  Many multichannel systems use a linear filter to retrieve a signal of
interest corrupted by noise whose statistics are partly unknown. The optimal
filter in Gaussian noise requires knowledge of the noise covariance matrix
$\Sigma$ and in practice the latter is estimated from a set of training
samples. An important issue concerns the characterization of the performance of
such adaptive filters. This is generally achieved using as figure of merit the
ratio of the signal to noise ratio (SNR) at the output of the adaptive filter
to the SNR obtained with the clairvoyant -- known $\Sigma$ -- filter. This
problem has been studied extensively since the seventies and this document
presents a concise overview of results published in the literature. We consider
various cases about the training samples covariance matrix and we investigate
fully adaptive, partially adaptive and regularized filters.

Signal Processing