Graduation date: 2007The main contributions of this thesis are the development and application of four computationally efficient solutions for least-squares-based (LS-based) minimum variance spectral estimation (MVSE). They are: (1) fast computational solution for the 1-D covariance LS-based MVSE, (2) fast computational solution for the 1-D modified covariance LS-based MVSE, (3) fast computational solution for the 2-D covariance LS-based MVSE, and (4) fast computational solution for the 2-D modified covariance LS-based MVSE. The four fast computational solutions not only significantly reduce computational complexity and save memory from array to vector sizing proportionalities, but they also inherit improved-feature details from the corresponding direct methods of 1-D and 2-D LS-based MVSEs. The two 2-D fast computational solutions numerically produce the same results as the corresponding 1-D fast solutions when the estimation order in one of the two dimensions is set to zero. MVSEs are high-resolution spectral estimators which have been used extensively in the sensor community (for example, radar, sonar, communication signal localization, and seismic velocity discrimination) for extracting and resolving more features from limited data collection apertures than traditional Fourier-based techniques. Least-squares-based MVSEs are especially applicable in the case that the autocorrelation is unknown and only 1-D or 2-D finite data acquisitions are available. However, LS-based minimum variance (MV) spectral estimators require intensive computational burdens which limit their operational use. This thesis proposes 1-D and 2-D fast computational solutions. The basis for the fast solutions is the exploitation of the special structures of the various inverse matrix relationships, which express the inverse of autocorrelation matrices (or autocorrelation-like quadratic-data-matrix product matrices in the case of the least-squares algorithms) in terms of the parametric autoregressive (AR) or linear prediction (LP) parameters. The fast algorithms also have the serendipitous feature that all lower-order solutions are obtained by the fast computational solutions without additional computations, unlike the non-fast approaches. This is useful especially when the correct order is unknown, requiring that a range of orders to be evaluated to determine the order that produces the best result using one algorithmic execution of a fast algorithm.