A review of matched and adaptive subspace detectors

Matched and adaptive subspace detectors apply to a wide range of problems in radar, sonar and data communication, wherein the signal is constrained to lie in a multidimensional linear subspace. These detectors generalize all known results in matched and adaptive detection theory. They have a rich set of algebraic invariances, and these invariances have evocative geometrical interpretations. Moreover their distributions are known, so that the performance may be predicted. We review new and old results in the theory of matched and adaptive subspace detection, and suggest new directions for the further development of the theory. In particular we suggest how rank reduction can be systematically introduced into the theory, and indicate where previous extensions to wireless communication on the multiaccess fading channel may be found.