Particle Filtering and Posterior Cramér-Rao Bound

IEEE SENSORS JOURNAL, VOL. 12, NO. 2, FEBRUARY 2012
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Particle Filtering and Posterior Cramér-Rao Bound for 2-D Direction of Arrival Tracking Using an Acoustic Vector Sensor
Xionghu Zhong, Member, IEEE, A. B. Premkumar, Senior Member, IEEE, and A. S. Madhukumar, Senior Member, IEEE
Abstract—Acoustic vector sensor (AVS) measures acoustic pressure as well as particle velocity, and therefore AVS signal contains 2-D (azimuth and elevation) DOA information of an acoustic source. Existing DOA estimation techniques assume that the source is static and extensively rely on the localization methods. In this paper, a particle filtering (PF) tracking approach is developed to estimate the 2-D DOA from signals collected by an AVS. A constant velocity model is employed to model the source dynamics and the likelihood function is derived based on a maximum likelihood estimation of the source amplitude and the noise variance. The posterior Cramér-Rao bound (PCRB) is also derived to provide a lower performance bound for AVS signal based tracking problem. Since PCRB incorporates the information from the source dynamics and measurement models, it is usually lower than traditional Cramér-Rao bound which only employs measurement model information. Experiments show that the proposed PF tracking algorithm significantly outperforms Capon beamforming based localization method and is much closer to the PCRB even in a challenging environment (e.g., SNR = 10 dB). Index Terms—Acoustic vector sensor, direction of arrival, localization and tracking, particle filtering, posterior Cramér-Rao bound.
I. INTRODUCTION OCALIZING the direction of arrival (DOA) of an acoustic source in a noisy environment plays an important role in many applications such as speech, seismology, sonar and radar. It is usually achieved by using arrays with several pressure sensors, together with estimation techniques based on the collected pressure measurements [1], [2]. Acoustic vector sensor (AVS) is a new technology developed in the recent past to measure the acoustic pressure as well as three components ( -, - and -coordinates) of the particle velocity [3], [4]. Compared with the traditional pressure sensor, it produces additional information that enables 2-D (azimuth and elevation) DOA estimation unambiguously with a single vector sensor.
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Manuscript received August 12, 2011; accepted August 29, 2011. Date of publication September 15, 2011; date of current version December 07, 2011. The associate editor coordinating the review of this paper and approving it for publication was Dr. V. R. Singh The authors are with the School of Computer Engineering, College of Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: xhzhong@.sg; asannamalai@.sg; asmadhukumar@.sg). Color versions of one or more of the figures in this paper are available online at . Digital Object Identifier 10.1109/JSEN.2011.2168204
Theoretical aspects and practical applications of AVS have been extensively studied [5]–[11]. In [5], an intensity based algorithm which uses both pressure and particle velocity vector, and a velocity covariance based algorithm which uses only the particle velocity vector are developed for localizing acoustic sources. A maximum likelihood based DOA estimation algorithm is developed in [6]. The conventional beamforming (Bartlett beamforming) and Capon beamforming (minimum-variance distortionless response beamforming) for 2-D DOA estimation using acoustic vector sensors are investigated in [8]. It shows that both the azimuth and elevation can be unambiguously estimated by using an AVS array. Further, the subspace based approaches such as MUSIC [9], [11] and ESPRIT [7], [9], [10], [12] have been used for AVS localization problem. The problem of 2-D DOA estimation using a single AVS is particularly addressed and investigated in [13]. More practically, AVS localization in impulsive noise environments and shallow ocean environments are investigated, by employing fractional lower order statistics [14] and subspace intersection method [15], respectively. In addition, its application in room acoustic environment is studied in [16]. Traditionally, Cramér-Rao bound (CRB) provides a lower performance bound for any unbiased estimation of a deterministic unknown parameter. The DOA estimation performance of the aforementioned localization approaches and the CRB of acoustic vector sensors have also been studied in [5], [8]–[10], [17]–[19]. A compact expression of CRB on the mean square error (MSE) of DOA estimation is derived in [5], in which an explicit expression for CRB of a single AVS is presented. The performance of DOA estimation based on different localization approaches is compared with CRB in [8]–[10], [17]. However, all these CRB analysis are only for ideal vector sensors that are able to produce the desired 2-D DOA spatial response. Recently, the CRBs for DOA estimation using an AVS which considers uncertainties of nonideal gain/phase responses were presented in [19] and [20]. The existing DOA estimation techniques and CRB analysis [5]–[11], [18], [19] are formulated for the localization problem, in which the source is assumed to be static and only spatial information from current measurements are employed. In real applications, the sources (e.g., submarines or robots) are in fact dynamic and move smoothly. The DOAs are highly correlated between adjacent time steps. Hence, it is desirable to exploit the information from both the previous DOA estimates and the current measurements to localize the source. DOA estimation via tracking is an approach where previous estimates can be
