A Particle Filter for Tracking Two Closely Spaced ... - Semantic Scholar
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 6, JUNE 2006
357
A Particle Filter for Tracking Two Closely Spaced Objects Using Monopulse Radar Channel Signals Atef Isaac, Xin Zhang, Peter Willett, and Yaakov Bar-Shalom
Abstract—For the case of a single resolved target, monopulsebased radar sub-beam angle and sub-bin range measurements carry errors that are approximately Gaussian with known covariances, and hence, a tracker that uses them can be Kalman based. However, the errors accruing from extracting measurements for multiple unresolved targets are not Gaussian. We therefore submit that to track such targets, it is worth the effort to apply a nonlinear (non-Kalman) filter. Specifically, in this letter, we propose a particle filter that operates directly on the monopulse sum/difference data for two unresolved targets. Significant performance improvements are seen versus a scheme in which signal processing (measurement extraction from the monopulse data) and tracking (target state estimation from the extracted measurements) are separated. Index Terms—Monopulse radar, particle filter, tracking, unresolved targets.
the ML numerical measurement extractor to Kalman filters for tracking. However, in [9], on which this letter is based, it was observed that care must be taken when the measurements from the multi-target ML estimator/extractor are inserted into a tracker, since their errors are in general neither Gaussian nor with an easily derived covariance. Thus, we propose a second approach in Section IV: an integrated two-target filter that operates directly on the monopulse sum- and difference-channel matched-filter outputs. This observation model is highly nonlinear, and consequently, a particle filter is appropriate, and it will be seen that the extra computation is worthwhile when targets remain unresolved for an extended time. We do not concern ourselves with data association (i.e., deciding which target claims which measurement [11]) in this letter, since the measurement extraction stage is bypassed.
I. INTRODUCTION
II. MODELS
MONOPULSE radar operates by comparing energy returns of four squinted sub-beams steered symmetrically around the expected single target location [10]. The real part of the monopulse ratio [3] can be used to provide excellent sub-beam angular measurements. Problems arise when multiple returns from several unresolved targets are present in the same range cell and beam dwell, as the monopulse ratio’s estimated direction of arrival (DOA) will be of an equivalent single target that best matches the observation. The position of any of those targets may differ considerably from this estimate, leaving no clue of any of the targets’ whereabouts. Several researchers approached this problem in [3]–[5], [7], [11], [12], but only one matched filter sampling point was used, and no use was made of the fact that targets have returns in neighboring matched filter samples (“bin straddling”). In [14], the authors made use of this information. They correlated consecutive matched filter samples (the sum, horizontal difference, and vertical difference channels) utilizing the models developed in [4]. They were able to demonstrate uniqueness of the maximum-likelihood (ML) estimation of the targets’ location parameters of up to five targets between two adjacent matched filters samples. This letter1 briefs two tracking approaches based on the monopulse radar models described in Section II. The first approach, in Section III, builds upon the work in [14] by coupling
Considering the origin of a spherical coordinate system at the radar location and matched filter samples taken at the dis, ) as the boresight azimuth crete time instant , we take ( and elevation angles, respectively. Call target ’s position in spherical coordinates (range, azimuth, and elevation). These can be easily converted to the usual Cartesian and back as in [9]. The corresponding azimuth/elevation electronic-angles and sub-bin range (the param) for target can be also calcueter vector lated as detailed in [9]. Assume that the radar pulse is rectangular with a triangular matched filter response [14, Fig. 4]. Then, with the matched filter sampling rate at one per pulse, and with the two targets lying between two consecutive sampling points, the observation vector [5], [14] at these two points consists of the in-phase components of the sum, horizontal difference, and vertical difference samples in (1) for pulse index
A
Manuscript received October 1, 2005; revised December 29, 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Arie Yeredor. The authors are with the Electrical and Computer Engineering Department, University of Connecticut, Storrs, CT 06269 USA (e-mail: [email protected]. edu; [email protected]; [email protected]). Digital Object Identifier 10.1109/LSP.2006.871714 1This letter is a focused version of [9]; in [9], several other schemes were addressed, with some emphasis on a Gibbs sampling approach.
