A Unified Method for Measurement and Tracking of Contacts From an Array of Sensors

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1 2950 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 A Unified Method for Measurement and Tracking of Contacts From an Array of Sensors Robert E. Zarnich, Kristine L. Bell, Senior Member, IEEE, and Harry L. Van Trees, Life Fellow, IEEE Abstract A multiple target track estimation method that operates directly from array data is presented. The maximum a-posteriori (MAP) estimator for contact states is derived for temporally uncorrelated signals and uncorrelated contact tracks, where the number of contacts is assumed known. This estimator is an iterative algorithm employing a nonlinear programming (NLP) penalty method in conjunction with an expectation maximization (EM) algorithm. The NLP technique is used to find the MAP track estimate based on the synthetic signal estimates produced by the EM algorithm. This method eliminates the data association step of traditional multitarget tracking approaches by conditioning the measurement process on individual target state distributions. It results in a process similar to the EM algorithm for direction finding, with an additional penalty term imposed by the track distributions. The algorithm is derived as a batch method. An extension to support sequential tracking is also developed. Simulation results for a relevant submarine towed array scenario are presented and discussed. Index Terms Antenna arrays, MAP estimation, nonlinear programming, target tracking. I. INTRODUCTION AN important and pervasive problem in the engineering of sensor systems is the detection and tracking of multiple contacts through observations made from an array of sensors. An optimal approach would estimate the tracks of objects directly from the array snapshot data; however, the solution to this estimation problem is quite difficult. Traditional solutions partition the track estimation operation into two isolated processes: direction-of-arrival (DOA) estimation from array snapshot data followed by track estimation from the DOA estimates. This partitioning results in procedures that are suboptimal and require data association to match DOA estimates to contacts. When only a single object is in the observation space, the tracking problem is in its simplest form, and data association is not required. This problem is discussed in detail in [1] and [2]. For multiple targets, many approaches have been offered to solve the data association problem [3]. The three Manuscript received August 10, 2000; revised August 23, This work was supported by the Department of the Navy s Program Executive Office for Undersea Warfare, Advanced Systems and Technology Office (PEO(USW) ASTO). The associate editor coordinating the review of this paper and approving it for publication was Dr. Fulvio Gini. R. E. Zarnich is with the Naval Sea System Command, Advanced Systems and Technology Office, Washington, DC USA ( ZarnichRE@navsea.navy.mil). K. L. Bell is with the Department of Applied and Engineering Statistics, George Mason University, Fairfax, VA USA ( kbell@gmu.edu). H. L. Van Trees is with the Department of Electrical and Computer Engineering, George Mason University, Fairfax, VA USA ( hlv@gmu.edu). Publisher Item Identifier S X(01) main bodies of research that have been pursued are joint probabilistic data association (JPDA) [4], multihypothesis filtering or tracking (MHF or MHT) [5], and probabilistic multihypothesis tracking (PMHT) [6]. Each of these methods can be employed in batch processing using all the data collected over a large observation interval or in sequential processing using a limited amount of data to update tracks as data is received. Each of these important works has been pursued and modified by many other researchers. Nevertheless, data association can be a major contributor to poor system performance and continues to be a significant area of research. Alternatively, full Bayesian approaches to multiple target tracking compute likelihood functions of the target state from which track parameters can be inferred [7], [8]. These Bayesian methods are extremely flexible and powerful but require significant computational resources. These techniques are typified by the structure of a prior likelihood of the target state over a discrete state space and a weighting of that prior by the quantized spatial spectrum in form of a likelihood to produce a posterior likelihood of the target state. They have been constructed as recursive methods and produce optimal estimates, in the discrete state space, of the current state. However, they do not generate optimal track estimates over a batch of observations. Multiple pass extensions have been formulated; however, this only increases the computational intensity of these methods. In between the partitioned processing approach and the full Bayesian approach, there are several techniques that use array data directly in forming track estimates or couple the DOA and track estimation procedures. In [9] [11], a direction-finding process is followed by a tracking process, where the tracking process provides predictions to initialize or aid the DOA estimation in a sequential fashion. In [12] and [13], a method is developed to estimate the trajectory parameters of multiple targets with a deterministic motion model and deterministic signals. A Newton algorithm is employed to determine the set of signals and trajectories that maximize the likelihood function. In [14] and [15], a maximum a-posteriori (MAP) solution for estimating the target states directly from the array data is proposed. The formulation assumes a discrete target state space, deterministic signals, and nondeterministic hidden Markov motion models. The expectation-maximization (EM) algorithm [16] is used to decompose the array data into individual signals and signal parameters, and the Viterbi algorithm is used to solve the dynamic programming problem in determining a set of tracks that maximize the joint likelihood function. Both batch and sequential versions are developed. They provide elegant but computationally expensive solutions X/01$ IEEE

