Geometric Interpretation of Theoretical Bounds for RSS-based Source Localization with Uncertain Anchor Positions

The Received Signal Strength based source localization can encounter severe problems originating from uncertain information about the anchor positions in practice. The anchor positions, although commonly assumed to be precisely known prior to the source localization, are usually obtained using previous estimation algorithm such as GPS. This previous estimation procedure produces anchor positions with limited accuracy that result in degradations of the source localization algorithm and topology uncertainty. We have recently addressed the problem with a joint estimation framework that jointly estimates the unknown source and uncertain anchors positions and derived the theoretical limits of the framework. This paper extends the authors previous work on the theoretical performance bounds of the joint localization framework with appropriate geometric interpretation of the overall problem exploiting the properties of semi-definiteness and symmetry of the Fisher Information Matrix and the Cram{\`e}r-Rao Lower Bound and using Information and Error Ellipses, respectively. The numerical results aim to illustrate and discuss the usefulness of the geometric interpretation. They provide in-depth insight into the geometrical properties of the joint localization problem underlining the various possibilities for practical design of efficient localization algorithms.Comment: 30 pages, 15 figure

On the Performance Limits of Map-Aware Localization

Establishing bounds on the accuracy achievable by localization techniques represents a fundamental technical issue. Bounds on localization accuracy have been derived for cases in which the position of an agent is estimated on the basis of a set of observations and, possibly, of some a priori information related to them (e.g., information about anchor positions and properties of the communication channel). In this paper, new bounds are derived under the assumption that the localization system is map-aware, i.e., it can benefit not only from the availability of observations, but also from the a priori knowledge provided by the map of the environment where it operates. Our results show that: a) map-aware estimation accuracy can be related to some features of the map (e.g., its shape and area) even though, in general, the relation is complicated; b) maps are really useful in the presence of some combination of low SNRs and specific geometrical features of the map (e.g., the size of obstructions); c) in most cases, there is no need of refined maps since additional details do not improve estimation accuracy.United States. Air Force Office of Scientific Research (Grant FA9550-12-0287)United States. Office of Naval Research (Grant N00014-11-1-0397)Massachusetts Institute of Technology. Institute for Soldier Nanotechnologie

Localization Optimal Multi-user Beamforming with multi-carrier mmWave MIMO

In this paper, we propose optimal beamforming strategies for a millimeter wave (mmWave) system consisting of multiple users based on the localization performance bounds. We consider a single base station (BS) with prior coarse knowledge of the users\u27 positions and formulate the optimal beamforming problem in order to minimize the localization error consisting of Cramer Rao Lower Bounds (CRLBs) of delay, angle of departure (AoD) and angle of arrival (AoA) estimation at the mobile users. We first formulate the simplified CRLB of estimation parameters, taking advantage of multiple sub-carriers, and then formulate the localization error for optimization of the beamformer. Finally, we evaluate the resulting position and orientation error bounds after optimization for several fairness strategies through Monte Carlo simulations

Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.Comment: 40 pages, 4 figure

Improved Target Localization in Multi-Waveform Multi-Band Hybrid Multistatic Radar Networks

This study proposes an algorithm to improve the target localization performance. This is implemented in a multi-waveform multi-band hybrid (passive and active) multistatic radar network scenario, that utilize broadcasting signals for radar sensing, in addition to the radar waveforms. Multi-waveform multi-band radar receivers can exploit the broadcast signals transmitted by non-cooperative transmitters, such as communication or broadcasting systems, for target sensing in addition to radar waveform. Hence, multiple measurements of the targets can be acquired and fused to improve the target detection and parameter estimation. Because of utilizing various waveforms, each transmitter-receiver (Tx-Rx) pair has a different range and velocity estimation accuracy, that is also affected by the bistatic geometry of the bistatic pairs. Taking this into account, this study proposes a target localization algorithm based on bistatic Cramér-Rao Lower Bounds (CRLBs) for multistatic multi-band radar networks. It is shown that modeling the entire network and evaluating the bistatic range CRLB of each bistatic pair in advance, and utilizing this information while estimating the target location significantly improves the localization accuracy. Moreover, the proposed algorithm also includes a target height estimation correction stage to achieve a better 3D localization accuracy

