The derived bound provides an approximation for the threshold SNR even in the case of a high-dimensional unknown parameter vector, for which computing the Barankin bound is usually formidable. We examine the performance of the derived bound for both wideband and narrowband DOA estimation problems. In addition, we examine the effect of the distance between the acoustic vector sensors (AVSs) in a uniform linear array on the narrowband DOA estimation error. Unlike scalar sensors, AVSs measure not only the sound pressure but also the components of particle velocity [0]. One of the advantages of the AVS arrays over the scalar ones is that the distance between adjacent AVSs can be larger than half of the signal wavelength without introducing spatial ambiguities in narrowband DOA estimation. Thus, we may design the distance between the AVSs to achieve better estimation results. Our examination of the derived BTB and actual MLE error shows that the threshold SNR increases with the distance between the AVSs, and that enlarging this distance does not always improve the estimation accuracy. Compared with the CRB, which consistently decreases with the increase of inter-avs distance, the derived BTB provides more accurate reference for the optimal distance determination, as it incorporates the effect of large estimation errors. The remainder of this correspondence is organized as follows. In Section II, we formulate the model for data collected by acoustic sensor arrays. In Section III, we derive the BTB for source localization. Section IV presents the numerical examples. The conclusion of this correspondence appears in Section V. II. MEASUREMENT MODEL In this section, we present the measurement models for both wideband and narrowband sources for uniform linear AVS arrays. The measurement models for other types of sensor arrays can be formulated in the same way using the corresponding steering vectors. We first present the measurement model for wideband sources. Consider signal waves from K distant wideband acoustic sources impinging on a uniform linear array of M AVSs, each of which consists of m components. In similar to [], [], using the wideband harmonic signal model, we approximate the incident signal from the kth source as x k (t) = c kj j (t) () where (t);...; (t) are the monochromatic waves representing Fourier series kernels and c k ;...;c k are their coefficients. Assuming the frequency of j (t) is fj, we write the measurement model as y(t) = = K k= a j ( k )c kj j (t) +e(t) A j ()c j j (t) +e(t); t =;...;N () where y(t) is the Mm measurement vector, k is the one- or two-dimensional DOA of the kth source, e(t) is the Mm noise vector, = [ T ;...;K] T T ;c j = [c j ;...;c Kj ] T, T denotes the transpose operation, N is the number of temporal measurements, A j () = [aj ()...aj(k)], and aj (k ) is the steering vector for the kth source given by a j (k )=h j(k ) (3) u( k ) where denotes the Kronecker product, h j ( k ) = [;e if ( ) ;...;e if (M0) ( ) T ] with ( k ) the differential time delay of the kth source signal between two adjacent AVSs, and u( k ) is the (m 0 ) unit-norm direction vector from the array to the kth source. We assume the noise is zero-mean circular Gaussian with known spatial covariance matrix 6 and is temporally white. For narrowband sources, the steering vectors can be assumed to be constant over the signal bandwidth with frequency fixed on the carrier frequency of the narrowband sources, i.e., a j () =a(), and the measurement model () then becomes y(t) = K k= a( k ) c kj j (t) + e(t) = A()C (t) +e(t); t =;...;N (4) where C is a K matrix whose (k; j)th entry is c kj ; (t) = [ (t);...; (t)] T ;A() =[a( )...a( K )], and a( k )=h( k ) u( k ) where h( k ) = [;e if( ) ;...;e if(m0) ( ) ] T and f is the carrier frequency of the narrowband signals. Note that in (4), c kj j (t) is the complex amplitude of the kth incident signal approximated by Fourier series expansion. III. BARANKIN-TYPE BOUND In this section, we derive a BTB on the DOA vector based on the measurement models in Section II. An analytical form of the Barankin bound is generally not available. We use the Barankin-type lower bound developed in [] [4] to approximate the actual Barankin Bound. For an unbiased estimator ^ of an unknown deterministic parameter vector, this lower bound can be formulated as (5) covf^g T (B 0 T ) 0 T T (6) where T =[ 0... L 0]; ;...; L are the test points selected from the parameter space of ; is a column vector of dimension L with all entries equal to, and B is the L L Barankin matrix whose (l; n)th entry is B ln =Efr(y; l ;)r(y; n ;)g (7) where Efg denotes the expectation operation, r(y; l ;) = f (y; l )=f (y; ), and f (y; ) is the probability density function (pdf) of y with parameters. The supremum of the BTB given in (6) approaches the actual Barankin bound when the number of test points tends to infinity [3]. In the following, we first derive the BTB on based on the wideband measurement model (). Stacking all measurements together, we rewrite () as Y = A j ()cj T j + E (8) where Y = [y()... y(n )];E = [e()... e(n)], and j = [ j ();...; j (N )] T. Vectorizing both sides of (8), we obtain y = ( j A j ())cj + e = R()c + e (9)

