Gabor's signal expansion and a modified Zak transform for a quincunx-type sampling geometry

Gabor's (1946) signal expansion and the Gabor transform are formulated on a quincunx lattice instead of on the traditional rectangular lattice; the representation of the quincunx lattice is based on the rectangular lattice via either a shear operation or a rotation operation. A modified Zak (1972) transformation is defined with the help of which Gabor's signal expansion and the Gabor transform can be brought into product forms that are identical to the ones that are well known for the rectangular sampling geometry. The shear operation on the lattice is associated with an operation on the synthesis and the analysis window, consisting of a multiplication by a quadratic-phase function. Following this procedure, the well-known biorthogonality condition for the window functions in the rectangular sampling geometry can be directly translated to the quincunx case.