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Quickest Detection and Tracking of Spawning Targets Using Monopulse Radar Channel Signals
Abstract
Recent advances have been reported in detecting and estimating the location of more than one target within a single monopulse radar beam. Successful tracking of those targets has been achieved with the aid of nonlinear filters that approximate the targets' states' conditional pdf, bypassing the measurement extraction stage, and operating directly on the monopulse sum/difference data, i.e., without measurement extraction. The problem of detecting a target spawn will be tackled in this paper. Particle filters will be employed as nonlinear tracking filters to approximate the posterior probability densities of the targets' states under different hypotheses of the number of targets, which in turn can be used to evaluate the likelihood ratio between two different hypotheses at subsequent time steps. Ultimately, a quickest detection procedure based on sequential processing of the likelihood ratios will be used to decide on a change in the underlying target model as an indication of a newly spawning target. Radar signal processing, data association, and target tracking are handled simultaneously.
... This assumption significantly simplifies the analysis since an important source of uncertainty has been removed. Isaac et al. [5,6] employ monopulse-radar signal processing techniques, combined with an auxiliary particle filter, to detect and track spawned objects. They assume that, during a sampling period, the number of objects can increase by at most one, which, again, is a significant simplification of a spawning event. ...
... Step 1: Time-update: At time t the predicted PHD D t + 1|t is a Gaussian-mixture whose weight, mean and covariance matrix can be derived as in (6). The predicted cardinality distribution is given as in (7). ...
... In our second simulation, the primary object is obscured by a single spawning event with 19 ancillary objects. Both scenarios are much more difficult, and general than Isaac et al.'s approach [5,6], where the number of objects can increase by at most one. Furthermore, Gardner et al. [4] assume which target generates which measurement. ...
Many multi-object tracking scenarios are complicated by the fact that an object of interest may spawn additional objects which, for some period of time, follow roughly the same trajectory as the original object and then fall away. The challenge is then to discriminate the original object from the spawned ancillaries in a timely fashion. This study proposes a solution to this problem based on the increasingly well-known multi-object track-before-detect algorithm called the cardinalised probability hypothesis density (CPHD) filter. Precisely, the authors assume zero false alarms (ZFA) in the CPHD filter, and apply the proposed scheme to linear and non-linear simulation scenarios based on widely used object-trajectory and sensor models. The authors have also demonstrated that a Gaussian mixture implementation of the ZFA-CPHD filter (i) establishes stable estimates of object number, (ii) rapidly eliminates the ancillary objects and (iii) detects and accurately estimates the trajectory of the original object of interest.
... In some applications, there may be targets that are spawned from other targets [14], for instance, a cell may undergo mitosis (cell division) [15], a skydiver may jump from an airplane, or a spacecraft may break up [16]. This possibility can be A. F. García-Fernández is with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, United Kingdom, and also with the ARIES Research Centre, Universidad Antonio de Nebrija, Madrid, Spain (angel.garcia-fernandez@liverpool.ac.uk). ...
... where n k|k is the number of potentially detected trees, A k|k is the set of global tree hypotheses, w a k|k is the weight of global tree hypothesis a = a 1 , ..., a n k|k , which contains indices to local tree hypotheses for each potentially detected tree and f i,a i k|k (·) is the Bernoulli density of the tree trajectory with local tree hypothesis a i . At each time step in each local hypothesis, we can associate more than one measurement to each Bernoulli tree, as each of its alive branches can generate one measurement, see (14). A global tree hypothesis a therefore indicates the subset of measurements that are associated to each Bernoulli tree at each time step. ...
This paper proposes a Poisson multi-Bernoulli mixture (PMBM) filter on the space of sets of tree trajectories for multiple target tracking with spawning targets. A tree trajectory contains all trajectory information of a target and its descendants, which appear due to the spawning process. Each tree contains a set of branches, where each branch has trajectory information of a target or one of the descendants and its genealogy. For the standard dynamic and measurement models with multi-Bernoulli spawning, the posterior is a PMBM density, with each Bernoulli having information on a potential tree trajectory. To enable a computationally efficient implementation, we derive an approximate PMBM filter in which each Bernoulli tree trajectory has multi-Bernoulli branches obtained by minimising the Kullback-Leibler divergence. The resulting filter improves tracking performance of state-of-the-art algorithms in a simulated scenario.
