Xin Zhang

Dynamic Cramer-Rao bound for target tracking in clutter

Recently, there have been several new results for an old topic, the Cramer-Rao lower bound (CRLB). Specifically, it has been shown that for a wide class of parameter estimation problems (e.g. for objects with deterministic dynamics) the matrix CRLB, with both measurement origin uncertainty (i.e., in the presence of false alarms or random clutter) and measurement noise, is simply that without measurement origin uncertainty times a scalar information reduction factor (IRF). Conversely, there has arisen a neat expression for the CRLB for state estimation of a stochastic dynamic nonlinear system (i.e., objects with a stochastic motion); but this is only valid without measurement origin uncertainty. The present paper can be considered a marriage of the two topics: the clever Riccati-like form from the latter is preserved, but it includes the IRF from the former. The effects of plant and observation dynamics on the CRLB are explored. Further, the CRLB is compared via simulation to two common target tracking algorithms, the probabilistic data association filter (PDAF) and the multiframe (N-D) assignment algorithm.

Detection and Localization of Multiple Unresolved Extended Targets via Monopulse Radar Signal Processing

When the size of targets is comparable to the range resolution of monopulse radars, these targets should be considered as extended rather than point targets. If several closely-spaced targets fall within the same radar beam and between adjacent matched filter samples in range, the full monopulse information from all of these samples can and should be used to resolve these targets, i.e., estimate the number of targets and their respective angles-of-arrival and ranges. To detect and localize multiple unresolved extended targets, we establish a model for monopulse radar returns from extended objects, and present a maximum likelihood estimator (MLE) to localize the targets. Rissanens minimum description length (MDL) criterion will be used to decide the number of existing extended objects. We also derive the upper limit on the number of targets and their scattering centers that can be resolved, and we provide necessary conditions for these targets to be uniquely identified. We compare the new extended target monopulse processing scheme with previously developed point-target monopulse techniques in the simulations.

The Multitarget Monopulse CRLB for Matched Filter Samples

It has recently been found that by jointly processing multiple (sum, azimuth-, and elevation-difference) matched filter samples, it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cramer-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimators efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLBs implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulses length, five targets can be isolated between two matched filter samples given that the targets signal-to-noise ratios (SNRs) are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of oversampling are given.

Monopulse radar detection and localization of multiple targets via joint multiple-bin processing

If several closely-spaced targets fall between two 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). Here, a model is established and a maximum likelihood (ML) extractor is developed. A limit of the number of targets that can be estimated is established, and indeed it will be shown that the new extractor can detect up to five targets in a range extent of one resolution cell, meaning from detections in two adjacent matched filter samples. A minimum description length (MDL) criterion is used to detect the number of targets between the two matched filter samples. Simulations are presented to show the performance of the new extractor.

A particle filter for tracking two closely spaced objects using monopulse radar channel signals

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.

Detection and localization of multiple unresolved extended objects via monopulse radar signal processing

With high resolution radars, realistic objects should be considered as extended rather than point targets. If several closely-spaced targets fall within the same radar beam and between adjacent matched filter samples in range, the full monopulse information from all of these samples can and should be used for estimation, both of angle and of range (with the range to sub-bin accuracy). To detect and localize multiple unresolved extended targets, we establish a model for monopulse radar returns from extended objects, and use a maximum likelihood estimator to localize the targets. Rissanens minimum description length (MDL) will be used to decide the number of existing extended objects. We compare the new extended target monopulse processing scheme with previously developed point target monopulse techniques to show the improvement in terms of the estimation of target locations, the detection of the number of existing targets, and the tracking performance with a multiple hypothesis tracker (MHT) using the output from the proposed extended target monopulse processor.

Simplification of the Dynamic Cramér-Rao Bound for Target Tracking in Clutter

A simplified dynamic Cramer-Rao lower bound (CRLB) is derived for the general case of target tracking in clutter with non-IID measurement noise components, discussed recently in.

Tracking of two closely spaced objects using monopulse measurements

The ultimate goal of this paper is to track two closely spaced and unresolved targets using monopulse radar measurements, the quality of such tracking being a determinant of successful detection of target spawn. It 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 amongst the various strategies discussed, a particle filter that operates directly on the monopulse measurements is especially promising.

IR pixel size optimization from a tracking perspective

This paper considers the optimal resolution cell (pixel) size in detection and tracking IR targets. Using refined resolution can help localizing the position of the targets precisely. However, along with a smaller resolution cell the signal power in each resolution cell becomes lower, because a point target is recorded as a blur according to the point spread function (PSF). Meanwhile, since the noise power is proportional to the area of the pixel, the noise is also lower. On the other hand, using coarse resolution (which is the result of opting for a high signal power in the resolution cell) renders less accurate target position estimates together with higher noise power. That is, as the pixel size changes there is a trade-off in terms of detection performance versus estimation accuracy. We submit that the only defensible way to rationalize this is from system level concerns: what is best for tracking? We will first look at the initial state estimation of a constant velocity target. Relationships between the Cramer-Rao lower bound for the initial state estimation and the resolution cell size will be established. Then, from a general target tracking perspective, the pixel-size effects on the probability of detection and the target location centroiding accuracy will be analyzed.

Tracking considerations in selection of radar waveform given range measurements

Previously, in a sequence of papers, there has been considerable analysis of the effects of waveforms on tracking; but these have always assumed that measurements both of range and of range-rate are deliverable. Many radar systems use what may be termed “conventional” processing, in which measurements only of range are available, and for which range-rate information must be inferred. Thus, in this paper, results are extended to that case. As an example result, we show that to allow for the bias to the observed range measurement via a linear function of the range-rate results in a considerable loss versus what would be possible with more accurate accounting.