Discrete multiwindow Gabor-type transforms

The discrete (finite) Gabor scheme is generalized by incorporating multiwindows. Two approaches are presented for the analysis of the multiwindow scheme: the signal domain approach and the Zak transform domain approach. Issues related to undersampling, critical sampling, and oversampling are considered. The analysis is based on the concept of frames and on generalized (Moore-Penrose) inverses. The approach based on representing the frame operator as a matrix-valued function is far less demanding from a computational complexity viewpoint than a straightforward matrix algebra in various operations such as the computation of the dual frame. DFT-based algorithms, including complexity analysis, are presented for the calculation of the expansion coefficients and for the reconstruction of the signal in both signal and transform domains. The scheme is further generalized and incorporates kernels other than the complex exponential. Representations other than those based on the dual frame and nonrectangular sampling of the combined space are considered as well. An example that illustrates the advantages of the multiwindow scheme over the single-window scheme is presented.