Modified Zak transform for the quincunx-type Gabor lattice

Gabor's 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 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