IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 3, MARCH

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1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH An Asymptotically Unbiased Estimator for Bearings-Only and Doppler-Bearing Target Motion Analysis K C Ho, Senior Member, IEEE, and Y T Chan, Senior Member, IEEE Abstract Bearings-only (BO) and Doppler-bearing (DB) target motion analysis (TMA) attempt to obtain a target trajectory based on bearings and on Doppler and bearing measurements, respectively, from an observer to the target The BO-TMA and DB-TMA problems are nontrivial because the measurement equations are nonlinearly related to the target location parameters The pseudolinear formulation provides a linear estimator solution, but the resulting location estimate is biased The instrumental variable method and the numerical maximum likelihood approach can eliminate the bias Their convergence behavior, however, is not easy to control This paper proposes an asymptotically unbiased estimator of the tracking problem The proposed method applies least squares minimization on the pseudolinear equations with a quadratic constraint on the unknown parameters The resulting estimator is shown to be solving the generalized eigenvalue problem The proposed solution does not require initial guesses and does not have convergence problems Sequential forms of the proposed algorithms for both BO-TMA and DB-TMA are derived The sequential algorithms improve the estimation accuracy as a new measurement arrives and do not require generalized eigenvalue decomposition for solution update The proposed estimator achieves the Cramer Rao Lower Bound (CRLB) asymptotically for Gaussian noise before the thresholding effect occurs Index Terms Bearings-only tracking, constrained least squares, Doppler-bearing tracking, sequential estimation I INTRODUCTION TARGET motion analysis (TMA) based on bearing measurements has been of considerable interest over the past several decades [1] [8] Bearings-only TMA (BO-TMA) often assumes the target is moving at a constant velocity, and the observer must maneuver in order to satisfy the observability condition to ensure a determinant solution The optimal maneuvering of an observer has been investigated in [9] and [10] If the source maneuvers, maneuver detection may be necessary to reinitialize a BO-TMA algorithm Yet, a better approach is to reformulate and solve the BO-TMA problem for maneuvering targets directly [11], [12] When the source emits harmonic components, the harmonic signals will experience Doppler shifts at the observer so that the frequency measurements can be explored Manuscript received November 29, 2003; revised October 17, 2004 The associate editor coordinating the review of this manuscript and approving it for publication was Dr Jonathan H Manton K C Ho is with the Department of Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO USA ( hod@missouriedu) Y T Chan is with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada ( chan-yt@rmcca) Digital Object Identifier /TSP to improve the estimation accuracy The use of both Doppler shifts and bearing angles to analyze a moving target is termed Doppler-bearing TMA (DB-TMA) [5], [6] DB-TMA has the advantage over BO-TMA in that DB-TMA does not require the observer to maneuver to obtain a determinant solution This paper considers both BO-TMA and DB-TMA when the target is moving at a constant velocity and does not maneuver The measurement equations in both BO-TMA and DB-TMA are nonlinear with respect to the unknown target parameters, which makes TMA a nontrivial task Lindgren and Gong [1] proposed a pseudolinear (PL) estimator for the BO-TMA problem, where the nonlinearities are all lumped into the noise term so that the TMA problem can be solved by Kalman filtering One undesirable consequence in the PL formulation is that the noisy bearing angles appear in the measurement matrix so that it is correlated with the noise in the measurements The correlation leads to a biased solution [2] The bias can be very significant, and it may not decrease as the number of measurements increases Several techniques have been proposed to overcome the bias problem in BO-TMA and DB-TMA [3] [6], [13], [14] Aidala and Hammel [3] proposed a formulation of the BO-TMA problem in the modified polar coordinates Interestingly enough, the state vector estimate in the modified polar coordinates is shown to be asymptotically unbiased Nonlinear transformation is necessary to obtain the target track parameters from the state vector The method proposed by Holtsberg [13] provides a nearly unbiased solution This method requires a time reference point in the middle of a data measurement record Hence it is more appropriate for batch processing than real-time sequential estimation of a target The technique by Rosenqvist [14] is for DB-TMA and also has a time reference point in the middle of the data record It employs a two-step approach, and the estimator may have a smaller noise threshold above which the estimation accuracy decreases dramatically Passerieux et al [5] and Chan et al [6] applied the instrumental variable (IV) technique to remove the bias The estimation accuracy of the IV technique can reach the Cramer Rao lower bound (CRLB) [6] for Gaussian noise at moderate noise levels Yet another approach to obtain an unbiased solution is by maximum likelihood (ML) estimation The ML function is highly nonlinear with respect to the unknowns, and gradient search techniques such as the quasi-newton method [15], [16] are often used A multipass approach [17] is also proposed to reduce the chance of local convergence in the gradient search Both IV and numerical ML methods are iterative in nature and X/$ IEEE

