We consider the decentralized detection problem, in whichN independent, identical sensors transmit a finite-valued function of their observations to a fusion center which then decides which one ofM hypotheses is true. For the case where the number of sensors tends to infinity, we show that it is asymptotically optimal to divide the sensors intoM(M-1)/2 groups, with all sensors in each group using the same decision rule in deciding what to transmit. We also show how the optimal number of sensors in each group may be determined by solving a mathematical programming problem. For the special case of two hypotheses and binary messages the solution simplifies considerably: it is optimal (asymptotically, asN→∞) to have all sensors perform an identical likelihood ratio test, and the optimal threshold is very easy to determine numerically.
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