IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER Bayesian Beamforming for DOA Uncertainty: Theory and Implementation Chunwei Jethro Lam, Student Member, IEEE, and Andrew C. Singer, Senior Member, IEEE Abstract A Bayesian approach to adaptive narrowband beamforming for uncertain source direction-of-arrival (DOA) is presented. The DOA is modeled as a random variable with prior statistics that describe its level of uncertainty. The Bayesian beamformer is the corresponding minimum mean-square error (MMSE) estimator, which can be viewed as a mixture of directional beamformers combined according to the posterior distribution of the DOA given the data. Under a deterministic DOA, the mean-square error (MSE) of the Bayesian beamformer becomes as low as that of the directional beamformer equipped with the DOA candidate in the prior set that is the closest to the true DOA at exponential rate, where closeness is defined in the Kullback Leibler sense. Two efficient algorithms using a uniform linear array (ULA) are presented. The first method utilizes the efficiency of the fast Fourier transform (FFT) to compute the posterior distribution on a large number of DOA candidates. The second method approximates the posterior distribution by a Gaussian distribution, which leads to a directional beamformer incorporated with a particular spreading matrix and an adjusted DOA. Numerical simulations show that the proposed beamformer outperforms other related blind or robust beamforming algorithms over a wide range of signal-to-noise ratios (SNRs). Index Terms Adaptive beamforming, Bayesian model, direction-of-arrival (DOA) uncertainty, minimum mean-square error (MMSE) estimation. I. INTRODUCTION ADAPTIVE beamforming is widely used in radar, sonar, seismology, speech processing, medical imaging and wireless communications for signal estimation and interference suppression using an array of sensors [1] [9]. It is well known that the performance of a beamformer degrades substantially when the presumed model does not match the environment [10] [15]. Model uncertainties can often be parameterized by a single variable that describes a physical phenomenon such as direction-ofarrival (DOA) of the source, carrier frequency, or propagation speed, for example. This paper considers DOA uncertainty, and the results can be generalized to other types of single variable uncertainties as well. Many adaptive beamforming algorithms have been developed to combat DOA uncertainty. Direction-finding methods use algorithms such as MUSIC [16] and ESPRIT [17] to estimate the Manuscript received February 1, 2005; revised November 30, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Tulay Adali. This work was supported in part by the National Science Foundation under Grant Number CCR (CAREER) and by NASA under Grant Number NAG The authors are with the Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign, Coordinated Science Laboratory, Urbana, IL USA ( clam@uiuc.edu; acsinger@uiuc.edu). Digital Object Identifier /TSP true DOA, and then use the estimate to construct the optimal beamformer. These DOA estimation methods are nearly optimal at high signal-to-noise ratio (SNR), but they often fail under low SNR or insufficient data size [18], [19]. Diagonal loading [20], linearly constrained minimum variance (LCMV) [21], and covariance matrix taper methods [22], [23], and [24], enhance DOA robustness by forming a wide mainlobe to tolerate small DOA error around the presumed DOA. These methods are typically ad hoc, and it is often unclear how to choose the design parameters. Robust beamformers against steering uncertainties are introduced in [25] [29]. These methods can also combat DOA uncertainties, but they are often overly conservative and thus suitable only for small DOA errors. Bayesian beamforming, introduced in [30], is able to estimate signals when the DOA is uncertain or completely unknown. Applying a Bayesian model, the uncertain DOA is assumed to be a random variable with a prior distribution that describes the level of uncertainty. The corresponding minimum meansquare error (MMSE) estimator can be viewed as a mixture of conditional MMSE estimators combined according to the data-driven posterior distribution function (PDF) of the DOA given the data. Bayesian beamforming can be characterized by its adaptive learning ability via the evolution of the posterior distribution under a wide range of SNRs and data sizes. When the SNR is low or the data size is short, the Bayesian beamformer emphasizes the prior information of the DOA. When the SNR is high or the data size is long, the beamformer adapts to the data and points at the implicit (via the Bayesian mixture) estimated DOA. By definition, the Bayesian beamformer is MMSE optimal. When the Bayesian model does not hold, i.e., the true DOA is deterministic and is not included in the prior, MMSE optimality is lost. In this paper, the Bayesian beamformer is shown to perform as well as the directional beamformer equipped with the DOA that is the closest to the underlying true DOA in the Kullback Leibler distance sense when the data size is sufficiently large. Under certain conditions, the Bayesian beamformer converges to the beamformer equipped with the true DOA. This shows that the Bayesian beamformer is feasible in practical applications even when the Bayesian model may not fit the true signal environment. The Bayesian beamformer is generally not trivial to implement because it is nonlinear and involves a complicated Bayesian integral. Based on some mild statistical assumptions and approximations, two efficient adaptive Bayesian beamforming algorithms using a uniform linear array (ULA) are derived. The first method uses the fast Fourier transform (FFT) to evaluate the posterior probability mass function (PMF) on X/$ IEEE