2 ZARNICH et al.: UNIFIED METHOD FOR MEASUREMENT AND TRACKING OF CONTACTS FROM AN ARRAY OF SENSORS 2951 is. We assume the motion of the objects is described by a first-order Gauss-Markov process, i.e., for the th contact (1) where is the time interval from to, and is a zero mean white Gaussian noise process with covariance, which is assumed known and fixed over the observation period and equal for all objects. Under these assumptions, the pdf of given is Fig. 1. Array observation geometry. (2) The focus of this paper is the development of an efficient MAP estimation technique for determining the tracks of multiple objects directly from the array data when the number of contacts is assumed known and remains the same during the observation period. The proposed algorithm is one component of a complete tracking system that would also include the enumeration of the number of contacts and the ebb and flow of contacts from the observation space. The problem formulation assumes a continuous state space, random signals, and hidden Markov motion models. We introduce a nonlinear programming (NLP) penalty method to ease the computation of the MAP track estimate and provide implicit data association; we then use the EM algorithm to separate the sources and provide measurements to the track estimator. The result is an iterative procedure for estimating the target states with a mechanism to control the tradeoff between convergence rate and estimation error. The problem formulation is very similar to [15]; however, the continuous state space used here is more natural and results in a solution that is more computationally efficient. The paper is organized as follows. In Section II, the signal and motion models are developed for the multiple target tracking problem of interest. A batch solution for continuous state space (CSS) track estimation is developed in Section III, and a sequential estimation version is presented in Section IV. Simulation results for the two CSS algorithms and a comparison with the discrete state space technique in [15] are presented in Section V. A summary and discussion of algorithm enhancements is given in Section VI. II. STATISTICAL MODEL AND ASSUMPTIONS We consider the multitarget tracking problem where there are contacts radiating narrowband signals received by an array of sensors. The number of objects is assumed known, and the trajectories of the objects are uncorrelated with the trajectories of other objects. We assume for simplicity that the targets and the array lie in the - plane as illustrated in Fig. 1. The twodimensional (2-D) state is defined as its bearing and bearing rate. Thus, the state of the th contact at snapshot where denotes the determinant of. There are snapshots in an observation batch. No data is available at, so we assume the prior distribution on initial object states is Gaussian with mean and covariance At the array, the complex envelope of the observations has the form where is a random narrowband signal from the th object at the th snapshot with. The vector is the array response vector for the DOA, and is a vector of uncorrelated sensor noise samples. The source signals and noise are assumed to be sample functions of independent zero-mean Gaussian random processes. It is assumed that the snapshots are sufficiently spaced that the observations are independent from snapshot to snapshot. The signal powers are assumed to be time varying and unknown. The noise covariance matrix is assumed to be constant and known with. In this model, any Doppler shift due to source motion is assumed small and is ignored. Note that the array data depends on the target state only through the bearing and not the bearing rate. We will use the notation with to denote the bearing component of the state in the subsequent derivation. We denote the collection of target states and powers at time as and. It is also useful to define the collection of target states over the batch (the track) for each target as. (3) (4) (5)

3 2952 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 The array data is then jointly complex Gaussian with zero mean and covariance matrix and the pdf of the array data conditioned on the target states is given by (6) we discuss implementation issues and convergence. The development expands on an earlier approach presented in [18] and [19]. A. MAP Track Estimation The MAP/ML estimates of and are the solutions to the optimization problem (7) We have used the notation to emphasize that this pdf is conditioned on the random vector but is also a function of the nonrandom but unknown parameter vector. The single snapshot joint pdf of the observations and contact state conditioned on the previous contact state is Let, and denote the batch collections,, and. The joint pdf of the array snapshot data and target states over the batch is then (8) (10) To assist in the solution, we introduce a set of auxiliary DOA variables for each target and snapshot and define collections of these variables as and. We replace with the new auxiliary variable in the term. In order to retain the original optimization problem, we then require the new variable to be equal to the old variable, i.e.,. The unconstrained optimization problem in (10) can be written as an equivalent constrained optimization problem as follows: (9) s.t. (11) where III. BATCH CONTINUOUS STATE-SPACE ALGORITHM In the classical single target tracking problem, the discrete time Kalman filter provides the minimum mean square error (MMSE) and MAP estimates of the target states given the observations. When a batch of observations is used, the MMSE and MAP estimates are obtained from the fixed interval Kalman smoother [1], [2]. In the problem considered here, the MMSE and MAP estimates will be different due to the nonlinear dependence of the observations on the target states. We also have the added complication of the unknown nuisance parameter. The MAP methodology provides a tractable framework for solving this problem. We can jointly find the MAP estimate of and the maximum likelihood (ML) estimate of by maximizing the joint pdf or, equivalently, its logarithm, with respect to both and. Traditional calculus-based optimization is not a tractable approach, and a brute force Newton-type algorithm has difficulties from both an analytic and an implementation perspective [17]. In [15], an exact solution was obtained by discretizing the state space; however, this results in a computationally intensive solution due to the size of the discrete state space. We now develop a solution that provides a tractable and efficient algorithm for the natural continuous state-space model using the penalty method. Following the algorithm derivation, (12) This formulation allows us to use the penalty method of NLP (e.g., [20], [21]) for constrained optimization problems. It is an iterative procedure that involves solving a sequence of easier unconstrained optimization problems. The easier problems are related to the original constrained problem by a continuous, differentiable penalty function that is equal to zero in the feasible region where the constraints are satisfied and is negative in the infeasible region. The penalty function relaxes the equality constraint, resulting in a problem that is an approximation to the original problem. With each iteration, a stronger penalty is imposed for infeasibility, and the solution to the unconstrained problem converges to the solution to the original constrained problem. An overview of the method and the convergence properties is provided in Appendix A. For this problem, we choose the quadratic penalty function (13) where is a parameter that affects the strength of the penalty and will be discussed in more detail later.

