  We analyze the distribution of the signal to noise ratio (SNR) loss at the
output of an adaptive filter which is trained with samples that do not share
the same covariance matrix as the samples for which the filter is foreseen. Our
objective is to find an accurate approximation of the distribution of the SNR
loss which has a similar form as in the case of no mismatch. We successively
consider the case where the two covariance matrices satisfy the so-called
generalized eigenrelation and the case where they are arbitrary. In the former
case, this amounts to approximate a central quadratic form in normal variables
while the latter case entails approximating a non-central quadratic form in
Student distributed variables. In order to obtain the approximate distribution,
a Pearson type approach is advocated. A numerical study show that this
approximation is rather accurate and enables one to assess, in a
straightforward manner, the impact of covariance mismatch.
