Product forms in Gabor analysis for a quincunx-type sampling geometry

Recently a new sampling lattice - the quincunx lattice - has been introduced [1] as a sampling geometry in the Gabor scheme, which geometry is different from the traditional rectangular sampling geometry. In this paper we will show how results that hold for rectangular sampling (see, for instance, [2,3]) can be transformed to the quincunx case. In particular we will concentrate on the well-known product forms [2] of Gabor's signal expansion and the Gabor transform, in terms of the Fourier transform of the expansion coefficients and the Zak transforms of the signal and the window functions; these product forms hold in the case of critical sampling, to which case we will confine ourselves. We will show that identical product forms can be formulated in the case of a quincunx sampling geometry, as well, but then in terms of a modified version of the Zak transform. [1] P. Prinz, "Calculating the dual Gabor window for general sampling sets," IEEE Trans. Signal Processing, 44 (1996) 2078-2082. [2] M.J. Bastiaans, "Gabor's signal expansion and its relation to sampling of the sliding-window spectrum," in Advanced Topics in Shannon Sampling and Interpolation Theory, R.J. Marks II, Ed., Springer, New York (1993) 1-35. [3] H.G. Feichtinger and T. Strohmer, Eds., Gabor Analysis and Algorithms: Theory and Applications. Berlin: Birkhauser (1998).