4 ZARNICH et al.: UNIFIED METHOD FOR MEASUREMENT AND TRACKING OF CONTACTS FROM AN ARRAY OF SENSORS 2953 To enforce a more costly penalty at each iteration, a term scales the penalty function, where is the iteration index. The penalized unconstrained maximization problem is given by reduces to solving separate track estimation problems. Expanding the pdfs, the problem becomes (17) (14) The sequence of penalty parameters have the property that for all, and. The scaled penalty function then drives the total squared error between the auxiliary variables in and the bearing components of down at each iteration. Generating a sequence of solutions to (14) will produce the optimal estimate of in the original constrained problem in the limit as. This is discussed in more detail in Appendix A. Expanding and rearranging the terms in (14), we have (15) Note that the first term is only a function of and, the second term is only a function of, and the third term provides the coupling between the parameter sets. First, consider that for a fixed, we can solve for both and by maximizing over the first and third terms in (15). These terms decouple with respect to the snapshots; therefore, this problem reduces to solving separate multiple source DOA estimation problems of the form (16) This is a maximum penalized likelihood (MPL) estimation problem that can be solved iteratively using the EM algorithm [22]. The EM algorithm is an iterative technique that decomposes the data into synthetic target signals and then estimates the DOA and power of each signal separately. The derivation follows the approaches in [23] and [24] and is given in Appendix B. Next, consider that for a fixed and, we can find by maximizing over the second and third terms in (15). These terms decouple with respect to the targets, and therefore, the problem This problem has the form of the classical single source tracking problem with acting as the noisy measurements and as the measurement variance. We show in Appendix C that the solution is the fixed-interval Kalman smoother. A simple alternating maximization approach iteratively finds the penalized DOAs and power estimates using the EM algorithm and then updates the track estimates via fixed-interval Kalman smoothers using the DOA estimates as synthetic measurements. B. Implementation An explicit pseudo-code description of the batch CSS target tracking algorithm is provided in Table I. There are three levels of iteration: 1) NLP iteration (indexed by ); 2) alternating maximization iteration (indexed by ); 3) EM iteration (indexed by ). Each of the iterations may be performed until a convergence criterion is satisfied or for a fixed number of cycles. The tradeoff is algorithm complexity versus estimation accuracy. The algorithm requires knowledge of the number of tracks and some initial track estimates, as well as the noise power. There are many approaches to provide an initialization. The one used here computes the minimum variance distortionless response (MVDR) spatial spectrum and counts the number of peaks per snapshot. The number of sources is determined from the median number of peaks. A set of coarse linear trajectories is matched with spectral peaks, and each trajectory-peak set is smoothed to provide an initialized set of tracks. The noise power is estimated from portions of the spatial spectrum where there are no contacts. A detailed discussion can be found in [17]. Other parameters that need to be specified are the state transition covariance matrix and the penalty parameters and. should be chosen to be consistent with the expected target dynamics. The motion model allows for a random walk acceleration process, specifying that has a structure of, where is the period between observations, and is the instantaneous random acceleration [25]. Together, and control the strength of the penalty function. The tuning of and the reduction schedule for have significant ramifications for estimation accuracy and convergence rate. If these values are chosen too small, the solution

5 2954 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 TABLE I EXPLICIT PSEUDO CODE can easily get trapped in local maxima or demonstrate slow improvement. If these values are chosen too large, the algorithm can diverge. In the Kalman smoother, has the interpretation of the measurement variance. Since the measurements provided to the Kalman smoother are the DOA estimates from the EM algorithm, it is logical to set their variance to be proportional to the Cramér Rao Bound (CRB) for the DOA estimation problem. The CRB will be smaller for stronger targets than weaker targets and will be larger when targets are close together. This provides a natural tightening or relaxation of the penalty for different tracks and different portions of a track. A similar idea is used in the tracker developed in [10], [11]. The CRB is given by [26]: CRB (18) where is the Hadamard product, and (19) (20) (21) diag (22) We chose to set equal to the CRB. The choice of the reduction schedule for then represents a tradeoff between overall convergence speed and estimation accuracy. One possibility is to follow an exponential decay, i.e.,, where and are positive parameters. We want to choose the initial value large enough so that the DOA estimation process and tracking process are only loosely coupled in the initial phases of the CSS algorithm. This allows the algorithm to make large corrections to the initialized track estimates as