M.Sc Thesis
M.Sc Student Kopaigorodski Alex On Signal and Image Reconstruction from Discrete Data Department of Electrical Engineering Professor Moshe Porat

Abstract

We introduce two methods for signal reconstruction from discrete data. The methods are based on Prolate Spheroidal Wave Functions (PSWF) of order zero. The first method is for sampling and reconstructing band-limited signals known only on finite intervals. Nyquist sampling theorem is widely accepted as a means for representing band-limited signals by their samples, however, its main drawback is that practical signals are time-limited and therefore not band-limited. As a consequence, in most cases Nyquist theorem cannot be used to perfectly reconstruct signals, though it can be used as an approximation for the acquisition of signals. When a coarse approximation is not sufficient, we propose a new representation approach, providing better approximation than uniform Nyquist-based sampling. This method is useful in cases where the product of the finite interval and the signal’s band-width is small. The method is based on a PSWF series expansion of the signal. It is shown that only few first terms of the series are significant, while other terms can be practically omitted in many cases. Furthermore, the theory described here for one-dimensional signals can be readily generalized to two-dimensional signals, i.e., images. The reconstruction error is derived and compared to Monte-Carlo simulation results. The derivation of the proposed method includes numerical method of sampling-reconstruction and error analysis. Simulations results are presented and discussed.

The second method is for band-limited approximation of general continuous signals. According to Riemann-Lebesgue lemma, Fourier transform of any Lebesgue integrable function tends to zero as the frequency goes to infinity. Furthermore, the smoothness of the function is reflected in the decay of its Fourier Transform. As a result, any continuous signal can be approximated by a band-limited signal. In order to approximate a continuous signal it is possible to filter it out and apply the sampling theorem. The drawback of this approach is that the filter's output is not time limited. We propose methods for approximating continuous signals by band limited signals in L1, L2 and L?_infinity norms using PSWFs. The core of the L_infinity approximation method is an exchange algorithm, where the optimal solution is found in an iterative manner. The proposed L1 minimization method is of a gradient decent type algorithm. Our conclusion is that the proposed approach to signal representation using PSWF could be superior to presently available methods and may be instrumental in practical cases of finite support signals.