  Source localization and spectral estimation are among the most fundamental
problems in statistical and array signal processing. Methods which rely on the
orthogonality of the signal and noise subspaces, such as Pisarenko's method,
MUSIC, and root-MUSIC are some of the most widely used algorithms to solve
these problems. As a common feature, these methods require both apriori
knowledge of the number of sources, and an estimate of the noise subspace. Both
requirements are complicating factors to the practical implementation of the
algorithms, and when not satisfied exactly, can potentially lead to severe
errors. In this paper, we propose a new localization criterion based on the
algebraic structure of the noise subspace that is described for the first time
to the best of our knowledge. Using this criterion and the relationship between
the source localization problem and the problem of computing the greatest
common divisor (GCD), or more practically approximate GCD, for polynomials, we
propose two algorithms which adaptively learn the number of sources and
estimate their locations. Simulation results show a significant improvement
over root-MUSIC in challenging scenarios such as closely located sources, both
in terms of detection of the number of sources and their localization over a
broad and practical range of SNRs. Further, no performance sacrifice in simple
scenarios is observed.
