A MIMO Version of the Reed-Yu Detector and Its Connection to the Wilks Lambda and Hotelling $T^2$ Statistics

In this paper we study the problem of detecting a known signal transmitted over a MIMO channel of unknown complex gains and additive noise of unknown covariance. The problem arises in many contexts, including transmit-receiver synchronization. We derive the exact probability distribution for a generalized likelihood ratio (GLR) statistic, and establish the connection between this statistic and the Wilks Lambda and Hotelling $T^2$ statistics. We give alternatives to the GLR statistic, which include the Bartlett-Nanda-Pillai trace, the Lawley-Hotelling trace, and the Roy maximum eigenvalue statistics, each of which is favored under special conditions on the actual MIMO channel. For example, if the channel is an incoherent scattering channel, then the competition for greatest power is among the Bartlett-Nanda-Pillai, Lawley-Hotelling, and GLR statistics. If it is a coherent channel that supports a propagating wave, then Roy's test is more powerful. We discuss the null distribution theory of the GLR at length to show how it may be used to accurately predict false alarm probabilities.