Multiantenna GLR detection of a Gaussian signal in spatially uncorrelated noise

The exact all-SNR Generalized Log-likelihood Ratio Test (GLRT) of a Gaussian rank-one signal impinging on an M-antenna receiver in unknown spatially uncorrelated white noise is derived and compared with some previous low-SNR approximations to this problem. It is confirmed that the coherence matrix used in the mentioned approximations is a sufficient statistic for the GLRT over the complete SNR range. In contrast, this exact, albeit more complex test, requires the unconstrained optimization of a highly nonlinear M-variate complex function which can be addressed using classical optimization techniques. A maximum eigenvalue problem combined with a univariate optimization problem serves the purpose of initialization. Results validate the yet unproven close to optimum performance of previous detectors under the rank-one signal model for the tested SNR's.