Analysis of Multiwindow Gabor-Type Schemes by Frame Methods

Abstract

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. Utilizing the piecewise Zak transform (PZT), the frame operator associated with the multiwindow Gabor-type frame is examined for a rational oversampling rate by representing the frame operator as a finite-order matrix-valued function in the PZT domain. Completeness and frame properties of the sequence of representation functions are examined in relation to the properties of the matrix-valued function. Calculations 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. Some examples and the issue of a different sampling rate for each window are also considered. The so-called Balian-Low theorem is generalized to consideration of a scheme of multiwindows which makes it possible to overcome in a way the constraint imposed by the original theorem in the case of a single Window. © 1997 Academic Press.