The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis-Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.

Riemann manifold Langevin and Hamiltonian Monte Carlo methods

Sharing of the frequency bands between radar and communication systems has attracted substantial attention, as it can avoid under-utilization of otherwise permanently allocated spectral resources, thus improving efficiency. Further, there is increasing demand for radar and communication systems that share the hardware platform as well as the frequency band, as this not only decongests the spectrum, but also benefits both sensing and signaling operations via the full cooperation between both functionalities. Nevertheless, the success of spectrum and hardware sharing between radar and communication systems critically depends on high-quality joint radar and communication designs. In the first part of this paper, we overview the research progress in the areas of radar-communication coexistence and dual-functional radar-communication (DFRC) systems, with particular emphasis on application scenarios and technical approaches. In the second part, we propose a novel transceiver architecture and frame structure for a DFRC base station (BS) operating in the millimeter wave (mmWave) band, using the hybrid analog-digital (HAD) beamforming technique. We assume that the BS is serving a multi-antenna user equipment (UE) over a mmWave channel, and at the same time it actively detects targets. The targets also play the role of scatterers for the communication signal. In that framework, we propose a novel scheme for joint target search and communication channel estimation, which relies on omni-directional pilot signals generated by the HAD structure. Given a fully-digital communication precoder and a desired radar transmit beampattern, we propose to design the analog and digital precoders under non-convex constant-modulus (CM) and power constraints, such that the BS can formulate narrow beams towards all the targets, while pre-equalizing the impact of the communication channel. Furthermore, we design a HAD receiver that can simultaneously process signals from the UE and echo waves from the targets. By tracking the angular variation of the targets, we show that it is possible to recover the target echoes and mitigate the resulting interference to the UE signals, even when the radar and communication signals share the same signal-to-noise ratio (SNR). The feasibility and efficiency of the proposed approaches in realizing DFRC are verified via numerical simulations. Finally, the paper concludes with an overview of the open problems in the research field of communication and radar spectrum sharing (CRSS).

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Joint Radar and Communication Design: Applications, State-of-the-Art, and the Road Ahead

Nonlinear filters can provide estimation accuracy that is vastly superior to extended Kalman filters for some important practical applications. We compare several types of nonlinear filters, including: particle filters (PFs), unscented Kalman filters, extended Kalman filters, batch filters and exact recursive filters. The key practical issue in nonlinear filtering is computational complexity, which is often called "the curse of dimensionality". It has been asserted that PFs avoid the curse of dimensionality, but this is generally incorrect. Well-designed PFs with good proposal densities sometimes avoid the curse of dimensionality, but not otherwise. Future research in nonlinear filtering will exploit recent progress in quasi-Monte Carlo algorithms (rather than boring old Monte Carlo methods), as well as ideas borrowed from physics (e.g., dimensional interpolation) and new mesh-free adjoint methods for solving PDEs. This tutorial was written for normal engineers, who do not have nonlinear filters for breakfast.

Nonlinear filters: beyond the Kalman filter

Massive MIMO is a reality—What is next?: Five promising research directions for antenna arrays

