Analysis of Adaptive Detectors Robustness to Mismatches on Noise Mean Value

We consider the problem of detecting a signal of interest corrupted by Gaussian noise with unknown mean and covariance matrix when the training samples available have a different mean. We evaluate the robustness of well-known adaptive detectors designed under the assumption of a same mean. More precisely, statistical representations of the generalized likelihood ratio test, the adaptive matched filter and the adaptive coherence estimator are derived for both an additive model with an arbitrary mismatch between the means, and a replacement model which is widely used in hyperspectral imaging. The new representations are given in terms of simple $F$ distributions and are shown to depend in a simple way on the norm of the whitened mean difference and its angle with the whitened signal of interest signature, or on the replacement factor. These new representations allow to identify the key parameters that impact most the probability of false alarm and probability of detection. Numerical simulations illustrate the theoretical results.