2 4436 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 a set of discrete DOA candidates, and then uses the inverse FFT to compute the Bayesian mixture. Due to the efficiency of the FFT algorithm, the algorithm can accommodate a large number of DOA candidates at a low computational cost. The second method approximates the posterior distribution by a Gaussian distribution the Bayesian mixture can be expressed in closed form. The resulting beamformer is a directional beamformer incorporated with an adjusted pointing DOA and a spreading matrix that controls the resolution of the beamformer. Related work on the implementation of the Bayesian beamformer has been addressed in the literature. The algorithm in [30] considers a small number of discrete DOA candidates around the true DOA without restriction to ULA. However, the algorithm is expensive when a large number of DOA candidates are considered. In [31], the authors apply a Bayesian model on the steering vector and obtain a closed-form approximation of the beamformer by replacing the posterior distribution by the prior distribution. This, however, discards the learning ability of the Bayesian method. The paper is organized as follows. Section II presents the derivation of the Bayesian beamformer. Section III presents its learning ability. Section IV introduces its approximate form. Sections V and VI present the two efficient algorithms and corresponding numerical simulations. Conclusions are given in Section VII. In this paper, boldface indicates column vectors, while, and denote complex conjugation, transpose, and Hermitian transpose, respectively. II. BAYESIAN BEAMFORMING The standard narrowband time-domain snapshot model is considered [30] in which a set of narrowband signals impinge upon an array of sensors. One signal is the desired signal and the rest are interferers. The received vector at time is given by where is the desired signal with known power, the DOA of the desired signal is denoted with being the corresponding steering vector, and is the interference-plusnoise component with covariance. Both and are assumed to be zero mean, white complex random processes that are mutually independent. In practice, the true DOA is often uncertain due to inaccurate source localization, array miscalibration, asynchronous timing, or source motion. Using a Bayesian approach, the DOA is modeled as a random variable with a prior distribution defined over a candidate set that describes the level of uncertainty. Such a random model is preferred over a deterministic model because it considers the average effect of the DOA error instead of a particular perturbed value that may not be representative enough to describe the uncertain scenario. Based on this model, the conditional covariance matrix of the received data is given by (1) (2) When conditioned on, the received vectors are independent and identically distributed. Given a data set of samples,, the MMSE estimate of is the conditional mean of given, which can be expanded as where is the posterior distribution of given. Since are independent and identically distributed given, the conditional MMSE estimate is equivalent to.defining the estimate in (3) becomes which can be viewed as a mixture of conditional (or directional) MMSE estimates combined according to the posterior distribution. The corresponding estimator is known as the Bayesian beamformer. When is assumed to be a discrete random variable over the prior set, the MMSE estimate becomes The discrete case is of interest because (6) is often easier to evaluate than (5). III. PERFORMANCE UNDER DETERMINISTIC DOA The Bayesian beamformer minimizes the mean-square error (MSE) under the average effect of a random DOA. However, it is more practical to assert the performance of the beamformer when the data is generated from a deterministic DOA. Note that if the prior set is discrete, the likelihood for the true DOA to coincide with any of the candidates in the discrete set is zero. When this happens, the Bayesian model does not hold, and the MMSE optimality is lost. The performance of the Bayesian beamformer under the above situations is studied in this section. It is shown that under a deterministic (but unknown) true DOA, the Bayesian beamformer possesses an adaptive learning ability, which enables it to asymptotically perform as well as the directional beamformer pointing at the DOA in the prior set that is the closest to the true one. Under certain conditions, the Bayesian beamformer asymptotically becomes the optimal beamformer equipped with the true DOA. In the sequel, the true DOA is denoted and can be viewed as a deterministic unknown parameter. The received data vector is generated from as. The performance of the Bayesian beamformer is measured by the conditional MSE, i.e., (3) (4) (5) (6) (7)

