Zak Transform Methods
Any Gabor system G(g, α,β ) possesses a doubly periodic structure. In a technical sense, we have exploited this property in Chapters 6 and 7 through periodization tricks and the Poisson summation formula. However, the sequential application to the variables x and ω separately leads to an unnatural asymmetry, which was most pronounced in Walnut’s representation. The Zak transform offers a convenient tool for a more symmetric treatment of Gabor systems. Since the Zak transform is just a version of the Poisson summation formula, it is related to Fourier series and can be discretized easily. In engineering it has therefore become the preferred tool for the analysis of Gabor frames. In many important special cases, the invertibility and the spectrum of the frame operator can be characterized explicitly and clearly using the Zak transform.
KeywordsOrthonormal Basis Riesz Basis Tight Frame Gabor Frame Frame Operator
Unable to display preview. Download preview PDF.