A Unified Approach to Dual Gabor Windows

摘要

In this paper, we describe a new method for studying the invertibility of Gabor frame operators. Our approach can be applied to both the continuous (on $BBR^{d}$) and the finite discrete setting. In the latter case, we obtain algorithms for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature. The framework we propose can also be used to derive other (known) results in Gabor theory in a unified way such as the Zibulski–Zeevi representation. The approach we suggest is based on an adequate splitting of the twisted convolution, which, in turn, provides another twisted convolution on a finite cyclic group. By analogy with the twisted convolution of finite discrete signals, we derive a mapping between the sequence space and a matrix algebra which preserves the algebraic structure. In this way, the invertibility problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. Using Cramer's rule and proving Wiener's lemma for this special class of matrices, we obtain an invertibility criterion that can be applied to a variety of different settings. This alternative approach provides further insight into Gabor frames, as well as a unified framework for Gabor analysis.