Design of Reduced-Rank MVDR Beamformers under Finite Sample-Support

The design of optimal reduced-rank minimum variance beamformers based on the Krylov-subspace spanned by the covariance matrix of the array observations is addressed. We concentrate on finite sample-support situations that naturally appear in practice when the number of antennas and the sample-size are comparable in magnitude. The design of the coefficients of the resulting polynomial expansion is approached by first approximating the signal-to-interference-plus-noise ratio (SINR) at the output of the antenna array in the small sample-size regime defined assuming that both the number of samples and the observation dimension grow together without bound at the same rate. Limiting SINR values in this double-limit context are very representative of the reality because, as it happens to be the case in realistic scenarios, both quantities are considered to be of the same order of magnitude. The building blocks of the (asymptotic) output SINR expression are spectral functions of the true covariance matrix that can be estimated using the statistical theory of large observations (or general statistical analysis) developed by Girko. This paper proposes a consistent estimator of the reduced-rank minimum variance beamformer that is consistent when both the number of antennas and the sample-support go to infinity with a constant ratio between them