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fed back into the system for the following estimation. Among all tracking approaches, particle filtering (PF) [21] is found effective in coping with nonlinear and non-Gaussian system models and has been widely employed for target tracking problems [22]–[24]. It employs a number of particles to represent the probability density function (PDF) of the unknown state vector, and evaluate the importance weights of these particles according to the source dynamic model and the likelihood. Subsequently, the particles are duplicated/discarded according to the high/low importance weights, and the resampled particles are able to represent the posterior PDF of the state. For more details of the PF and its application to the tracking problem, the reader is referred to two books [24], [25]. PF has been successfully used for underwater acoustic source and geoacoustic tracking problems [26], [27], and room acoustic source tracking problem [28], [29]. In this paper, a PF is developed to track the 2-D DOA of an acoustic source using an AVS in noisy environments. Since the movement of an underwater/room acoustic source can usually be assumed to be slow, the constant velocity (CV) model [30] is used to model the source dynamics. The source amplitude and the variance of measurement process are unknown in practice. These parameters are regarded as nuisance parameters and estimated by using a maximum likelihood estimator. The likelihood of particles is then formulated based on the measurement sequence and these estimated parameters. Due to a sample-based representation of the posterior PDF of the state vector, PF is able to cope with the nonlinear measurement model and well suited for DOA estimation. It is observed that the mainlobe of the likelihood function is spread by low signal-to-noise ratio (SNR) environments. The likelihood function is further exponentially weighted to generate a sharper peak and to emphasize the particles sampled at high likelihood area. The key advantage of the proposed PF tracking algorithm is that it incorporates both the temporal and spatial information. Hence it is able to estimate the DOA accurately and efficiently even though the source is dynamic in a low SNR noisy environment. Traditional CRB is only suitable for static source and localization problem. Since the source can be dynamic and the PF tracking approach models the source dynamics as well as the measurement process, the bound should take both the prior information (from the source dynamic model) and the measurement information into account. The posterior Cramér-Rao bound (PCRB) [17], [31] on the MSE of sequential Bayesian estimator provides a lower performance bound for such a case. To the best of authors’ knowledge, no such a bound has yet been derived for AVS tracking problem. In this paper, the PCRB for the DOA tracking based on a single AVS is derived. The Fisher information matrix (FIM) for PCRB is formulated by taking the expectation with respect to the second derivative of the joint PDF of the system states and the measurements up to current time step, while for CRB, the FIM is obtained from the PDF of the measurement process. Unlike the traditional CRB in which the 2-D DOA is assumed to be fixed and time-invariant [5], [8], [19], the PCRB considers the uncertainties from the source dynamics as well as the measurements, and is calculated recursively as a function of time. The PF tracking performance and the PCRB bound are then fully investigated under different simulated experiments. Although more practical online conditional
PCRB which are based on real measurements up to current time step [32] and previous state estimates [33] have been proposed very recently, we focus on the PCRB in [17] to provide a theoretical bound based on the system models since our aim here is to analyze the performance of tracking approach and compare it with such a theoretical bound rather than to develop a sensor selection or deployment scheme. The core contributions of this work are twofold: 1) a PF tracking approach is formulated for DOA tracking by using an AVS and the tracking performance is compared with the Capon beamforming approach; and 2) the lower bound, PCRB for AVS tracking problem is derived. Comparisons between PCRB and traditional CRB demonstrate that how much performance gain can be obtained theoretically by using a tracking approach rather than using a localization approach. The rest of this paper is organized as follows. In Section II, the AVS signal model and Capon beamforming method are