necessary. Once settling on improved tracks, the reduction in allows the algorithm to refine the estimates to arrive at the final solution. C. Convergence A proof of convergence to a local maximum for the general NLP penalty method is provided in [21], and outlined in Appendix A. There are three conditions required for convergence: 1) nonincreasing likelihood function; 2) nonincreasing penalized likelihood function; 3) nondecreasing penalty function. The convergence conditions are satisfied when an exact maximization of the penalized likelihood function is found at each step. Since we employ an alternating maximization scheme with an embedded EM algorithm, we do not necessarily find the maximum of the penalized likelihood function at each step; therefore, convergence is not guaranteed in general. However, we have verified that in our simulations, the conditions for convergence are satisfied after the first few iterations; therefore, convergence is achieved in practice. This is discussed in more detail in Section V. D. Summary The algorithm has a great intuitive appeal. Given an assumed track for each contact, we decompose the array data into the synthetic signals for each snapshot, find their directions constrained to the neighborhood of the current track, then adjust the current track estimate using the fixed-interval Kalman smoother. We repeat the process, and at every iteration, we enforce a stricter relationship between the estimated DOA and the estimated track.

6 ZARNICH et al.: UNIFIED METHOD FOR MEASUREMENT AND TRACKING OF CONTACTS FROM AN ARRAY OF SENSORS 2955 Fig. 2. Sequential algorithm flow. The outstanding feature of this approach is that the association of the measurements to tracks is implicit. The penalty function couples the DOA estimates to the tracker as synthetic measurements and creates a feedback mechanism to enhance the DOA estimation process with the strength of the coupling controlled by the Cramér Rao bound. IV. SEQUENTIAL CONTINUOUS-STATE SPACE ALGORITHM The full batch method can be extended to a sequential method by moving through the data with a shorter batch window of length and a stride of length. This is illustrated in Fig. 2. We assume that is odd and that the index of the snapshot at the center of the window is. For each window, the data snapshots for are used to produce state estimates over the same time indices, which we denote as. In the full batch method, and, whereas a fully sequential method would use and.in between, we have batch-sequential methods where is determined by how often state updates are needed, and is chosen to balance estimation accuracy with algorithmic complexity. If the stride is less than the window size, multiple estimates are produced for the states where the windows overlap. The batch MAP estimation algorithm can be used as the processing technique on each window of data. We also need to specify an initialization procedure for each batch and a method for combining state estimates for snapshots where windows overlap. For small window sizes, the first window can be initialized by starting with a set of target positions at and assuming that their positions are constant over the window. For subsequent windows, state estimates for the overlapping snapshots obtained from the previous window are used as the initial values for the current window. For the new state estimates, we simply project out in time with the motion model from the most recent state estimate. Designate as the index of the last estimate produced. The initial estimates for the new states are given by (23) Fig. 3. True target tracks overlaid on spatial spectrum estimate. The measurement error variances must also be initialized. In the initial iteration, we use the CRB as in the full batch method for the snapshots for which a previous estimate exists. For the new time steps, we set the variance to be larger to allow the tracker the freedom to adapt to minor maneuvers. For subsequent iterations, the CRB is used at all time steps in the window, and the measurement error variances all follow the same reduction schedule. In each window, the Kalman smoother provides the conditional mean of the states given the synthetic measurements and the conditional covariance matrix. Let and denote the state estimates for the th object at the th snapshot from two overlapping windows, and let and denote their covariance matrices. We find the combined state estimate from the weighted average of the state estimates as follows: (24) Other recombination schemes are possible and are an area for further development. V. SIMULATION RESULTS A typical submarine towed array scenario was simulated to test the algorithms. An array of ten elements was used. There are four contacts, with tracks shown in Fig. 3. Target 1 is a weak dynamic target that starts as the left-most trace (forward endfire) and ends as the right-most trace (aft endfire). Target 2 is a strong dynamic target that starts as the right-most trace (aft endfire) and gradually moves to the left toward forward endfire. Target 3 is a weak static target at broadside, and Target 4 is a strong static target near forward endfire. Using a time step of s, the maximum acceleration of Targets 1 and 2 is about -units/s. The MVDR spatial spectrum is also shown in Fig. 3. It gives an indication of the relative strengths of the targets with respect to the background noise and was used to obtain initial track estimates for the batch CSS algorithm [17].

