On One Problem in Multichannel Signal Detection

TL;DR

This paper investigates the statistical problem of detecting an unknown-energy signal across multiple channels in Gaussian noise, comparing maximum posterior probability and Bayes tests, and analyzing their properties and limitations.

Contribution

It introduces a detailed comparison of the maximum posterior probability test and the optimal Bayes test for multichannel signal detection, including their limiting distributions and non-detectable signals.

Findings

Derived limiting distributions of test statistics.

Identified sets of non-detectable signals.

Compared statistical properties of two detection methods.

Abstract

We consider a statistical problem of detection of a signal with unknown energy in a multi-channel 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. In our work we describe and compare statistical properties of 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 non-detectable signals.