3 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO., ANUARY where y = [y T ();...;y T (N )] T ;e = [e T ();...;e T (N)] T ; c = [c T ;...;c T ] T, and R() =[ A ()... A ()]. The unknown parameter vector in (9) is = [ T ; Refc T g; Imfc T g] T, in which Refc T g and Imfc T g are the real and imaginary parts of c T, respectively. The pdf of y with parameters is f (y; ) = expf0(y 0 R()c) (I N 6 0 )(y 0 R()c)g (0) where det( ) denotes the determinant of a matrix and I N is the unit matrix of size N. Substituting (0) into (7), we obtain B ln = expf(y 0 R()c) (I N 6 0 ) (y 0 R()c) 0 (y 0 R( l )c l ) (I N 6 0 ) (y 0 R( l )c l ) 0 (y 0 R( n )c n ) (I N 6 0 ) (y 0 R( n )c n )g dy = A expfref(r()c 0 R( l )c l ) (I N 6 0 ) (R()c 0 R( n )c n )gg () where [ T l ; Refc T l g; ImfcT l g]t is the lth selected test point for and A = Thus, we have expf(y 0 R( l )c l 0 R( n)c n + R()c) (I N 6 0 )(y 0 R( l )c l 0 R( n )c n + R()c)gdy =: () B ln = expfref(r()c 0 R( l )c l ) (I N 6 0 ) (R()c 0 R( n )c n )gg: (3) To obtain a tight bound, we ought to choose the test points that maximize the right side of (6). Observe that in many cases is a high-dimensional vector, which generally requires a huge amount of test points to produce a tight bound. This would inevitably augment the burden in computing the inverse of B 0 T. We employ the approximation method proposed in [7] to reduce the required number of test points. This method is adapted as follows for DOA estimation. Select L candidate test points ;...; L for the DOA parameter. For each selected l ; l = ;...; L, select test point c jl for c j ; j =;...; such that B ll is minimized at l =[ T l ; Refc l T g; Imfc l T g]t =[ T l ; Ref[c l; T...; c l]g; T Imf[c l; T...; c l]g] T T. From ;...; L, choose the L test points producing the L smallest values in B;...;B LL as the test points to compute the bound [7]. Alternatively, choose the test points at the L highest lobe peaks of B 0 ;...;B0 L L provided that the values of B 0 ;...;B 0 L L exhibit evident sidelobes. To obtain the BTB according to the above method, we first find the test point c l that minimizes B ll for a given l. Taking the derivative of B ll with respect to c l, we have the test point c l minimizing B ll is c l = G 0 ( l ; l )G( l ;)c (4) where G( l ;) =R ( l )(I N 6 0 )R(). The general result provided by (4) can be further simplified if we have k j = kj (5) with kj the Kronecker delta function, which is a general property of the problem at hand since (t);...; (t) can be chosen using the kernels of the Fourier series or by Karhunen Loève transform. Using (4) and (5), we can further obtain (see the Appendix) the test point c jl that minimizes B ll c jl = P 0 j ( l ; l )P j ( l ;)c j ; j =;...; (6) where P j ( l ;) =A j ( l )6 0 A j (). Substituting (6) into (3), we obtain the following expression for the (l; n)th entry of B: B ln = expftrfrefq ( l ;)6 0 Q( n ;)ggg (7) where trfg denotes the trace of a matrix, Q( l ;) = [q ( l ;)...q ( l ;)], and q j ( l ;) =(A j () 0 A j ( l )P 0 j ( l ; l ) P j ( l ;))c j. Assuming the noise covariance matrix 6 = I and there is only one source, we are able to simplify the formula in (7) as B ln = exp where Ref(A j ()c j ) 3 j ( l ; n )(A j ()c j )g (8) 3 j ( l ; n )= I 0A j ( l )A j ( l )= I 0A j ( n )A j ( n )= : (9) In (9), is the square of the Euclidean norm of the steering vector. For scalar-sensor arrays, we have = M. For vector-sensor arrays, we have = M (note that the Euclidean norm of u() is ). The diagonal entries of B can be further simplified as B ll = exp cj c j 0 A j ()A j ( l ) = (0) where j jdenotes the amplitude of a complex number. Using the derived expressions of B, we are able to compute the BTB on an unbiased estimator ^ of, which is given by covf^g T (B 0 T ) 0 T T () where T =[ 0... L 0 ]. The bound based on the narrowband model (4) can be derived similarly and herein we only present the results. For the narrowband model in (4), the test point C l minimizing B ll is C l = P 0 ( l ; l )P ( l ;)C () where P ( l ;) =A ( l )6 0 A(). Under the condition in (5), the formula of B for the narrowband model can be written as B ln = expftrfrefq ( l ;)6 0 Q( n ;)ggg (3) where Q( l ;) =(A() 0 A( l )P 0 ( l ; l )P ( l ;))C. Note that though our test-point selection method is based on the idea in [7], our models and derived results are more general than their deterministic counterparts in [7] and thus have broader applications. Our derived bounds can be used as well for some other array-processing problems such as DOA estimation for signals of known waveforms [4], [5], in which ;...; denote the known signal waveforms.

