Multiwindow Gabor-type transform for signal representation and analysis
Meir Zibulski, Yehoshua Y. Josh Zeevi · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1995
The Gabor scheme is generalized to incorporate several window functions as well as kernels other than the exponential. The properties of the sequence of representation functions are characterized by an approach based on the concept of frames. the frame operator associated with the multi-window Gabor-type frame, is examined for a rational oversampling rate by representing the frame operator as a finite order matrix-valued function in the Zak Transform domain. Completeness and frame properties of the sequence of representation functions are examined in relation to the properties of the matrix-valued function. Calculation of the frame bounds and the dual frame, as well as the issue of tight frames are considered. It is shown that the properties of the sequence of representation functions are essentially not changed by replacing the widely-used exponential kernel with other kernels. The issue of a different sampling rate for each window is also considered. The so-called Balian-Low theorem is generalized to consideration of a scheme of multi-windows, which makes it possible to overcome the constraint imposed by the original theorem in the case of a single window.