Distribution-free detection under complex elliptically symmetric clutter distribution

We study the constant false alarm rate matched subspace detector (CFAR MSD) of a signal observed under additive noise following a complex elliptically symmetric (CES) distribution which include the class of compound-Gaussian (CG) distributions as special cases. We prove that the detector is distribution-free under the null (signal free) hypothesis and derive simple expressions for the probability of detection assuming CG-distributed clutter. The derived theoretical results are then illustrated by contrasting them with the performance of a practical adaptive detector which estimates the shape matrix (normalized clutter covariance matrix) from the set of secondary data using complex Tyler's M-estimator of scatter under small sample lengths. We also prove that the complex Tyler's M-estimator is the maximum likelihood estimator (MLE) of the shape matrix under the assumption that the secondary data are independent random vectors from a possibly different CES distributions but which share the same shape matrix parameter.