  In this paper, we consider the problem of testing equality of the covariance
matrices of L complex Gaussian multivariate time series of dimension $M$ . We
study the special case where each of the L covariance matrices is modeled as a
rank K perturbation of the identity matrix, corresponding to a signal plus
noise model. A new test statistic based on the estimates of the eigenvalues of
the different covariance matrices is proposed. In particular, we show that this
statistic is consistent and with controlled type I error in the
high-dimensional asymptotic regime where the sample sizes $N_1,\ldots,N_L$ of
each time series and the dimension $M$ both converge to infinity at the same
rate, while $K$ and $L$ are kept fixed. We also provide some simulations on
synthetic and real data (SAR images) which demonstrate significant improvements
over some classical methods such as the GLRT, or other alternative methods
relevant for the high-dimensional regime and the low-rank model.
