Fast algorithms for least-squares-based minimum variance spectral estimation

The minimum variance (MV) spectral estimator is a robust high-resolution frequency-domain analysis tool for short data records. The traditional formulation of the minimum variance spectral estimation (MVSE) depends on the inverse of a Toeplitz autocorrelation matrix, for which a fast computational algorithm exists that exploits this structure. This paper extends the MVSE approach to two data-only formulations linked to the covariance and modified covariance cases of least-squares linear prediction (LP), which require inversion of near-to-Toeplitz data product matrices. We show here that the near-to-Toeplitz matrix inverses in the two new fast algorithms have special representations as sums of products of triangular Toeplitz matrices composed of the LP parameters of the least-squares-based formulations. Fast algorithm solutions of the LP parameters have been published by one of the authors. From these, we develop fast solutions of two least-squares-based minimum variance spectral estimators (LS-based MVSEs). These new MVSEs provide additional resolution improvement over the traditional autocorrelation-based MVSE.