Graham W. Pulford

Joint target tracking and classification using radar and ESM sensors

Bayesian target classification methods using radar and electronic support measure (ESM) data are considered. A joint treatment of target tracking and target classification problems is introduced. First, a method for target classification using radar data and class-dependent kinematic models is presented. Second, a target classification method using ESM data is presented. Then, a Bayesian radar and ESM data fusion algorithm, referred to as direct identity fusion (DIF), for target classification is presented. This algorithm exploits the dependence of target state on target class via the use of class dependent kinematic models but fails to exploit the dependence of target class on target state. We then introduce a method, referred to as joint tracking and classification (JTC), for treating target tracking and classification problems jointly, by exploiting the dependence of target class on target state via flight-envelope-dependent classes and the dependence of target state on target class via class dependent kinematic models. A two-dimensional example demonstrates the relative merits of these methods. It is shown that, while the incorporation of the two-way dependence between target state and class (i.e., JTC) promises some benefits over the method that incorporates only a one-way dependence (i.e., DIF), there are severe filter implementation difficulties for the former. The results also demonstrate that the fusion of information from radar and ESM sensors via the DIF approach results in improvements over classification methods based on either of the individual sensors.

Block decision feedback equalization

A natural generalization of the conventional decision feedback equalizer (DFE) based on block processing and maximum a posteriori decisions is presented. This block DFE is indexed by two parameters depending on the block length p and the number of decisions q >

A multipath data association tracker for over-the-horizon radar

A new algorithm, multipath probabilistic data association (MPDA), for initiation and tracking in over-the-horizon radar (OTHR) is described. MPDA is capable of exploiting multipath target signatures arising from discrete propagation modes that are resolvable by the radar. Nonlinear measurement models exhibiting multipath target signatures in azimuth, slant range, and Doppler are used. Tracking is performed in ground coordinates and therefore depends on the provision of estimates of virtual ionospheric heights to achieve coordinate registration. Although the propagation mode characteristics are assumed to be known, their correspondence with the detections is not required to be known. A target existence model is included for automatic track maintenance. Numerical simulations for four resolvable propagation modes are presented that demonstrate the ability of the technique to initiate and maintain track at probabilities of detection of 0.4 per mode in clutter densities for which conventional probabilistic data association (PDA) has a high probability of track loss, and suffers from track bias. A nearest neighbor version of MPDA is also presented.

Probabilistic data association for systems with multiple simultaneous measurements

We consider the problem of estimating the state of a discrete-time, linear stochastic system whose observation process consists of a finite set of known, linear measurement models. The correspondence of the measurements with the models is unknown. Assuming all the measurements relate to the state, we derive a recursive algorithm, the multiple simultaneous measurement filter (MSMF), which provides a fixed-complexity, sub-optimal solution to this problem. The MSMF is a generalization of the discrete-time Kalman filter and reduces to the latter when a single measurement model applies. Simulations are provided that demonstrate the superior performance of the MSMF over the probabilistic data association filter for tracking a single target in an environment where up to three measurements of the state are received simultaneously.

Multihypothesis Viterbi Data Association: Algorithm Development and Assessment

Two algorithms for tracking in clutter, based on the Viterbi algorithm are presented: single-target Viterbi data association (ST-VDA) and multihypothesis VDA (MH-VDA). MH-VDA is designed specifically for multiple-target tracking (MTT), although ST-VDA still achieves good performance on MTT problems. The basic philosophy of both methods is to set up an optimisation problem for the sequence of measurement-to-target associations rather than directly seeking the target state estimates. The joint optimisation problem for the data association sequence is decomposed into a sequence of scalar optimisation problems by means of an approximate forward dynamic programming recursion to which the Viterbi algorithm is applicable. Once the data association problem is solved, the target state estimates can be retrieved by backtracking. The operation of the algorithms is easily visualised as a search on a trellis for the optimal path. For ST-VDA, nodes in the trellis correspond to measurements. For MH-VDA, nodes correspond to multitarget data association hypotheses. Conventional measurement gating is extended to work within this context. Results from simulations that compare the performance of ST-VDA and MH-VDA with four, standard, zero-scan-back tracking approaches are given. The performance assessment includes metrics for track loss and track swaps in a multiple crossing target context. The Viterbi data association (VDA) algorithms are shown to outperform the alternative algorithms. In particular the ST-VDA is found to have the best track swap performance, while MH-VDA has the lowest track loss figure. Average state estimation errors for both VDA algorithms are only about 10% larger than a Kalman filter with known data associations. While both variants of VDA are essentially batch processing approaches, the simulation results indicate that the algorithms can be implemented with a fixed processing lag of only a few scans without significant loss in performance.