introduced. Section III presents the source motion and likelihood models. The enhanced likelihood model and the tracking algorithm are also formulated. The RCRB is derived and studied in Section IV. Simulated experiments are organized in Section V. Finally, conclusions are drawn and future directions of this work are discussed in Section VI. II. AVS SIGNAL MODEL AND CAPON BEAMFORMING This section provides a brief review on AVS acoustic source localization. The signal model for an AVS is introduced first. A traditional localization approach, namely Capon beamforming is then reviewed. A. Signal Model for an AVS Consider a narrow band acoustic source signal , with a center frequency , impinging on an AVS. At discrete time instance , assume that the signal is emitted by a source in a 2-D direction given by (1) with and denoting the azimuth angle and the elevation angle, respectively. Throughout this paper the wave propagation environment is assumed to be isotropic, quiescent and homogeneous. denote the particle velocity of acoustic Let wave at position in a three dimensional space, and be the acoustic pressure. The relationship between the acoustic pressure and the particle velocity is stated by the Euler’s equation [5], and is given by (2) where is the ambient density, and is the propagation speed is the unit direction of the acoustic wave in the medium. vector pointing from the origin toward the source position, given by (3) where the superscript denotes the matrix transpose. Further assume that the AVS is located at , which is known and is fixed during all time steps. The acoustic pressure and particle velocity


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will only depend on the time . Using the phasor representation, the received signal model can be written as (4) where is the complex pressure envelope of the source signal, and and represent the corresponding pressure and velocity noise terms separately. is the time delay between the sensor and the origin of the coordinate . system, i.e., For an acoustic source which moves relatively slowly, the DOA can be assumed to be stable if a small number of snapshots are processed at each time step. Assume that snapshots are taken into account at time step , and let (5) denote the snapshots of the source signal where The noise and received data matrices can be expressed as .
Fig. 1. Response of the Capon beamforming under the environments (a) SNR = 10 dB; (b) SNR = 10 dB. The source signal is located at ( 44:4 ; 14:4 ). The estimated DOA is labeled at the top of each figure.
0
0
0
where is an th order identity matrix, and and are the noise variances for the pressure and velocity components, respectively. The Capon spectra estimation for signal model (8) is [8]
(6) (7) where . Accordingly, is used to express the DOA at time step . Equation (4) can thus be written as
(12) where is the covariance matrix given as
(13) (8) where (9) is the steering vector. The received signal includes both the the azimuth and elevation information, and can be used for 2-D DOA estimation. B. Capon Beamforming for DOA Estimation The localization approaches such as Capon beamforming, MUSIC and ESPRIT methods have been developed for 2-D DOA estimation. Among all these existing methods, Capon beamformer, which is also known as the minimum variance distortionless response (MVDR) filter, is popular due to its simplicity and its ability to minimize the power contributed by noise and other interference signals. Assume that: 1) the noise terms in (8) are independent identically distributed (i.i.d.), zero-mean complex circular Gaussian processes and are independent from different channels; and 2) the source signal and the noise are not correlated. The PDF of the measurements can be written as (10) where represents a multivariate complex Gaussian distribution with mean and covariance matrix . The covariis given as ance matrix (11) where is the expectation operation. In practice, the expectation is calculated by (14) denotes the conjugate transpose. The where the superscript DOA estimation can easily be obtained by implementing a 2-D search over the potential which maximizes the output of Capon beamformer, stated as (15) denotes the amplitude of a complex value. In noisy where environments where the SNR is relatively high, Capon spectra is able to present the source DOA by a sharp peak as shown in Fig. 1(a). However, when the SNR is low, the peak may be distorted and the estimated DOA may diverge from the ground truth, as shown in Fig. 1(b). III. DOA ESTIMATION VIA PARTICLE FILTERING DOA estimation based on the localization approaches only use the spatial information from current measurements. Since the DOAs between adjacent time steps are highly correlated, it is desirable to estimate the source DOA exploiting both the spatial and the temporal information (implied in the source dynamic model). In this section, the constant velocity (CV) source dynamic model and the PF tracking framework are introduced first. A likelihood function which is based on a maximum likelihood estimation of the source amplitude and the variance matrix of measurement noise process is then formulated.


