Study of Sum of Complex Exponential Signals

Abstract

This thesis focuses mainly on a finite sum of complex exponential signals. This kind of signal model usually captures a variety of applications involving damped signals such as magnetic resonance spectroscopy, gravitational wave bursts, synchronized neuronal hyppocampal rhythms, financial modelling and the acoustic localization of underground oil field. Each coefficient and complex exponential function in the sum signal can be uniquely characterized by the z-transform of the sampling signal of the continuous-time sum signal as its poles and residues. We study the estimation of all these parameters using Pade approximation theory. The Pade approximation theory deals primarily with the optimally asymptotic approximation of a rational fraction with each of the denominator and numerator having an allowable degree to the z- transform of a given discrete-time signal. The poles and the residues of the rational fraction normally provide relatively precise information on the poles and the residues of the original z-transform. Particularly for the sum of complex exponential signals, its poles and the residues can be completely characterized by its Pade approximation rational fraction with a proper degree constraint. Hence, the Pade approximation theory becomes a very strong mathematical tool for studying such sum of complex exponential signals. We make two kinds of contributions: (a) When the residues are all equal to each other, we derive a closed-form formula for determining the Pade approximation rational fraction as well as for predicting the future data points. (b) A co-prime sampling scheme is proposed, with a closed-form algorithm being provided for efficiently determining the original poles and residues using elementary diophantine theory.