Discrete Subspace Multiwindow Gabor Frames and Their Duals

This paper addresses discrete subspace multiwindow Gabor analysis. Such a scenario can model many practical signals and has potential applications in signal processing. In this paper, using a suitable Zak transform matrix we characterize discrete subspace mixed multi-window Gabor frames (Riesz bases and orthonormal bases) and their duals with Gabor structure. From this characterization, we can easily obtain frames by designing Zak transform matrices. In particular, for usual multi-window Gabor frames (i.e., all windows have the same time-frequency shifts), we characterize the uniqueness of Gabor dual of type I (type II) and also give a class of examples of Gabor frames and an explicit expression of their Gabor duals of type I (type II).

[1]  B. Hirosaki,et al.  An Orthogonally Multiplexed QAM System Using the Discrete Fourier Transform , 1981, IEEE Trans. Commun..

[2]  Christopher Heil,et al.  A DISCRETE ZAK TRANSFORM , 1989 .

[3]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[4]  Jason Wexler,et al.  Discrete Gabor expansions , 1990, Signal Process..

[5]  Izidor Gertner,et al.  The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals , 1991, IEEE Trans. Signal Process..

[6]  Richard S. Orr,et al.  Derivation of the finite discrete Gabor transform by periodization and sampling , 1993, Signal Process..

[7]  Joel M. Morris,et al.  Discrete Gabor expansion of discrete-time signals in l2(Z) via frame theory , 1994, Signal Process..

[8]  A. Janssen.  From continuous to discrete Weyl-Heisenberg frames through sampling , 1997 .

[9]  T. Strohmer,et al.  Gabor Analysis and Algorithms: Theory and Applications , 1997 .

[10]  Helmut Bölcskei,et al.  Discrete Zak transforms, polyphase transforms, and applications , 1997, IEEE Trans. Signal Process..

[11]  Y. Zeevi,et al.  Analysis of Multiwindow Gabor-Type Schemes by Frame Methods☆ , 1997 .

[12]  Martin Vetterli,et al.  Tight Weyl-Heisenberg frames in l2(Z) , 1998, IEEE Trans. Signal Process..

[13]  O. Christensen,et al.  Weyl-Heisenberg frames for subspaces of L^2(R) , 1998, math/9811146.

[14]  Yehoshua Y. Zeevi,et al.  Multi-window Gabor schemes in signal and image representations , 1998 .

[15]  Shidong Li.  Discrete multi-Gabor expansions , 1999, IEEE Trans. Inf. Theory.

[16]  O. Christensen,et al.  Weyl-Heisenberg frames for subspaces of ²() , 2000 .

[17]  Jean-Pierre Gabardo,et al.  Subspace Weyl-Heisenberg frames , 2001 .

[18]  Yehoshua Y. Zeevi,et al.  Multiwindow Gabor-type Representations and Signal Representation by Partial Information , 2001 .

[19]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[20]  Deguang Han,et al.  The uniqueness of the dual of Weyl-Heisenberg subspace frames , 2004 .

[21]  Deguang Han,et al.  Balian–Low phenomenon for subspace Gabor frames , 2004 .

[22]  Peter L. Søndergaard,et al.  Gabor frames by sampling and periodization , 2007, Adv. Comput. Math..

[23]  Bruno Torrésani,et al.  Time-Frequency Jigsaw Puzzle: Adaptive Multiwindow and Multilayered Gabor Expansions , 2007, Int. J. Wavelets Multiresolution Inf. Process..

[24]  C. Heil.  History and Evolution of the Density Theorem for Gabor Frames , 2007 .

[25]  J. Gabardo,et al.  Oblique duals associated with rational subspace Gabor frames , 2008 .

[26]  Qiaofang Lian,et al.  Gabor systems on discrete periodic sets , 2009 .

[27]  Yun-Zhang Li,et al.  Density results for Gabor systems associated with periodic subsets of the real line , 2009, J. Approx. Theory.

[28]  Qiao-Fang Lian,et al.  The duals of Gabor frames on discrete periodic sets , 2009 .

[29]  H. Feichtinger,et al.  Constructive reconstruction from irregular sampling in multi-window spline-type spaces , 2010 .

[30]  Hans G. Feichtinger,et al.  Constructive realization of dual systems for generators of multi-window spline-type spaces , 2010, J. Comput. Appl. Math..

[31]  Qiao-Fang Lian,et al.  Gabor frame sets for subspaces , 2011, Adv. Comput. Math..

[32]  Qiaofang Lian,et al.  Multi-window Gabor frames and oblique Gabor duals on discrete periodic sets , 2011 .

[33]  Gabor families in $l^{2}(\mathbb{Z}^{d})$ , 2012 .

[34]  Hans G. Feichtinger,et al.  Advances in Gabor Analysis , 2012 .

[35]  Yan Zhang,et al.  Rational time-frequency multi-window subspace Gabor frames and their Gabor duals , 2014 .