Fred Daum

MIMO radar: Snake oil or good idea?

MIMO communication is theoretically superior to conventional communication under certain conditions, and MIMO communication also appears to be practical and cost-effective in the real world for some applications. It is natural to suppose that the same is true for MIMO radar, but the situation is not so clear. Researchers claim many advantages of MIMO radar relative to phased array radars (e.g., better detection performance, better angular resolution, better angular measurement accuracy, improved robustness against RFI, ECM, multipath, etc.). We will evaluate such assertions from a system engineering viewpoint. In particular, there are serious trade-offs of MIMO vs. phased array radars relative to cost, system complexity, and risk considering numerous real world effects that are not included in most theoretical analyses. Moreover, in many cases one can achieve essentially the same radar system improvement with phased array radars using simpler, less expensive, and less risky algorithms. We evaluate roughly a dozen asserted advantages of MIMO radar relative to phased arrays.

Particle degeneracy: root cause and solution

We have solved the well known and important problem of particle degeneracy for particle filters. Our filter is roughly seven orders of magnitude faster than standard particle filters for the same estimation accuracy. The new filter is four orders of magnitude faster per particle, and it requires roughly three orders of magnitude fewer particles to achieve the same accuracy as a standard particle filter. Typically we beat the EKF or UKF accuracy by approximately two orders of magnitude for difficult nonlinear problems.

Particle flow with non-zero diffusion for nonlinear filters

We derive several new algorithms for particle flow with non-zero diffusion corresponding to Bayes’ rule. This is unlike all of our previous particle flows, which assumed zero diffusion for the flow corresponding to Bayes’ rule. We emphasize, however, that all of our particle flows have always assumed non-zero diffusion for the dynamical model of the evolution of the state vector in time. Our new algorithm is simple and fast, and it has an especially nice intuitive formula, which is the same as Newton’s method to solve the maximum likelihood estimation (MLE) problem (but for each particle rather than only the MLE), and it is also the same as the extended Kalman filter for the special case of Gaussian densities (but for each particle rather than just the point estimate). All of these new flows apply to arbitrary multimodal densities with smooth nowhere vanishing non-Gaussian densities.

Zero curvature particle flow for nonlinear filters

We derive a new algorithm for computing Bayes’ rule using particle flow that has zero curvature. The flow is computed by solving a vector Riccati equation exactly in closed form rather than solving a PDE, with a significant reduction in computational complexity. Our theory is valid for any smooth nowhere vanishing probability densities, including highly multimodal non-Gaussian densities. We show that this new flow is similar to the extended Kalman filter in the special case of nonlinear measurements with Gaussian noise. We also outline more general particle flows, including: constant curvature, geodesic flow, non-constant curvature, piece-wise constant curvature, etc.

Seventeen dubious methods to approximate the gradient for nonlinear filters with particle flow

We have investigated more than 17 distinct methods to approximate the gradient of the loghomotopy for nonlinear filters. This is a challenging problem because the data are given as function values at random points in high dimensional space. This general problem is important in optimization, financial engineering, quantum chemistry, chemistry, physics and engineering. The best general method that we have developed so far uses a simple idea borrowed from geology combined with a fast approximate k-NN algorithm. Extensive numerical experiments for five classes of problems shows that we get excellent performance.

Particle flow with non-zero diffusion for nonlinear filters, Bayesian decisions and transport

We derive a new algorithm for particle flow with non-zero diffusion corresponding to Bayes’ rule, and we report the results of Monte Carlo simulations which show that the new filter is an order of magnitude more accurate than the extended Kalman filter for a difficult nonlinear filter problem. Our new algorithm is simple and fast to compute, and it has an especially nice intuitive formula, which is the same as Newton’s method to solve the maximum likelihood estimation (MLE) problem (but for each particle rather than only the MLE), and it is also the same as the extended Kalman filter for the special case of Gaussian densities (but for each particle rather than just the point estimate). All of these particle flows apply to arbitrary multimodal densities with smooth nowhere vanishing non- Gaussian densities.

Fourier transform particle flow for nonlinear filters

We derive five new algorithms to design particle flow for nonlinear filters using the Fourier transform of the PDE that determines the flow of particles corresponding to Bayes’ rule. This exploits the fact that our PDE is linear with constant coefficients. We also use variance reduction and explicit stabilization to enhance robustness of the filter. Our new filter works for arbitrary smooth nowhere vanishing probability densities.

Particle flow inspired by Knothe-Rosenblatt transport for nonlinear filters

We derive a new algorithm for particle flow corresponding to Bayes’ rule that was inspired by Knothe- Rosenblatt transport, which is well known in transport theory. We emphasize that our flow is not Knothe- Rosenblatt transport, but rather it is a completely different algorithm for particle flow. In particular, we pick a nearly upper triangular Jacobian matrix, but the meaning of the word “Jacobian” as used here is completely different than used in Knothe-Rosenblatt transport.

Small curvature particle flow for nonlinear filters

We derive five new particle flow algorithms for nonlinear filters based on the small curvature approximation inspired by fluid dynamics. We find it extremely interesting that this physically motivated approximation generalizes two of our previous exact flow algorithms, namely incompressible flow and Gaussian flow. We derive a new algorithm to compute the inverse of the sum of two linear differential operators using a second homotopy, similar to Feynmans perturbation theory for quantum electrodynamics as well as Gromovs h-principle.

A Cramér-Rao Bound for Multiple Target Tracking

This paper describes a new theoretical bound on performance for tracking in a dense multiple target environment with clutter, false alarms, missed detections, target maneuvers, etc. It is a Cramer-Rao bound on the estimation error covariance matrix. In contrast to the recent bound reported in [1], the bound described in this paper requires no Monte Carlo simulation. The new bound is based on Kamens formulation of the multiple target tracking (MTT) problem as a nonlinear estimation problem (see [3]-[4]). Kamens model of MTT eliminates the combinatorial aspect of the problem, and replaces it with an analytical problem, which allows the use of the Cramer-Rao bound ([8]-[9]).