... Zhang [9] considered a more common case, that the targets were located between range profile sampling points (also referred to as range profile samples or samples hereafter), and developed a maximum likelihood (ML)-based algorithm to estimate angles of multiple unresolved targets. Isaac [10] and Nandakumaran [11] incorporated the detection of unresolved targets into tracking and proposed particle filter-based algorithms to improve detection performance, but the computational load was high, especially when trying to increase the number of particles to achieve higher performance. Glass [12] proposed the JMBP GLRT algorithm based on Zhang's model to detect unresolved Rayleigh targets. ...
... In addition, it is the magnitude of and instead of their value that rises in the presence of unresolved targets [1]. Therefore, it is reasonable to have the magnitude of and weighted by as the feature, which means the feature vector of the th sample at time is , (10) where and is calculated by ...
Detecting unresolved targets is very important for radars in their target tracking phase. For wideband radars, the unresolved target detection algorithm should be fast and adaptive to different bandwidths. To meet the requirements, a detection algorithm for wideband monopulse radars is proposed, which can detect unresolved targets for each range profile sampling points. The algorithm introduces the Gaussian mixture model and uses a priori information to achieve high performance while keeping a low computational load, adaptive to different bandwidths. A comparison between the proposed algorithm and the latest unresolved target detection algorithm Joint Multiple Bin Processing Generalized Likelihood Ratio Test (JMBP GLRT) is carried out by simulation. On Rayleigh distributed echoes, the detection probability of the proposed algorithm is at most 0.5456 higher than the JMBP GLRT for different signal-to-noise ratios (SNRs), while the computation time of the proposed algorithm is no more than two 10,000ths of the JMBP GLRT computation time. On bimodal distributed echoes, the detection probability of the proposed algorithm is at most 0.7933 higher than the JMBP GLRT for different angular separations of two unresolved targets, while the computation time of the proposed algorithm is no more than one 10,000th of the JMBP GLRT computation time. To evaluate the performance of the proposed algorithm in a real wideband radar, an experiment on field test measured data was carried out, in which the proposed algorithm was compared with Blair GLRT. The results show that the proposed algorithm produces a higher detection probability and lower false alarm rate, and completes detections on a range profile within 0.22 ms.
... In [18], nonparametric change detection algorithms were developed for detecting a change in the spatial distribution of alarmed sensors in large-scale sensor networks. In [19], the CUSUM algorithm was adopted for detecting spawning targets and was used jointly with particle filters to handle radar signal processing, data association , and target tracking simultaneously. In the context of cognitive radio for opportunistic spectrum access, the CUSUM algorithm was applied in [20] and [21] for detecting the return of primary users (i.e., the starting point of a busy period) in a given single channel. ...
... Extensions to is straightforward. The19) where and are the value functions for a given current action. From the cost structure, we obtain the following: ...
We consider the quickest detection of idle periods in multiple ON-OFF processes. At each time, only one process can be observed, and the observations are random realizations drawn from two different distributions depending on the current state (ON or OFF) of the chosen process. The objective is to catch an idle period in any of the ON-OFF processes as quickly as possible subject to a reliability constraint. We show that this problem presents a fresh twist to the classic problem of quickest change detection that considers only one stochastic process. A Bayesian formulation of the problem is developed for both infinite and finite number of processes based on the theory of partially observable Markov decision process (POMDP). While a general POMDP is PSPACE-hard, we show that the optimal decision rule has a simple threshold structure for the infinite case. For the finite case, basic properties of the optimal decision rule are established, and a low-complexity threshold policy is proposed which converges to the optimal decision rule for the infinite case as the number of processes increases. This problem finds applications in spectrum sensing in cognitive radio networks where a secondary user searches for idle channels in the spectrum.
... In [18], nonparametric change detection algorithms were developed for detecting a change in the spatial distribution of alarmed sensors in large-scale sensor networks. In [19], the CUSUM algorithm was adopted for detecting spawning targets and was used jointly with particle filters to handle radar signal processing, data association , and target tracking simultaneously. In the context of cognitive radio for opportunistic spectrum access, the CUSUM algorithm was applied in [20] and [21] for detecting the return of primary users (i.e., the starting point of a busy period) in a given single channel. ...
... Extensions to is straightforward. The19) where and are the value functions for a given current action. From the cost structure, we obtain the following: ...