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A. Particle Filtering To formulate the general framework for DOA tracking problem, the state-space model has to be defined first. Consider that a source is currently at DOA and moving with a velocity (in rad/s). Since we consider the DOA tracking problem rather than the exact position tracking, the source state is and the motion velocity , i.e., constructed by the DOA . Such a polar coordinate system is widely used for DOA tracking problem [34]–[36]. In this paper, the CV model [30] is used to model the DOA dynamics, given as (16) where the coefficient matrix and are defined by (17) with representing the time period in seconds beis a tween the previous and current time step, and zero-mean real Gaussian process (i.e., ) with ) used to model the turbulence on the represents a diagonal matrix source velocity. Here and 0 elsewhere. The CV model with main diagonal entry has been employed for joint DOA and array shape tracking in [35]. Since only the source DOA and the velocity of DOA are considered, it is appropriate for source motion with small angle-velocity. For more complicated trajectories and faster moving sources, an acceleration component can be cascaded to construct an constant acceleration model to model the source dynamics [34], [36]. A natural choice for the measurement function is the AVS denote all meadata model (8). Let surements obtained until time step . The task here is to estirecursively. The solution based mate the posterior on Bayesian recursive estimation towards to this problem can be given as • Predict (18) • Update (19) is the posterior distribution In this recursion, is the prior disestimated at the last time step, and tribution for the current time step. The Bayesian recursion states that given both the posterior distribution of the state estimated and system model, the current at the previous time step probability distribution of the state can be obtained recursively. Although Kalman filter can be used to solve the Bayesian recursion in (18) and (19), its usage is limited in the case of linear and Gaussian system models. Here, the PF that provides an excellent solution to the nonlinear problem is employed [21]. Given , for at previous time step the state particles , the particles are sampled at the current time step according to the source dynamic model (16), stated as (20)
The importance weights of the particles at current time step are given by (21)
stands for importance function. Since the particles where are drawn according to the source dynamic model, we have (22) The particles are thus weighted according to (23) where is the normalized weight given as
(24) After the resampling scheme, the posterior distribution of the state is
(25) where is a Dirac-delta function, and particles. B. Tracking Algorithm To formulate a PF tracking algorithm, the task remaining here is deriving the likelihood according to the AVS data model (8). Since the measurement noise process is assumed to be Gaussian, the likelihood function can be written as is the number of the
(26) such that outputs the DOA part of where the state, and denotes the determinant, and repand resents the trace operation. Since the source signal of measurement noise processes are the variance matrix unknown in practice, they are estimated by using a maximum likelihood estimator. Analytical solutions are obtained by solving the gradient equations and , respectively. The results are given as (27)


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(28)
and Inserting the maximum likelihood estimation back into the (26), the likelihood for the parameters of interest can be obtained as
(29) Given the transition and likelihood models derived above, the PF algorithm for AVS source tracking is summarized in Algorithm 1. Since the initial DOA is unknown, the particles for the DOA state are initialized as a uniform distribution over the possible DOA range, given as (30) where is a uniform distribution over the range for
Fig. 2 MSE in log scale under different exponential weighting factor r .