(1) For notational simplicity, we will be dealing with only the in-phase components. Quadrature components are statistically the same as in-phase components, such that a signal with complex pulses can be thought of as one with real pulses. The in-phase components are evaluated using [14, Eq. (23)] by . setting Under the Swerling II model [10], assumed hereafter, target returns have pulse-to-pulse independent Rayleigh distributed . This magnitude and uniformly distributed phase in means that the noise-free sum and difference signals will each
1070-9908/$20.00 © 2006 IEEE
358
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 6, JUNE 2006
be Gaussian, with independent in-phase and quadrature components. Their additive noises will be assumed to be independent and Gaussian, with zero mean and known variance, expressed , for the in-phase and quadrature sum as , , , for channel noises and the in-phase and quadrature horizontal and vertical difference channel noises. Fairly accommodating , the signal-to-noise ratio (SNR) is justified to be
SNR
(2)
The observation vector is constructed as in (1), and the crosscovariance between its components are the elements of the (6 6) covariance matrix [14, Eq. (27)]. The state and the observations of the targets of interest are assumed to follow a nearly constant velocity (white-noise acceleration) kinematic model [1] given as
(3) (4) are the targets’ state transition and input-output where , is target ’s state at time , a six-dimensional matrices, vector with the first to third elements representing the target’s positional coordinates, and the fourth to sixth representing the corresponding velocities along these coordinates. In (3) and (4), and are the process and measurement noise that is considered zero-mean white with corresponding covariances and and uncorrelated between the targets. III. APPROACH 1: ML MEASUREMENT EXTRACTION KF FOLLOWED BY TWO KALMAN FILTERS MLE This approach MLE KF dedicates a separate Kalman filter to track each of the two targets based on the measurements extracted using the signal processor that is an ML estimator operating on the monopulse channel data. That is, the input to each of the Kalman filters is the Cartesian measurement that corre, obtained by numersponds to the ML position estimate ically maximizing the logarithm of the likelihood function as in [14], where a Levenberg–Marquardt approach is used. It is
assumed that the fact that there are two unresolved targets is known—a means to decide this is available in [14]. Unfortunately, by itself, this solution does not reveal any information about the goodness of its estimate. Now, for a single (the covariresolved measurement, the expression for ance of the monopulse-ratio-based electronic angles estimates given the sum-channel SNR for target ) was given in [13, Eq. (26)]. The sub-bin range variance is taken usually as that of a , where is uniform random variable, i.e., the radar bin range width. The corresponding covariance matrix in Cartesian coordinates is
(5) where is the transformation Jacobian given in (6) and (7), shown at the bottom of the page. There is no particular reason to assume that this expression is accurate—indeed, since it is designed for a single resolved target, there is every reason to expect that it is optimistic, at least as regards angle—but for multi-target ML measurement extraction, there does not seem to be a better alternative. IV. APPROACH 2: INTEGRATED PARTICLE FILTER (PF) TRACKING Approach 1 relies on a separation of duties between signal processing—i.e., measurement extraction—and tracking. There is a tacit assumption that the measurements the first stage delivers to the second have Gaussian errors (or nearly so); and consequently, the second stage ought, at least in the absence of measurement-origin uncertainty and under the assumption of a linear/Gaussian motion model, to be of a Kalman form. However, the monopulse measurement model in [14] is not linear: although the thermal noise perturbations are additive, the moreimportant skin returns, while themselves Gaussian, appear multiplicatively with the desired parameters. Thus, target tracking with monopulse measurements really ought to be a matter of nonlinear filtering. With a single target, there is some evidence that the measurement errors are nearly Gaussian so that a Kalman filter is acceptable. With multiple unresolved targets, this may not be an appropriate assumption. The Chapman–Kolmogorov equation (see, e.g., [1]) of course
(6)
(7)
ISAAC et al.: PARTICLE FILTER FOR TRACKING TWO CLOSELY SPACED OBJECTS
Fig. 1. X and Y estimates for a single run of the two trackers SNR = 20 dB, M = 10.