Mutual Information in Rank-One Matrix Estimation

We consider the estimation of a n-dimensional vector x from the knowledge of noisy and possibility non-linear element-wise measurements of xxT , a very generic problem that contains, e.g. stochastic 2-block model, submatrix localization or the spike perturbation of random matrices. We use an interpolation method proposed by Guerra and later refined by Korada and Macris. We prove that the Bethe mutual information (related to the Bethe free energy and conjectured to be exact by Lesieur et al. on the basis of the non-rigorous cavity method) always yields an upper bound to the exact mutual information. We also provide a lower bound using a similar technique. For concreteness, we illustrate our findings on the sparse PCA problem, and observe that (a) our bounds match for a large region of parameters and (b) that it exists a phase transition in a region where the spectum remains uninformative. While we present only the case of rank-one symmetric matrix estimation, our proof technique is readily extendable to low-rank symmetric matrix or low-rank symmetric tensor estimationComment: 8 pages, 1 figure

Cramer-Rao Bounds for Joint RSS/DoA-Based Primary-User Localization in Cognitive Radio Networks

Knowledge about the location of licensed primary-users (PU) could enable several key features in cognitive radio (CR) networks including improved spatio-temporal sensing, intelligent location-aware routing, as well as aiding spectrum policy enforcement. In this paper we consider the achievable accuracy of PU localization algorithms that jointly utilize received-signal-strength (RSS) and direction-of-arrival (DoA) measurements by evaluating the Cramer-Rao Bound (CRB). Previous works evaluate the CRB for RSS-only and DoA-only localization algorithms separately and assume DoA estimation error variance is a fixed constant or rather independent of RSS. We derive the CRB for joint RSS/DoA-based PU localization algorithms based on the mathematical model of DoA estimation error variance as a function of RSS, for a given CR placement. The bound is compared with practical localization algorithms and the impact of several key parameters, such as number of nodes, number of antennas and samples, channel shadowing variance and correlation distance, on the achievable accuracy are thoroughly analyzed and discussed. We also derive the closed-form asymptotic CRB for uniform random CR placement, and perform theoretical and numerical studies on the required number of CRs such that the asymptotic CRB tightly approximates the numerical integration of the CRB for a given placement.Comment: 20 pages, 11 figures, 1 table, submitted to IEEE Transactions on Wireless Communication

Position and Orientation Estimation through Millimeter Wave MIMO in 5G Systems

Millimeter wave signals and large antenna arrays are considered enabling technologies for future 5G networks. While their benefits for achieving high-data rate communications are well-known, their potential advantages for accurate positioning are largely undiscovered. We derive the Cram\'{e}r-Rao bound (CRB) on position and rotation angle estimation uncertainty from millimeter wave signals from a single transmitter, in the presence of scatterers. We also present a novel two-stage algorithm for position and rotation angle estimation that attains the CRB for average to high signal-to-noise ratio. The algorithm is based on multiple measurement vectors matching pursuit for coarse estimation, followed by a refinement stage based on the space-alternating generalized expectation maximization algorithm. We find that accurate position and rotation angle estimation is possible using signals from a single transmitter, in either line-of- sight, non-line-of-sight, or obstructed-line-of-sight conditions.Comment: The manuscript has been revised, and increased from 27 to 31 pages. Also, Fig.2, Fig. 10 and Table I are adde

Cramer-Rao Bound for Target Localization for Widely Separated MIMO Radar

In this paper, we derive the Cramer-Rao Bounds (CRBs) for the 2-dimensional (2D) target localization and velocity estimations for widely separated Multiple-Input Multiple-Output (MIMO) radar. The transmitters emit signals with different frequencies and the receivers receive these signals with amplitude fluctuations and with Doppler shifts due to the target motion. The received signal model is constructed using the Swerling target fluctuations to take into account the undesired effects of target amplitude and phase fluctuations. Moreover, the time delays and the Doppler frequencies are included in the signal model to get a more realistic model. Then, the Cramer-Rao Bounds are derived for the proposed signal model for the target position and velocity estimations. Contrary to known models of CRBs, we derived the CRBs jointly and using the Swerling target fluctuations