variable . The velocity part of the state are initialized as a around the Gaussian distribution with the covariance matrix actual velocity , i.e., . Although Algorithm 1 is developed for narrowband acoustic source tracking. However, its extension to wideband acoustic source tracking scenario is straightforward. The likelihood model of wideband acoustic signal is a production of the likelihood of each frequency component as described in (26). can be Consequently, the likelihood of particles obtained by taking the product of the likelihood over all frequencies in (29). Algorithm 1: PF for AVS 2-D DOA tracking Initialization: for , draw particles ; ; set the initial weight for for to do to do according to (16); from (29); ;
C. Enhanced Likelihood It can be observed in Section III-B that the particles are actually weighted by the likelihood when the prior importance function is employed. It is desired that the particles around the ground truth will present a large likelihood and weighted significantly larger than those far away from the ground truth. Hence, these particles which contribute more to our estimation can be replicated. However, the mainlobe of likelihood function (29) is usually spread and flat due to the low SNR noise environment. It is similar to the results presented in Fig. 1: when the noise is heavy, the likelihood is spread and the peak is not very sharp. The particles can thus not be weighted effectively under such a scenario. To eliminate this effect due to heavy noise, we further exponentially weight the likelihood by a constant value . The likelihood function will thus be more peaky and the weight of particles which are located at high likelihood area can be enhanced. The likelihood function (29) can now be written as (31) where . This exponentially weighting step is important since it is able to help the subsequent resampling algorithm to select and replicate the particles more efficiently. In our algorithms, the enhanced likelihood (31) will be employed to replace the original one in (29). Preliminary study of tracking performance under different weighting factor . The weighting factor can be determined based on experimental study. Fig. 2 gives our preliminary study about the tracking performance versus different weighting factor . The detailed simulation setup is provided in Section V. are emDifferent values ployed to test the tracking performance. The number of , and SNR varies from 10 dB snapshots used here is to 0 dB with a 2 dB increment. The mean-square-error (MSE) over 500 Monte Carlo runs is employed here to evaluate the tracking performance. It can be observed that when is set larger than 5, better tracking performance can be achieved. Hence, will be chosen throughout our experiments in
1) calculate the covariance matrix according to (13); 2) draw particles end 4) compute the enhanced likelihood according to (31); 5) compute the importance weight from (23); 6) normalise the weight 8) output the estimates end . ; 7) resample the particles according to the weights;
3) compute the likelihood


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next sections. Note that using a factor slightly larger than 5 will not lead to a significant differentce of the tracking performance. Also, is not necessary to be an integer and exactly to be 5. For and will result in very similar example, using tracking performance. IV. PCRB FOR DOA TRACKING The PCRB provides a lower performance bound on the MSE matrix for sequential Bayesian estimation of random parameters. This section presents the derivation of PCRB for AVS signal based DOA tracking. A brief introduction of PCRB formulation is given first. A. General Formulation Let denote an unbiased estimator of the state vector based on the aforementioned measurement sequence . The PCRB (lower bound) on the estimation error has the form [17]
which is a linear function of inverse of the state noise covariance denote the covariance matrix of state process matrix. Let can be written as noise. According to the CV model (16),
(42)
(32) The Fisher information matrix (FIM) is give by
is singular, and It is obvious that the covariance matrix does therefore, the transition probability density not exist. The PCRB is thus not available under such a scenario. In [37], the CV model is also employed for tracking one dimensional target bearing. Based on the recursion (36), the authors have presented another equation for FIM estimation which is able to bypass the inversion of the covariance matrix by using the matrix inversion lemma (Woodbury identity [38]). However, it is unclear how the equation is derived under an undefined transition probability density and a non-invertible . matrix B. PCRB for AVS Acoustic Source Tracking Following the development of PCRB for a singular covariance matrix case in [17], we divide the state vector into two blocks so that the covariance matrix of each block is nonsingular, given as (43) where represents the DOA part and is the velocity part. The innovation process of the DOA and velocity components (16) can be addressed as
(33) where is the second order partial derivative. The notations of gradient and second order derivative are given as
(34) (35) Here we assume that the derivatives and expectations in (33) exist. According to [17], the FIM can be calculated recursively given as
(44) (36) where The second part of the first one (37) (38) (39) (40) (41) The transition PDF and the likelihood can be obtained according to the state (16) and measurement (8), respectively. Note that calculations of (37) to (41) include a second partial derivative of the transition PDF , (45) can also be written as a linear combination , given as (46) The aim here is to estimate the FIM for the state at the current time step recursively. In following derivation, we will show that the FIM can be derived based on this state partition and by which the transition PDF can be defined. Let . Consider a state vector and define the joint PDF at the previous time step as (47)


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The FIM for the state vector can thus be derived according to (33). Assume that the FIM for the state vector can be written as
at current time step The state vector holds a relationship with the auxiliary vector as
(48) (54) (which is called as the information submatrix The FIM for in [17]) is defined as the inverse of the right-lower for . Assume it has the block form block of (49) (55) where all the entries , for , 2 are given in (64) in Appendix A. To formulate a recursive estima, we consider an auxiliary vector tion of the FIM for , whose joint PDF is denoted by . Using the chain rule, the PDF of the auxiliary vector can be written as According to the modified CV model (44), the transition PDF for DOA part of the state can be written as
Since is an invertible matrix, the corresponding information submatrix for can be given as [39]
(56) where tries in . Substituting (56) into (53), the encan be obtained as following:
(50) According to the system model, we can write
(51) . According The joint PDF for the auxiliary vector is thus to the derivation in Appendix A, the information submatrix for can be written as
(57) For a complex-valued Gaussian noise process in measurements , the likelihood function is given in (26). The expectation can thus be obtained as [40] term in
(52) where
(58) and denotes the power of the signal and the where noise, respectively. The detailed derivation of (58) is given in Appendix B.
