  This paper addresses the passive detection of a common rank-one subspace
signal received in two multi-sensor arrays. We consider the case of a
one-antenna transmitter sending a common Gaussian signal, independent Gaussian
noises with arbitrary spatial covariance, and known channel subspaces. The
detector derived in this paper is a generalized likelihood ratio (GLR) test.
For all but one of the unknown parameters, it is possible to find closed-form
maximum likelihood (ML) estimator functions. We can further compress the
likelihood to only an unknown vector whose ML estimate requires maximizing a
product of ratios in quadratic forms, which is carried out using a trust-region
algorithm. We propose two approximations of the GLR that do not require any
numerical optimization: one based on a sample-based estimator of the unknown
parameter whose ML estimate cannot be obtained in closed-form, and one derived
under low-SNR conditions. Notably, all the detectors are scale-invariant, and
the approximations are functions of beamformed data. However, they are not
GLRTs for data that has been pre-processed with a beamformer, a point that is
elaborated in the paper. These detectors outperform previously published
correlation detectors on simulated data, in many cases quite significantly.
Moreover, performance results quantify the performance gains over detectors
that assume only the dimension of the subspace to be known.