7 2956 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 Fig. 4. Batch CSS algorithm after ten iterations. Fig. 5. Batch DSS algorithm after ten iterations. TABLE II BATCH CSS VERSUS BATCH DSS A. Performance Comparisons In the simulations, a fixed number of iterations was used with for the EM algorithm, for the alternating maximization between DOA estimates and track estimates, and for the NLP iterations. The penalty parameter was set to 10 for the first iteration, with an exponential decay where the decay rate would approximately double the penalty every five iterations. This expression is given by, where and. These values were determined experimentally based on estimation error improvement. was chosen to provide adequate coverage of the expected target motion over the observation period without unnecessary spread. For these simulations, we chose via experimentation, which is somewhat larger than the maximum acceleration. This translates to ; therefore,. Fig. 4 shows the true tracks (marked with dashes) and the estimated tracks (lines) after ten iterations of the batch CSS algorithm. The overall root mean square error (RMSE) for each track is given in Table II. The estimated tracks for the strong targets are very close to the true tracks, even in the regions where tracks cross. The weak target tracks show more variability, particularly near the beginning and end of the batch. The results were compared with the discrete state space (DSS) technique proposed in [15]. This required developing a straightforward extension of their algorithm to the stochastic signal model used here. The DSS algorithm was initialized with the same track estimates as the CSS algorithm. Fig. 5 shows the estimated tracks versus true tracks for DSS algorithm, and Table II shows a comparison of the estimation accuracy performance of the CSS and DSS algorithms. The DSS technique had higher RMSE for all four targets, particularly for the weak dynamic target in regions where it crossed other target tracks. This behavior is not observed with the CSS method. It is difficult to absolutely identify the root cause. One of the reasons may be that the DSS method employs only one iteration of the EM signal decomposition prior to using the forward component of the Viterbi algorithm to compute the intermediate single contact likelihood values. The CSS algorithm allows for multiple EM iterations to compute the conditional DOA estimates, prior to using these DOA estimates in the fixed-interval Kalman smoothers to provide intermediate track estimates. Multiple iterations of the EM algorithm permit greater separation of signals during crossing periods and is probably the reason for the different performance for crossing tracks. The DSS technique also had a significantly higher computational complexity than the CSS method, even though great care was taken to tune it to the data and to make it as computationally efficient as possible. The CSS algorithm is dominated by a matrix inversion; therefore, it is an operation, where tracking is negligible, and is the number of operations to perform the maximization operation. The DSS algorithm is an operation where the tracking component is no longer trivial. The value is the size of the discrete state space. In the simulations performed, this state space was kept as small as possible while still retaining sufficient resolution to allow the DSS algorithm to have competitive RMS error. In the DSS algorithm, the tracking dominates the computationally complexity. The execution time for the CSS algorithm was about 4 min, whereas the execution time for the DSS algorithm was approximately 20 h. (These times were on a 550 MHz Dell PC using MATLAB v5.3.) The computational burden can be attributed mainly to the size of the discrete state space required to obtain accurate track estimates in conjunction with the Viterbi algorithm s complexity. In comparison, a traditional beam-scan technique with fine bearing inter-

8 ZARNICH et al.: UNIFIED METHOD FOR MEASUREMENT AND TRACKING OF CONTACTS FROM AN ARRAY OF SENSORS 2957 (a) Fig. 6. Sequential CSS algorithm. TABLE III BATCH CSS VERSUS SEQUENTIAL CSS (b) polation followed by tracking is an operation, where the tracking operations are negligible in comparison with the DOA estimation. However, the traditional method also requires a data association step, which employs considerable programming logic but not necessarily a great deal of computation. The sequential CSS method was implemented with a window length of and a stride of. For the first window, iteration limits were, and. For all subsequent windows,, and. The track estimates are shown in Fig. 6. Numerical results are summarized in Table III. The sequential algorithm produced lower RMSE for the weak dynamic target than the batch method but also produced higher RMSE for the two relatively static contacts. Most of the difference is attributable to the beginning of the target track, where the sequential algorithm s initialization procedure was able to obtain a better initial position than the batch algorithm s initialization procedure. Furthermore, since the sequential algorithm is self-initializing, it has a less complex system implementation. The fact that the algorithm worked well with the self-initialization process indicates it should be useful as a real time tracker. B. Convergence Properties of the Batch Algorithm The batch CSS algorithm was allowed to iterate for 100 