2 810 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 their proper convergence are sensitive to initial conditions and step sizes [12] Le Cadre and Jauffret [12] have established the theoretical properties and sufficient conditions to ensure convergence of iterative methods for the BO-TMA problem Nevertheless, IV and numerical ML methods may have difficulty reaching the global minimum solution if the initial solution guess is not close to the true solution, the number of data points is not sufficient, or the geometry between the target and observer becomes unfavorable as the target proceeds [14] Recently, a closed-form solution for BO-TMA has been proposed by Nardone and Graham [18] Bias remains in the solution when the observability conditions deteriorate It may be useful, however, for initialization in iterative methods such as IV and numerical ML This paper proposes an alternate solution for the BO-TMA and DB-TMA problems It uses constrained least squares (CLS) minimization on the PL equations It is algebraic and avoids the difficulties of initial condition sensitivity and possible convergence problem associated with the IV or numerical ML techniques [12] The CLS takes a generalized eigenvector as the estimate and is proven to be asymptotically unbiased While the unbiased estimators in [12] are batch processors, sequential algorithms are derived for the CLS Furthermore, simulation results show that the CLS attains the CRLB for Gaussian measurement noise, and has a lower signal-to-noise threshold [22] than the IV counterpart The preliminary results of the proposed algorithms based on simulations were presented in [23] This paper provides the theoretical development of the algorithms, derives the sequential implementations, and presents more complete comparison with existing techniques Section II derives the unbiased solution to the BO-TMA problem and shows analytically that the solution is asymptotically unbiased Section III contains the asymptotically unbiased estimator for the DB-TMA problem Section IV presents the sequential algorithms for the proposed BO-TMA and DB-TMA methods Section V contains simulations to compare the performance of the proposed mthod with the IV technique, the numerical ML method and the CRLB, and Section VI is the conclusions We shall briefly summarize the notations used in this paper and represent the position and velocity The subscript or in a symbol means that the symbol quantity belongs to the target or the observer, respectively For a noisy variable represents the true value without noise and is the component due to noise so that The bearing measurements are and the tonal frequency measurements are, where is the time index and denote the measurement matrix and measurement vector and is the augmented measurement matrix is the unknown target parameter vector and is the augmented parameter vector to be estimated II BEARINGS-ONLY TMA Let be the initial position of the target at time 0 Assuming the target is moving at a constant velocity, then at time, the target s position is (1) Fig 1 Bearing angle, observer, and target positions where is the observation period and is the time index To determine the target trajectory, the observer measures the bearing angle with respect to the axis where is the true bearing and is the independent identically distributed measurement noise with power Given the observer positions and the bearing measurements, we wish to obtain the target position at instant Since the target is moving at a constant velocity, the problem becomes one of estimating the target s initial position and velocity As shown in Fig 1, the true bearing from the observer to the target is Cross-multiplying gives The true bearing angles are not known and only the noisy bearing measurements are available Replacing by in (4) results in the pseudolinear equation error [6] where and is the target range with respect to the observer at instant Let be the unknown parameter vector be the measurement vector, and be the measurement matrix Putting (1) into (5) and collecting the equation errors as a vector yields (2) (3) (4) (5) (6) (7) (8)

3 HO AND CHAN: ASYMPTOTICALLY UNBIASED ESTIMATOR FOR TARGET MOTION ANALYSIS 811 Minimizing with respect to yields the least squares solution where (9) (17) This LS solution is biased [2], [6] Putting into (9) and taking expectation yields (10) where is the true solution and is the equation error vector evaluated at Note that the noisy bearing measurements appear in both and, and hence As a result, the second term on the right of (10) is not zero, which gives rise to a bias The bias can be very significant It may not decrease as the number of bearing measurements increases, and it increases with the measurement noise power We shall propose a method that will give an asymptotically unbiased solution This method defines an augmented measurement matrix and an augmented solution vector (11) (12) where is a scaling constant whose role will be discussed after (19) and before (22) To proceed, we shall assume that the bearing measurement errors are small, as in the PL formulation and the IV method If the errors are large, the thresholding effect where the estimation accuracy deviates sharply from the CRLB will occur [22] This is a common phenomenon for using a linear estimator to solve a nonlinear estimation problem The simulation section has more discussion on the thresholding effect Under the small measurement noise condition, and From (2) and expanding the sum of angles identities, we have (13) (14) Putting them into the elements of and in (6) and (7) yields (15) where is with the noisy bearings replaced by the true bearings, and (16) is the component of the augmented matrix due to noise We define the constraint matrix as (18) Note from (16) that Hence represents the correlation matrix of the noise component in, and its role will be described in the sequel The equation error vector (8) can be rewritten in terms of and defined in (11) and (12) as The proposed method forms the augmented correlation matrix and finds by minimizing subject to (19) The constant 1 in the constraint is arbitrary Using a different value than 1 simply changes the constant in Constrained minimization is necessary here because we have expanded the parameter vector from four unknowns in to five in The constraint reduces one degree of freedom in the search space so that the number of free parameters remains four To justify that (19) provides a good solution, let us first assume that the measurement noise is absent In this case, From (4) it is easy to verify that attains its minimum value of zero at the true solution The constraint avoids the trivial solution and is satisfied when is equal to, where represents the true solution of In the presence of noise, from (15), when the number of measurements is large Under the constraint, the noise component in does not affect minimization so that minimizing also yields the true solution The constraint in this case not only avoids the trivial solution, but also restricts the search space in to be within the hypersphere defined by the constraint so that the noise component in does not affect minimization Appendix A provides a more rigorous justification of (19) and shows theoretically that it yields an asymptotically unbiased solution The Lagrange multiplier method can solve the constrained minimization problem by forming the auxiliary cost function (20) where is the Lagrange multiplier Taking the partial derivative of with respect to gives (21) which indicates that is the generalized eigenvector Premultiplying (21) by gives, which is the quantity to be minimized Hence is the generalized eigenvector of