A Bayesian formulation of quickest change detection in multiple on- off processes is obtained within a decision-theoretic framework. For geometrically distributed busy and idle times, we show that the op- timal joint design of channel switching and change detection has a simple threshold structure under a mild condition. Extensions to arbitrarily distributed busy and idle times, in particular, heavy tail distributions, are discussed. We show that this problem presents a fresh twist to the classic problem of quickest change detection that considers only one stochastic process. We demonstrate that the key to quickest change detection in multiple processes is to abandon the current process when its state is unlikely to change in the near future (as indicated by the measurements obtained so far) and seek oppor- tunities in a new process to avoid realizations of long busy periods. This problem arises in spectrum opportunity detection in cognitive radio networks where a secondary user searches for idle channels in the spectrum. Index Termsó Quickest change detection, heavy tail distribu- tion, spectrum opportunity detection, cognitive radio.
... This paper is an outgrowth of material presented in [11]. Some details of the particle filter used for tracking are reported in [12], and its application to detection of target spawn is given in [13]. ...
... The approach above makes explicit the particle filter used in [11], and is also given in [12]. The work in [13] uses an auxiliary particle filter, which seems to work better in that application. ...
Four techniques to successfully track two closely spaced and unresolved targets using monopulse radar measurements have been discussed, the quality of such tracking being a determinant of successful detection of target spawn. The paper explores statistical estimation techniques based on the maximum likelihood criterion and Gibbs sampling, and addresses concerns about the accuracy of the measurements delivered thereby. In particular, the Gibbs approach can deliver joint measurements (and the associated covariances) from both targets, and it is therefore natural to consider a joint filter. The ideas are compared, and among the various strategies discussed, a particle filter that operates directly on the monopulse measurements seems to be the best.
... Particle filtering techniques that have the advantage of providing computational tractability are applicable under most general circumstances since there is no assumption made on the form of the density. Some methods utilizing particle filter were introduced to detect and track two closely spaced targets [44,45], and they focused on constructing hypothesis testing for detection based on Akaike information criterion (AIC) and using particles to achieve state estimation [46,47]. ...
... Unlike the KF, which do not guarantee optimum solution, we employ the use of PF for target tracking with the measurement collaboration given in (3) [9]. Furthermore, we adopt the Markov chain Monte Carlo (MCMC)-based resampling method for estimating the more accurate posterior distribution of the target state, [10,11]. ...
In this paper, we investigate the tracking of the 3D moving target when multiple measurements are available. We propose a particle filter-based 3D target tracking algorithm with measurement of time difference of arrival (TDoA), frequency difference of arrival (FDoA), and angle of arrival (AoA). We analyze the performance of the proposed scheme by MATLAB simulations and show the efficiency of the particle filter for 3D moving target tracking with various types of measurements. Keywords: 3D mobility tracking, AoA, FDoA, TDoA, Particle filter
Unresolved targets detection is an important problem for monopulse radar to avoid false angular tracking. This article proposes an unresolved targets detection algorithm for the case where the direction-of-arrival (DOA) of one of the targets is known. The detector is derived using the generalized likelihood ratio test (GLRT) with some approximations. The simulation results show that significant improvement in detection performance is obtained by exploiting the a priori knowledge of the DOA. An application of the proposed algorithm to the noise jamming scenario is also presented.
Monopulse radar accurately determines the target direction. However, multiple target echoes and dense false signals lead to unresolved targets and false detections. This makes target tracking challenging. The aircraft is equipped with a towed radar active decoy at a fixed distance, forming the leader-follower configuration. This study treats the aircraft and the decoy as a single target in the Bayesian state model. Subsequently, constraints from the configuration are employed to jointly estimate their angles. Monopulse measurements from the leader-follower configuration are a mixture of potentially unresolved measurement, dense false signals and background clutter, and they are modeled as target area measurements, state-dependent clutter and state-independent clutter, respectively. Subsequently, these different measurement types are combined using a random finite set approach, enabling a holistic tracking solution. The proposed method is implemented by particle filtering. In the simulations, the proposed method was compared with other methods and found to be efficient in tracking unresolved aircraft and towed decoy. The results indicate that the performance of the method improves as the follower-to-leader ratio increases within a certain range, highlighting its suitability for practical applications.
This paper proposes a Poisson multi-Bernoulli mixture (PMBM) filter on the space of sets of tree trajectories for multiple target tracking with spawning targets. A tree trajectory contains all trajectory information of a target and its descendants, which appear due to the spawning process. Each tree contains a set of branches, where each branch has trajectory information of a target or one of the descendants and its genealogy. For the standard dynamic and measurement models with multi-Bernoulli spawning, the posterior is a PMBM density, with each Bernoulli having information on a potential tree trajectory. To enable a computationally efficient implementation, we derive an approximate PMBM filter in which each Bernoulli tree trajectory has multi-Bernoulli branches, obtained by minimising the Kullback-Leibler divergence. The resulting filter improves tracking performance of state-of-the-art algorithms in a simulated scenario.