cycles to investigate its convergence properties. Recall from Section III and Appendix A that there are three conditions required for convergence of the NLP technique: a nonincreasing constrained log-likelihood function (11), a nonincreasing penalized (c) Fig. 7. Convergence behavior for batch CSS algorithm. (a) Total RMS error. (b) Original log-likelihood, penalized NLP log-likelihood, and constrained (nonpenalized) NLP log-likelihood. (c) penalty function. log-likelihood function (14), and a nondecreasing penalty function (13). We cannot guarantee convergence of our algorithm because an exact maximization of the penalized likelihood function is not found at each step; however, we can monitor these functions to verify that the convergence conditions are satisfied. We also monitored the original log-likelihood function (10) and the total RMS error. The RMS error is shown in Fig. 7(a), the three log-likelihood functions are shown in Fig. 7(b), and the penalty function is shown in Fig. 7(c). In the first few iterations, all three likelihood functions are increasing, and the RMS error is decreasing as the algorithm

9 2958 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 makes large corrections to improve the initial estimates. Then, both the constrained log-likelihood function and the penalized log-likelihood function decrease as the solution is forced into the feasible region, whereas the original log-likelihood function continues to increase. All three functions eventually converge as the penalty function approaches zero. The total RMS error reaches a minimum at about the 19th iteration, prior to the likelihood being maximized. However, as the likelihood is maximized, the increase in RMS error is small. VI. SUMMARY AND DISCUSSION OF ENHANCEMENTS We have developed an efficient algorithm for the MAP estimation of multiple contact tracks directly from array data, when the number of contacts is assumed known and remains the same during the observation period. The proposed algorithm is one component of a complete tracking system that would also include the enumeration of the number of contacts and the ebb and flow of contacts from the observation space. To construct this algorithm, we introduced an auxiliary DOA parameter and an NLP penalty technique to decouple the Gauss Markov motion model and the array data model. Convergence of the artificial problem to the original problem was enforced through a stronger penalty with each iteration. In decoupling the problem, DOA and power estimates were obtained from an EM algorithm solution to a maximum penalized likelihood problem, and track estimates were obtained from single target fixed-interval Kalman smoothers, with the DOA estimates serving as noisy measurements. The batch algorithm provided accurate track estimates in about ten iterations, with a substantial reduction in computation compared with a similar discrete state space solution. A sequential version of the algorithm with a less-complex system implementation performed nearly as well, demonstrating its potential for use as a real-time tracker. The algorithm presented here used the simplest possible data model to keep the notation and derivation as straightforward as possible. Many enhancements are possible to account for more complex scenarios. Different matrices could easily be specified for each target to model different kinematics. The noise power could be treated as an unknown parameter to be estimated. This would affect the EM portion of the algorithm and would result in different formulas for estimating the DOAs and powers. The state variable could include both range and bearing or target location in coordinates. Again, this would affect the state variable and nuisance parameter estimates in the EM algorithm. Broadband signals could be modeled. The assumption about tracks being uncorrelated could be relaxed to allow for formations of targets. The above-mentioned enhancements are fairly straightforward extensions of the basic data model. The same methodology could be applied to derive a modified tracking algorithm for the model of interest. APPENDIX A NONLINEAR PROGRAMMING PENALTY METHOD The penalty method of NLP (e.g., [20], [21]) is a technique for solving constrained optimization problems of the form s.t. (A.1) Let denote the feasible region consisting of all values of for which the constraints are satisfied. Now, define an ideal penalty function as follows: (A.2) An equivalent unconstrained problem is obtained by adding this penalty function to the objective function (A.3) This problem is difficult to solve because of the discontinuity in the penalty function and the numerical issues associated with trying to implement the negative infinity term. The penalty method gets around these difficulties by solving a sequence of easier unconstrained optimization problems using a nonideal but well-behaved penalty function. The penalty function must be a continuous, differentiable, nonpositive function with the property that (A.4) A commonly used penalty function with these properties is the quadratic penalty function (A.5) The strength of the penalty function is controlled by a sequence of positive penalty parameters such that for all, and. The penalty method then consists of solving a sequence of unconstrained problems of the form (A.6) Initially, the penalty is quite loose, and infeasible solutions are obtained. With each iteration, the penalty function is forced closer to the ideal penalty function, the unconstrained problem is forced closer to the original constrained problem, and the solution is forced into the feasible region. The optimization problems, as well as their solutions, converge in the limit. A proof of convergence to a local maximum for the penalty method is provided in [21]. The proof relies on three properties. Let denote the unconstrained objective function of (A.6): The properties that ensure convergence are 1) a nonincreasing unconstrained objective function 2) a nonincreasing constrained objective function 3) a nondecreasing penalty function (A.7) (A.8) (A.9) (A.10)

