Consistent Reduced-Rank LMMSE Estimation With a Limited Number of Samples per Observation Dimension

An improved construction of the optimal reduced- rank linear minimum mean-square error (MMSE) estimator of a signal waveform of interest is derived that is consistent under a limited number of samples per filtering degree-of-freedom. The new filter design generalizes traditional filter realizations based on directly replacing the theoretical covariance matrix by its sample estimate, and being consistent when all dimensions in the model but the number of samples remain bounded. Our solution not only generalizes the conventional estimator, but also turns out to appropriately characterize model mismatch constraints due to finite sample-size limitations of fundamental importance in practical situations. The proposed implementation results from a generalized consistent estimation of the set of MMSE filter subspace coefficients on the reduced-dimensional subspace. Results are based on the theory of the spectral analysis of large-dimensional random matrices. In particular, we build on the analytical description of the asymptotic spectrum of sample-covariance-type matrices in the limiting regime defined as both the number of samples and the observation dimension grow without bound at the same rate. As a result, the proposed MMSE signal waveform estimator is shown to present a superior mean-square error performance under a finite sample-size by avoiding the breakdown experienced as the selected rank increases.