Aiming at the problem of the parameter estimation of multiple unresolved targets within the radar beam, using the joint bin processing model, a method of jointly estimating the number and the position of the targets is proposed based on reversible jump Markov Chain Monte Carlo (RJ-MCMC). Reasonable assumptions of the prior distributions and Bayesian theory are adopted to obtain the posterior probability density function of the estimated parameters from the conditional likelihood function of the observation, and then the acceptance ratios of the birth, death and update moves are given. During the update move, a hybrid Metropolis-Hastings (MH) sampling algorithm is used to make a better exploration of the parameter space. The simulation results show that this new method outperforms the method of ML-MLD [11] proposed by X.Zhang for similar estimation accuracy is achieved while fewer sub-pulses are needed.
The Miniature Air Launched Decoy (MALD) flies cooperatively with a true target and construct unresolved targets within the radar beam. The unresolved targets and decoy result in echoes aliasing and observation merging, which causes serious measurement error and traditional track processing failure. Based on the analysis of jamming patterns and characteristics, the signal model of the unresolved targets is proposed. Combining jamming detection, echoes observation, and likelihood function characteristics, the proposed approach directly operates on the monopulse sum and difference returns and uses the particles propagating and evolving in the state space to obtain the conditional state probability density of the unresolved targets and decoy. This method effectively bypasses measurement extraction and parameter measurement, and realizes the joint tracking of the target and decoy. Furthermore, it provides the necessary conditions for target selection to the radar based on track and trajectory data. Simulation experiments illustrate the effectiveness of the proposed method.
A genetic resampling particle filter based on the fast mutation is proposed for removing the degeneracy phenomenon and alleviating the sample impoverishment problem in the particle filter. The algorithm chooses the fast Metropolis-Hastings (MH) moving as a mutation operator, and the traditional crossover and the traditional selection are combined with it to form a new resampling algorithm. The fast MH mutation moves the particles so that their stationary distribution is a target posterior density. The fast mutation can solve the divergence problem of the traditional mutation and produce the typical particle to reflect the features of a target density more quickly. The simulation results show that the genetic resampling based on the fast MH mutation can improve the variety of particles, remove the degeneracy phenomenon and reduce the tracking error.
The parameter estimation of the target and towed radar active decoy (TRAD) within the radar beam is the necessary condition for radar seeker to realize target selection and achieve the boresight steering adjust and accurate tracking. The "closely spaced" target and decoy make radar echoes aliased and the traditional method for parameter measurement and estimation fail. Based on the signal model, that the energy of single target will spill over to adjacent matched filter sampling points when the output of matched filter of radar seeker is sampled actually. Bayesian theory is adopted to obtain the posterior probability density function of estimated parameters from the conditional likelihood function of the observation and the prior distribution of unknown parameters in this paper. Then the Metropolis-Hastings (M-H) sampling algorithm belonged to Markov Chain Monte Carlo (MCMC) is utilized to estimate parameters of target and decoy jointly. At the same time, based on the jamming model and characteristics of TRAD, the M-H sampling is improved. The simulation results under different jamming conditions and dynamic attack scenarios illustrate the performance of proposed method.
In ballistic missile defense, the situation of unresolved targets is of great threaten to the tracking, identification and guidance of ground based radars, and some special efforts must be adopted to deal with it, among which the detection of the presence of unresolved targets is the precondition and basis of all the others. It is indicated by tests that, when multiple unresolved targets exist, the main lob's phase of LFM pulse compress output changes tempestuously, other than single target case where the phase changes tardily. The difference between those two cases above can provide a new way to detect the presence of unresolved targets. In this paper, the model of sample phase error is put forward, and its statistic characteristics under the condition of single target and multiple unresolved targets are derived. Based on what above, a general likelihood ratio test is developed to detect the presence of unresolved targets. Finally, the Monte Carlo simulation result identifies the validity of the models.
The modelling of angular estimation methods for unresolvable targets is mostly based on Swerling II type targets. In this case, however, frequency agility is usually used to ensure that the amplitudes of the targets for pulse-to-pulse are irrelevant. Considering that the target echoes observed by non-frequency agility radars are slowly fluctuant, the influence of targets' radar cross section (RCS) fluctuation on the direction finding of monopulse radars is investigated and an angular resolution approach for two slow-fluctuation unresolvable targets based on statistical characteristics of monopulse angular estimation is provided. Monte Carlo simulations confirm the validity of the proposed approach.
