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Asymptotic analysis of reduced rank Wiener filters

We revisit recent papers of M.L. Honig and W. Xiao (see IEEE Trans. on Inf. Theory, vol.47, no.5, p.1928-46, 2001) and L.G.F. Trichard et al. (see Proc. IEEE Int. Conf. on Commun., p.1461-5, 2002; Proc. ISIT 2002) devoted to the asymptotic analysis of reduced rank Wiener filters. Appropriate connections between the asymptotic behavior of the signal-to-noise ratios (SNRs) at the outputs of these filters and the theory of orthogonal polynomials for the power moment problem are established. Using some classical results of this theory, it can be established, in particular, that the reduced rank filter output SNR converges exponentially in the filter rank toward the full rank Wiener filter output SNR. The convergence rate is given. Interestingly, it depends only on the support of the limiting eigenvalue distribution of the observation covariance matrix, but not on its particular form.