Jammer placement algorithms for wireless localization systems
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The optimal jammer placement problem is proposed and analyzed for wireless localization systems. In particular, the optimal location of a jammer node is obtained by maximizing the minimum of the Cram´er-Rao lower bounds (CRLBs) for a number of target nodes under location related constraints for the jammer node. For scenarios with more than two target nodes, theoretical results are derived to specify conditions under which the jammer node is located as close to a certain target node as possible, or the optimal location of the jammer node is determined by two of the target nodes. Also, explicit expressions are provided for the optimal location of the jammer node in the presence of two target nodes. In addition, in the absence of distance constraints for the jammer node, it is proved, for scenarios with more than two target nodes, that the optimal jammer location lies on the convex hull formed by the locations of the target nodes and is determined by two or three of the target nodes, which have equalized CRLBs. Numerical examples are presented to provide illustrations of the theoretical results in different scenarios. Furthermore, an iterative algorithm is proposed for numerically determining the optimal jammer location. At each iteration of the algorithm, the jammer node is moved one step along a straight line with the purpose of increasing the CRLB(s) of the target node(s) with the minimum CRLB in the system. It is shown that the algorithm converges almost surely to the optimal jammer location under certain conditions for an infinitesimally small step size in the absence of location constraints for the jammer node. Simulations illustrate the effectiveness of the proposed algorithm in finding the optimal jammer location and its superiority in terms of the computational complexity compared to the exhaustive search over all feasible locations.