The parameter estimation and identity recognition of targets and decoys under the towed radar active decoy jamming are the precondition for a guided radar to realize precision strike. The unresolved target and decoy result in the radar echoes aliasing and lead the traditional parameter measurement method to fail. When the closely spaced target and decoy fall between two adjacent matched filter samples, the monopulse information from both of these samples can be used and is helpful for parameter estimation. The probability distribution of received echoes is obtained after utilizing the information contained in two consecutive matched filter sampling points. Then the particle swarm optimization method is used to estimate the parameters of targets and decoys. At the same time, based on the characteristic of jamming, the optimization is improved. With the estimation results the identity recognition of target and decoy is realized. Simulation results with different conditions illustrate the performance of the proposed method.
An innovative implementation platform based no structured application specific integrated circuit (ASIC) design method for digital receiver of monopulse tracking radar applications applying linear frequency modulation (LFM) signal is given in this paper. In order to meet the increased requirements of system miniaturization and low power consumption, system functions are thereby partitioned into software and hardware implementation way respectively, and the HW implemented functions verified in field programmable gate array (FPGA) are integrated in a structured ASIC chip. In the verification part, the experimental results prove the validity and practical applicability of the new platform.
A system-on-chip (SoC) architecture employed in the digital receiver for monopulse tracking radar applications applying linear frequency modulation (LFM) signals is proposed. The signal processing flow will then be decomposed into several time-division stages, and the corresponding system function will be achieved with software or hardware implementation way respectively. Thereafter, an optimized SoC architecture for digital receiver is illustrated, and the prototype structure in field programmable gate array (FPGA) is designed so that all the processing modules can be integrated in single FPGA chip. Finally, the experimental results prove the validity and engineering applicability of the proposed architecture.
The arithmetic idea of infrared sensor and radar tracking fusion is : getting solar by infrared image processing, time matching, synchronization fusion with radar data, updating object state by fusion data filtering. The fusion combines matching scores of individual obstacle recognition modules in order to improve the system's final decision. The obtained bimodal feature vector is used as input to an SVM-based classification scheme. We can found from computer simulation results that the tracking precision improved by information fusion, and mobility was responded in short time.
Chaff centroid jamming released by vessel makes the target indicated azimuth angle of radar seeker of anti-ship missile bias off the target. The failure to detect the presence of chaff centroid jamming is catastrophic to target tracking for the radar seeker. In this study, the authors take the azimuthal in-phase monopulse ratio of radar seeker as test statistic and detect the presence of chaff centroid jamming with the aid of GPS/INS. Probability density functions of the azimuthal in-phase monopulse ratio when jamming is absent and present are derived at first. Then the authors propose a method using the measured information of both GPS/INS and radar seeker to estimate the target off-boresight azimuth angle which is vital for detection threshold setting. Finally, the authors show the performance of the chaff centroid jamming detection by simulations and analyse the effects of some key jamming parameters on detection performance.
This work investigates the various criteria for track-to-track association/fusion (T2TA/F): likelihood ratios and distance criteria. Procedures to obtain the quantities needed by the LR criterion from the limited information available from the real world communication networks are developed. Algorithms for T2TA/F with heterogeneous sensors and investigation of several assignment algorithms for the T2TA problem are carried out. Procedures for simultaneous handling of continuous valued (kinematic and feature) states and discrete valued ones (attribute/classification) for an integrated approach to the Track Association and Fusion problem are presented.
A problem of optimal stopping is formulated and simple rules are proposed which are asymptotically optimal in an appropriate sense. The problem is of central importance in quality control and also has applications in reliability theory and other areas.
This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.
This paper analyses the recently suggested particle approach to filtering time series. We suggest that the algorithm is not robust to outliers for two reasons: the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution. Both problems are tackled in this paper. We believe we have largely solved the first problem and have reduced the order of magnitude of the second. In addition we introduce the idea of stratification into the particle filter which allows us to perform on-line Bayesian calculations about the parameters which index the models and maximum likelihood estimation. The new methods are illustrated by using a stochastic volatility model and a time series model of angles. Some key words: Filtering, Markov chain Monte Carlo, Particle filter, Simulation, SIR, State space. 1 1
For missing data problems, Tanner and Wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors. Their method of constructing the complete data sets is closely related to the Gibbs sampler. Both required iterations, and, similar to the EM algorithm, convergence can be slow. We introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights. In many applications this new procedure works very well without the need for iterations. Sensitivity analysis, influence analysis. and updating with new data can be performed cheaply. Bayesian prediction and model selection can also be incorporated. Examples taken from a wide range of applications are used for illustration.
