Joseph Shmuel Picard

Network Localization with Biased Range Measurements

We discuss the challenge of node localization in ad-hoc wireless networks using range measurements and a few anchors. The measurements are assumed to be noisy with unknown bias. We propose an iterative Maximum Likelihood algorithm that is statistically efficient under these conditions. The algorithm is related to phase retrieval and can be viewed as an adaptation of the Gerchberg-Saxton Iterations alternating projection procedure. As a byproduct the algorithm estimates the bias of the range measurements. Closed form expressions for the Fisher Information Matrix are derived. Finally, simulations are used to corroborate the analysis.

Bounds on the Number of Identifiable Outliers in Source Localization by Linear Programming

Precise localization have attracted considerable interest in the engineering literature. Most publications consider small measurement errors. In this work we discuss localization in the presence of outliers, where several measurements are severely corrupted while sufficient other measurements are reasonably precise. It is known that maximum likelihood or least squares provide poor results under these conditions. On the other hand, robust regression can successfully handle up to 50% outliers but is associated with high complexity. Using the ¿ 1 norm as the penalty function provides some immunity from outliers and can be solved efficiently with linear programming methods. We use linear equations to describe the localization problem and then we apply the ¿ 1 norm and linear programming to detect the outliers and avoid the wild measurements in the final solution. Our main contribution is an exploitation of recent results in the field of sparse representation to obtain bounds on the number of detectable outliers. The theory is corroborated by simulations and by real data.

Direction finding of multiple emitters by spatial sparsity and linear programming

The problem of multiple emitters direction finding using an array of sensors is addressed. We describe a sparsity-based covariance-matrix fitting method. The procedure consists of finding a sparse representation of the sample covariance matrix, using an over-complete basis obtained from array manifold samples. Sparsity is encouraged by an ℓ1-norm penalty function. The penalty function is minimized efficiently by linear programming. The proposed method is simple enough to provide useful insight and it does not require the identification of the signal and noise subspaces. Therefore, the method does not rely on a good estimate of the number of emitters. Some of the approach properties are super-resolution, robustness to noise, robustness to emitter correlation, and no sensitivity to initialization. Special emphasis is given to uncorrelated sources and uniform linear arrays.

Localization of multiple emitters by spatial sparsity methods in the presence of fading channels

The problem of multiple emitters geolocation using sensor arrays is addressed, in the case of fading channels. A sparsity-based covariance-matrix fitting method is described. The procedure consists of finding a sparse representation of the sample covariance matrices obtained at the arrays, by representing each matrix by an over-complete basis. Sparsity is encouraged by an ℓ1-norm based penalty function. The penalty function is minimized by semi-definite programming. The proposed method provides useful insight and it does not require the identification of the signal and noise subspaces. Therefore, the method does not rely on a good estimate of the number of emitters. Some of the approach properties are super-resolution, robustness to noise, robustness to emitter correlation, no sensitivity to initialization and no need for synchronizing the arrays. Special emphasis is given to uncorrelated sources and uniform linear arrays.

Improvement of Location Accuracy by Adding Nodes to Ad-Hoc Networks

We discuss the effect of adding nodes on the location accuracy of Ad-Hoc networks. All results are obtained by analyzing the Cramér-Rao Lower bound. We show that for planar network the additional node must have at least 3 connections in order to have any effect on the existing nodes accuracy. Further, we identify the nodes whose accuracy will be improved. Finally, we show that the accuracy cannot be improved without limit by adding more and more nodes to an existing network.

Maximum Likelihood Positioning of Network Nodes Using Range Measurements

Given a network of stations with incomplete and possibly imprecise inter-station range measurements, it is required to find the relative positions of the stations. Due to its asymptotic properties maximum likelihood estimation is discussed. Although the problem is quadratic, the proposed solution is based on solving a linear set of equations. For precise measurements we obtain explicitly the exact solution with a small number of operations. For noisy measurements the method provides an excellent initial point for the application of the Gerchberg-Saxton iterations. Proof of convergence is provided. The case of planar geometry is coached using complex numbers which reveals a strong relation to the celebrated problem of phase retrieval. We provide a compact, matrix form of the Cramer-Rao bound, small error analysis and evaluation of the computational load. Numerical examples are provided to corroborate the results.

Theoretical facts on RSSI-based geolocation

We address the problem of locating a stationary emitter using the Received Signal Strength (RSS) at receivers with known locations. The Maximum Likelihood estimator for the emitter location requires the minimization of a non-convex cost function. Since this cost function exhibits numerous local minima, its global minimization is usually realized by means of a grid search and is therefore computationally expensive. In this document, we prove three novel theoretical properties of RSS-based cost functions for Maximum Likelihood localization. First, we show that local maxima of RSS-based cost functions occur at receivers locations. Thus, unlike local minima, the locations of local maxima are a-priori known since the receivers locations are known. Second, we show that the smallest local maximum is necessarily closer to the global minimum than any other local minimum. Third, we show that the global minimum of the non-convex cost function lies within a triangular area defined by the smallest local maxima. Combining these theoretical facts, we propose a procedure for delimiting a small geographical area that contains the global minimum of the cost function, with high probability. Therefore, localization can be achieved by grid search over this reduced area only, which significantly reduces computational costs.

Error bounds for convex parameter estimation

We evaluate the accuracy of sparsity-based estimation methods inspired from compressed sensing. Typical estimation approaches consist of minimizing a non-convex cost function that exhibits local minima, and require excessive computational resources. A tractable alternative relies on a sparse representation of the observation vector using a large dictionary matrix and a convex cost function. This estimation approach converts the intractable high-dimensional non-convex problem into a simpler convex problem with reduced dimension. Unfortunately, the advantages come at the expense of increased estimation error. Therefore, an evaluation of the estimation error is of considerable interest. We consider the case of estimating a single parameter vector, and provide upper bounds on the achievable accuracy. The theoretical results are corroborated by simulations.

Maximum Likelihood localization of wireless networks using biased range measurements

Localization of ad-hoc wireless networks is useful for services, management and routing. Localization is frequently based on station-to-station range measurements and a few reference sensors. We address the localization problem in the case of incomplete set of noisy range measurements with unknown bias. A statistically efficient, maximum likelihood algorithm, inspired by the Gerchberg-Saxton procedure for phase retrieval, is presented. In addition, a compact explicit expression for the Fisher Information matrix is provided. A set of numerical examples demonstrates the bias effect on the localization accuracy. As expected, the localization accuracy improves when the unknown bias is estimated.

Time-delay and Doppler-shift based geolocation in the presence of outliers

We address the problem of locating a stationary transmitter using receivers embedded in fast moving platforms. It is required to estimate the emitter location using time-delay and Doppler-shift measurements realized by the receivers at successive locations along their trajectories. Most publications consider small error measurements only, and overlook the possible presence of outliers in the set of time-delay and Doppler-shift measurements. In this work we discuss localization in the presence of outliers, where several measurements are severely corrupted while sufficient other measurements are reasonably precise. Following convexity and sparsity principles, we propose a localization algorithm with robustness against outliers.