359
. Predict the state vector pertaining to using (3). Set spherical coordinates . 2) Radar observations acquisition: Set the radar boresight3 , ) to the azimuth and elevation components ( of . Acquire M-subpulse radar observations , as in (1). 3) Propagate each particle in time: Each particle is propagated one step in time, such that the conditional satisfies (3). For density function , set . Get the from the corresponding positional coordinates . Using the rejection method, set state vector , such that lies within the radar beam width and bin width. , convert 4) Importance sampling: For to the corresponding targets parameters . Evaluate the importance weights vector . Normalize the importance weights: for , set . to particle . Sample the particles Assign with replacement from the set according to the latter set’s weights . Convert
to . Predict
using (3). 5) Time advance: Set
. Set pertaining to
and go to step 2).
V. SIMULATION RESULTS
Fig. 2. Position RMSE for target 1 SNR = 20 dB, M = 4.
provides the true means for an integrated nonlinear filter, one that tracks directly based on the monopulse sum-and-difference-channel observations, but the complex high-dimensional integrations involved render it infeasible. A particle filter provides an attractive means to realize a nearly optimal recursive estimator [6]. Below, we propose a bootstrap-filter [6], [8] tracking algorithm that combines the measurement extraction and tracking into a single stage. It takes the monopulse radar sum-and-difference-channel returns directly and generates the tracks for both targets, as follows. Particle filtering algorithm 1) Initialization: Define as the total number of particles. and ,2, set , For , and . Get the . Cartesian positional coordinates 2 . Set the cartesian stacked Set 2Velocity
is assumed to be known, or else two-point initialization used.
The tracking scenario for all algorithms starts when the two targets are at the coordinates (in metric units with and the origin at the radar) . They are travelling in-bound with . The radar beam width is 10 mrad in both azimuth and elevation, and the range-bin width is 100 m. The radar’s scan rate is 1 Hz. The basic dynamic model matrices in (3) and (4) and their 12-state counterparts used for the particle filter are given in (7). The two straight lines in Fig. 1 are the true trajectories of the two targets, projected on the X-Y and X-Z planes, respectively. Tracking results for KF and IV the two trackers discussed in Sections III ML (PF) are also shown. For moderate SNRs (20 dB), Figs. 2 and 3 show the position root mean square error (RMSE), based on 50 Monte Carlo runs. The normalized estimation error squared (NEES) [1] for the particle filter, along with its 95% confidence distribution is shown in Fig. 4, while those region, of the driven by the ML estimator are shown in Fig. 5. As is evident from the figure, the particle filter is more consistent than the ML-based Kalman filter. The evolution of particles through time steps is shown in Fig. 6 with the scale 50:5:2 along the , , and axis to render the two targets tracks distinguishable while they are too close. We do not show the corresponding results for very low SNRs (lower than 5 dB), since in such cases, the performance is poor for even a single target. 3Defined
in Section II.
360
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 6, JUNE 2006
Fig. 3. Position RMSE for target 2 SNR
= 20 dB, M = 4. Fig. 6. Particles evolution and the particle filter estimated tracks for SNR
20 dB, M = 10.
=
VI. CONCLUSION We have introduced a particle filter as a direct tracker of two closely spaced and unresolved targets. It operates directly on the sum and difference radar channel data. Results showed a considerable improvement over the Kalman filter tracker fed by an ML measurement extractor. REFERENCES
Fig. 4. NEES for the particle filter SNR
Fig. 5. NEES for the MLE
= 20 dB, M = 4.
+ 2KF approach SNR = 20 dB, M = 4.
