On Minimax Robust Detection of Stationary Gaussian Signals in White Gaussian Noise

TL;DR

This paper analyzes the minimax robust detection of stationary Gaussian signals in white Gaussian noise with spectral density uncertainty, establishing conditions for optimal detection strategies as observation length increases.

Contribution

It introduces a dominance condition for spectral density uncertainty sets, proving the existence of a saddle point and optimal likelihood ratio tests without convexity assumptions.

Findings

Existence of a saddle point under the dominance condition

Likelihood ratio tests matched to a dominated spectral density are optimal

No convexity condition needed for the uncertainty set

Abstract

The problem of detecting a wide-sense stationary Gaussian signal process embedded in white Gaussian noise, where the power spectral density of the signal process exhibits uncertainty, is investigated. The performance of minimax robust detection is characterized by the exponential decay rate of the miss probability under a Neyman-Pearson criterion with a fixed false alarm probability, as the length of the observation interval grows without bound. A dominance condition is identified for the uncertainty set of spectral density functions, and it is established that, under the dominance condition, the resulting minimax problem possesses a saddle point, which is achievable by the likelihood ratio tests matched to a so-called dominated power spectral density in the uncertainty set. No convexity condition on the uncertainty set is required to establish this result.