On one problem in multichannel signal detection

We consider a statistical problem of detecting a signal with unknown energy in a multichannel system, observed in a Gaussian noise. We assume that the signal can appear in the $k$th channel with a known small prior probability $\bar\pi_k$. Using noisy observations from all channels, we would like to detect whether the signal is presented in one of the channels or we observe pure noise. We describe and compare statistical properties of the maximum posterior probability test and optimal Bayes test. In particular, for these tests we obtain limiting distributions of test statistics and define sets of their undetectable signals.