3 LAM AND SINGER: BAYESIAN BEAMFORMING FOR DOA UNCERTAINTY: THEORY AND IMPLEMENTATION 4437 where denotes the conditional expectation. The following conditions are required in the performance analysis. R1) The conditional MSE (on ) of the directional MMSE estimate,, is finite for all, i.e., Note that the conditional MSE is independent of because the data vectors are independent and identically distributed when conditioned on the true DOA. R2) The directional MMSE estimate,, is continuous in the conditional mean-square sense at the point, i.e., for every, there exists for all,wehave (8) (9) (10) R3) The regularity conditions in [32] are satisfied is continuous in on which the mean value theorem can be applied. R4) The Kullback Leibler distance [33] (11) is continuous in and possesses a unique global minimum over the positive support of. These conditions hold for practical settings, e.g., those in which the signals are complex Gaussian processes and the steering vector is continuous in. A. Discrete Bayesian Beamformer In a discrete Bayesian beamformer given by (6), the prior PMF is defined on the candidate set. The positive support of the prior is defined as, which does not necessarily include. The variable is defined as the closest DOA to in in the sense of (11), i.e., where (14) follows from the MMSE optimality of the Bayesian beamformer under any arbitrary, (15) follows from Jensen s inequality and the convexity of the square function, and (16) follows from the independence of when conditioned on. The following theorem, introduced by Liporace in [34], describes a convergence property of the expected posterior PMF. The proof is given in the Appendix with slight modifications. Theorem 1: If exists for all and there is an interval exists, then (18) where and are constants independent of and is defined in (12). The theorem states that the expected probabilistic weights on all (except on ) converge to zero at exponential rate. As a result, the weight on the exceptional converges to one due to the normalization of the PMF. Substituting into (17) and using that and, we obtain (19) As grows unbounded, becomes as small as at exponential rate. The Bayesian beamformer thus performs asymptotically as well as the directional MMSE estimator equipped with. Note that this does not imply that the MSE converges at an exponential rate, but, rather, only that the excess MSE above does so. B. Continuous Bayesian Beamformer In a continuous Bayesian beamformer, the prior probability density function (PDF),,isdefined on the real line. Similarly, is defined as the positive support of. In the continuous case, it is reasonable to assume. Similar to the discrete case, an upper bound on can be derived as Note that if and only if. An upper bound on can be derived as follows: (12) (20) (13) (21) (14) (15) (16) (17) (22) where is a closed interval containing, and is its compliment. Step (22) follows from R3) and the mean value theorem on the two regions and, with the interior points defined as and, respectively. The following theorem describes a convergence property of the probabilistic mass outside a specific -neighborhood.

4 4438 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 Theorem 2: If exists for all and let be sufficiently small is a single closed interval, and if there is an interval exists, then processes. Adopted from [30], the posterior distribution is given by (23) where, and are constants independent of. Consider an arbitrary, the value of can be further reduced is completely contained in where is defined in R2). By the orthogonality principle, the MSE term in (22) can be written as (27) where and. The exponential term can be expressed as (24) and it follows that the second term is upper bounded by by R2) since. Substituting (24) in (22) and using the same techniques as in the discrete case, the bound is obtained as (25) It follows immediately that converges to since by definition and is arbitrary. C. Remarks The learning rate of the Bayesian beamformer depends on for (26) where and are positive as shown in the Appendix. For each that is far from in the Kullback Leibler sense, is small, in effect causing the posterior weight to vanish at a faster rate. The learning rate also depends on SNR. Since noise does not carry any DOA information, it reduces the discrepancy between the two densities and. When SNR is low, is closer to 0 and thus closer to its minimal value, in effect inducing larger and a slower convergence rate. On the other hand, when SNR is high, the Kullback Leibler distance is more sensitive to the deviation between and, thereby giving a smaller. Note that the above argument also holds for Theorem 2 that involves and with from the Appendix. IV. APPROXIMATE BAYESIAN BEAMFORMER The Bayesian beamformer can be approximated to facilitate efficient adaptive algorithms. In the sequel, the desired signal, the interferers and noise are assumed to be mutually