Optimum sensor placement for localization in three dimensional under log normal shadowing

Optimum sensor placement for localization of a hazardous source under log normal shadowing

“…This brings us to a key difference between the development here and that in [14] and [2]. Those papers relied on the fact that in those settings…”

“…We further assume that the source position y ∈ R N has a radially symmetric probability density function (pdf). In particular, the pdf of y is given by, [7], f Y (y) = g( y ) for y ∈ B(r 1 ) 0 else (2) We further assume that there are n > N sensors located at positions x i ∈ R N , which must localize this source by measuring the received signal strength (RSS) of a signal emanating from it. We assume that the RSS experiences log-normal shadowing.…”

“…Similarly our conference papers [13], [15], are in 2-dimensions, and assume the pdf f Y (y) = δ(y) and uniform respectively. To the best of our knowledge, the only papers dealing with three dimensions are [14] and [2], but are again restricted to the pdfs f Y (y) = δ(y) and uniform respectively. It turns out that they exploit a special property of the FIM that holds for these two pdfs but not for the general radially symmetric pdf considered here.…”

“…What is more [2] lacks proof. Moreover all these references, [13], [15], [14] and [2] only give sufficient conditions for the [Q, x 1 ] pairs defining the canonical solutions to be optimizing while not residing on an (N − 1)-dimensional hyperplane.…”

“…What is more [2] lacks proof. Moreover all these references, [13], [15], [14] and [2] only give sufficient conditions for the [Q, x 1 ] pairs defining the canonical solutions to be optimizing while not residing on an (N − 1)-dimensional hyperplane. Finally we note that though the N > 3 cases are not directly relevant to localization, as argued in [3], the f Y (y) = δ(y) case impacts the design of optimal sensing matrices for compressed sensing.…”