  We consider the distributed detection problem of a temporally correlated
random radio source signal using a wireless sensor network capable of measuring
the energy of the received signals. It is well-known that optimal tests in the
Neyman-Pearson setting are based on likelihood ratio tests (LRT), which, in
this set-up, evaluate the quotient between the probability density functions
(PDF) of the measurements when the source signal is present and absent. When
the source is present, the computation of the joint PDF of the energy
measurements at the nodes is a challenging problem. This is due to the
statistical dependence introduced to the received signals by the radio source
propagated through fading channels. We deal with this problem using the
characteristic function of the (intractable) joint PDF, and proposing an
approximation to it. We derive bounds for the approximation error in two
wireless propagation scenarios, slow and fast fading, and show that the
proposed approximation is exponentially tight with the number of nodes when the
time-bandwidth product is sufficiently high. The approximation is used as a
substitute of the exact joint PDF for building an approximate LRT, which
performs better than other well-known detectors, as verified by Monte Carlo
simulations.
