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.