Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry

Gabor's signal expansion and the Gabor transform are formulated on a non-orthogonal time-frequency lattice instead of on the traditional rectangular lattice. The reason for doing so is that a non-orthogonal sampling geometry might be better adapted to the form of the window functions (in the time-frequency domain) than an orthogonal one. Oversampling in the Gabor scheme, which is required to have mathematically more attractive properties for the analysis window, then leads to better results in combination with less oversampling. The new procedure presented in this paper is based on considering the non-orthogonal lattice as a sub-lattice of a denser orthogonal lattice that is oversampled by a rational factor. In doing so, Gabor's signal expansion on a non-orthogonal lattice can be related to the expansion on an orthogonal lattice (restricting ourselves, of course, to only those sampling points that are part of the non-orthogonal sub-lattice), and all the techniques that have been derived for rectangular sampling can be used, albeit in a slightly modified form

Gabor's signal expansion for a non-orthogonal sampling geometry

Gabor's signal expansion and the Gabor transform on a rectangular lattice have been introduced, along with the Fourier transform of the array of expansion coefficients and the Zak transforms of the signal and the window functions. Based on these Fourier and Zak transforms, the sum-of-products forms for the Gabor expansion and the Gabor transform, which hold in the rationally oversampled case, have been derived. We have then studied Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry. We have done this by considering the non-orthogonal lattice as a sub-lattice of an orthogonal lattice. This procedure allows us to use all the formulas that hold for the orthogonal sampling geometry. In particular we can use the sum-of-products forms that hold in the case of a rationally oversampled rectangular lattice

Gabor's signal expansion for a non-orthogonal sampling geometry

Gabor's signal expansion and the Gabor transform on a rectangular lattice have been introduced, along with the Fourier transform of the array of expansion coefficients and the Zak transforms of the signal and the window functions. Based on these Fourier and Zak transforms, the sum-of-products forms for the Gabor expansion and the Gabor transform, which hold in the rationally oversampled case, have been derived. We have then studied Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry. We have done this by considering the non-orthogonal lattice as a sub-lattice of an orthogonal lattice. This procedure allows us to use all the formulas that hold for the orthogonal sampling geometry. In particular we can use the sum-of-products forms that hold in the case of a rationally oversampled rectangular lattice