Direction-of-arrival estimation in applications with multipath and few snapshots

This paper analyzes two direction-of-arrival (DOA) estimation algorithms used in the presence of multipath propagation and with very few snapshots. The conditional maximum likelihood (CML) algorithm and the method of direction estimation (MODE) are discussed. The estimates provided by these algorithms are shown to coincide for large number of snapshots or large signal-to-noise ratio. Necessary and sufficient conditions are derived for the algorithms to yield unique estimates. It is shown that their uniqueness conditions coincide with the minimal uniqueness condition on the array, that is independent of the algorithm used (if the array does not satisfy this minimal condition, no DOA estimation method can give unique estimates). Numerical examples are presented to demonstrate the theoretical results.The work of A. Nehorai and D. Starer was supported by the Air Force Office of Scientific Research under Grant No. AFOSR-90-0164 and by the Office of Naval Research under Grant No. N00014-91-J-1298.     

MEG forward problem formulation using equivalent surface current densities

We present a formulation for the magnetoencephalography (MEG) forward problem with a layered head model. Traditionally the magnetic field is computed based on the electric potential on the interfaces between the layers. We propose to express the effect of the volumetric currents in terms of an equivalent surface current density on each interface, and obtain the magnetic field based on them. The boundary elements method is used to compute the equivalent current density and the magnetic field for a realistic head geometry. We present numerical results showing that the MEG forward problem is solved correctly with this formulation, and compare it with the performance of the traditional formulation. We conclude that the traditional formulation generally performs better, but still the new formulation is useful in certain situations.     

EEG/MEC error bounds for a static dipole source with a realistichead model

We derive Cramer-Rao bounds (CRBs) on the errors of estimating the parameters (location and moment) of a static current dipole source using data from electro-encephalography (EEG), magneto-encephalography (MEG), or the combined EEG/MEG modality. We use a realistic head model based on knowledge of surfaces separating tissues of different conductivities obtained from magnetic resonance (MR) or computer tomography (CT) imaging systems. The electric potentials and magnetic field components at the respective sensors are functions of the source parameters through integral equations. These potentials and field are formulated for solving them by the boundary or the finite element method (BEM or FEM) with a weighted residuals technique. We present a unified framework for the measurements computed by these methods that enables the derivation of the bounds. The resulting bounds may be used, for instance, to choose the best configuration of the sensors for a given patient and region of expected source location. Numerical results are used to demonstrate an application for showing expected accuracies in estimating the source parameters as a function of its position in the brain, based on real EEG/MEG system and MR or CT images     

Array response kernels for EEG and MEG in multilayer ellipsoidal geometry

We present forward modeling solutions in the form of array response kernels for electroencephalography (EEG) and magnetoencephalography (MEG), assuming that a multilayer ellipsoidal geometry approximates the anatomy of the head and a dipole current models the source. The use of an ellipsoidal geometry is useful in cases for which incorporating the anisotropy of the head is important but a better model cannot be defined. The structure of our forward solutions facilitates the analysis of the inverse problem by factoring the lead field into a product of the current dipole source and a kernel containing the information corresponding to the head geometry and location of the source and sensors. This factorization allows the inverse problem to be approached as an explicit function of just the location parameters, which reduces the complexity of the estimation solution search. Our forward solutions have the potential of facilitating the solution of the inverse problem, as they provide algebraic representations suitable for numerical implementation. The applicability of our models is illustrated with numerical examples on real EEG/MEG data of N20 responses. Our results show that the residual data after modeling the N20 response using a dipole for the source and an ellipsoidal geometry for the head is in average lower than the residual remaining when a spherical geometry is used for the same estimated dipole.     

Mutual coupling of two collocated orthogonally oriented circular thin-wire loops

Coupling between two collocated orthogonal circular thin-wire loops is analyzed in this paper. Two coupled integral equations for the loop currents are derived and their solution in general form is found in terms of Fourier series. An analytical expression for currents induced through the mutual coupling is obtained for the case when all loop current harmonics higher than first can be ignored. It is found that strong coupling can exist for all loop current harmonics, except for the fundamental. It is also found that coupling for orthogonal collocated loop antennas depends on the relative locations of the loop terminals.     

