Discrete Gabor transform

The Gabor expansion, which maps the time domain signal into the joint time and frequency domain, has long been recognized as a very useful tool in signal processing. Its applications, however, were limited due to the difficulties associated with selecting the Gabor coefficients. Because timeshifted and frequency-modulated elementary functions in general do not constitute an orthogonal basis, the selections of the Gabor coefficient are not unique. One solution to this problem, developed by Bastiaans, is to introduce an auxiliary biorthogonal function. Then, the Gabor coefficient is computed by the usual inner product rule. Unfortunately, it is not easy to determine the auxiliary biorthogonal function for an arbitrary given synthesis function and sampling pattern. While less success was found in the continuous case, we present a discrete solution in this paper, which is named the discrete Gabor transform (DGT). For a given synthesis window and sampling pattern, computing the auxiliary biorthogonal function of the DGT is nothing more than solving a linear system. The DGT presented applies for both finite as well as infinite sequences. Using the advantages of the nonuniqueness of the auxiliary biorthogonal function at oversampling, we further introduce the so-called orthogonal-like DGT. As the DFT (a discrete realization of the continuous-time Fourier transform), the DGT introduced provides a feasible vehicle to implement the useful Gabor expansion.