Signal Reconstruction and Analysis Via New Techniques in Harmonic and Complex Analysis

Abstract

Abstract : We have used tools from theory of harmonic analysis and number theory to extend existing theories and develop new approaches to problems. This work has focused on several areas. We have developed algorithms on extensions of Euclidean domains which have led to new computationally straightforward algorithms for parameter estimation for periodic point processes, and in particular, for sparse, noisy data. This is useful in the analysis of radar and sonar data, among other things. We have shown why Fourier analytic methods, e.g., Wiener s periodogram, do not produce maximum likelihood estimates for the sparse data sets on which our methods work. We are also working on extending our work to multiply periodic processes. We have also extended our work on multichannel deconvolution. We have used the tools from multichannel deconvolution to develop a new procedure for multi-rate sampling. We have investigated applying these techniques to develop a new procedure for A D conversion. We have also extended the work on both deconvolution and sampling to radial domains, exploiting coprime relationships among zero sets of Bessel functions. We have also discussed applications of these ideas to specific applied problems, including radar and sonar. We have developed interlinked wavelet bases, interlinked via number-theoretic conditions that proved useful for both multi-channel deconvolution and multi-rate sampling. We have extended these ideas to operator theory, creating sets of strongly coprime chirp and chirplet operators. Finally, we have developed a new way of visualizing mappings of the complex plane. We have a given function evolve in time, starting from the unaltered complex plane and ending with the range of the function. We use a computer to develop a frame-by-frame movie of the evolution, which appears to be a continuous deformation. We are using this technique to create an electronic dictionary of conformal mappings.