Using the amplitude-conditioned Probability Density Function (PDF) for monopulse measurements, the Cramer-Rao Lower Bound (CRLB) is developed for any unbiased estimator of the Direction-of-Arrival (DOA). Maximum Likelihood (ML) and Method of Moments (MM) estimators of the DOA are developed along with estimates of the variances associated with the monopulse ratios and DOA estimate. The case of an unknown target amplitude is addressed by using the ML estimate of the target amplitude parameter formed from the sum-signal. Using simulation results, the performances of the ML and MM estimators of the DOA are compared with the standard monopulse ratio and the performance of the variance estimators is studied.
A monopulse radar is able to derive accurate angular measurements via intelligent processing of its sum and difference channel returns. Recently there have emerged techniques for angular estimation of several unresolved targets, meaning targets that are, in principle, merged within the same radar beam, can be extracted separately. The key is the joint exploitation of information in several range bins. Here we show the performance of this approach in a high-fidelity simulation: we observe considerable improvement in track RMSE, but little corresponding gain in track completeness. Coupled with a hidden Markov model on target number, however, the performance is impressive.
Hidden Markov models have become a widely used class of statistical models with applications in diverse areas such as communications engineering, bioinformatics, finance and many more. This book is a comprehensive treatment of inference for hidden Markov models, including both algorithms and statistical theory. Topics range from filtering and smoothing of the hidden Markov chain to parameter estimation, Bayesian methods and estimation of the number of states.
In a unified way the book covers both models with finite state spaces, which allow for exact algorithms for filtering, estimation etc. and models with continuous state spaces (also called state-space models) requiring approximate simulation-based algorithms that are also described in detail. Simulation in hidden Markov models is addressed in five different chapters that cover both Markov chain Monte Carlo and sequential Monte Carlo approaches. Many examples illustrate the algorithms and theory. The book also carefully treats Gaussian linear state-space models and their extensions and it contains a chapter on general Markov chain theory and probabilistic aspects of hidden Markov models.
This volume will suit anybody with an interest in inference for stochastic processes, and it will be useful for researchers and practitioners in areas such as statistics, signal processing, communications engineering, control theory, econometrics, finance and more. The algorithmic parts of the book do not require an advanced mathematical background, while the more theoretical parts require knowledge of probability theory at the measure-theoretical level.
Olivier Cappé is Researcher for the French National Center for Scientific Research (CNRS). He received the Ph.D. degree in 1993 from Ecole Nationale Supérieure des Télécommunications, Paris, France, where he is currently a Research Associate. Most of his current research concerns computational statistics and statistical learning.
Eric Moulines is Professor at Ecole Nationale Supérieure des Télécommunications (ENST), Paris, France. He graduated from Ecole Polytechnique, France, in 1984 and received the Ph.D. degree from ENST in 1990. He has authored more than 150 papers in applied probability, mathematical statistics and signal processing.
Tobias Rydén is Professor of Mathematical Statistics at Lund University, Sweden, where he also received his Ph.D. in 1993. His publications include papers ranging from statistical theory to algorithmic developments for hidden Markov models.
From the Publisher:
"Estimation with Applications to Tracking and Navigation treats the estimation of various quantities from inherently inaccurate remote observations. It explains state estimator design using a balanced combination of linear systems, probability, and statistics." "The authors provide a review of the necessary background mathematical techniques and offer an overview of the basic concepts in estimation. They then provide detailed treatments of all the major issues in estimation with a focus on applying these techniques to real systems." "Suitable for graduate engineering students and engineers working in remote sensors and tracking, Estimation with Applications to Tracking and Navigation provides expert coverage of this important area."--BOOK JACKET.
