Finite discrete Gabor transform relations under periodization and sampling

Finite, discrete equations are derived for the digital computation of the Gabor transform. Using the method of periodizing and sampling, many of the significant Gabor formulas of the continuous parameter theory are reduced to their finite, discrete counterparts in a way that preserves much of the nature of the original theory. The entities treated include the: (i) signal, or data, to be expanded, (ii) window function, (iii) biorthogonal function, (iv) Gabor coefficients, (v) Zak transforms, (vi) inner products in the time and Zak domains, (vii) sampled auto- and cross-ambiguity functions of the signal and window, and (viii) the Fourier transforms of the signal and window