Asymptotically Optimal Linear Shrinkage of Sample LMMSE and MVDR Filters

Abstract

Abstract-Conventional implementations of the linear minimum mean-square (LMMSE) and minimum variance distortionless response (MVDR) estimators rely on the sample matrix inversion (SMI) technique, i.e., on the sample covariance matrix (SCM). This approach is optimal in the large sample size regime. Nonetheless, in small sample size situations, those sample estimators suffer a large performance degradation. Thus, the aim of this paper is to propose corrections of these sample methods that counteract their performance degradation in the small sample size regime and keep their optimality in large sample size situations. To this aim, a twofold approach is proposed. First, shrinkage estimators are considered, as they are known to be robust to the small sample size regime. Namely, the proposed methods are based on shrinking the sample LMMSE or sample MVDR filters towards a variously called matched filter or conventional (Bartlett) beamformer in array processing. Second, random matrix theory is used to obtain the optimal shrinkage factors for large filters. The simulation results highlight that the proposed methods outperform the sample LMMSE and MVDR. Also, provided that the sample size is higher than the observation dimension, they improve classical diagonal loading (DL) and Ledoit-Wolf (LW) techniques, which counteract the small sample size degradation by regularizing the SCM. Finally, compared to state-of-the-art DL, the proposed methods reduce the computational cost and the proposed shrinkage of the LMMSE obtains performance gains.

Low‐complexity cross‐validation design of a linear estimator

“…It is seen that our proposed CV choices of parameters for (5) significantly improve the performance compared to the sample LMMSE estimator (2) and can approach the oracle performance achieved with (7). The method of [2], which is based on RMT and can perform very well for perfect knowledge of , does not work well here with directly replaced by .…”

“…Other similar schemes have been discussed in [3–8]. In particular, Serra et al [2–4] assume , while the schemes of [5–8] rely on a grid search to tune the parameters, which incurs higher complexities.…”

“…may significantly outperform (2), as shown in [2]. However, the coefficients (r, t) must be carefully chosen to optimise performance.…”

“…It has a low complexity as no grid search is required. We note that [2] provides an RMT-based analytical solution to choose the composite filter (5) for the case r = r.…”

Linear shrinkage estimation of covariance matrices using low-complexity cross-validation

“…We assume a N = 30-element uniform linear array (ULA) with halfwavelength spacing between neighboring antennas. As in [33], we assume that the desired complex Gaussian signal has an angle of arrival (AoA) of θ 0 = 0 • and there are 8 complex Gaussian interferences in the directions {θ m } = {8 • , −15 • , 23 • , −21 • , 46 • , −44 • , −85 • , 74 • }, all with an average power 10 dB higher than the desired signal. The noise is assumed to be additive white Gaussian noise (AWGN) with an average power 10 dB lower than the desired signal.…”