Discrete Gabor transform

Abstract

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.

Comparison of the Cross Deleted Wigner Representation and the Matching Pursuit Distribution (Via Adaptive Signal Decomposition)

“…The major problem with the Gabor expansion is how to find the biorthogonal analysis function. In this paper we consider the orthogonal-like Discrete Gabor Transform (DGT) for an infinite signal [9] -as the real signal is usually very long sequence. In the orthogonal-like DGT, the Gabor coefficients can be thought of as the measure of similarity between the underlying signal, ( ) x i , and the individual basis functions, } { ,n m h…”

“…As long as the synthesis window function, ) (i h , is localized, the orthogonal-like DGT will well reflect signal local behaviors because Δ . So the synthesis window is selected to be Gaussian and the biorthogonal analysis function that has the most similarity to the synthesis function is determined according to the algorithm described in [9]. Fig.3 shows the calculated analysis and synthesis functions.…”

Lung Sound Noise Reduction Using Gabor Time-Frequency Masking

“…This is called the oversampled case and the synthesis functions are no longer lineary independent. At the critical sampling case 2 T π Ω = , the logons are linearly independent, but are not orthogonal in general (Balian-Low obstruction) [4,5]. This means that the Gabor coefficients c m,n are not simply the projections of f(t) onto the corresponding logons g(t) (i.e.…”

“…In this work, we used the generalized Gabor expansion for lung sound signal modeling. For finite discrete-time signals Gabor synthesis and analysis equations are of the form respectively [5] ( ) ( ) ( ) ( )…”

“…Biorthogonal sequence was computed using algorithms in [5]. Both the window and the biothogonal sequence are illustrated in Figure 1.…”