4 812 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 the pair that yields the minimum generalized eigenvalue Once is found, then and the estimated target parameter vector is (22) The proposed solution requires the constraint matrix given in (18) It is dependent on the true bearings that are not known The true bearings can be replaced by the noisy angle measurements and the resulting errors are negligible when the noise is small In fact, one can derive, using (13) and (14) and expressing as in (17), that is, then a tonal from the target will experience a Doppler shift at the observer so that the Doppler shifted frequency at time index is (25) where is the propagation speed of the tone signal and is positive when observer and target are closing and negative when receding Expressing the radial velocity in terms of the target and observer velocities [6], (25) becomes (26) Since the true Doppler shifted frequencies and bearing angles are not available, replacing them by the noisy measurements and gives the equation error (23) Hence when is small, can be approximated by using the noisy bearings Alternatively, (23) can be used to form so that it becomes (24) Another remedy for high noise situation is to use the noisy measurements in to obtain an initial estimate and then update to obtain a more accurate solution To summarize, the proposed estimator: 1) forms the correlation matrix, where is defined in (11) and the constraint matrix given in (18) with the true bearings replaced by the noisy measurements; 2) performs the generalized eigendecomposition of the pair ; 3) selects the generalized eigenvector of the smallest generalized eigenvalue; 4) applies (22) to obtain the target initial position and velocity vector The proposed solution is algebraic and removes the noise effects by imposing a quadratic constraint in optimization The proposed solution is different from the IV method because it does not require initialization and is not iterative, and it does not reply on any instrumental variable to decorrelate the noise The proposed estimator is asymptotically unbiased Details of the proof are in Appendix A Although the proof requires asymptotic conditions, simulation indicates that the bias decreases quickly to zero as the number of measurements increases, and the solution accuracy approaches the CRLB for Gaussian measurement noise III DOPPLER-BEARING TMA When the target and observer have relative motions and the target s radial velocity with respect to the observer at time index (27) Combining the frequency and bearing measurement equations (27) and (5) yields (28) where the vectors and matrices in (28) are defined in (29) (32), shown at the bottom of the next page Note that the parameter vector contains an additional unknown 1 As in the BO-TMA case, direct minimization of will lead to a biased solution because the bearing measurement noise appears in both and The proposed asymptotically unbiased solution follows a similar derivation as in the BO-TMA case Defining the augmented measurement matrix and solution vector (33) (34) where is a constant The equation error (28) can be rewritten as Putting (13) and (14) into (31) and (32) yields (35) where is with the noisy measurements replaced by their true values and where (36) (37)

5 HO AND CHAN: ASYMPTOTICALLY UNBIASED ESTIMATOR FOR TARGET MOTION ANALYSIS 813 and is defined in (17) Since the true bearing angles are not known, they are replaced by the noisy measurements in (37) and (17) The constraint matrix for the DB-TMA case is (38) where is the frequency measurement noise power As in the BO-TMA case, depends on the true bearings They will be replaced by their noisy values and the resulting error is negligible, especially for small measurement noise When the noise is large, one can use the obtained solution to form a better to generate a more accurate solution Note that the constraint matrix for DB-TMA in (38) also requires the knowledge of the bearing and frequency measurement noise powers Defining, the proposed DB-TMA solution vector is obtained by minimizing subject to (39) The resulting solution is the generalized eigenvector of the pair that has the smallest generalized eigenvalue Note from (34) that the last element of is Hence (40) It can be shown by following the same argument as in Appendix A for the BO-TMA case that is asymptotically unbiased Unlike the BO-TMA case, the proposed solution in DB-TMA requires the noise powers of the frequency and bearing measurements to form the constraint matrix (38) The measurement noise powers are usually known in practice If they are not known, we can estimate them as follows At the beginning, since the constraint matrix is not available, we start with the biased solution by minimizing, where is given in (28), to obtain Substituting the values in bearing angles noise is then given by (41) to (3) produces estimates of the An estimate of the bearing (42) Similarly, putting the estimated values from and also into (26) gives estimates of the Doppler shifted frequencies, An estimate of the frequency measurement noise power is (43) and in (38) are then replaced by their estimates to form the constraint matrix so that the constrained solution can be found This can be used to generate better estimates of the noise powers to improve the accuracy of the target location estimate Simulation results indicate that repeating the noise power estimation using the generated solution two to three times is sufficient to arrive at an accurate solution, and using the noise estimates sometimes gives better results than using the true noise power values To summarize, the proposed estimator: (29) (30) (31) (32)