independent, zero-mean, white circularly symmetric complex Gaussian (28) as shown in the Appendix. On both sides, the term can be approximated by as the latter is the -sample estimate of the covariance matrix when presents. Assuming that the interferers are far away from the positive support of the prior, the quadratic functional can be approximated by the constant [30]. Using these approximations, the posterior distribution can be alternatively approximated as (29) where is a normalization factor that ensures the function integrates to one. For Gaussian signals and noise, the directional MMSE estimate is the output of the directional Wiener filter (30) (31) (32) Since can be approximated by, can be approximated closely by, which can be further approximated by to obtain (33) Putting (29) and (33) into (5), the approximate Bayesian beamformer has the form where as (34) (35) is the approximate Bayesian steering vector defined (36)

5 LAM AND SINGER: BAYESIAN BEAMFORMING FOR DOA UNCERTAINTY: THEORY AND IMPLEMENTATION 4439 Note that can also be viewed as an approximation of the MMSE estimate of the steering vector, evaluated using the approximate posterior PDF. The approximate Bayesian beamformer in (35) can be implemented using the sample matrix inversion (SMI) algorithm without knowledge of the high dimensional covariance matrix. The variable amplifies the variations of the posterior PDF along. It can be replaced by a constant without affecting the asymptotic performance of the beamformer. For simplicity, is assumed to be known in this paper. The prior PDF should be chosen the DOA candidates are far away from the interferers in order for the approximation of to be valid. In practice, if both the desired signal and interferer are covered by the prior, the algorithm may confuse the interference DOA as the desired DOA and proceed to estimate the interferer. To enhance the performance in the discrete case, a large number of densely located candidates should be used in the presumed range of the true DOA is sufficiently close to. V. BAYESIAN BEAMFORMING USING THE FFT A. Derivation The steering vector of a ULA with half-wavelength spacing has the form (37) where is the angle-of-arrival. For the rest of this paper, the DOA is described by the spatial frequency, where. This yields (38) Using the discrete Bayesian model, the DOA is assumed to be a discrete random variable in the set, where are uniformly spaced grid points between and (which correspond to for ) with, i.e., The Bayesian steering vector in (26) for the discrete set is (39) (40) Define. Since is Hermitian and is a sequence of complex exponentials, can be written as a harmonic series, given by where (41) (42) Denoting as the vector, the function can be written as (43) where stands for the real part. Since is a sequence of complex exponentials, the term is equivalent to the discrete time Fourier transform (DTFT) of. Taking an -point discrete Fourier transform (DFT) on yields critical samples along evaluated at. Likewise, the samples along where can be obtained by performing an -point DFT on with zero-padding. Note that arbitrary grid points can be evaluated using the chirp transform [35]. With all the samples, the approximate posterior PMF can be computed as (44) where is computed to normalize the PMF. The -point inverse DFT of the approximate posterior PMF has the form (45) It follows immediately that the first elements, indexed by, yield. The efficiency of the algorithm arises from a significant reduction in the cost of computing the posterior distribution when is chosen to be a power of 2 and the FFT is used to compute the DFT. Regardless of the value of can be generated via one -point FFT operation and one -point inverse FFT operation. If, the cost can be reduced further by pruning the FFT structure. B. Numerical Simulations An ULA of ten sensors with half-wavelength spacing is considered. The DOA candidates are defined as in (40) with. The prior PMF is chosen to be uniform over (which corresponds to for ). The desired signal power is. The true DOA is 0.9, which is not included in. Two interferers arrive at 40 and 50, respectively, with power 30 db above the noise level. For comparison purposes, the sample covariance matrix involved in the Wiener filter in (35) is replaced by the true covariance in order to eliminate the performance discrepancy induced by imperfect covariance estimation. Note that the sample covariance matrix is still used in the computation of. The SNR is defined as the ratio between the desired signal power and the noise power at each array element. Fig. 1 shows the evolution of the beampatterns and the corresponding posterior PMFs of the Bayesian beamformer at different values of. The desired signal and the interferers are indicated by the vertical solid and dashed lines, respectively. Initially, the beamformer is a uniform combination of directional Wiener filters and the beampattern possesses a broad mainlobe. As the size of data snapshots increases, the Bayesian

