This paper analyzes the limiting behavior of the uncertainty in localizing an unknown number of transmitters within a given geographical area. The set-up consists of n binary sensors that are deployed uniformly at random locations within the area. These sensors detect for the presence of a transmitter within their radio range, and their individual decisions are combined to estimate the number of transmitters as well as their locations. With the mean sum absolute error in transmitter localization as the metric, the optimal scaling of the radio range and the necessary minimum transmitter separation is determined, as n gets large. It is shown that both the localization error and the radio range optimally scale as log(n) . The analysis is extended to the case of unreliable sensors, where, surprisingly, the optimal scaling is found to still be log(n) . The cognitive radio problem of identifying the available whitespace, i.e., the regions that do not contain any transmitter, emerges as a special case. Finally, the optimal distribution of sensor deployment is determined, given the distribution of the transmitters. Simulation results illustrate the significant performance benefit that can be obtained by optimally scaling the radio range, compared to existing fixed sensing range based designs.
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