Gabor's signal expansion on a quincunx lattice and the modified Zak transform

Gabor's expansion of a signal on a quincunx lattice with oversampling by a rational factor is presented for continuous-time signals. It is shown how a modified Zak transform instead of the ordinary Zak transform can be helpful in determining Gabor's signal expansion coefficients and how it can be used in finding the dual window. Furthermore, some examples of dual windows for the quincunx case are given and compared with dual windows for the rectangular case.