Time Frequency Split Zak Transform for Finite Gabor Expansion

The relationship between the finite discrete Zak transform and the finite Gabor expansion are discussed in this paper. We present two DFT-based algorithms for computing Gabor coefficients. One is based upon the time-split Zak transform, the other upon the frequency-split Zak transform. These two methods are time and frequency dual pairs. Furthermore, we extend the relationship between the finite discrete Zak transform and the Gabor expansion to the 2-D case and compute 2-D Gabor expansion coefficients through the 2-D discrete Zak transform and 4-D DFT. Four methods can be applied in the 2-D case. They are time-time-split, time-frequency-split, frequency-time-split and frequency-frequency-split.