We consider the problem of optimum sensor placement for localizing a hazardous source located inside an $ N $-dimensional hypersphere centered at the origin with a known radius $ r_1 $. All one knows about the probability density function (pdf) of the source location is that it is spherically symmetric, i.e. it is a function only of the distance from the center. The sensors must be placed at a safe distance of at least $ r_2>r_1 $ from the center, to avoid damage. Localization must be effected from the strength of a signal emanating from the source, as received by a set of sensors that do not lie on an $ (N-1) - $ dimensional hyperplane. Under the assumption that this signal strength experiences log normal shadowing, we characterize non-coplanar sensor positions that optimize three distinguished parameters associated with the underlying Fisher Information Matrix (FIM): maximizing its smallest eigenvalue, maximizing its determinant, and minimizing the trace of its inverse. We show that all three have the same set of optimizing solutions and involve placing the sensors on the surface of the hypersphere of radius $ r_2. $ As spherical symmetry of the pdf precludes uniqueness we provide certain canonical optimizing solutions where the $ i $-th sensor position $ x_i = Q^{i-1}x_1 $, with $ Q $ an orthogonal matrix. We provide necessary and sufficient conditions on $ Q $ and $ x_1 $ for $ x_i $ to be non-coplanar and optimizing. In addition, we provide a geometrical interpretation of these solutions. We observe the $ N $-dimensional solutions for $ N>3 $ have implications for optimal design of sensing matrices in certain compressed sensing problems.
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(a) Illustration of optimum sensor placement in two dimensions using four sensors. (b) Illustration of optimum sensor placement in three dimensions using six sensors and sphere of radius of 2 (i.e.r_{2} = 2)
Plot of determinant of FIM Versus Number of sensors in the network
Plot of Minimum Eigenvalue of FIM Versus Number of sensors in the network
Plot of $10log_{10}$(Average Normalized Mean square error in the source location) Versus Signal to Noise Ratio (dB). Red dotted line represents the performance of the random placement. Blue line represents the performance of the proposed optimum sensor placement