On Minimax Detection of Gaussian Stochastic Sequences and Gaussian Stationary Signals

TL;DR

This paper characterizes the optimal detection performance for Gaussian stochastic sequences with unknown covariances, identifying the largest set of covariances allowing simplified testing without loss of detection efficiency.

Contribution

It precisely describes the maximal set of covariance matrices enabling robust minimax detection to be replaced by simple hypothesis testing.

Findings

Complete description of the maximal covariance set for minimax detection

Corollaries on detecting signals in white Gaussian noise

Results applicable to stationary Gaussian signals

Abstract

Minimax detection of Gaussian stochastic sequences (signals) with unknown covariance matrices is studied. For a fixed false alarm probability (1-st kind error probability), the performance of the minimax detection is being characterized by the best exponential decay rate of the miss probability (2-nd kind error probability) as the length of the observation interval tends to infinity. Our goal is to find the largest set of covariance matrices such that the minimax robust testing of this set (composite hypothesis) can be replaced with testing of only one specific covariance matrix (simple hypothesis) without any loss in detection characteristics. In this paper, we completely describe this maximal set of covariance matrices. Some corollaries address minimax detection of the Gaussian stochastic signals embedded in the White Gaussian noise and detection of the Gaussian stationary signals.