OPTIMAL RECOVERY OF SIGNALS FROM LINEAR MEASUREMENTS AND PRIOR KNOWLEDGE (EXTRAPOLATION, DETERMINISTIC, BAND-LIMITED, SPECTRAL, ESTIMATION)

The problem of band-limited extrapolation is studied in a general framework of estimation of a signal in an ellipsoidal signal class from the value of a linear transformation. The dissertation deals with finite-length sequences and consequently with the Discrete Fourier Transform for our frequency domain. An algorithm is proposed for defining the signal class from the data. Optimal Recovery theory is described for estimating the value of a desired linear transformation from a given linear transformation and a bound on the norm in a Hilbert space. The optimal estimation procedure requires that we find the minimum norm signal that satisfies the linear measurements. With additive errors, we require a regularized solution to the minimum norm problem. A filter class is an ellipsoidal signal class defined for band-limited sequences with a weighted frequency domain norm. This weight is the squared magnitude of a filter function that defines the class. The minimum norm signal in the filter class that satisfies a given set of samples is the signal estimate. It wil usually have frequency contents that resembles that of the filter function. We next develop a procedure to define the filter from the given samples in a recursive manner. The estimate found at one iteration is used to define the filter of the class that is used to estimate at the next iteration. The new filter is a windowed version of the previous estimate, where the window is placed in the region of the given samples. At each iteration, this provides a smoothing of the previously estimated spectrum as well as a dependence of the filter on the data. A convergence analysis for the case where no windowing is done shows a tendency to obtain narrow-band spectra. The extension to two-dimensional signals is described and examples to illustrate this signal class modification algorithm as an interpolator/extrapolator and as a spectral estimator are provided