6 814 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 TABLE I SEQUENTIAL ALGORITHM OF THE PROPOSED BO-TMA ESTIMATOR 1) forms the correlation matrix, where is defined in (33) and constraint matrix given in (38); 2) performs the generalized eigendecomposition of the pair ; 3) selects the generalized eigenvector of the smallest generalized eigenvalue; 4) applies (40) to obtain the target initial position and velocity vector Our discussion so far considers only a single tone Typically, a single tone comes from the propeller noise of a vessel, when it is traveling at a constant speed There could be other tones that can come from machinery onboard, such as compressors It is straightforward to extend the proposed technique to multiple tones, either through the tonal summation technique [6] or through the use of a parameter vector that contains the reciprocals of all the tonal frequencies [5] In a multicontact environment, the particular tones may help in distinguishing the different contacts, since each may radiate different tones IV SEQUENTIAL ESTIMATION In practice, the measurements come in sequentially and it is beneficial to improve the target location estimate in the previous instant as a new measurement is available Since this paper considers a constant velocity target only, the sequential algorithm developed here is a way to implement a batch algorithm to

7 HO AND CHAN: ASYMPTOTICALLY UNBIASED ESTIMATOR FOR TARGET MOTION ANALYSIS 815 reduce computation and increase efficiency, and is not intended for tracking time-varying parameters Denote the augmented correlation matrix for BO-TMA up to time as With a new bearing measurement, it is updated as where from (6), (7), and (11) The constraint matrix is updated similarly as (44) (45) where is defined in (17) with the true bearings replaced by the bearing measurements The updates of and for DB-TMA are similar The generalized eigendecomposition can then be applied to the pair to obtain an improved estimate Repeating the generalized eigendecomposition procedure after each measurement is costly Also, processing in this manner does not take advantage of the solution obtained in the previous instant We shall derive here a computationally efficient sequential processing scheme to improve the target location estimate based on the previous one as a new measurement arrives, and no generalized eigendecomposition is needed We shall begin with the BO-TMA case and then extend the solution to DB-TMA The sequential update scheme is based on some modification of a previous work [20] Suppose we have the solution up to time instant In the next instant 1, and are updated according to (44) and (45), and satisfies (21) Premultiplying the inverse of on both sides of (21) and replacing by on the right gives the sequential estimation equation Normalizing (46) to ensure that gives the new (46) (47) In practice, iterating (46) and (47) a few times may be needed to reach a stable solution In such a case, (46) and (47) are modified to (48) where corresponds to the iteration index Our investigation finds that reaches the solution quickly and no divergence behavior has been observed The good convergence behavior of is also supported by the study in [20] Note that the factor is removed in the first equation of (48) because it is not needed due to the normalization step The iteration of (48) is a numerical technique to avoid generalized eigendecomposition to save computation Its purpose is different from the iteration in iterative methods such as IV and numerical ML where they depend on good initial solution guess to start iteration to arrive at a good solution The computation can further be reduced by applying the matrix inversion lemma [21] in (44) to update directly from Denoting as, then [21] (49) Table I summarizes the sequential processing algorithm for the proposed BO-TMA solution At the beginning, a number of measurements are collected and the generalized eigendecomposition or the direct LS minimization is used to obtain an initial solution In each time step afterwards, the sequential solution update consists of (49), (45), and (48) The sequential processing algorithm for DB-TMA is similar In each time step, is updated by two vectors corresponding to frequency and bearing measurements, so that the matrix inversion lemma will be applied twice If the measurement noise powers are not known, they are estimated from (42) and (43) using the target location solution obtained at the present instant and will be used to determine the solution in the next instant If computation saving is desired, the noise power estimates can be updated through exponential averaging Using the current solution in (3) and (26) produces and Then the noise power estimates are updated as (50) where is the forgetting factor that is close to and less than unity Experimental results show that using (42) and (43) gives a more accurate solution than using (50) initially This is probably because the noise power estimates are inaccurate at the beginning due to a poor initial target location estimate, and exponential update takes a few iterations to reach accurate noise power estimates Table II summarizes the proposed sequential processing algorithm for DB-TMA The measurement noise powers are assumed not known The sequential update stage consists of updating and followed by measurement noise powers estimation The estimation of noise powers is controlled by UpdateNoisePowersAllMeasurements, which uses either (42) and (43) or (50), depending on the available computation resources and the desirable performance accuracy We now examine the computational complexity of the proposed sequential algoithms After the initial solution, the computation burden in each time step is in matrix-vector multiplication of size, where is the number of estimating parameters plus one It is equal to five for BO-TMA and six for DB-TMA Hence the complexity in each time step is only In the DB-TMA case, additional computation in each time step is needed to estimate the noise powers if they are not known The sequential IV method also requires computational complexity [6], assuming the noise powers are