6 4440 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 Fig. 3. Output SINR versus SNR of various blind beamforming methods at SNR = 0 db and k = trials. Fig. 1. Beampattern (left) and posterior distribution (right) of the discrete Bayesian beamformer at different data sizes k. SNR = 0 db. performance between the two beamformers is induced by the mismatch between and. The Bayesian beamformer is compared against several beamforming methods including the optimal beamformer, the directional Wiener filter equipped with the maximum a posteriori (MAP) DOA estimate, and the same filter equipped with the DOA estimate obtained from the MUSIC algorithm based on [16]. The MAP DOA estimate is computed by, where is the approximate posterior distribution given in (29). The MUSIC spectrum is computed along the 64 candidate points in. Fig. 3 shows the output signal-to-interference-noise ratio (SINR) of these beamformers, and the output SINR is defined as SINR (46) Fig. 2. MSE of the Bayesian beamformer and the optimal Wiener filter with versus k and (a) SNR = 10 db, (b) SNR = 3 db, and (c) SNR = 0 db. 100 trials. beamformer becomes a single directional Wiener filter pointing towards. Note that the Bayesian beamformer tries to nullify the two interferers regardless of since each Wiener filter in the Bayesian mixture preserves the adaptive degrees of freedom [30]. Fig. 2 shows the MSE of the Bayesian beamformer and that of the optimal beamformer, i.e., the directional Wiener filter in (33) equipped with the true DOA. The plot verifies that high SNR induces a faster learning rate. The discrepancy in the asymptotic At low SNR, the Bayesian beamformer outperforms other methods as it forms a broad mainlobe to cover all possible DOAs, while MAP and MUSIC suffer from incorrect DOA selections. At medium SNR, the Bayesian, MAP, and MUSIC beamformers give nearly optimal performance as they are equipped with the closest DOA. At high SNR, the three beamformers start to degrade because the high SNR amplifies the effect of the small mismatch between and. A. Derivation VI. BAYESIAN BEAMFORMING USING LOCAL PDF APPROXIMATION It is often practical to deal with the case when the signal is known to arrive within a small range of uncertainty near a presumed DOA or prior DOA, denoted. For this case, the Bayesian steering vector is given by (47)

7 LAM AND SINGER: BAYESIAN BEAMFORMING FOR DOA UNCERTAINTY: THEORY AND IMPLEMENTATION 4441 where is a small region around. Given that the presumed DOA is located sufficiently close to the true DOA, the peak of the posterior distribution can typically be well approximated by a Gaussian distribution [32]. Letting be such an approximation, (47) can be approximated by (48) The choice of a Gaussian distribution as an approximation kernel is motivated by properties of the asymptotic form of the posterior distribution [32]. Moreover, when a ULA is used, the resulting Bayesian beamformer can be expressed in closed form. Using a Bayesian model, the DOA is modeled as a continuous random variable on the real line as. The prior is assigned to be a Gaussian distribution whose peak is sufficiently far away from the interferers. From (29), the approximate posterior PDF has the form (49) where the data-driven function is given in (41). The function can be approximated by a second-order Taylor expansion at to give (50) where the first and second derivative of can be obtained directly from (41). Substituting (50) into (49) and rearranging the terms yields, up to a scaling constant, a Gaussian distribution, where (51) (52) The mean is equivalent to one step of Newton s search towards the local maximum of, where is located [36]. Using the above Gaussian distribution gives the approximate posterior Gaussian distribution, with (53) (54) The Gaussian distribution is the local approximation of the posterior PDF near. When is large, then, i.e., the beamformer points to the data-driven DOA estimate,. When is small,, which resembles the prior. With this approximation, the Bayesian steering vector is (55) Assuming a ULA where consists of complex exponentials, the above integral is equivalent to the inverse DTFT of a Gaussian PDF, which can be expressed in closed form as (56) where is an diagonal matrix with diagonal elements (57) Putting (56) into (34), the Bayesian beamformer is an SMI-type algorithm as it has the form (58) The structure of the Bayesian beamformer is similar to that of a directional Wiener filter except for the presence of the matrix and the adjusted pointing direction. The matrix controls the resolution of the beamformer by weighting the relative contribution of each element in the sensor array, where a shorter array results in lower resolution, and vice versa [1]. Thus, it acts as a spreading matrix that enhances DOA robustness by forming a wider mainlobe. The pointing direction is an adjustment from towards using one iteration of Newton search. B. Numerical Simulations A ULA of ten sensors with half-wavelength spacing is considered. The source power is 0.1. Two interferers arrive at 40 and 50 and are 30 db relative to the noise. The presumed DOA is. Similar to the previous section, the covariance estimate in (32) is replaced by,but is still used to compute and in the Bayesian beamformer. Fig. 4 shows the beampatterns of the Bayesian beamformer and the Gaussian approximations of the posterior