A Unified Approach to Dual Gabor Windows

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 R^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