8 816 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 TABLE II SEQUENTIAL ALGORITHM OF THE PROPOSED DB-TMA ESTIMATOR

9 HO AND CHAN: ASYMPTOTICALLY UNBIASED ESTIMATOR FOR TARGET MOTION ANALYSIS 817 Fig 2 Geometry for the BO-TMA simulation Fig 3 Performance of the proposed BO-TMA method, x-position, = =360 rad known Hence the proposed algorithms have complexity comparable to the sequential IV method V SIMULATION The simulations for both BO-TMA and DB-TMA follow the target-observer scenarios described in [6] The sequential forms of the proposed algorithms given in Tables I and II were used, where the instant for initialization is 50, the stopping criteria constant for finding the minimum generalized eigenvector is 10, and the selection flag UpdateNoisePowerAllMeasurements is 1 (using (42) and (43) instead of (50) The number of ensemble runs is 200 The proposed method is compared with a) the sequential IV method derived in [6] and b) the batch mode algorithms batch IV and numerical ML [12], [16] A Comparison With Sequential IV For BO-TMA, the observer starts at (0, 0) and is moving along a zigzag path at a constant speed of 127, as shown in Fig 2 It maneuvers at time instances with 90 turns, The target is initially at (127 km, 127 km) and is moving at a constant speed of 9 m/s at an angle of 30 with respect to the axis The observation period is s The bearing noise is zero mean, white, and assumed Gaussian for simplicity In practice, the bearing noise may not be Gaussian Note that the bearing-to-noise ratio varies with the target s trajectory because the bearing changes Fig 3 gives the estimation accuracy of the proposed method in terms of the root mean square error (RMSE) and bias in the -position of the target trajectory The bearing noise power is rad The target s -position estimate at time index was generated according to (1) as, where and are the target s initial -position and -velocity estimates at time index At the beginning, the bias is large However, the bias decreases to zero quickly at around time index 100 The CRLB for the BO-TMA problem [6] is shown as a reference The RMSE of the proposed method converges quickly to the CRLB Fig 4 Performance of the proposed BO-TMA method, x-velocity, = =360 rad Fig 4 depicts the results for the target s -velocity estimate Again, the bias drops quickly to zero and the proposed method yields an asymptotically unbiased estimate The estimation accuracy approaches the CRLB as the number of measurement increases The results for the target s -position and -velocity are similar to those in Figs 3 and 4 and are not shown The sequential IV method [6] was implemented for comparison It did not converge when it was initialized to the biased LS solution (9) at When it was initialized with the true target s position and velocity at to start the iteration, it converged and reached the CRLB as increased To better examine the optimality of the proposed method, Fig 5 gives the RMSE of the target s and positions at time index as a function of the noise power The bearing noise power was set to rad NoiseFactor, where NoiseFactor is the noise scaling factor and it varies from 20 to 10 db The solid line represents the CRLB and the cross-solid line denotes the RMSE of the proposed method It is clear that over a wide range of measurement noise, the proposed