4 434 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO., ANUARY 0 Fig.. Square roots of the derived Barankin-type bound and maximum likelihood estimation MSE versus SNR for narrowband direction of arrival estimation using uniform linear scalar-sensor array. Fig. 3. Square roots of the derived Barankin-type bound, Cramér Rao bound, and maximum likelihood estimation MSE versus signal-to-noise ratio for narrowband direction of arrival estimation using uniform linear acoustic vectorsensor arrays with d = ;, and 3, respectively. Fig.. Same as in Fig., but for the wideband direction of arrival estimation. IV. NUMERICAL EXAMPLES To demonstrate the performance of the derived bound, we present numerical examples of DOA estimation for both scalar- and vectorsensor arrays. For simplicity, we assume the noise covariance matrix 6 = I in all examples. A. Examples for Scalar-Sensor Array We present results of two examples for scalar-sensor array, one for narrowband and the other for wideband signal. For the narrowband example, we consider a sinusoidal signal with elevation angle of 0 degree arriving at a uniform linear scalar-sensor array of 40 sensors. The distance d between the sensors equals half of the signal s wavelength. We took 0 samples of measurement and uniformly selected 30 test points from [0=;=] to compute the bound. Note that uniformly selecting test points is a special case of the proposed test point selection method with L = L. Also for a uniform linear scalar-sensor array, the distance between adjacent sensors should not be larger than half of the signal wavelength to avoid ambiguity in the DOA estimation of a narrowband signal. For the wideband example, we assume the distance between the sensors is meter and the signal consists of 0 harmonic components, with elevation angle of 0 degree. The frequencies of the 0 harmonic components of the signal are 0, 40,...; 80, and 00 z, respectively. The linear array consists of 40 sensors. The sound propagation speed is c =500 m/ s. We took ten measurements at a sampling period of 5 ms and uniformly selected 60 test points to compute the bound. In this problem, we have ten unknown signal parameters, which would make approximating the Barankin bound challenging. We present the computed square roots (SRs) of the derived BTB and actual MLE mean-square error (MSE) in Figs. and. It can be seen that the derived bounds predict the true threshold SNRs with differences about to 3 db in these two examples. B. Examples for Vector-Sensor Array AVS arrays outperform the standard scalar-sensor arrays, as they utilize measurements from the components of the acoustic particle velocity as well as the pressure [0]. They resolve end-fire and conical ambiguities [6] and a single AVS can identify up to two sources in three-dimensional space [7]. Clearly, since a single AVS is able to provide DOA information, the distance d between the AVSs in a uniform linear array is not limited to be smaller than half of the wavelength of the narrowband impinging signal. Since d can be larger than half the signal wavelength, it is possible to improve the estimation accuracy through adjusting the distance between the AVSs. The lower bounds on the mean-square estimation error should provide useful information about the choice of the optimal distance. We examined the effect of the distance between the AVSs on the derived BTB, CRB, and actual MLE error for a narrowband one-dimensional DOA estimation problem (see Fig. 3). In our example, we consider a narrowband signal with DOA = 0 degree impinges on a uniform linear array consisting of ten AVSs. The vector sensors measure the pressure and only two velocity components such that h() = [;e i(=)d sin ;...;e i(=)(m0)d sin ] T and u() = [cos; sin ] T in (5), where is the one-dimensional DOA and is the wavelength of the narrowband signal. We selected the test points at the peaks of the Bll 0 values over 00 uniformly sampled candidate points, and computed the bounds and MLE errors using one measurement for d = ; and 3, respectively. According to [8], selecting test points at the peaks of Bll 0 is equivalent to selecting test points at the lobes of the maximum likelihood ambiguity function (MLAF). Observe from the results of the BTB and MLE error in