Monte Carlo methods are revolutionizing the on-line analysis of data in fields as diverse as financial modeling, target tracking and computer vision. These methods, appearing under the names of bootstrap filters, condensation, optimal Monte Carlo filters, particle filters and survival of the fittest, have made it possible to solve numerically many complex, non-standard problems that were previously intractable. This book presents the first comprehensive treatment of these techniques, including convergence results and applications to tracking, guidance, automated target recognition, aircraft navigation, robot navigation, econometrics, financial modeling, neural networks, optimal control, optimal filtering, communications, reinforcement learning, signal enhancement, model averaging and selection, computer vision, semiconductor design, population biology, dynamic Bayesian networks, and time series analysis. This will be of great value to students, researchers and practitioners, who have some basic knowledge of probability. Arnaud Doucet received the Ph. D. degree from the University of Paris-XI Orsay in 1997. From 1998 to 2000, he conducted research at the Signal Processing Group of Cambridge University, UK. He is currently an assistant professor at the Department of Electrical Engineering of Melbourne University, Australia. His research interests include Bayesian statistics, dynamic models and Monte Carlo methods. Nando de Freitas obtained a Ph.D. degree in information engineering from Cambridge University in 1999. He is presently a research associate with the artificial intelligence group of the University of California at Berkeley. His main research interests are in Bayesian statistics and the application of on-line and batch Monte Carlo methods to machine learning. Neil Gordon obtained a Ph.D. in Statistics from Imperial College, University of London in 1993. He is with the Pattern and Information Processing group at the Defence Evaluation and Research Agency in the United Kingdom. His research interests are in time series, statistical data analysis, and pattern recognition with a particular emphasis on target tracking and missile guidance.
Bayesian methods provide a rigorous general framework for dynamic state estimation problems. We describe the nonlinear/non-Gaussian tracking problem and its optimal Bayesian solution. Since the optimal solution is intractable, several different approximation strategies are then described. These approaches include the extended Kalman filter and particle filters. For a particular problem, if the assumptions of the Kalman filter hold, then no other algorithm can out-perform it. However, in a variety of real scenarios, the assumptions do not hold and approximate techniques must be employed. The extended Kalman filter approximates the models used for the dynamics and measurement process, in order to be able to approximate the probability density by a Gaussian. Particle filtering approximates the density directly as a finite number of samples. A number of different types of particle filter exist and some have been shown to outperform others when used for particular applications. However, when designing a particle filter for a particular application, it is the choice of importance density that is critical. These notes are of a tutorial nature and so, to facilitate easy implementation, 'pseudo-code' for algorithms are included at relevant points.
An algorithm, the bootstrap filter, is proposed for implementing
recursive Bayesian filters. The required density of the state vector is
represented as a set of random samples, which are updated and propagated
by the algorithm. The method is not restricted by assumptions of
linearity or Gaussian noise: it may be applied to any state transition
or measurement model. A simulation example of the bearings only tracking
problem is presented. This simulation includes schemes for improving the
efficiency of the basic algorithm. For this example, the performance of
the bootstrap filter is greatly superior to the standard extended Kalman
filter
For the case of a single resolved target, monopulse-based 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.
In this paper, we present a new result that can be used for detection purposes. We show that when estimating the a posteriori probability density of a possible signal in noise by means of a particle filter, the output of the filter, i.e., the unnormalized weights, can be used to approximately construct the likelihood ratio, which arises in many different detection schemes.
If several closely spaced targets fall within the same radar beam and between two adjacent matched filter samples in range, monopulse information from both of these samples can and should be used for estimation, both of angle and of range (i.e., estimation of the range to sub-bin accuracy). Similarly, if several closely spaced targets fall within the same radar beam and among three matched filter samples in range, monopulse information from all of these samples should be used for the estimation of the angles and ranges of these targets. Here, a model is established, and a maximum likelihood (ML) extractor is developed. The limits of the number of targets that can be estimated are given for both case A, where the targets are in a beam and in a range "slot" between the centers of two adjacent resolution cells (that is, from detections in two adjacent matched filter samples), and case B, where the targets are in two or more adjacent slots (among three or more adjacent samples). A minimum description length (MDL) criterion is used to detect the number of targets between the matched filter samples, and simulations support the theory.
