Abstract.
In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. The invertibility condition then follows from Cramer’s rule and Wiener’s lemma for this special class of matrices. The problem results from a well known approach of studying the invertibility properties of the Gabor frame operator in the rational case. The presented approach gives further insights into Gabor frames. In particular, it can be applied for both the continuous (on \({\Bbb R}^d\)) and the finite discrete setting. In the latter case, we obtain algorithmic schemes for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature.
Similar content being viewed by others
References
O Christensen (2003) An introduction to frames and Riesz bases Birkhäuser Basel Occurrence Handle1017.42022
I Daubechies HJ Landau Z Landau (1995) ArticleTitleGabor time-frequency lattices and the Wexler-Raz Identity J Four Anal Appl 1 437–478 Occurrence Handle0888.47018 Occurrence Handle10.1007/s00041-001-4018-3 Occurrence Handle1350701
PJ Davis (1994) Circulant Matrices EditionNumber2 Chelsea Publ New York Occurrence Handle0898.15021
HG Feichtinger W Kozek (1998) Quantization of TF–lattice invariant operators on elementary LCA groups HG Feichtinger T Strohmer (Eds) Gabor Analysis and Algorithms: Theory and Applications Birkhäuser Boston 233–266
HG Feichtinger T Strohmer (1998) Gabor Analysis and Algorithms SeriesTitleTheory and Applications Birkhäuser Boston Occurrence Handle0890.42004
HG Feichtinger T Strohmer (2003) Advances in Gabor Analysis Birkhäuser Boston Occurrence Handle1005.00015
GB Folland (1989) Harmonic Analysis in Phase Space Univ Press Princeton, NJ Occurrence Handle0682.43001
K Gröchenig (2001) Foundations of Time-Frequency Analysis Birkhäuser Boston Occurrence Handle0966.42020
Gröchenig K, Kozek W (1997) Weyl-Heisenberg Systems and Wiener’s Lemma. Unpublished notes
K Gröchenig M Leinert (2003) ArticleTitleWiener’s Lemma for twisted convolution and Gabor frames J Amer Math Soc 17 1–18 Occurrence Handle10.1090/S0894-0347-03-00444-2
AJEM Janssen (1995) ArticleTitleDuality and biorthogonality for Weyl-Heisenberg Frames J Fourier Anal Appl 1 403–436 Occurrence Handle0887.42028 Occurrence Handle10.1007/s00041-001-4017-4 Occurrence Handle1350700
T Werther YC Eldar N Subbanna (2005) ArticleTitleDual Gabor Frames: Theory and computational aspects IEEE Transactions on Signal Processing 53 4147–4158 Occurrence Handle10.1109/TSP.2005.857049 Occurrence Handle2241120
Werther T, Matusiak E, Subbanna N, Eldar YC (2005) A unified approach to dual Gabor windows. IEEE Transactions on Signal Processing, accepted for publication
M Zibulski YY Zeevi (1997) ArticleTitleAnalysis of multiwindow Gabor-type schemes by frame methods Appl Comp Harm Anal 4 188–221 Occurrence Handle0885.42024 Occurrence Handle10.1006/acha.1997.0209 Occurrence Handle1448221
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eldar, Y., Matusiak, E. & Werther, T. A Constructive Inversion Framework for Twisted Convolution. Mh Math 150, 297–308 (2007). https://doi.org/10.1007/s00605-006-0438-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-006-0438-0