Complete Gabor transformation for signal representation

An analytical method for transforming a function into complete Gabor representation was proposed based on biorthogonal functions by M. Bastiaans [6]. However, no efficient numerical algorithm has been developed for discrete Gabor transformation because the Gabor elementary functions are not orthogonal and there is no efficient numerical method for computing the biorthogonal functions. In this work, properties of the Gabor transformation are discussed. The discrete complete Gabor transformation can be expressed in matrix notation. Complete Gabor coefficients can be found by multiplying the inverse of the Gabor matrix and the signal vector. The Gabor matrix can be decomposed into the product of a sparse constant complex matrix (which has known inverse) and another sparse matrix which depends only on the window function. A fast algorithm is suggested to compute the inverse of the window function matrix. So, discrete signals can be transformed into generalized nonorthogonal Gabor representations efficiently. A comparison is made between this method and the analytical method. The relation between the window function matrix and the biorthogonal functions is demonstrated. A numerical computation method for the biorthogonal functions is proposed.