An expectation-maximisation tracker for multiple observations of a single target in clutter

We consider the estimation the state of a discrete-time, linear stochastic system whose observation process consists of a finite set of known, linear measurement models with additive white noise. Unlike conventional data fusion and tracking problems, the correspondence between the measurements and the models is assumed to be unknown. In addition, some of the measurements may be false alarms which convey no information about the state of the system. The expectation maximisation (EM) algorithm is applied as a MAP estimator of the sequence of measurement-to-model associations, with state sequence estimates obtained through fixed-interval Kalman smoothing conditioned on the association sequence. Each pass uses a Viterbi algorithm to provide updated data association estimates. The new technique is called expectation maximisation data association and represents an optimal fusion of dynamic programming and Kalman smoothing for data association.

When has a decision-directed equalizer converged?

Adaptive decision-directed equalizers can become locked for long periods onto incorrect equilibria. A test involving data available at the equalizer output is proposed for determining whether an equilibrium is correct or not, up to a fixed overall delay. If an independent sequence of random variables taking values +or-1 is the input to a finite impulse response filter, and the output of the filter is passed through a slicer, then the slicer output is uncorrelated if and only if it is a delayed version of the filter input. An analogous result for M-ary rather than binary data is outlined. >

Analysis of a Nonlinear Least Squares Procedure Used in Global Positioning Systems

Iterated least squares (ILS) is a widely used parameter estimation algorithm for nonlinear least squares problems. The ILS estimation error covariance is usually written (GTR-1G)-1, where G is the Jacobian matrix at the solution and R is the noise covariance. Using a first-order expansion of the “gain matrix” in ILS, we provide a rigorous justification for the covariance formula. The analysis includes uncertainty in the initial estimate and is capable of modeling the transient performance of the algorithm. Given convergence, the usual ILS covariance is obtained asymptotically. The analysis makes use of matrix differential calculus to obtain the first differential of the ILS gain matrix, and includes, as a special case, the R=I case, where the gain matrix is the pseudoinverse of the Jacobian matrix. The results are harnessed to obtain a sensitivity analysis of the ILS algorithm to additional random parametric variations. The analysis is then applied to a Global Positioning System problem to characterize the effect of ephemeris errors on the ILS position estimates. Results from a comparative Monte Carlo simulation demonstrate the approachs effectiveness.

Multi-Target Viterbi Data Association

Viterbi data association (VDA) is an approximate likelihood-based approach for solving the data association problem arising in target tracking. Here the VDA algorithm, previously developed for single target tracking in clutter, is extended to the multiple target case. The multi-target VDA (MT-VDA) formulation retains the main features of the VDA approach while generalizing the notion of a trellis to cover multiple target to measurement data association hypotheses. The cost metric for the problem is developed in terms of the multi-target association likelihood. Application of the Viterbi algorithm results in an efficient pruning strategy that can be represented graphically on a trellis. The MT-VDA algorithm is compared with a number of conventional zero-scan-back approaches including JPDA and is shown to provide significantly lower track loss, track swaps and tracking errors on a 1-D test scenario. The trellis merge depth is also investigated

Manoeuvring target tracking with the IMM-VDA algorithm

This paper describes an algorithm for tracking a manoeuvring target in heavy clutter and/or with a low probability of detection. It is known that when tracking under such adverse conditions multi-scan tracking algorithms, such as the multi-hypothesis tracker (MHT), provide improved performance over single-scan trackers. This paper uses a computationally efficient algorithm for multi-scan target tracking based on the Viterbi algorithm, known as the Viterbi data association (VDA) algorithm. In this paper it is shown how the VDA algorithm can be combined with the well-known interacting multiple model (IMM) method to create an effective multi-scan manoeuvring target tracker. The performance of the IMM-VDA algorithm is shown by simulation. It is compared to another manoeuvring target algorithm based on the VDA approach which uses a hard decision manoeuvre detection scheme. In addition, it is compared to a single-scan tracking algorithm based on the probabilistic data association method.