No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Detecting and Tracking Separating Objects Using Direct Monopulse Measurements
Abstract
A significant complication to the ballistic missile tracking problem is the tendency for target "spawn": one ballistic object becomes two, several or several hundred, of which only one or a few are of concern (warheads), while uninteresting objects such as chaff and boost-rockets comprise the rest. To discern which is which takes time; and during that time it remains vital to maintain surveillance on all objects. Unfortunately these objects are closely-spaced (CSOs), and detection/tracking for CSOs is notoriously difficult. One approach is to use radar waveforms of such high resolution that all objects are "resolved" by a sharp ambiguity function peak. However, not all radar systems have the required bandwidth or operational modes, and inclusion of those modes in new radars would impact the cost. It appears that with unresolved targets there is little that can be done to improve tracking until the targets become resolved. Nonetheless, recent work with advanced statistical modeling of monopulse radar returns has questioned the need for such high resolution: signal processing techniques can "see" inside a previously-closed resolution cell in a similar way that a mis-formed Hubble mirror can be corrected to see clearly.In this presentation we proffer two approaches. In the first, we discuss the maximum-likelihood approach: it turns out that it can be shown that up to 5 targets can be resolved from between two matched filter samples, rather than the one-per- sample case that might otherwise be thought. We also will present an even more advanced statistical approach known as particle filtering. The standard tracking approach is to work with detection "hits"; that is, to separate the signal processing and tracking. The particle filter is a more natural integrated approach, and we have found tracking errors reduced by 50% or more by its use along with significant increase in speed of detection of multiple target spawn.
Forward-looking imaging plays a significant role in navigation, self-landing, etc. In this chapter, an improved monopulse forward-looking imaging algorithm is presented. Firstly, the angle of the target is estimated by monopulse angle measurement. Secondly, the radar return energy is placed to the azimuth bin indicated by the estimate angle. Then, upon completion of beam scan, a high-resolution image is gained. Finally, the mask technique is used to make the target's boundary clear. Simulation results validate the effectiveness of this algorithm.
In anti-ship terminal guidance radar tracking system, the target echo is always corrupted by the sea clutter. When sea clutter is large at high sea state, the signal -to-noise ratio (SNR) of the target echo is usually relatively low as a result of the effect of sea clutter. The precision of angle error extraction in radar tracking servo is low, while the fluctuating error of angle measure is large in low SNR region. The radar antenna has to adjust itself very frequently to keep tracking its target. While it leads that the radar can not always hold on tracking the target steadily. To improve the precision of angle error extraction and steady off-boresight angle measure of the target, a novel angle measure method based on particle filter is proposed, which achieves off-boresight angle estimation by approximating the posterior probability diversity with particle set, and the steps of algorithm are presented. Finally the simulation experiment confirms the valid of the proposed algorithm.
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.
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