Can one evaluate the Gabor expansion using Gabor's iterative algorithm?

Abstract

D. Gabor's original (1946) approach for calculation of expansion coefficients was examined and it was found that convergence of the suggested algorithm is conditional on the window function chosen. The method of analysis enables a process of synthesis, whereby eligible windows can be defined, ensuring convergence of the algorithm for every represented signal. For windows having a finite support of width D, the coefficients calculated according to the algorithm are obtained after a single iteration. It is proved that in the special case of a Gaussian window the algorithm converges only for specific signals. On the basis of the conclusions drawn, conditions are formulated which ensure convergence of the Gabor algorithm, permitting examination of alternative windows. >