Finding the best set of K paths through a trellis withapplication to multitarget tracking

A solution is presented to the problem of finding the best set of K completely unmerged paths through a trellis with M _i⩾K states at depth i in the trellis, i=0, 1, 2, . . ., N. Here, `best set' means that the sum of the metrics of all K paths in the set is minimized, and `completely unmerged' means that no two paths pass through a common state. The solution involves using the Viterbi algorithm on an expanded trellis. This result is then used to separate the tracks of K targets optimally in a simplified model of a multitarget radar system. The model includes measurement errors and false alarms, but it does not include the effects of missing detections or merged measurements     

Convergence bounds of an SMI/Gram-Schmidt canceler in colored noise

The performance of the sampled matrix inversion (SMI) adaptive algorithm in colored noise is investigated using the Gram-Schmidt (GS) canceler as an analysis tool. Lower and upper bounds of average convergence are derived, indicating that average convergence slows as the input time samples become correlated. When the input samples are uncorrelated, the fastest SMI algorithm convergence occurs. When the input samples are correlated then the convergence bounds depend on the number of channels N, the number of samples per channels K , and the eigenvalues associated with K×K correlation matrix of the samples in a given channel. This matrix is assumed identical for all channels     

Adaptive canceler and pulse compressor interactions

Performance results for the sidelobe level of a compressed pulse that has been preprocessed through an adaptive canceler are obtained. The adaptive canceler is implemented using the sampled matrix inversion algorithm. Because of finite sampling, the quiescent compressed pulse sidelobe levels are degraded due to the preprocessing of the main channel input data stream (the uncompressed pulse) through an adaptive canceler. It is shown that if N is the number of input canceler channels (main and auxiliaries) and K is the number of independent samples per channel, then K/N can be significantly greater than one in order to retain sidelobes that are close to the original quiescent sidelobe level (with no adaptive canceler). Also it is shown that the maximum level of degradation is independent of whether pulse compression occurs before or after the adaptive canceler if the uncompressed pulse is completely contained within the K samples that are used to calculate the canceler weights. This same analysis can be used to predict the canceler noise power level that is induced by having the desired signal present in the canceler weight calculation     

Tracking multiple targets with correlated measurements andmaneuvers

The problem of tracking N targets with correlation in both measurement and maneuver statistics is solved by transforming to a coordinate frame in which the N targets are decoupled. For the case of N identical targets, the decoupling is shown to coincide with a transformation to a set of nested center-of-mass coordinates. Absolute and differential tracking accuracies are compared with suboptimal results to show the improvement that is achieved by properly exploiting the correlation between targets     

Optimal speckle reduction in polarimetric SAR imagery

Speckle is a major cause of degradation in synthetic aperture radar (SAR) imagery. With the availability of fully polarimetric SAR data, it is possible to use the three complex elements (HH, HV, VV) of the polarimetric scattering matrix to reduce speckle. The optimal method for combining the elements of the scattering matrix to minimize image speckle is derived, and the solution is shown to be a polarimetric whitening filter (PWF). A simulation of spatially correlated, K-distributed, fully polarimetric clutter is then used to compare the PWF with other, suboptimal speckle-reduction methods. Target detection performance of the PWF, span, and single-channel |HH|² detectors is compared with that of the optimal polarimetric detector (OPD). A novel, constant-false-alarm-rate (CFAR) detector (the adaptive PWF) is as a simple alternative to the OPD for detecting targets in clutter. This algorithm estimates the polarization covariance of the clutter, uses the covariance to construct the minimum-speckle image, and then tests for the presence of a target. An exact theoretical analysis of the adaptive PWF is presented; the algorithm is shown to have detection performance comparable with that of the OPD     

CFAR data fusion center with inhomogeneous receivers

Detection systems with distributed sensors and data fusion are increasingly used by surveillance systems. A system formed by N inhomogeneous constant false alarm rate (CFAR) detectors (cell-averaging (CA) and ordered statistic (OS) CFAR detectors) is studied. A recursive formulation of an algorithm that permits a fixed level of false alarms in the data fusion center is presented, to set the optimum individual threshold levels in the CFAR receivers and the optimum `K out of N' decision rule in order to maximize the total probability of detection. The algorithm also considers receivers of different quality or with different communication channel qualities connecting them with the fusion center. This procedure has been applied to several hypothetical networks with distributed CA-CFAR and OS-CFAR receivers and for Rayleigh targets and interference, and it was seen that in general the fusion decision OR rule is not always the best     