[1] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation With Application to Tracking and Navigation. New York: Wiley, 2001. [2] Y. Bar-Shalom and X. Li, Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS, 1995. [3] W. D. Blair and M. Brandt-Pearce, “Statistical description of monopulse parameters for tracking Rayleigh targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 597–611, Apr. 1998. [4] ——, “Unresolved Rayleigh target detection using monopulse measurements,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 543–552, Apr. 1998. [5] ——, “Monopulse DOA estimation of two unresolved Rayleigh targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 2, pp. 452–469, Apr. 2001. [6] A. Doucet, N. Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice. New York: Springer-Verlag, 2001. [7] F. Gini, M. Greco, and A. Farina, “Multiple radar targets estimation by exploiting induced amplitude modulation,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 4, pp. 1316–1332, Oct. 2003. [8] N. Gordon, D. Salmond, and A. Smith, “Novel approach to Bayesian nonlinear non-Gaussian state estimation,” Proc. Inst. Elect. Eng. F, no. 140, pp. 107–113, 1993. [9] A. Isaac, X. Zhang, P. Willett, and Y. Bar-Shalom, “Tracking of two closely-spaced objects using monopulse measurements,” in Proc. SPIE Conf. Signal Data Processing Small Targets, San Diego, CA, Aug. 2005. [10] N. Levanon, Radar Principles. New York: Wiley, 1988. [11] A. Sinha and Y. Bar-Shalom, “Maximum likelihood angle extractor of two closely spaced targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 1, pp. 183–203, Jan. 2002. [12] Z. Wang, A. Sinha, P. Willett, and Y. Bar-Shalom, “Angle estimation for two unresolved targets with monopulse radar,” IEEE Trans. Aerosp. Electron. Syst., vol. 40, no. 3, pp. 998–1019, Jul. 2004. [13] P. Willett, W. Blair, and Y. Bar-Shalom, “On the correlation between horizontal and vertical monopulse measurements,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 533–549, Apr. 2003. [14] X. Zhang, P. Willett, and Y. Bar-Shalom, “Monopulse radar detection and localization of multiple unresolved targets via joint bin processing,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1225–1236, Apr. 2005.
357
A Particle Filter for Tracking Two Closely Spaced Objects Using Monopulse Radar Channel Signals Atef Isaac, Xin Zhang, Peter Willett, and Yaakov Bar-Shalom
Abstract—For the case of a single resolved target, monopulsebased radar sub-beam angle and sub-bin range measurements carry errors that are approximately Gaussian with known covariances, and hence, a tracker that uses them can be Kalman based. However, the errors accruing from extracting measurements for multiple unresolved targets are not Gaussian. We therefore submit that to track such targets, it is worth the effort to apply a nonlinear (non-Kalman) filter. Specifically, in this letter, we propose a particle filter that operates directly on the monopulse sum/difference data for two unresolved targets. Significant performance improvements are seen versus a scheme in which signal processing (measurement extraction from the monopulse data) and tracking (target state estimation from the extracted measurements) are separated. Index Terms—Monopulse radar, particle filter, tracking, unresolved targets.
the ML numerical measurement extractor to Kalman filters for tracking. However, in [9], on which this letter is based, it was observed that care must be taken when the measurements from the multi-target ML estimator/extractor are inserted into a tracker, since their errors are in general neither Gaussian nor with an easily derived covariance. Thus, we propose a second approach in Section IV: an integrated two-target filter that operates directly on the monopulse sum- and difference-channel matched-filter outputs. This observation model is highly nonlinear, and consequently, a particle filter is appropriate, and it will be seen that the extra computation is worthwhile when targets remain unresolved for an extended time. We do not concern ourselves with data association (i.e., deciding which target claims which measurement [11]) in this letter, since the measurement extraction stage is bypassed.