PDFs. In this example, SNR 0 db, the prior DOA is 0 and the true DOA is 1.5. The Bayesian beamformer initially forms a broad mainlobe to cover the possible DOAs. As increases, the Bayesian beamformer becomes a directional Wiener filter with the adjusted direction. In Fig. 5, Bayesian beamformers with different values of are compared against the optimal beamformer, the directional Wiener filter with the presumed DOA, and the robust beamformers introduced in [23] and [24]. The last two beamformers are closely related to the proposed Bayesian method. Er and Ng s [23] beamformer applies a first-order Taylor approximation to, while the proposed method applies a second-order Taylor approximation to ; Riba s [24] beamformer incorporates a nondata-driven prior PDF on the beamformer, while the proposed beamformer uses a data-driven posterior PDF. In Er and Ng s method, the one-step approach is used. The angular spreading variance in Riba s method is set to 0.2. In both beamformers, the look direction is. For demonstration purposes, diagonal loading is not incorporated in the beamformers since it introduces a similar detuning effect on all SMI-type beamformers. Due to the same reason, the covariance loading algorithms in [25] [28] are not considered. As shown in Fig. 5, the Bayesian beamformers give some characteristic flat responses

8 4442 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 that none of the other methods can produce the same characteristic flat and narrow response possessed by the Bayesian beamformer. Fig. 5 also shows the variations of the Bayesian beamformers at different values of prior variance. This parameter controls the flatness and the width of the SINR response, e.g., for large prior variance, the mainlobe appears to be flat and narrow, while for small prior variance, the mainlobe is wide but not as flat. VII. CONCLUSION A Bayesian approach was used to mitigate uncertainty in the DOA for adaptive narrowband beamforming. The Bayesian beamformer was shown to perform asymptotically as well as the directional beamformer equipped with the closest DOA to the true DOA when the Bayesian model does not hold. The excess MSE vanishes exponentially at a rate that is dependent on the SNR. Numerical simulations showed that the proposed Bayesian beamformer enhances DOA robustness via its learning ability under a wide range of SNRs and data sizes. Between the two proposed implementation methods, the first is suitable when the DOA is completely unknown, while the second one is suitable when the level of DOA uncertainty is small. Without extra computational cost, the two methods can be combined to form a hybrid version that shares the virtues of both methods, i.e., uses the FFT method to find the closest DOA and switches to the second method by using the closest DOA as the prior DOA. Fig. 4. Beampattern (left) and Gaussian PDF approximation (right) of the Bayesian beamformer at different data sizes k. A. Proof of Theorem 1 Because APPENDIX,wehave (59) From Bayes rule (60) (61) The last step is obtained by removing all terms in the denominator except the term. This is permissible because by definition. Taking the conditional expectation given,wehave Fig. 5. Output SINR versus DOA error of various robust beamforming methods at SNR = 0 db, N = 10; k= 100; and (a) =0:1, (b) =10, and (c) = trials. that accommodate a wider range of DOA error around the presumed DOA. However, when the DOA error is too large, the Taylor approximation fails, and the Bayesian beamformer deteriorates more rapidly than the other robust beamformers. Note (62) (64) (64)

9 LAM AND SINGER: BAYESIAN BEAMFORMING FOR DOA UNCERTAINTY: THEORY AND IMPLEMENTATION 4443 It can be shown that when is strictly less than 1. Consider the difference between two Kullback Leibler distances and, as follows: (65) (66) Note that since. Consider the ratio between the probability masses outside and inside, as follows: Applying the mean value theorem, there exists a that the denominator of (78) can be written as (78) such (67) According to the Lebesgue dominated-convergence theorem [37], the expectation and limit can be interchanged From the definition of the limit, for any (68), there exists DEN (79) The numerator of (78) is divided into integrals according to the previously defined partition. Applying the mean value theorem on each partition, there exists for the numerator can be written as NUM (80) or (69) For any outside, the expected value of the probability mass is bounded as (81) Fixing to be, there exist (70) (82) (83) (71) By the definition of, wehave. Thus, the right-hand side is strictly less than 1 and larger than 0. Define and as (72) (73) and note that and. Substituting and into (64) for yields To obtain a common bound for all and,wedefine (74) NUM DEN Applying the bound in (71) yields where (84) (85) (86) (87) (75) and and. B. Proof of Theorem 2 The set is partitioned into disjoint (except the end points) intervals the prior probability mass over each interval is no larger than that over, that is (76) (77) (88) From the definition of for, implying that is strictly less than 1. A common bound for every can be obtained by defining and (89) where and.

