DERIVATION OF THE FINITE DISCRETE GABOR TRANSFORM BY PERIODIZATION AND SAMPLING

Recent interest in the Gabor transform for time-frequency signal analysis can be attributed in part to increased knowledge about the accuracy, stability and complexity of algorithms for computing the transforms. Behavior of the computations depends on. among other things, the manner in which the continuous-parameter equations are made discrete and finite. The most straightforward means, truncating the time functions to compact support and sampling, relinquishes some control and blurs the relationship of the discrete equations to the original transforms. A more satisfying discretization and finitization process that preserves relations to the continuous parameter case is found by periodization and sampling, the same method used to obtain the finite, discrete Fourier transform from the Fourier integral. By this method we derive the finite, discrete Gabor transform equations from their continuous parameter counterparts. in the process explicitly exhibiting the aliasings that permit one periodic sequence to be the finite, discrete Gabor transform of the other. By examining the various forms in which the Gabor equations can be expressed, we discover how the input, window, biorthogonal function, Gabor coefficients and Zak transforms map under periodization and sampling.