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This paper discusses the problem of accurate estimation of the bearings or directions of multiple narrowband point signal sources from data collected at a spatially distributed array of sensors. A class of high resolution algorithms based upon covariance matrix eigenstructure are reviewed and compared. The paper then presents a new eigenstructure based bearing algorithm that offers improved resolution and stability performance in high noise, small sample scenarios. The new algorithm is based on an approximate maximum likelihood criterion, that provides a more realistic measure of orthogonality between signal and noise subspaces in a stochastic environment. The algorithms to be discussed may be applied equally well to time-series spectral analysis, and other parameter estimation problems.