Fig. 3 that the SNR threshold increases with the distance between the AVSs, and that increasing the distance between the AVSs does not always improve the estimation accuracy. From the MLAF plots in Fig. 4, we can see that with the increase of d, the main lobe width of the MLAF decreases whereas the number and level of side lobes increase. Therefore, when the d value increases, the asymptotic ML estimation error decreases while the threshold SNR increases. This explains the phenomena observed in Fig. 3. Since the MLE search includes ambiguities from side lobes, we must use a global error bound like the BTB to capture the threshold effect impact of these ambiguities often observed at medium to low SNRs. From Fig. 3, we can see that the derived BTB provides

5 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO., ANUARY where [ j A j ( l )] is a matrix with blocks whose jth block is j A j ( l ) and [( k j ) (A k ( l )6 0 A j ())] is a matrix with blocks whose (k; j)th block is ( k j ) (A k ( l )6 0 A j ()). Since k j = kj, wehave G( l ;) =diag[a j ( l )6 0 A j ()] (5) which is a block-diagonal matrix of blocks with the jth block equal to A j ( l )6 0 A j (). Using (4), we obtain c l = G 0 ( l ; l )G( l ;)c = diag[(a j ( l )6 0 A j ( l )) 0 (A j ( l )6 0 A j ())] c = diag[p 0 j ( l ; l )P j ( l ;)] c (6) from which we derive (6). Fig. 4. Variation of MLAFs with respect to the inter-avs distance d. better estimation of the optimal distance than the CRB, which is a local error bound and consistently decreases with the increase of inter-avs distance. Note that although the derived bound correctly predicts that the threshold SNR increases with the distance between the sensors, the true threshold SNRs are about 6 to 7 db larger than the predicted ones due to the approximation in computing the bound. We found similar differences between the true and predicted threshold SNRs in the wideband DOA estimation with an AVS array of ten sensors and the same setup as above for the wideband signal. V. CONCLUSION We derived a BTB on DOA estimation using acoustic sensor arrays. Numerical examples show that this bound closely predicts the threshold SNR for DOA estimation using scalar-sensor arrays. For the narrowband DOA estimation using AVS arrays, the distance between the sensors is not required to be lower than half a wavelength to avoid ambiguity. Numerical results from the derived BTB demonstrate that increasing the distance between the AVSs improves the DOA estimation accuracy consistently only at high SNRs, and it increases the threshold SNR as well. As a result, at medium SNR values the estimation accuracy does not necessarily improve with this distance. This predicts the behavior of the actual MLE error. The derived BTB exhibits evident advantage over the CRB in determining the optimal distance between the AVSs. Our future work will aim at developing more accurate computation of the Barankin bound for high-dimensional unknown parameter vectors. We will aim to improve the approximation of the Barankin bound on DOA estimation by developing new test point selection methods for high-dimensional unknown parameter vectors and considering alternative approximation approaches of the Barankin bound such as the one in [3]. Note that APPENDIX DERIVATION OF EQUATION (6) G( l ;) =R ( l )(I N 6 0 )R() =[ j A j ( l )] (I N 6 0 )[ j A j ()] =[ j A j ( l )] [ j (6 0 A j ())] = k j (A k ( l )6 0 A j ()) (4) REFERENCES [] E. W. Barankin, Locally best unbiased estimates, Ann. Math. Statist., vol. 0, pp , 949. []. M. ammersley, On estimating restricted parameters,. Roy. Statist. Soc., Series B, vol., no., pp. 9 40, 950. [3] D. G. Chapman and. Robbins, Minimum variance estimation without regularity assumptions, Ann. Math. Statist., vol., no. 4, pp , Dec. 95. [4] R. McAulay and L. P. Seidman, A useful form of the Barankin lower bound and its application to PPM threshold analysis, IEEE Trans. Inf. Theory, vol. 5, no., pp , Mar [5] A. Zeira and P. M. Schultheiss, Realizable lower bounds for time delay estimation, IEEE Trans. Signal Process., vol. 4, pp , Nov [6] A. Zeira and P. M. Schultheiss, Realizable lower bounds for time delay estimation: Part Threshold phenomena, IEEE Trans. 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