10 818 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 Fig 5 Accuracy of the target s x and y positions using the proposed BO-TMA method as the measurement noise power increases, =(=180 rad) 2 NoiseFactor RMSE of the proposed method (x-solid line), RMSE of the IV method (o-dashed line), CRLB (solid line) algorithm reaches the CRLBs in both and positions The accuracy in -position is higher than that in -position because the target s angle is 30 and the displacement in is less than the displacement in In any nonlinear estimation, there is a threshold effect whereby when the signal-to-noise ratio (SNR) is below a certain value, the estimates will become unreliable [22] This SNR value is dependent on the particular problem and the estimator used, and is usually difficult to predict In the simulation scenario, at, the proposed CLS solution (cross solid line) has a threshold value at rad, while the sequential IV (circle-dashed line) has a threshold at rad As Fig 5 shows that, even with true value initialization when, at, the IV estimates diverge away from the CRLB when rad Similar results for the velocity estimates appear in Fig 6 The conclusion is that the CLS has a lower SNR threshold than the IV method For the DB-TMA simulation, the observer is stationary at the origin The target is initially at (20 km, 18 km) and proceeds due north at 9 m/s The target radiates a single tone at Hz The observation period in this simulation is s The frequency measurements are corrupted with zero mean white Gaussian noise of power Hz The zero-mean white Gaussian noise in the bearing measurements has a power of rad Fig 7 gives the results in the -position estimate of the target using the proposed method The results for -velocity are in Fig 8 No prior knowledge about the measurement noise powers was used in the proposed method It is clear that the proposed solution is able to track the target very well and produces unbiased position and velocity estimates after with accuracy close to the CRLB The sequential IV method was initialized to the true solution at The IV method diverged right at the beginning and failed to reach a solution Figs 9 and 10 depict the accuracy of the proposed method in terms of RMSE at time index as the measure- Fig 6 Accuracy of the target s x and y velocities using the proposed BO-TMA method as the measurement noise power increases, = (=180 rad) 2 NoiseFactor RMSE of the proposed method (x-solid line), RMSE of the IV method (o-dashed line), CRLB (solid line) Fig 7 Performance of the proposed DB-TMA method, x-position, = 0:3 Hz, = =180 rad ment noise powers increase Fig 9 is for the target positions and Fig 10 is for the target velocities No prior knowledge about the measurement noise powers was used in the proposed method The measurement noise powers in this simulation are Hz NoiseFactor and rad NoiseFactor, where the NoiseFactor varies from 20 to 10 db When the NoiseFactor is below 2 db, the proposed method reaches the CRLB in both positions and velocities Above this NoiseFactor, thresholding effect due to nonlinear nature of the problem occurs and the accuracy deviates from the CRLB For the sequential IV method, it was initialized at with the true target location parameters to generate its most favorable results Also, the exact bearings and frequency measurement noise powers were used in the IV algorithm The IV method provides good solution only at very low noise level When the NoiseFactor is above 10 db, the IV method diverges sharply from the CRLB An explanation of the poor performance of the sequential IV is

11 HO AND CHAN: ASYMPTOTICALLY UNBIASED ESTIMATOR FOR TARGET MOTION ANALYSIS 819 Fig 8 Performance of the proposed DB-TMA method, x-velocity, = 0:3 Hz, = =180 rad Fig 10 Accuracy of the target s x and y velocities using the proposed BO-TMA method as the measurement noise powers increase, = (0:3 Hz) 2 NoiseFactor; =(=180 rad) 2 NoiseFactor RMSE of the proposed method (x-solid line), RMSE of the IV method (o-dashed line), CRLB (solid line) Fig 9 Accuracy of the target s x and y positions using the proposed DB-TMA method as the measurement noise powers increase, = (0:3 Hz) 2 NoiseFactor; =(=180 rad) 2 NoiseFactor RMSE of the proposed method (x-solid line), RMSE of the IV method (o-dashed line), CRLB (solid line) that the bias remains high and becomes larger as the target-observer geometry becomes worse, and it is not sufficiently cancelled by the instrumental variables A better choice of the IV could make the results better B Comparison With Batch Algorithms The simulations in the previous section use sequential IV where the solution from the previous time instant is used to form the instrumental variables to compute the solution in the current instant The batch IV method is expected to give better results because it uses all the available measurements together and iterates several times to obtain a solution [6], [12] It was initialized using the biased estimates (9) for BO-TMA and (41) for DB-TMA to generate the results The batch algorithm numerical ML [12] based on quasi-newton iteration is also implemented for comparison Numerical ML is quite sensitive to initialization and it does not converge if it is initialized with (9) In the results presented, it was initialized with the true solution vector added with zero-mean Gaussian white noise vector whose elements have power equal to 2 times the CRLBs for the individual parameters to be estimated Both IV and numerical ML used five iterations, and increasing the number of iterations did not improve performance Both batch IV and numerical ML are batch algorithms and their complexity is, where is the number of measurements The proposed technique performs similarly in batch mode that requires generalized eigendecomposition or in sequential mode that is given in Tables I and II The complexity is only in sequential processing, and it is used in the simulation with initialization at Fig 11 shows the total RMSE comparison for BO-TMA The simulation scenario is the same as the BO-TMA in the previous section, except that the observer speed is decreased to 2 m/s and the bearing noise power is reduced to rad Since the observer speed is reduced, the TMA scenario is far worse than before The IV is converging in batch mode, but having accuracy higher than the CRLB The numerical ML has difficulty converging to a solution most of the time The proposed method does not require initial solution guesses, behaves relatively stable, and reaches the CRLB accuracy as the number of measurements increases Fig 12 gives the total RMSE comparison for the DB-TMA case The DB-TMA simulation scenario and noise powers are exactly the same as those in generating Figs 7 and 8 In contrast to the sequential IV, which fails to converge, the batch IV does converge As the target-observer geometry becomes worse