I. INTRODUCTION
II. MODELS
MONOPULSE radar operates by comparing energy returns of four squinted sub-beams steered symmetrically around the expected single target location [10]. The real part of the monopulse ratio [3] can be used to provide excellent sub-beam angular measurements. Problems arise when multiple returns from several unresolved targets are present in the same range cell and beam dwell, as the monopulse ratio’s estimated direction of arrival (DOA) will be of an equivalent single target that best matches the observation. The position of any of those targets may differ considerably from this estimate, leaving no clue of any of the targets’ whereabouts. Several researchers approached this problem in [3]–[5], [7], [11], [12], but only one matched filter sampling point was used, and no use was made of the fact that targets have returns in neighboring matched filter samples (“bin straddling”). In [14], the authors made use of this information. They correlated consecutive matched filter samples (the sum, horizontal difference, and vertical difference channels) utilizing the models developed in [4]. They were able to demonstrate uniqueness of the maximum-likelihood (ML) estimation of the targets’ location parameters of up to five targets between two adjacent matched filters samples. This letter1 briefs two tracking approaches based on the monopulse radar models described in Section II. The first approach, in Section III, builds upon the work in [14] by coupling
Considering the origin of a spherical coordinate system at the radar location and matched filter samples taken at the dis, ) as the boresight azimuth crete time instant , we take ( and elevation angles, respectively. Call target ’s position in spherical coordinates (range, azimuth, and elevation). These can be easily converted to the usual Cartesian and back as in [9]. The corresponding azimuth/elevation electronic-angles and sub-bin range (the param) for target can be also calcueter vector lated as detailed in [9]. Assume that the radar pulse is rectangular with a triangular matched filter response [14, Fig. 4]. Then, with the matched filter sampling rate at one per pulse, and with the two targets lying between two consecutive sampling points, the observation vector [5], [14] at these two points consists of the in-phase components of the sum, horizontal difference, and vertical difference samples in (1) for pulse index
A
Manuscript received October 1, 2005; revised December 29, 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Arie Yeredor. The authors are with the Electrical and Computer Engineering Department, University of Connecticut, Storrs, CT 06269 USA (e-mail: [email protected]. edu; [email protected]; [email protected]). Digital Object Identifier 10.1109/LSP.2006.871714 1This letter is a focused version of [9]; in [9], several other schemes were addressed, with some emphasis on a Gibbs sampling approach.
(1) For notational simplicity, we will be dealing with only the in-phase components. Quadrature components are statistically the same as in-phase components, such that a signal with complex pulses can be thought of as one with real pulses. The in-phase components are evaluated using [14, Eq. (23)] by . setting Under the Swerling II model [10], assumed hereafter, target returns have pulse-to-pulse independent Rayleigh distributed . This magnitude and uniformly distributed phase in means that the noise-free sum and difference signals will each
1070-9908/$20.00 © 2006 IEEE
358
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 6, JUNE 2006
be Gaussian, with independent in-phase and quadrature components. Their additive noises will be assumed to be independent and Gaussian, with zero mean and known variance, expressed , for the in-phase and quadrature sum as , , , for channel noises and the in-phase and quadrature horizontal and vertical difference channel noises. Fairly accommodating , the signal-to-noise ratio (SNR) is justified to be
SNR
(2)
The observation vector is constructed as in (1), and the crosscovariance between its components are the elements of the (6 6) covariance matrix [14, Eq. (27)]. The state and the observations of the targets of interest are assumed to follow a nearly constant velocity (white-noise acceleration) kinematic model [1] given as