10 ZARNICH et al.: UNIFIED METHOD FOR MEASUREMENT AND TRACKING OF CONTACTS FROM AN ARRAY OF SENSORS 2959 In trying to maximize the objective function over the feasible region, we start with infeasible solutions that, in general, have a larger objective function than the optimal feasible solution. Therefore, both the original objective function and the penalized objective function will decrease as the solutions approach the feasible region. Furthermore, the penalty function is negative and should be moving closer to zero (getting larger) as the solutions move closer to the feasible region. In [21], it is shown that these properties are satisfied, provided that the optimum solution to the unconstrained optimization problem in (A.6) is found at each step, i.e., if APPENDIX B EM ALGORITHM FOR MPL ESTIMATION (A.11) We start from the penalized likelihood function in (16), which is repeated here for convenience (B.1) For notational simplicity, let denote DOA and power of the th source at the th snapshot, and let denote the entire collection of signal parameters at the th snapshot. In addition, define the penalty function where In this notation, the problem becomes (B.2) (B.3) (B.4) The EM algorithm maximizes the penalized likelihood function by solving a sequence of easier maximization problems [16], [22]. We define a set of hidden or complete data that is related to the observed or incomplete data by a many-to-one mapping. Following the approach in [23] and [24], we decompose the observed data vector into distinct uncorrelated data vectors, each corresponding to one of the sources plus noise (B.5) The noise vectors are uncorrelated, zero-mean Gaussian random vectors that satisfy (B.6) We assume that each hidden data vector has an equal fraction of noise; therefore,. We then have and (B.8) (B.9) Let denote the collection of hidden data vectors. The joint pdf of the hidden data has the form where (B.10) (B.11) At each iteration of the EM algorithm, there are two steps: the expectation or E-step and the maximization or M-step. In the E-step, we find the expected value of the log-likelihood function of the hidden data conditioned on the observed data and the current estimates of the parameters of interest, i.e., (B.12) where denotes the iteration index. In the M-step, new estimates of the parameters are obtained by maximizing this function plus the penalty term (B.13) At each iteration, the penalized likelihood function increases, and convergence to a stationary point is guaranteed. Because of the way we constructed the hidden data, the E-step decouples into separate estimation problems with separate maximization prob- and the M-step decouples into lems (B.14) (B.15) (B.16) This problem is identical to the problem considered in [24], except for the additional penalty term. Following their derivation, the E-Step involves finding the conditional expectation of the single snapshot sample covariance matrix of the hidden data, which is given by Therefore (B.7) (B.17)

11 2960 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 where is the single snapshot sample covariance matrix of the observed data (B.18) and and are the theoretical covariance matrices for and as a function of (B.19) (B.20) The M-step involves maximizing the complete data log-likelihood using the quantities computed in the E-step in place of their unknown values. The M-step, which is modified by the penalty function, becomes (B.21) where is a vector of zero mean Gaussian measurement noise with covariance matrix that is independent from snapshot to snapshot, and is an transformation matrix. Let denote the batch of measurements, and let denote the batch of system states. The joint pdf of and is given by (C.3) where is a multivariate Gaussian pdf with mean and covariance matrix is a multivariate Gaussian pdf with mean and covariance matrix, and is a multivariate Gaussian pdf with mean and covariance matrix. We wish to find the MMSE/MAP estimate of the states given the measurements. The MMSE estimate is the mean of the a posteriori pdf, and the MAP estimate is the value for which maximized. In this problem, the maximum of the a posteriori pdf is at the mean, and the MMSE and MAP estimates are the same. The MAP estimate also maximizes the joint pdf or, equivalently, the joint log-likelihood function. Ignoring constant terms, has the form (B.22) (C.4) A complete derivation is given in [24]. APPENDIX C FIXED INTERVAL KALMAN SMOOTHER The fixed interval Kalman smoother is the optimal MMSE and MAP estimator for the state of a first-order Gauss Markov process given a batch of measurements. In this Appendix, we briefly review the general problem statement and solution. The derivation can be found in many textbooks, including [1] and [2]. Let denote a vector of state variables