Complete Gabor transformation for signal representation

Properties of the Gabor transformation used for image representation are discussed. The properties can be expressed in matrix notation, and the complete Gabor coefficients can be found by multiplying the inverse of the Gabor (1946) matrix and the signal vector. The Gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function. A fast algorithm is suggested to compute the inverse of the window function matrix, enabling discrete signals to 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