(3) (4) are the targets’ state transition and input-output where , is target ’s state at time , a six-dimensional matrices, vector with the first to third elements representing the target’s positional coordinates, and the fourth to sixth representing the corresponding velocities along these coordinates. In (3) and (4), and are the process and measurement noise that is considered zero-mean white with corresponding covariances and and uncorrelated between the targets. III. APPROACH 1: ML MEASUREMENT EXTRACTION KF FOLLOWED BY TWO KALMAN FILTERS MLE This approach MLE KF dedicates a separate Kalman filter to track each of the two targets based on the measurements extracted using the signal processor that is an ML estimator operating on the monopulse channel data. That is, the input to each of the Kalman filters is the Cartesian measurement that corre, obtained by numersponds to the ML position estimate ically maximizing the logarithm of the likelihood function as in [14], where a Levenberg–Marquardt approach is used. It is
assumed that the fact that there are two unresolved targets is known—a means to decide this is available in [14]. Unfortunately, by itself, this solution does not reveal any information about the goodness of its estimate. Now, for a single (the covariresolved measurement, the expression for ance of the monopulse-ratio-based electronic angles estimates given the sum-channel SNR for target ) was given in [13, Eq. (26)]. The sub-bin range variance is taken usually as that of a , where is uniform random variable, i.e., the radar bin range width. The corresponding covariance matrix in Cartesian coordinates is
(5) where is the transformation Jacobian given in (6) and (7), shown at the bottom of the page. There is no particular reason to assume that this expression is accurate—indeed, since it is designed for a single resolved target, there is every reason to expect that it is optimistic, at least as regards angle—but for multi-target ML measurement extraction, there does not seem to be a better alternative. IV. APPROACH 2: INTEGRATED PARTICLE FILTER (PF) TRACKING Approach 1 relies on a separation of duties between signal processing—i.e., measurement extraction—and tracking. There is a tacit assumption that the measurements the first stage delivers to the second have Gaussian errors (or nearly so); and consequently, the second stage ought, at least in the absence of measurement-origin uncertainty and under the assumption of a linear/Gaussian motion model, to be of a Kalman form. However, the monopulse measurement model in [14] is not linear: although the thermal noise perturbations are additive, the moreimportant skin returns, while themselves Gaussian, appear multiplicatively with the desired parameters. Thus, target tracking with monopulse measurements really ought to be a matter of nonlinear filtering. With a single target, there is some evidence that the measurement errors are nearly Gaussian so that a Kalman filter is acceptable. With multiple unresolved targets, this may not be an appropriate assumption. The Chapman–Kolmogorov equation (see, e.g., [1]) of course
(6)
(7)
ISAAC et al.: PARTICLE FILTER FOR TRACKING TWO CLOSELY SPACED OBJECTS
Fig. 1. X and Y estimates for a single run of the two trackers SNR = 20 dB, M = 10.
359
. Predict the state vector pertaining to using (3). Set spherical coordinates . 2) Radar observations acquisition: Set the radar boresight3 , ) to the azimuth and elevation components ( of . Acquire M-subpulse radar observations , as in (1). 3) Propagate each particle in time: Each particle is propagated one step in time, such that the conditional satisfies (3). For density function , set . Get the from the corresponding positional coordinates . Using the rejection method, set state vector , such that lies within the radar beam width and bin width. , convert 4) Importance sampling: For to the corresponding targets parameters . Evaluate the importance weights vector . Normalize the importance weights: for , set . to particle . Sample the particles Assign with replacement from the set according to the latter set’s weights . Convert
to . Predict
using (3). 5) Time advance: Set
. Set pertaining to
and go to step 2).
V. SIMULATION RESULTS
Fig. 2. Position RMSE for target 1 SNR = 20 dB, M = 4.
provides the true means for an integrated nonlinear filter, one that tracks directly based on the monopulse sum-and-difference-channel observations, but the complex high-dimensional integrations involved render it infeasible. A particle filter provides an attractive means to realize a nearly optimal recursive estimator [6]. Below, we propose a bootstrap-filter [6], [8] tracking algorithm that combines the measurement extraction and tracking into a single stage. It takes the monopulse radar sum-and-difference-channel returns directly and generates the tracks for both targets, as follows. Particle filtering algorithm 1) Initialization: Define as the total number of particles. and ,2, set , For , and . Get the . Cartesian positional coordinates 2 . Set the cartesian stacked Set 2Velocity
is assumed to be known, or else two-point initialization used.
