  The ensemble covariance matrix of a wide sense stationary signal spatially
sampled by a full linear array is positive semi-definite and Toeplitz. However,
the direct augmented covariance matrix of an augmentable sparse array is
Toeplitz but not positive semi-definite, resulting in negative eigenvalues that
pose inherent challenges in its applications, including model order estimation
and source localization. The positive eigenvalues-based covariance matrix for
augmentable sparse arrays is robust but the matrix is unobtainable when all
noise eigenvalues of the direct augmented matrix are negative, which is a
possible case. To address this problem, we propose a robust covariance matrix
for augmentable sparse arrays that leverages both positive and negative noise
eigenvalues. The proposed covariance matrix estimate can be used in conjunction
with subspace based algorithms and adaptive beamformers to yield accurate
signal direction estimates.