Increasingly, for many application areas, it is becoming important
to include elements of nonlinearity and non-Gaussianity in order to
model accurately the underlying dynamics of a physical system. Moreover,
it is typically crucial to process data on-line as it arrives, both from
the point of view of storage costs as well as for rapid adaptation to
changing signal characteristics. In this paper, we review both optimal
and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking
problems, with a focus on particle filters. Particle filters are
sequential Monte Carlo methods based on point mass (or "particle")
representations of probability densities, which can be applied to any
state-space model and which generalize the traditional Kalman filtering
methods. Several variants of the particle filter such as SIR, ASIR, and
RPF are introduced within a generic framework of the sequential
importance sampling (SIS) algorithm. These are discussed and compared
with the standard EKF through an illustrative example
Most present-day radar systems use monopulse techniques to extract angular measurements of sunbeam accuracy. The familiar "monopulse ratio" is a very effective means to derive the angle of a single target within a radar beam. For the simultaneous estimation of the angles of two closely-spaced targets, a modification on the monopulse ratio was derived in (Blair and Pearce, 2001), while (Sinha et al., 2002) presented a maximum likelihood (ML) technique via numerical search. In this paper it is shown that the ML solution can in fact be found explicitly, and the numerical search of ((Sinha et al., 2002) is unnecessary. However, the ML solution requires the signal to noise ratio (SNR) for each target to be known, and hence we generalize it so it requires only the relative SNR. Several versions of expectation maximization (EM) joint angle estimators are also derived, these differing in the degree to which prior information on SNR and on beam pattern are assumed. The performances of the different direction-of-arrival (DOA) estimators for unresolved targets are studied via Monte Carlo, and it is found that most have similar performance: this is remarkable since the use of prior information (SNR, relative SNR, beam pattern) varies widely between them. There is, however, considerable performance variability as a function of the two targets' off-boresight angles. A simple combined technique that fuses the results from different approaches is thus proposed, and it performs well uniformly.
In a scenario of closely spaced targets special attention has to
be paid to radar signal processing. We present an advanced processing
technique, which uses the maximum likelihood (ML) criterion to extract
from a monopulse radar separate angle measurements for unresolved
targets. This processing results in a significant improvement, in terms
of measurement error standard deviations, over angle estimators using
the monopulse ratio. Algorithms are developed for Swerling I as well as
Swerling III models of radar cross section (RCS) fluctuations. The
accuracy of the results is compared with the Cramer Rao lower bound
(CRLB) and also to the monopulse ratio technique. A novel technique to
detect the presence of two unresolved targets is also discussed. The
performance of the ML estimator was evaluated in a benchmark scenario of
closely spaced targets - closer than half power beamwidth of a monopulse
radar. The interacting multiple model probabilistic data association
(IMMPDA) track estimator was used in conjunction with the ML angle
extractor
This paper provides for new approaches to the processing of
unresolved measurements as two direction-of-arrival (DOA) measurements
for tracking closely spaced targets rather than the conventional single
DOA measurement of the centroid. The measurements of the two-closely
spaced targets are merged when the target echoes are not resolved in
angle, range, or radial velocity (i.e., Doppler processing). The
conditional Cramer Rao lower bound (CRLB) is developed for the DOA
estimation of two unresolved Rayleigh targets using a standard monopulse
radar. Then the modified CRLB is used to give insight into the boresight
pointing for monopulse DOA estimation of two unresolved targets.
Monopulse processing is considered for DOA estimation of two unresolved
Rayleigh targets with known or estimated relative radar cross section
(RCS). The performance of the DOA estimator is studied via Monte Carlo
simulations and compared with the modified CRLB
The conditional probability density function (pdf) is developed
for each monopulse measurement of a Rayleigh target by conditioning the
pdf of the complex monopulse ratio on the measured amplitude of the sum
signal. The conditional pdf is used to develop the conditional
Cramer-Rao Lower Bound (CRLB) for any unbiased estimator of the
direction-of-arrival (DOA). Conditional maximum likelihood (CML) and
conditional method of moments (CMM) estimators of the DOA are developed
along with estimates of the variances associated with the monopulse
ratio and DOA estimate. Using simulation results, the performances of
the CML and CMM estimators of the DOA are compared with the performance
of standard monopulse ratio and the performances of the variance
estimators are also studied
When the returns from two or more targets interfere (i.e., the
signals are not resolved in the frequency or time domains) in a
monopulse radar system, the direction-of-arrival (DOA) estimate
indicated by the monopulse ratio can wander far beyond the angular
separation of the targets. Generalized maximum likelihood (GML)
detection of the presence of unresolved Rayleigh targets is developed
with probability density functions (pdfs) conditioned on the measured
amplitude of the target echoes. The Neyman-Pearson detection algorithm
uses both the in-phase and quadrature portions of the monopulse ratio
and requires no a priori knowledge of the signal-to-noise ratio (SNR) or
DOA of either target. Receiver operating characteristic (ROC) curves are
given along with simulation results that illustrate the performance and
application of the algorithm