The tracking scenario for all algorithms starts when the two targets are at the coordinates (in metric units with and the origin at the radar) . They are travelling in-bound with . The radar beam width is 10 mrad in both azimuth and elevation, and the range-bin width is 100 m. The radar’s scan rate is 1 Hz. The basic dynamic model matrices in (3) and (4) and their 12-state counterparts used for the particle filter are given in (7). The two straight lines in Fig. 1 are the true trajectories of the two targets, projected on the X-Y and X-Z planes, respectively. Tracking results for KF and IV the two trackers discussed in Sections III ML (PF) are also shown. For moderate SNRs (20 dB), Figs. 2 and 3 show the position root mean square error (RMSE), based on 50 Monte Carlo runs. The normalized estimation error squared (NEES) [1] for the particle filter, along with its 95% confidence distribution is shown in Fig. 4, while those region, of the driven by the ML estimator are shown in Fig. 5. As is evident from the figure, the particle filter is more consistent than the ML-based Kalman filter. The evolution of particles through time steps is shown in Fig. 6 with the scale 50:5:2 along the , , and axis to render the two targets tracks distinguishable while they are too close. We do not show the corresponding results for very low SNRs (lower than 5 dB), since in such cases, the performance is poor for even a single target. 3Defined
in Section II.
360
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 6, JUNE 2006
Fig. 3. Position RMSE for target 2 SNR
= 20 dB, M = 4. Fig. 6. Particles evolution and the particle filter estimated tracks for SNR
20 dB, M = 10.
=
VI. CONCLUSION We have introduced a particle filter as a direct tracker of two closely spaced and unresolved targets. It operates directly on the sum and difference radar channel data. Results showed a considerable improvement over the Kalman filter tracker fed by an ML measurement extractor. REFERENCES
Fig. 4. NEES for the particle filter SNR
Fig. 5. NEES for the MLE
= 20 dB, M = 4.
+ 2KF approach SNR = 20 dB, M = 4.
[1] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation With Application to Tracking and Navigation. New York: Wiley, 2001. [2] Y. Bar-Shalom and X. Li, Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS, 1995. [3] W. D. Blair and M. Brandt-Pearce, “Statistical description of monopulse parameters for tracking Rayleigh targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 597–611, Apr. 1998. [4] ——, “Unresolved Rayleigh target detection using monopulse measurements,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 543–552, Apr. 1998. [5] ——, “Monopulse DOA estimation of two unresolved Rayleigh targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 2, pp. 452–469, Apr. 2001. [6] A. Doucet, N. Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice. New York: Springer-Verlag, 2001. [7] F. Gini, M. Greco, and A. Farina, “Multiple radar targets estimation by exploiting induced amplitude modulation,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 4, pp. 1316–1332, Oct. 2003. [8] N. Gordon, D. Salmond, and A. Smith, “Novel approach to Bayesian nonlinear non-Gaussian state estimation,” Proc. Inst. Elect. Eng. F, no. 140, pp. 107–113, 1993. [9] A. Isaac, X. Zhang, P. Willett, and Y. Bar-Shalom, “Tracking of two closely-spaced objects using monopulse measurements,” in Proc. SPIE Conf. Signal Data Processing Small Targets, San Diego, CA, Aug. 2005. [10] N. Levanon, Radar Principles. New York: Wiley, 1988. [11] A. Sinha and Y. Bar-Shalom, “Maximum likelihood angle extractor of two closely spaced targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 1, pp. 183–203, Jan. 2002. [12] Z. Wang, A. Sinha, P. Willett, and Y. Bar-Shalom, “Angle estimation for two unresolved targets with monopulse radar,” IEEE Trans. Aerosp. Electron. Syst., vol. 40, no. 3, pp. 998–1019, Jul. 2004. [13] P. Willett, W. Blair, and Y. Bar-Shalom, “On the correlation between horizontal and vertical monopulse measurements,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 533–549, Apr. 2003. [14] X. Zhang, P. Willett, and Y. Bar-Shalom, “Monopulse radar detection and localization of multiple unresolved targets via joint bin processing,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1225–1236, Apr. 2005.