12 820 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 VI CONCLUSION Fig 11 Comparison of performance with batch processing algorithms for BO-TMA, =0:5(=180 rad) Total RMSE of the batch IV method (o), total RMSE of the numerical ML method (square), total RMSE of the proposed method (x), CRLB (solid line) This paper proposes asymptotically unbiased estimators for BO-TMA and DB-TMA of a constant velocity target It applies a constrained least squares minimization on the pseudolinear measurement equations to obtain the asymptotically unbiased solution The solution is shown to be equal to the generalized eigenvector of the augmented measurement matrix that has the smallest generalized eigenvalue The proposed method does not require accurate initial solution guesses as in the IV and the numerical ML method and has no divergence problems, when the noise power is below the threshold value Sequential forms of the proposed BO-TMA and DB-TMA algorithms to update the solution estimates as a new measurement arrives, useful for on-line target monitoring and complexity reduction, are derived After computing a solution at the beginning, the sequential algorithms do not require generalized eigendecomposition in each subsequent time step, which reduces significantly the computation complexity compared to applying generalized eigendecomposition The sequential algorithms have the same order of complexity as the sequential IV method Simulation confirms that the proposed estimator produces an asymptotically unbiased solution, and it achieves accuracy close to the CRLB for Gaussian measurement noise as the number of measurements increases APPENDIX A We shall show in this Appendix that the proposed BO-TMA estimator defined in (19) is asymptotically unbiased The solution of the proposed estimator is the eigenvector of the smallest eigenvalue from the generalized eigendecomposition (21) We begin by computing the minimum value of, under the quadratic constraint, which is also the minimum generalized eigenvalue The correlation matrix is, after substituting (15) Fig 12 Comparison of performance with batch processing algorithms DB-TMA, =(0:3 Hz) ; =(=180 rad) Total RMSE of the batch IV method (o), total RMSE of the numerical ML method (square), total RMSE of the proposed method (x), CRLB (solid line) When using (16) (51) when the time index increases, IV diverges from the CRLB Numerical ML is not able to converge before, and attains CRLB afterwards The proposed algorithm reaches the CRLB earlier than the numerical ML and behaves very well as increases Better initialization of IV and numerical ML may give better results However, better initialization close to the true solution is not easy to find, and their convergence is not easy to control unless certain conditions are satisfied [12] (52) where is the th column of and is defined in (17) When is large and the measurement noise is ergodic, taking

13 HO AND CHAN: ASYMPTOTICALLY UNBIASED ESTIMATOR FOR TARGET MOTION ANALYSIS 821 probability limits [19] and utilizing the fact [19] that the probability limit of a random matrix is also its asymptotic expectation, we have ACKNOWLEDGMENT The authors would like to thank the reviewers for their valuable comments and suggestions to improve the quality of this paper REFERENCES (53) where denotes the probability limit of the variable and (18) has been used Under the quadratic constraint (54) The first term on the right-hand side is a quadratic form and its minimum value is zero when, where is chosen to satisfy the constraint Hence the minimum value of is under the quadratic constraint, when the number of measurements is large We now prove that the proposed solution is asymptotically unbiased through (21), when the generalized eigenvalue has its minimum value of After substituting in (51) and rearranging terms, (21) becomes (55) When the number of measurements is large, from (16), we have plim and plim, where plim stands for probability limits [19] From asymptotic distribution theory [19] where the number of measurements, taking probability limits in (55) and invoking Slutsky s theorem [19] in plim yields plim (56) where plim (56) implies that falls into the null space of, ie, If we let, then satisfies Now denote as the vector that contains the true values of the target s initial position and velocity It is obvious from (8) that when putting in the true bearing values Since is full column rank, we have Furthermore, because the plim of a random vector is also its asymptotic expectation [19], therefore is asymptotically unbiased [1] A G Lindgren and K F Gong, Position and velocity estimation via bearing observations, IEEE Trans Aerosp Electron Syst, vol AES-14, pp , Jul 1978 [2] V J Aidala, Kalman filter behavior in bearings-only tracking applications, IEEE Trans Aerosp Electron Syst, vol AES-15, pp 29 39, Jan 1979 [3] V J Aidala and S E Hammel, Utilization of modified polar coordinates for bearings-only tracking, IEEE Trans Autom Control, vol AC-28, pp , Mar 1983 [4] S C Nardone, A G Lindgren, and K F Gong, Fundamental properties and performance of conventional bearings-only target motion analysis, IEEE Trans Autom Control, vol AC-29, pp , Sep 1984 [5] J M Passerieux, D Pillon, P Blanc-Benon, and C Jauffret, Target motion analysis with bearings and frequencies measurements via instrumental variable estimation, in Proc IEEE Int Conf Acoustics, Speech, Signal Processing, 1989, pp [6] Y T Chan and S W Rudnicki, Bearings-only and Doppler-bearing tracking using instrumental variables, IEEE Trans Aerosp Electron Syst, vol 28, pp , Oct 1992 [7] M Gavish and A J Weiss, Performance analysis of bearings-only target location algorithm, IEEE Trans Aerosp Electron Syst, vol 26, pp 22 26, Oct 1992 [8] D T Pham, Some quick and efficient methods for bearings-only target motion analysis, IEEE Trans Signal Process, vol 41, pp , Sep 1993 [9] R J Evans, A Isaksson, and A Logothetis, An information theoretic approach to observer path design for bearings-only tracking, in Proc 36th IEEE Conf Decision Control, Dec 1997, pp [10] J M Passerieux and D Van Cappel, Optimal observer maneuver for bearings-only tracking, IEEE Trans Aerosp Electron Syst, vol 34, pp , Jul 1998 [11] J P Le Cadre and O Tremois, Bearings-only tracking for maneuvering sources, IEEE Trans Aerosp Electron Syst, vol 34, pp , Jan 1998 [12] J P Le Cadre and C Jauffret, On the convergence of iterative methods for bearings-only tracking, IEEE Trans Aerosp Electron Syst, vol 35, pp , Jul 1999 [13] A Holtsberg and J H Holst, A nearly unbiased inherently stable bearings-only tracker, IEEE J Ocean Eng, vol 18, pp , Apr 1993 [14] P A Rosenqvist, Passive Doppler-bearing tracking using pseudo-linear estimator, IEEE J Ocean Eng, vol 20, pp , Apr 1995 [15] N G Chernoguz, A smoothed Newton-Gauss method with application to bearings-only position location, IEEE Trans Signal Process, vol 43, pp , Aug 1995 [16] T Kirubarajan and Y Bar-Shalom, Low observable target motion analysis using amplitude information, IEEE Trans Aerosp Electron Syst, vol 32, pp , Oct 1996 [17] C Jauffret and Y Bar-Shalom, Track formation with bearing and frequency measurements in clutter, IEEE Trans Aerosp Electron Syst, vol 26, pp , Nov 1990 [18] S C Nardone and M L Graham, A closed-form solution to bearingsonly target motion analysis, IEEE J Ocean Eng, vol 22, pp , Jan 1997 [19] A S Goldberger, Econometric Theory New York: Wiley, 1964 [20] Y N Rao and J C Principe, An RLS type algorithm for generalized eigendecomposition, in Proc IEEE Signal Processing Workshop: Neural Networks Signal Processing, 2001, pp [21] S Haykin, Adaptive Filter Theory, 4th ed Upper Saddle River, NJ: Prentice-Hall, 2002 [22] A J Weiss and E Weinstein, Fundamental limits in passive time delay estimation Part I: Narrow-band systems, IEEE Trans Signal Process, vol 31, pp , Apr 1983 [23] K C Ho and Y T Chan, An unbiased estimator for bearings-only tracking and Doppler bearing tracking, in IEEE Proc ICASSP 03, Hong Kong, China, Apr 3, 2003, pp V169 V172

