Spectral Estimators that Extend the Maximum Entropy and Maximum Likelihood Methods
Charles L. Byrne, Raymond M. Fitzgerald · SIAM Journal on Applied Mathematics · 1984
The theory of best linear approximation in weighted $L^2 $ spaces is used to obtain a general procedure, the PDFT, for linearly reconstructing the Fourier transform from sampled data. The PDFT can be used either directly to reduce sidelobe structure and to extrapolate the data or indirectly to obtain high resolution spectral estimators. The direct and indirect PDFT include as special cases many of the commonly used spectral techniques, including Burg’s maximum entropy method, Capon’s maximum likelihood method, the spectral estimators based on bandlimited extrapolation, the eigenvalue/eigenvector methods for detecting sinusoids in noise (Pisarenko method, Schmidt’s MUSIC, eigenvector power beamforming), and the best linear unbiased estimator (BLUE) for regression coefficients. By exploiting their relationship to the linear PDFT, these nonlinear techniques can be analyzed in terms of linear approximation theory. In addition to providing a unifying formulation for many different spectral estimators, the PDFT approach provides new techniques which expand the class of available high resolution spectral estimators.