This paper addresses the behaviour of a classical multi-antenna GLRT test that allows to detect the presence of a known signal corrupted by a multi-path propagation channel and by an additive white Gaussian noise with unknown spatial covariance matrix. The paper is focused on the case where the number of sensors M is large, and of the same order of magnitude as the sample size N, a context which is modeled by the large system asymptotic regime M goes to infinity, N goes to infinity in such a way that M/N goes to c for c in (0, infinity). The purpose of this paper is to study the behaviour of a GLRT test statistics in this regime, and to show that the corresponding theoretical analysis allows to accurately predict the performance of the test when M and N are of the same order of magnitude.