(53)


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Substituting (58) and (57) into (52), and using (55), the FIM can be obtained. Asfor the objective vector sume that the obtained FIM has following block form
(59) The PCRB for current state can thus be obtained as
is fixed once it converges. noise environment, Algorithm 2 summarizes the steps involved in calculating the PCRB for AVS based 2-D DOA tracking. In this section, numerical studies about the PCRB are conducted. The system parameters are setup as follows: initial velocity diswith and tribution , the variance for state inno, and the number vation noise process of snapshots unless otherwise stated. The PCRBs are demonstrated based on this parameter setup since it will be used in simulated experiments in next section. In the first experiment, PCRBs under different SNRs are presented. The elevation changes from to with 50 time steps. A comparison with the traditional CRB for localization problem is also presented. The CRB can be obtained from , given by [19]
(60) where denotes an element which is not of our interest. A recursive estimation of PCRB for AVS tracking is thus achieved. The detailed derivation of (60) is given in Appendix C. Since the FIM for Bayesian tracking can also be interpreted as two additive parts: information obtained from the measurement sequence and the source dynamic models, the available information for tracking is more than the localization approaches which only take the current measurement into account. Hence, the PCRB is usually lower than the traditional CRB. In the next section, numerical experiments will be organized to demonstrate this observation. Algorithm 2: PCRB calculation Initialization: from (61). (62) The CRB can be regarded as a special case of PCRB where no prior information is available from the source dynamic model. Fig. 3 gives the PCRB and CRB under different noisy environments for 2-D DOA tracking. It shows that the PCRB is higher than CRB at initial steps. However, once it converges, PCRB is generally lower than CRB in all simulated noisy environments. This is because the tracking approach incorporates the prior information by modeling the source dynamics. The comparison illustrates that theoretically, better 2-D DOA estimation accuracy can be expected by using Bayesian tracking approach than that by using localization approaches. Generally, the higher the increases as the time step SNR, the lower the PCRB. (and becomes larger since it is a function of source elevation thus it is a function of time step ). This means that the variance of azimuth estimates will be larger at the end of the predefined trajectory. It can also be observed that once the PCRB for becomes stable, it will be a constant since is determined by the initial information matrix and the SNR. The PCRB and CRB are also investigated under different number of snapshots: . The SNR is fixed at 0 dB. Fig. 4 shows the PCRB and CRB of the experiment. The result is very similar as that under different SNRs: the PCRB is usually lower than CRB due to incorporating the temporal and spatial information. As the number of snapshots increases, the bounds are lower and PCRB is able to converge faster. The numerical results also show that the lower the SNR and the fewer the number of snapshots, the larger the difference between the PCRB and CRB. In next section, different simulated experiments will be organized to demonstrate the performance of the proposed PF tracking approach and the PCRB. V. SIMULATION EXPERIMENTS In this section, several experiments are organized to investigate the performance of the PF tracking algorithm and the effectiveness of the PCRB. To demonstrate the advantages of PF tracking approach, the experimental results are compared with that of localization approach obtained by using Capon
to do for — compute according to (58); — compute all the entries for matrix from (57); — compute FIM for auxiliary state according to (52); — compute the FIM for the objective state from (55); — obtain the PCRB according to (60). end C. Numerical Study are the two diagonal eleThe PCRBs for 2-D DOA and ments in . As previously mentioned, the DOA part of the state is initialized as a uniform distribution as described in (30) and the velocity part as a Gaussian distribution. The initial information matrix is thus a diagonal matrix given as (61) where the diagonal elements are the inverse of the corresponding variance matrix. Given a measurement sequence, the PCRB is a function of SNR and time step . According to information submatrix (58), varies as the elevation of the source changes. However, under the same

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