Preface Bibliography 1. Interpolation. Introduction The Taylor expansion The finite Taylor series with the remainder term Interpolation by polynomials The remainder of Lagrangian interpolation formula Equidistant interpolation Local and global interpolation Interpolation by central differences Interpolation around the midpoint of the range The Laguerre polynomials Binomial expansions The decisive integral transform Binomial expansions of the hypergeometric type Recurrence relations The Laplace transform The Stirling expansion Operations with the Stirling functions An integral transform of the Fourier type Recurrence relations associated with the Stirling series Interpolation of the Fourier transform The general integral transform associated with the Stirling series Interpolation of the Bessel functions 2. Harmonic Analysis. Introduction The Fourier series for differentiable functions The remainder of the finite Fourier expansion Functions of higher differentiability An alternative method of estimation The Gibbs oscillations of the finite Fourier series The method of the Green's function Non-differentiable functions Dirac's delta function Smoothing of the Gibbs oscillations by Fejer's method The remainder of the arithmetic mean method Differentiation of the Fourier series The method of the sigma factors Local smoothing by integration Smoothing of the Gibbs oscillations by the sigma method Expansion of the delta function The triangular pulse Extension of the class of expandable functions Asymptotic relations for the sigma factors The method of trigonometric interpolation Error bounds for the trigonometric interpolation method Relation between equidistant trigonometric and polynomial interpolations The Fourier series in the curve fitting 3. Matrix Calculus. Introduction Rectangular matrices The basic rules of matrix calculus Principal axis transformation of a symmetric matrix Decomposition of a symmetric matrix Self-adjoint systems Arbitrary n x m systems Solvability of the general n x m system The fundamental decomposition theorem The natural inverse of a matrix General analysis of linear systems Error analysis of linear systems Classification of linear systems Solution of incomplete systems Over-determined systems The method of orthogonalisation The use of over-determined systems The method of successive orthogonalisation The bilinear identity Minimum property of the smallest eigenvalue 4. The Function Space. Introduction The viewpoint of pure and applied mathematics The language of geometry Metrical spaces of infinitely many dimensions The function as a vector The differential operator as a matrix The length of a vector The scalar product of two vectors The closeness of the algebraic approximation The adjoint operator The bilinear identity The extended Green's identity The adjoint boundary conditions Incomplete systems Over-determined systems Compatibility under inhomogeneous boundary conditions Green's identity in the realm of partial differential operators The fundamental field operations of vector analysis Solution of incomplete systems 5. The Green's Function. Introduction The role of the adjoint equation The role of Green's identity The delta function -- The existence of the Green's function Inhomogeneous boundary conditions The Green's vector Self-adjoint systems The calculus of variations The canonical equations of Hamilton The Hamiltonisation of partial operators The reciprocity theorem Self-adjoint problems Symmetry of the Green's function Reciprocity of the Green's vector The superposition principle of linear operators The Green's function in the realm of ordinary differential operators The change of boundary conditions The remainder of the Taylor series The remainder of the Lagrangian interpolation formula

Linear Differential Operators. By C. Lanczos. Pp. 580. 80s. 1961. (Van Nostrand, London)

We have invented a new theory of exact particle flow for nonlinear filters. This
generalizes our theory of particle flow that is already many orders of magnitude faster than
standard particle filters and which is several orders of magnitude more accurate than the
extended Kalman filter for difficult nonlinear problems. The new theory generalizes our recent
log-homotopy particle flow filters in three ways: (1) the particle flow corresponds to the exact flow
of the conditional probability density; (2) roughly speaking, the old theory was based on
incompressible flow (like subsonic flight in air), whereas the new theory allows compressible flow
(like supersonic flight in air); (3) the old theory suffers from obstruction of particle flow as well as
singularities in the equations for flow, whereas the new theory has no obstructions and no
singularities. Moreover, our basic filter theory is a radical departure from all other particle filters in
three ways: (a) we do not use any proposal density; (b) we never resample; and (c) we compute
Bayes' rule by particle flow rather than as a point wise multiplication.

Exact particle flow for nonlinear filters

MIMO communication is theoretically superior to conventional comm. under certain conditions, and MIMO comm. 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 tradeoffs 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.

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.

We derive and test a new nonlinear filter that implements Bayes' rule using an ODE
rather than with a pointwise multiplication of two functions. This avoids one of the
fundamental and well known problems in particle filters, namely "particle collapse" as
a result of Bayes' rule. We use a log-homotopy to construct this ODE. Our new
algorithm is vastly superior to the classic particle filter, and we do not use any
proposal density supplied by an EKF or UKF or other outside source. This paper was
written for normal engineers, who do not have homotopy for breakfast.

Nonlinear filters with log-homotopy

We describe a new nonlinear filter that is vastly superior to the classic particle filter.
In particular, the computational complexity of the new filter is many orders of
magnitude less than the classic particle filter with optimal estimation accuracy for
problems with dimension greater than 2 or 3. We consider nonlinear estimation
problems with dimensions varying from 1 to 20 that are smooth and fully coupled (i.e.
dense not sparse). The new filter implements Bayes' rule using particle flow rather
than with a pointwise multiplication of two functions; this avoids one of the
fundamental and well known problems in particle filters, namely "particle collapse" as
a result of Bayes' rule. We use a log-homotopy to derive the ODE that describes
particle flow. This paper was written for normal engineers, who do not have
homotopy for breakfast.

Jim Huang

Papers

Exact particle flow for nonlinear filters

MIMO radar: Snake oil or good idea?

MIMO radar: Snake oil or good idea?

Nonlinear filters with log-homotopy

Particle flow for nonlinear filters with log-homotopy