Gabor's signal expansion 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: the set of shifted and modulated versions of the usual Gaussian synthesis window, for instance, corresponding to circular contour lines in the time-frequency domain, can be arranged more tightly in a hexagonal geometry than in a rectangular 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 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 - including an optical means of generating Gabor's expansion coefficients via the Zak transform in the case of integer oversampling - can be used, albeit in a slightly modified form