  Covariance matrix tapers have a long history in signal processing and related
fields. Examples of applications include autoregressive models (promoting a
banded structure) or beamforming (widening the spectral null width associated
with an interferer). In this paper, the focus is on high-dimensional setting
where the dimension $p$ is high, while the data aspect ratio $n/p$ is low. We
propose an estimator called Tabasco (TApered or BAnded Shrinkage COvariance
matrix) that shrinks the tapered sample covariance matrix towards a scaled
identity matrix. We derive optimal and estimated (data adaptive) regularization
parameters that are designed to minimize the mean squared error (MSE) between
the proposed shrinkage estimator and the true covariance matrix. These
parameters are derived under the general assumption that the data is sampled
from an unspecified elliptically symmetric distribution with finite 4th order
moments (both real- and complex-valued cases are addressed). Simulation studies
show that the proposed Tabasco outperforms all competing tapering covariance
matrix estimators in diverse setups. A space-time adaptive processing (STAP)
application also illustrates the benefit of the proposed estimator in a
practical signal processing setup.
