TIME AND FREQUENCY SPLIT ZAK TRANSFORM FOR FINITE GABOR EXPANSION

The relationship between finite discrete Zak transform and finite Gabo r expansion are well discussed in this paper. In this paper, we presen t two DFT-based algorithms for computing Gabor coefficients. One is ba sed upon the time-split Zak transform, the other is based upon the fre quency-split Zak transform. These two methods are lime and frequency d ual pairs. With the help of Zak transform, the closed-form solutions f or analysis basis can also be derived while the oversampling ratio is an integer. Moreover, we extend the relationship between finite discre te Zak transform and Gabor expansion to the 2-D case and compute 2-D G abor expansion coefficients through 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-freque ncy-split.