  In this paper, we address the problem of detection, in the frequency domain,
of a M-dimensional time series modeled as the output of a M x K MIMO filter
driven by a K-dimensional Gaussian white noise, and disturbed by an additive
M-dimensional Gaussian colored noise. We consider the study of test statistics
based of the Spectral Coherence Matrix (SCM) obtained as renormalization of the
smoothed periodogram matrix of the observed time series over N samples, and
with smoothing span B. To that purpose, we consider the asymptotic regime in
which M, B, N all converge to infinity at certain specific rates, while K
remains fixed. We prove that the SCM may be approximated in operator norm by a
correlated Wishart matrix, for which Random Matrix Theory (RMT) provides a
precise description of the asymptotic behaviour of the eigenvalues. These
results are then exploited to study the consistency of a test based on the
largest eigenvalue of the SCM, and provide some numerical illustrations to
evaluate the statistical performance of such a test.
