Derivation of Gabor transform relations using Bessel's equality

Abstract

Derivation of the key formulas of Gabor representation is usually facilitated by relationships taken from Fourier and Zak transform theory. We show that several of these fundamental formulas can be obtained directly from Bessel's equality for biorthogonal systems, aided only by the shift theorem for inner products of Gabor basis functions. The method is brought to the attention because it simplifies derivations in some cases and may be useful in finding new results in time-frequency representation theory.