14 822 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 K C Ho (S 89 M 91 SM 00) was born in Hong Kong He received the BSc degree in electronics (with first class honors) and the PhD degree in electronic engineering from the Chinese University of Hong Kong, Hong Kong, in 1988 and 1991, respectively He was a Research Associate with the Royal Military College of Canada from 1991 to 1994 He joined Bell-Northern Research, Montreal, PQ, Canada, in 1995 as a Member of Scientific Staff He was with the Faculty of the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, Canada, from September 1996 to August 1997 Since September 1997, he has been with the University of Missouri-Columbia, where he is currently an Associate Professor in the Electrical and Computer Engineering Department He is also an Adjunct Associate Professor with the Royal Military College of Canada His research interests are in statistical signal processing, source localization, subsurface object detection, wavelet transform, wireless communications, and the development of efficient adaptive signal-processing algorithms for various applications including landmine detection, echo cancellation, and time delay estimation He has been active in the development of ITU Standard Recommendation GI68: Digital Network Echo Cancellers since 1995 He is the editor of the ITU Standard Recommendation G168 He has received three US patents, three Canadian patents, and three European patents in the area of mobile communications Dr Ho is currently an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING and IEEE SIGNAL PROCESSING LETTERS He received the Croucher Foundation Studentship from 1988 to 1991 He received the Junior Faculty Research Award from the College of Engineering, University of Missouri-Columbia, in 2003 Y T Chan was born in Hong Kong He received the Bachelor s and Master s degrees from Queen s University, Kingston, ON, Canada, and the PhD degree from the University of New Brunswick, Fredericton, NB, Canada He was an Engineer with Nortel Networks and has been a Professor in the Electrical and Computer Engineering Department of the Royal Military College of Canada, Kingston, ON, Canada, serving as head of the department from 1994 to 2000 From 2002 to 2005, he was a visiting Professor at the Electronic Engineering Department of the Chinese University of Hong Kong He is currently an Adjunct Professor at the Royal Military College His research interests are in detection, estimation, localization and tracking He is author of Wavelet Basics (Norwell, MA: Kluwer, 1994) Dr Chan was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, the Technical Chair of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 84, General Chair of ICASSP 91, Vice-Chair of ICASSP 03, and Social Chair of ICASSP 04 He directed a NATO ASI in 1988

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