Detection performance in K-distributed and correlatedRayleigh clutters

Performance prediction for a detection system employing noncoherent integration is carried out for a chi-square family of fluctuating targets in K-distributed clutter plus noise. The detection performance for Swerling 11 targets in the K-distributed clutter plus noise is compared with that in exponentially correlated Rayleigh clutter. The results show that the performance prediction based on N pulses integrated in clutter plus noise using the K-distributed clutter model may be approximately equivalent to that using the exponentially correlated Rayleigh-distributed clutter model     

Efficient algorithms for finding the K best paths througha trellis

A set of algorithms is presented for finding the best set of K mutually exclusive paths through a trellis of N nodes, with worst-case computation time bounded by N³log n for a fixed-precision computation. The algorithms are based on a transformation of the K-path trellis problem into an equivalent minimum-cost network flow (MCNF) problem. The approach allows the application of efficient MCNF algorithms, which can obtain optimal solutions orders of magnitude faster than the algorithm proposed by J.K. Wolf et al. (1989). The resulting algorithms extend the practicality of the trellis formulation (in terms of required computations) to multiobject tracking problems with much larger numbers of targets and false alarms. A response by Wolf et al. is included     

Multiple target angle tracking using sensor array outputs

The use of the output of an array of sensors to track multiple independently moving targets is reported. The output of each sensor in the array is the sum of signals received from each of the targets. The results of direction-of-arrival estimation by eigenvalue analysis are extended to derive a recursive procedure based on a matrix quadratic equation. The solution of this matrix quadratic equation is used to provide updated target positions. A linear approximation method for estimating the solution of the matrix equation is presented. The algorithm is demonstrated by the simulated tracking of two targets. The main advantage of the algorithm is that a closed-form solution for updating the target angle estimates has been obtained. Also, its application is straightforward, and the data association problem due to uncertainty in the origin of the measurements is avoided. However, it requires the inversion of an N×N as well as other linear operations, so that the computational burden becomes substantial as N becomes very large     

Multiple target tracking using products of position measurements

The continued development of the symmetric measurement equation (SME) filter for track maintenance in multiple target tracking (MTT) is considered, focusing on the case in which the SMEs are generated by forming sums of products of the original position measurements. The SME filter is developed for the case of N targets whose motions consist of random perturbations about constant-velocity trajectories. It is assumed that measurements of x-coordinate positions are available, and that the number of measurements is equal to the number of targets. Various analytical properties of the SME filter are studied. It is shown that under a very weak condition, the estimation error equation is locally exponentially stable. The performance of the SME filter is investigated by comparing it with an optimal (minimum-variance) estimator and by generating a computer simulation in the six-target case     

Frequency diversity in millimeter wave satellite communications

Dual-band frequency diversity seems to be an effective fade countermeasure to rain-induced attenuation in satellite communications above 20 GHz. This method, particularly suited for satellites operating in two frequency bands, typically K_a band and C or K_u band, achieves very low levels of outage probability especially where the rain fades are severe. A theoretical analysis of a frequency diversity system is performed to evaluate the amount of reserve capacity needed to counteract fading in a satellite network. The problems that arise in implementing adaptive fade countermeasures because of the dynamic characteristics of fading are discussed, and the results of the simulations performed using the attenuation time series at 11.6 GHz, measured with the Sirio satellite throughout four years, are presented     

Target tracking in maneuver-centered coordinates

Aircraft targets normally maneuver on circular paths, which has led to tracking filters based on circular turns. A coordinate system to track circular maneuvers with a simple Kalman filter is introduced. This system is a polar coordinate system located at the center of the maneuver. It leads to a tracking filter with range, angle, and angular velocity in the state vector. Simulation results are presented, showing that the algorithm displays improved performance over methods based on constant x-y acceleration when tracking circular turns     

A parallel square-root algorithm for modified extended Kalmanfilter

A parallel square-root algorithm and its systolic array implementation are proposed for performing modified extended Kalman filtering (MEKF). The proposed parallel square-root algorithm is designed based on the singular value decomposition (SVD) and the Faddeev algorithm, and a very large scale integration (VLSI) systolic array architecture is developed for its implementation. Compared to other square root Kalman filtering algorithms, the proposed method is more numerically stable. The VLSI architecture described has good parallel and pipelining characteristics in applying to the MEKF and achieves higher efficiency. For n-dimensional state vector estimations, the proposed architecture consists of O(2n²) processing elements and uses O ((s+17)n) time-steps for a complete iteration at each instant, in contrast to the complexity of O((s+6) n³) time-steps for a sequential implementation, where s≈log n     