at time. The state at time is assumed to be a Gaussian random vector with mean and covariance matrix. In a first-order Gauss Markov process, the state at time is related to the state at time as follows: (C.1) where is a vector of zero mean Gaussian process noise with a known covariance, which is independent from snapshot to snapshot. is a linear transformation matrix. The states are observed indirectly through a measurement vector, which has the form (C.2) In the case where and the measurement is a scalar measurement of the first state variable, reduces to a scalar variance, and (C.4) can be rewritten as (C.5) which matches the form of (17). The MAP estimate can be computed using the fixed-interval Kalman smoother, which consists of the standard forward Kalman filter, followed by a backward smoothing filter. The forward Kalman filter is initialized with and. The state estimates and their error covariance matrices are computed sequentially for, using (C.6) (C.7) (C.8) (C.9) (C.10)

12 ZARNICH et al.: UNIFIED METHOD FOR MEASUREMENT AND TRACKING OF CONTACTS FROM AN ARRAY OF SENSORS 2961 Once the forward filtering pass has completed, the backward smoothing pass updates the state estimates and covariance matrices for, using (C.11) (C.12) (C.13) Note that on the backward pass, the updated error covariance matrix is not used in the state update and may be omitted from the computations. [20] S. G. Nash and A. Sofer, Linear and Nonlinear Programming. New York: McGraw-Hill, [21] W. I. Zangwill, Nonlinear Programming: A Unified Approach. Englewood Cliffs, NJ: Prentice-Hall, [22] P. J. Green, On use of the EM algorithm for penalized likelihood estimation, J. Roy. Statist. Soc., ser. B, vol. 52, no. 3, pp , [23] M. Feder and E. Weinstein, Parameter estimation of superimposed signals using the EM algorithm, IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp , Apr [24] M. I. Miller and D. R. 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Zarnich, K. L. Bell, and H. L. Van Trees, A unified method for passive measurement and tracking of contacts from an array of sensors, in Proc. 8th Annu. Adaptive Sensor Array Processing Workshop. Lexington, Mar [19], A sequential extension of the unified MAP track estimation method, Proc. 1st IEEE Sensor Array Multichannel Signal Process. Workshop, Mar Robert E. Zarnich received the B.S. degree in electrical engineering from Gannon University, Erie, PA, in 1985, the M.S. degree in electrical engineering from George Mason University (GMU), Fairfax, VA, in 1990, and the Ph.D. degree in information technology from GMU, in 2000, specializing in statistical signal and information processing. He has worked in a variety of capacities for the U.S. Navy in sonar and signal processing related technology. He is currently the Undersea Technology Chief Scientist with the Naval Sea System Command (NAVSEA), Advanced Systems and Technology Office (ASTO), Washington, DC. Kristine L. Bell (M 88 S 91 M 96 SM 01) received the B.S. degree from Rice University, Houston, TX, in 1985 and the M.S. and Ph.D. degrees from George Mason University (GMU), Fairfax, VA, in 1990 and From 1985 to 1990, she was with M/A-COM Government Systems and SAIC, Vienna, VA, where she was involved in analysis and development of military satellite communications systems. From 1990 to 1995, she was a Research Instructor with the C I Center, GMU. She joined the department of Applied and Engineering Statistics, GMU, in 1996 as an Assistant Professor. From 1995 to 1997, she was also a Research Associate at the Naval Research Laboratory, Washington, DC. Her research interests are in statistical signal processing, array processing, and wireless communications. Dr. Bell is an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING. Harry L. Van Trees (M 57 SM 73 F 74 LF 94) received the B.Sc. degree from the U.S. Military Academy, West Point, NY, in 1952 and the Sc.D. degree from the Massachusetts Institute of Technology (MIT), Cambridge, MA, in From 1961 to 1975, he was with the Electrical Engineering Department at MIT, becoming a Full Professor in During this period, he was active in graduate course development and was the leader of a research group working in detection and estimation theory and radar/sonar theory. Since 1988, he has been with George Mason University, Fairfax, VA, where he is currently a Distinguished Professor of Electrical Engineering. In June of 1989, he founded the Center of Excellence in Command, Control, Communications, and Intelligence. The Center has a research program that includes work in sensing and data fusion, command support systems, communications and signal processing, modeling and simulation, C3 architectures, and information systems. He has served as Chief Scientist of the Defense Communications Agency and Chief Scientist of the U.S. Air Force. He was Principle Deputy Assistant Secretary of Defense (C I). He served as an Executive Vice President and President of M/A-COM Government Systems Division. He is the author of a three-volume set of books on detection, estimation, and modulation theory. These books contain a unified approach to communications, radar, sonar, and seismic applications. The first volume is a classic in its field and is used in graduate schools throughout the world. Dr. Van Trees has received the Presidential Award for Meritorious Executive Service, the Distinguished Civil Service Award, and the AFCEA Gold Medal for Engineering.