A Biologically Inspired Compound-Eye Detector Array—Part II: Statistical Performance Analysis

This is the second part of our paper. In this paper, we propose, model, and analyze the performance of a detector array for localizing far-field particle-emitting sources, which is inspired by but generalizes the compound eye of insects. The array consists of multiple eyelets, each having a conical module with a lens on its top and an inner subarray containing multiple particle detectors. Using a parametric measurement model introduced for the array in Part I, in this part we analytically and numerically analyze the statistical performance of the array. First, we compute the statistical Cramer-Rao bounds (CRBs) on the errors in estimating the direction of arrival of the incident particles; then we derive a lower bound on the mean-square angular error (MSAE) of source localization for any specific array configuration; thirdly, we consider two source-direction estimators, the maximum likelihood estimator (MLE) and the weighted direction estimator (WDE), and analyze their MSAE performance. In the numerical examples, we quantitatively compare the performance of the proposed array with the biological compound eye; show the array performance as a function of the array configuration variables; optimally design the array configuration; illustrate that the MLE asymptotically attains the performance bound, whereas the WDE is nearly optimal for sufficiently large SNR; and analyze the hardware efficiency by comparing the two MSAE bounds. Potential applications of this work include artificial vision in medicine or robotics, astronomy assisted, security, and particle communications.     

An asymptotically efficient ARMA estimator based on sample covariances

An asymptotically efficient autoregressive moving-average (ARMA) spectral estimator is presented, based on the sample covariances of observed time series. The estimate of the autoregressive (AR) part is shown to be identical to the optimal instrumental variable (IV) estimator in [7] although derived here using a different approach. The moving-average (MA) spectral parameter estimate is new.     

Effect of Head Shape Variations Among Individuals on the EEG/MEG Forward and Inverse Problems

We study the effect of the head shape variations on the EEG/magnetoencephalography (MEG) forward and inverse problems. We build a random head model such that each sample represents the head shape of a different individual and solve the forward problem assuming this random head model, using a polynomial chaos expansion. The random solution of the forward problem is then used to quantify the effect of the geometry when the inverse problem is solved with a standard head model. The results derived with this approach are valid for a continuous family of head models, rather than just for a set of cases. The random model consists of three random surfaces that define layers of different electric conductivity, and we built an example based on a set of 30 deterministic models from adults. Our results show that for a dipolar source model, the effect of the head shape variations on the EEG/MEG inverse problem due to the random head model is slightly larger than the effect of the electronic noise present in the sensors. The variations in the EEG inverse problem solutions are due to the variations in the shape of the volume conductor, while the variations in the MEG inverse problem solutions, larger than the EEG ones, are caused mainly by the variations of the absolute position of the sources in a coordinate system based on anatomical landmarks, in which the magnetometers have a fixed position.     

Magnetoencephalography with diversely oriented and multicomponentsensors

To locate endocranial current sources, a magnetoencephalography (MEG) system usually measures the magnetic field at many points around the skull with an array of radial sensors. Despite the success of using radial components of the field, the authors show that using nonradial components may potentially also be beneficial. They demonstrate some benefits of using diversely oriented and multicomponent sensors to measure the nonradial components. A framework is provided for analyzing the accuracy of a system that estimates the location and direction of a current dipole inside a spherical skull. The framework is then used to determine the effect on accuracy of varying the orientations of sensors in an array and, as a consequence, it is found that the radial orientations commonly used in practice are suboptimal for locating dipoles near the array's center. A diversely oriented array that improves performance is presented. The authors show how a single multicomponent sensor can locate a dipole, and derive a simple algorithm for locating a dipole near the sensor     

A meshless method for solving the EEG forward problem

We present a numerical method to solve the quasistatic Maxwell equations and compute the electroencephalography (EEG) forward problem solution. More generally, we develop a computationally efficient method to obtain the electric potential distribution generated by a source of electric activity inside a three-dimensional body of arbitrary shape and layers of different electric conductivities. The method needs only a set of nodes on the surface and inside the head, but not a mesh connecting the nodes. This represents an advantage over traditional methods like boundary elements or finite elements since the generation of the mesh is typically computationally intensive. The performance of the proposed method is compared with the boundary element method (BEM) by numerically solving some EEG forward problems examples. For a large number of nodes and the same precision, our method has lower computational load than BEM due to a faster convergence rate and to the sparsity of the linear system to be solved.