Given a measurement value that has been corrupted with additive multivariate Gaussian noise with nonzero correlation in its covariance matrix, this paper derives two new Taylor series approximations to estimating an unbiased mean and a consistent covariance matrix. The conversion produces mean and covariance estimates that are more consistent than other expansions in the literature when the covariance matrix is not a diagonal matrix.

The paper first derives expressions for the MSE and bias of a mismatched linear filter applied to an arbitrary trajectory and describes how such expressions can be used for process noise selection. The performance of such algorithms when applied to the selection of process noise terms for a 1D sinusoidal dynamic model and for a 2D weaving dynamic model is then considered. Previously, the only algorithm specifically designed for process noise selection of 1D sinusoidal models is a 1995 paper by Sudano. Optimal gains are compared with suboptimal techniques in 1D and 2D, including with Sudano’s algorithm. It is shown that the algorithm optimizing over the explicit MSE outperforms the other techniques both in minimizing the MSE and also in guaranteeing that the peak error is never worse than “connecting the dots” between measurements.

When using long-range sensors to track targets located on the Earth's surface, such as ships at sea or cars on land, the literature often discusses the use of a local 2D tangent-plane coordinate system. This paper derives and compares a number of methods of propagating target motion over short and long distances on an ellipsoidal Earth under particle filter or tangent-plane Gaussian approximations.

The first two moments of a converted bistatic range (delay)-direction-cosine-range-rate measurement are derived taking into account the Doppler information and maximum bounds on the target velocity in orthogonal unobservable directions. Cubature integration is used to very efficiently evaluate the necessary multivariate integrals allowing correlations between measurement components to be easily taken into account. Such single-point track initialization is useful in many tracking algorithms, such as some variants of the joint integrated probabilistic data association filter (JIPDAF).

A considerable number of papers in the literature provide solutions for fixed-gain filters, such as the α- β filter, that minimize the mean squared error under various dynamic models. This note demonstrates that those specialized models are unnecessary as the tools to explicitly find optimal fixed gains for arbitrary observable systems are already in the literature. The results produced by this method agree with published results for specialized algorithms. The solution of this letter also obviates the need for special “aeolotropic” filter design. Code implementing the algorithm and simulations is available in Matlab on IEEEXplore.

Recent research has focussed on the use of compressive sensing for detection in both monostatic and bistatic radar systems. However, such work has not considered many of the practical aspects of implementing a detection algorithm. This letter looks at how the geometry of pulse chasing in a bistatic radar system can be used to reduce the dimensionality of the estimation problem for compressive sensing by only considering the region illuminated by the transmitter that is visible to the receiver. Specifically, a method of determining the extent of the transmit beam in the receiver's local$u-v$coordinate system is presented. Given this extent, the compressive sensing dictionary matrix can be formed over the discretized region.

Minimum mean squared error estimates generally are not optimal in terms of a common track error statistic used in tracking benchmarks, namely a form of the Mean Optimal Sub-pattern Assignment (MOSPA) metric. We derive an explicit solution for the MOSPA-optimal estimates for two scalar targets. We also generalize previous work on permutation variant and invariant PDF decompositions by Blom and Bloem (avoiding the use of measure theory), demonstrating how the means of these PDFs may be used to approximate minimum MOSPA estimates. These methods based upon PDF manipulation may be used with general PDFs for an arbitrary number of targets having states of arbitrary dimensionality. The results are also applicable within the context of channel estimation.

Optimizing over a variant of the Mean Optimal Subpattern Assignment (MOSPA) metric is equivalent to optimizing over the track accuracy statistic often used in target tracking benchmarks. Past work has shown how obtaining a Minimum MOSPA (MMOSPA) estimate for target locations from a Probability Density Function (PDF) outperforms more traditional methods (e.g. maximum likelihood (ML) or Minimum Mean Squared Error (MMSE) estimates) with regard to track accuracy metrics. In this paper, we derive an approximation to the MMOSPA estimator in the two-target case, which is generally very complicated, based on minimizing a Bhattacharyya-like bound. It has a particularly nice form for Gaussian mixtures. We thence compare the new estimator to that obtained from using the MMSE and the optimal MMOSPA estimators.

The information filter is a form of the Kalman filter that, in many of its realizations, allows optimal, unbiased, recursive state estimation without an initial state estimate. We review a number of forms of the information filter. We then derive the coefficients for the sliding-window Kalman finite impulse response (FIR) smoother (also known as a receding or moving horizon Kalman FIR smoother) starting from the equations for the information filter. The resulting FIR smoother has a simple, recursive form for calculating the coefficients, allowing them to be calculated with$O(N)$complexity versus the$O(N^{2})$to$O(N^{3})$complexity of previous approaches, where$N$is the length of the batch. It also allows for a control input, something not present in previous algorithms. This method is only limited in the assumption that the state transition matrix is invertible, which, however, is satisfied in most practical problems.

We present a correction to Algorithm 515 [Buckles and Lybanon 1977].