Eigenstructure-Based Performance Analysis and Toeplitz Approximation for Direction-Of Estimators.

Details Eigenstructure-Based Performance Analysis and Toeplitz Approximation for Direction-Of Estimators. Eigenstructure-Based Performance Analysis and Toeplitz Approximation for Direction-Of Estimators.

Jan 1990

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1990phdt........48w&link_type=abstract

Thesis (PH.D.)--THE UNIVERSITY OF MICHIGAN, 1990.Source: Dissertation Abstracts International, Volume: 51-07, Section: B, page:

Computer Science

Performance

Scientific paper

Two problems are considered in this dissertation: (1) Direction-of-arrival estimation analysis, and (2) Incorporating a priori knowledge about covariance matrices, including Toeplitz matrix structure, matrix definiteness, and matrix rank. The first problem is studied based on Wilkinson eigenstructure perturbation analysis for covariance matrices. The second problem is formulated as an optimal Toeplitz approximation problem for observed covariance matrices with constraints including matrix definiteness and matrix rank. For performance analysis, compact formulas for the expectation of products of linear functionals of a complex Gaussian vector and the expectation of products of bilinear forms of a complex Wishart matrix are derived. Applications of the formulas to multivariate statistics are demonstrated, including the calculation of the moments of complex Wishart distributions as well as the moments of Blackman-Tukey's power spectrum estimates. The performance analysis is conducted for small -sample conditions. The analysis predicts that the bias absolute value and the variance of direction-of-arrival estimation for MUSIC are inversely proportional to signal -to-noise ratio and the number of snapshots. The prediction errors are shown to be within 3 dB based on Monte Carlo simulations. For optimal Toeplitzation, it is shown that the usual unbiased estimates of autocorrelation lags in spectrum estimation coincide with the entries of unconstrained least -squares Toeplitz approximates of observed covariance matrices. For matrix positive definiteness, the constrained optimal Toeplitz approximation problem is reduced to an unconstrained problem with a follow-up linear test using Fast Fourier Transform (FFT) for the unconstrained solution. For nonnegative definiteness, an optimization problem constrained by a set of linear inequalities is proposed, whose solution, if it exists, is a suboptimal solution of the nonnegative definite optimal Toeplitz approximation problem. Two methods, AP and RQ Toeplitzation, for rank constrained problems are proposed. The averaging principle developed in least-square Toeplitz approximation is extended to the least-squares structured covariance approximation for minimum redundancy arrays. Based on the extension, Pillai and Haber's direction -of-arrival estimation using exact covariance matrices for minimum redundancy arrays is extended for finite sample conditions.

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