10 4444 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 C. Derivation of (28) The minimum variance distortionless response (MVDR) and minimum power distortionless response (MPDR) beamformer are considered [1]. Suppose the true DOA is and the pointing DOA for both beamformers are, the beamformers are given, respectively, by (90) (91) Using the matrix inverse formula [1], the average output power terms of these beamformers can be expressed as where The two power terms can be related by (92) (93) (94) (95) (96) (97) (98) Substituting the expressions of the two beamformers into the power terms and multiplying both sides by, the desired equality is obtained. REFERENCES [1] H. L. Van Trees, Optimum Array Processing. New York, NY: Wiley Interscience, [2] B. D. Van Veen and K. M. Buckley, Beamforming: A versatile approach to spatial filtering, IEEE Acoust., Speech, Signal Process. Mag., vol. 5, no. 2, pp. 4 24, Apr [3] J. E. Hudson, Adaptive Array Principles. London, U.K.: Peregrinus, [4] L. E. Brennan, J. D. Mallet, and I. S. Reed, Adaptive arrays in airborne MTI radar, IEEE Trans. Antennas Propag., vol. 24, pp , Sep [5] A. B. Gershman, E. Neheth, and J. F. Böhme, Experimental performance of adaptive beamforming in a sonar environment with a towed array and moving interfering sources, IEEE Trans. Signal Process., vol. 48, no. 1, pp , Jan [6] J. Capon, R. J. Greenfield, and R. J. Kolker, Multidimensional maximum-likelihood processing for a large aperture seismic array, Proc. IEEE, vol. 55, pp , Feb [7] S. Gannot, D. Burshtein, and E. Weinstein, Signal enhancement using beamforming and nonstationarity with applications to speech, IEEE Trans. Signal Process., vol. 49, no. 8, pp , Aug [8] B. D. Steinberg, Digital beamforming in ultrasound, IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 39, pp , Nov [9] L. C. Godara, Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations, Proc. IEEE, vol. 85, pp , Aug [10] C. L. Zahm, Effects of errors in the direction of incidence on the performance of an adaptive array, Proc. IEEE, vol. 60, pp , Aug [11] L. C. Godara, Error analysis of the optimal antenna array processors, IEEE Trans. Aerosp. Electron. Syst., vol. 22, pp , Jul [12] H. Cox, R. M. Zeskind, and M. H. Own, Effects of amplitude and phase errors on linear predictive array processors, IEEE Trans. Acoust., Speech, Signal Process., vol. 36, pp , Jan [13] M. Viberg and A. Swindlehurst, Analysis of the combined effects of finite samples and model errors on array processing performance, IEEE Trans. Signal Process., vol. 42, no. 11, pp , Nov [14] M. Wax and Y. Anu, Performance analysis of the minimum variance beamformer, IEEE Trans. Signal Process., vol. 44, no. 4, pp , Apr [15], Performance analysis of the minimum variance beamformer in the presence of steering vector errors, IEEE Trans. Signal Process., vol. 44, no. 4, pp , Apr [16] R. O. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., vol. 34, pp , Mar [17] R. Roy and T. Kailath, ESPRIT: Estimation of signal parameters via rotational invariance techniques, IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp , Jul [18] B. D. Rao and K. V. S. Hari, Performance analysis of Root-MUSIC, IEEE Trans. Acoustics, Speech, Signal Process., vol. 43, pp , Dec [19], Performance analysis of ESPRIT and TAM in determining the direction of arrival of plane waves in noise, IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp , Dec [20] B. Carlson, Covariance matrix estimation errors and diagonal loading in adaptive arrays, IEEE Trans. Aerosp. Electron. Syst., vol. 24, pp , Jul [21] I. O. L. Frost, An algorithm for linearly constrained adaptive processing, Proc. IEEE, vol. 60, pp , Aug [22] J. R. Guerci, Theory and application of covariance matrix tapers for robust adaptive beamforming, IEEE Trans. Signal Process., vol. 47, no. 4, pp , Apr [23] M. H. Er and B. C. Ng, A new approach to robust beamforming in the presence of steering vector errors, IEEE Trans. Signal Process., vol. 42, no. 7, pp , Jul [24] J. Riba, J. Goldberg, and G. Vazquez, Robust beamforming for interference rejection in mobile communications, IEEE Trans. Signal Process., vol. 45, no. 1, pp , Jan [25] S. Shahbazpanahi, Z.