Low-altitude target detection by coastline operated marine radar

Relevant to a Richian family of fluctuating targets with a composite background of sea-plus-land clutter, the performance prediction of a radar operating in near-coastal regions is elucidated by assuming noncoherent integration of the pulses. Considering the dominance of land clutter, a modified K-distributed statistic is indicated for the overall clutter envelope; and the corresponding probability of false alarm and probability of detection are deduced for fixed threshold detection (s) based on N pulses integrated in the presence of the sea-plus-land clutter and the noise. Even when the target offers a dominant scattered echo, the worst situations of the land clutter affecting the detection performance are indicated     

An optimal reduced-order stochastic observer-estimator

An optimal reduced-order observer-estimator (filter) is developed which can provide a full-dimensional vector of state estimates for systems where the dimension of the measurement vector is smaller than that of the state vector and none of the measurements are noise free. The reduced-order filter consists of two subfilters each of which provides a subset of the optimal estimate. A two-step L-K transformation is employed to minimize the estimate error variance of each subfilter. The optimal reduced-order filter developed is computationally efficient     

Collapsing losses of peak signal integrators

Collapsing losses are computed for systems in which the peak return of K samples of noise plus one sample of signal-plus-noise are integrated over N looks. The statistical approach, collapsing losses, and an application are described. The peak integrator is found to have substantially lower collapsing losses than conventional systems in which the average, not the peak, is integrated     

Bearings-only tracking of maneuvering targets using abatch-recursive estimator

We present a new batch-recursive estimator for tracking maneuvering targets from bearings-only measurements in clutter (i.e., for low signal-to-noise ratio (SNR) targets), Standard recursive estimators like the extended Kalman Iter (EKF) suffer from poor convergence and erratic behavior due to the lack of initial target range information, On the other hand, batch estimators cannot handle target maneuvers. In order to rectify these shortcomings, we combine the batch maximum likelihood-probabilistic data association (ML-PDA) estimator with the recursive interacting multiple model (IMM) estimator with probabilistic data association (PDA) to result in better track initialization as well as track maintenance results in the presence of clutter. It is also demonstrated how the batch-recursive estimator can be used for adaptive decisions for ownship maneuvers based on the target state estimation to enhance the target observability. The tracking algorithm is shown to be effective for targets with 8 dB SNR     

Distribution of general noncentral positive definite quadratic formin K-dimensions

The authors present a series solution using Hermite polynomials to the long-standing problem of computing the probability P that positive definite noncentral quadratic form d(x) of a Gaussian random vector x∈R satisfies d( x)⩽r² for any given r∈R. This problem has wide applications in radar, tracking, air traffic control, etc. The fast-converging series solution presented is very accurate and can be performed rapidly using the recursion relations for Hermite polynomials     

Adaptive Estimation for a System with Unknown Measu rement Bias

An adaptive state estimator for passive underwater tracking of maneuvering targets is developed. The state estimator is designed specifically for a system containing unknown or randomly switching biased measurements. In modeling the stochastic system, it is assumed that the bias sequence dynamics can be modeled by a semi-Markov process. By incorporating the semi-Markovian concept into a Bayesian estimation technique, an estimator consisting of a bank of parallel, adaptively weighted, Kalman filters has been developed. Despite the large and randomly varying measurement biases, the proposed estimator, provides an accurate estimate of the system states.     

Estimating the Doppler centroid of SAR data

After reviewing frequency-domain techniques for estimating the Doppler centroid of synthetic-aperture radar (SAR) data, the author describes a time-domain method and highlights its advantages. In particular, a nonlinear time-domain algorithm called the sign-Doppler estimator (SDE) is shown to have attractive properties. An evaluation based on an existing SEASAT processor is reported. The time-domain algorithms are shown to be extremely efficient with respect to requirements on calculations and memory, and hence they are well suited to real-time systems where the Doppler estimation is based on raw SAR data. For offline processors where the Doppler estimation is performed on processed data, which removes the problem of partial coverage of bright targets, the ΔE estimator and the CDE (correlation Doppler estimator) algorithm give similar performance. However, for nonhomogeneous scenes it is found that the nonlinear SDE algorithm, which estimates the Doppler-shift on the basis of data signs alone, gives superior performance