Status of effort

Spectral analysis of a time series often reveals information about the underlying mechanism generating the process. For instance, the spectral content of the echo of a radar provides important clues about the nature, location, and motion of potential targets. Similarly, cross correlation between signals can help identify models for their interaction. Nonlinear techniques and the use of second order statistics allow a certain improvement in robustness and resolution compared to classical linear Fourier analysis. Most important are the so-called modern nonlinear methods (Maximum entropy, Capon, MUSIC, ESPRIT, etc.) which rely primarily on exploiting the structure of Toeplitz matrices and their connection to the autocorrelation function. The research under the present grant has led to a natural evolution of these methods with substantial benefits. Our current aim is to study limits of resolution and robustness in spectral analysis and, in parallel, to develop reliable high resolution methods. The starting point has been to consider statistics other than the ordinary partial autocorrelation sequence. It turns out that, in the most general form, such statistics can be cast in the form of the state covariance of a linear filter driven by the stochastic process. We have shown that a state covariance has a very special structure and can be identified with the Pick matrix of an associated analytic interpolation problem. The fertile connection between generalized analytic interpolation and stochastic processes allows an elegant theory for identifying power spectra consistent with such