-Q. L. A. B. Gershman, and K. M. Wong, Robust adaptive beamforming for general-rank signal models, IEEE Trans. Signal Process., vol. 51, no. 9, pp , Sep [26] P. S. J. Li and Z. Wang, Robust capon beamforming and diagonal loading, IEEE Trans. Signal Process., vol. 51, no. 7, pp , Jul [27] S. Vorobyov, A. B. Gershman, and Z.-Q. Luo, Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem, IEEE Trans. Signal Process., vol. 51, no. 2, pp , Feb [28] R. G. Lorenz and P. Boyd, Robust minimum variance beamforming, IEEE Trans. Signal Process., vol. 53, no. 5, pp , May [29] C. J. Lam and A. C. Singer, Adaptive Bayesian beamforming for steering vector uncertainties under order recursive implementation, presented at the IEEE Conf. Acoustics, Speech, Signal Processing, Toulouse, France, May [30] K. L. Bell, Y. Ephraim, and H. L. Van Trees, A Bayesian approach to robust adaptive beamforming, IEEE Trans. Signal Process., vol. 48, no. 2, pp , Feb [31] O. Besson, A. A. Monakov, and C. Chalus, Signal waveform estimation in the presence of uncertainties about the steering vector, IEEE Trans. Signal Process., vol. 52, no. 9, pp , Sep [32] A. M. Walker, On the asymptotic behaviour of posterior distributions, J. Roy. Stat. Soc., Ser. B, no. 3, pp , May [33] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley Interscience, [34] L. Liporace, Variance of Bayes estimates, IEEE Trans. Inf. Theory, vol. 17, pp , Nov [35] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, [36] D. P. Bertsekas, Nonlinear Programming. Belmont, MA: Athena Scientific, [37] W. Rudin, Principles of Mathematical Analysis. New York: Mc- Graw-Hill, 1964.

11 LAM AND SINGER: BAYESIAN BEAMFORMING FOR DOA UNCERTAINTY: THEORY AND IMPLEMENTATION 4445 Chunwei Jethro Lam (S 03) was born in Hong Kong in He received the B.S. and M.S. degrees in electrical and computer engineering with a minor in computer science from the University of Illinois at Urbana-Champaign in 2001 and 2003, respectively. He is currently working towards the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign. In 2000 and 2002, he was with the Department of Electronic Engineering at the Chinese University of Hong Kong as a summer Research Assistant. Currently, he is a Research Assistant at the Coordinated Science Laboratory at the University of Illinois at Urbana-Champaign. His research interests include statistical signal and array processing, robust beamforming, signal estimation and detection, with applications to wireless communications, multiantenna systems, and sensor networks. Andrew C. Singer (S 92 M 95 SM 05) was born in Akron, OH, in He received the S.B., S.M., and Ph.D. degrees, all in electrical engineering and computer science, from the Massachusetts Institute of Technology (MIT), Cambridge, in 1990, 1992, and 1996, respectively. During the academic year 1996, he was a Postdoctoral Research Affiliate in the Research Laboratory of Electronics at MIT. From 1996 to 1998, he was a Research Scientist at Sanders, A Lockheed Martin Company, Manchester, NH. Since 1998, he has been on the faculty of the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign, where he is currently an Associate Professor in the Electrical and Computer Engineering Department, Research Associate Professor in the Coordinated Science Laboratory, and Director of the Technology Entrepreneur Center in the College of Engineering. His research interests include statistical signal processing and communication, universal prediction and data compression, and machine learning. Dr. Singer was a Hughes Aircraft Masters Fellow and was the recipient of the Harold L. Hazen Memorial Award for excellence in teaching in In 2000, he received the National Science Foundation CAREER Award, in 2001 he received the Xerox Faculty Research Award, and in 2002 he was named a Willett Faculty Scholar. He serves as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and is a member of the MIT Educational Council and of Eta Kappa Nu and Tau Beta Pi.