Efficient Algorithms for Discrete Gabor Transforms on a Nonseparable Lattice

The Discrete Gabor Transform (DGT) is the most commonly used transform for signal analysis and synthesis using a linear frequency scale. It turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup. Most of the literature focuses on separable subgroups, in this paper we will survey existing methods for a generalization to arbitrary groups, as well as present an improvement on existing methods. Comparisons are made with respect to the computational complexity, and the running time of optimized implementations in the C programming language. The new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download. By summarizing general background information on the state of the art, this article can also be seen as a research survey, sharing with the readers experience in the numerical work in Gabor analysis.

[1]  Steven G. Johnson,et al.  A Modified Split-Radix FFT With Fewer Arithmetic Operations , 2007, IEEE Transactions on Signal Processing.

[2]  B. Liu,et al.  Implementation of the Digital Phase Vocoder Using the Fast Fourier Transform , 2022 .

[3]  Bruno Torrésani,et al.  Time-Frequency and Time-Scale Analysis , 1999 .

[4]  Yehoshua Y. Zeevi,et al.  The Generalized Gabor Scheme and Its Application in Signal and Image Representation , 1998 .

[5]  Ronald W. Schafer,et al.  Design and simulation of a speech analysis-synthesis system based on short-time Fourier analysis , 1973 .

[6]  van Aj Arno Leest,et al.  Non-separable Gabor schemes : their design and implementation , 2001 .

[7]  Peter Prinz.  Calculating the dual Gabor window for general sampling sets , 1996, IEEE Trans. Signal Process..

[8]  Mj Martin Bastiaans,et al.  Gabor's discrete signal expansion and the discrete Gabor transform on a non-separable lattice , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

[9]  Antoine Petitet,et al.  Minimizing development and maintenance costs in supporting persistently optimized BLAS , 2005 .

[10]  Thomas Strohmer,et al.  Optimal OFDM design for time-frequency dispersive channels , 2003, IEEE Trans. Commun..

[11]  Yehoshua Y. Zeevi,et al.  Discrete multiwindow Gabor-type transforms , 1997, IEEE Trans. Signal Process..

[12]  Norbert Kaiblinger,et al.  Metaplectic operators for finite abelian groups and ℝd , 2009 .

[13]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[14]  Martin J. Bastiaans,et al.  Implementations of non-separable Gabor schemes , 2004, 2004 12th European Signal Processing Conference.

[15]  Markus Püschel,et al.  Offline library adaptation using automatically generated heuristics , 2010, 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS).

[16]  H. Feichtinger,et al.  Metaplectic Operators on C N , 2006 .

[17]  A. Weil.  Sur certains groupes d'opérateurs unitaires , 1964 .

[18]  H. Helms.  Fast Fourier transform method of computing difference equations and simulating filters , 1967, IEEE Transactions on Audio and Electroacoustics.

[19]  Yehoshua Y. Zeevi,et al.  Signal- and image-component separation by a multi-window Gabor-type scheme , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[20]  Peter G. Casazza,et al.  Fourier Transforms of Finite Chirps , 2006, EURASIP J. Adv. Signal Process..

[21]  R. C. Whaley,et al.  Minimizing development and maintenance costs in supporting persistently optimized BLAS , 2005, Softw. Pract. Exp..

[22]  Peter L. Søndergaard,et al.  Iterative Algorithms to Approximate Canonical Gabor Windows: Computational Aspects , 2006 .

[23]  L. D. Abreu,et al.  An inverse problem for localization operators , 2012, 1202.5841.

[24]  Mj Martin Bastiaans,et al.  Gabor's signal expansion for a non-orthogonal sampling geometry , 2003 .

[25]  M.J. Bastiaans,et al.  Modified Zak transform for the quincunx-type Gabor lattice , 1998, Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380).

[26]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[27]  M. Bastiaans,et al.  From the rectangular to the quincunx Gabor lattice via fractional Fourier transformation , 1998, IEEE Signal Processing Letters.

[28]  Thomas Strohmer,et al.  Numerical algorithms for discrete Gabor expansions , 1998 .

[29]  Peter L. Søndergaard,et al.  Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window , 2012 .

[30]  Hans G. Feichtinger,et al.  POCS approach to Gabor analysis , 1998, Other Conferences.

[31]  Thomas G. Stockham,et al.  High-speed convolution and correlation , 1966, AFIPS '66 (Spring).

[32]  Nicki Holighaus,et al.  LTFAT: The Large Time-Frequency Analysis Toolbox v. 1.4 , 2012 .

[33]  Sergios Theodoridis,et al.  A Novel Efficient Cluster-Based MLSE Equalizer for Satellite Communication Channels with-QAM Signaling , 2006, EURASIP J. Adv. Signal Process..

[34]  J.B. Allen,et al.  A unified approach to short-time Fourier analysis and synthesis , 1977, Proceedings of the IEEE.

[35]  Martin J. Bastiaans,et al.  Gabor's signal expansion and the Gabor transform based on a non-orthogonal sampling geometry , 2001, Proceedings of the Sixth International Symposium on Signal Processing and its Applications (Cat.No.01EX467).

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

[37]  L. T'oth,et al.  Representing and Counting the Subgroups of the Group , 2012, 1211.1797.

[38]  Bruno Torrésani,et al.  The Linear Time Frequency Analysis Toolbox , 2012, Int. J. Wavelets Multiresolution Inf. Process..

[39]  Marc Geilen,et al.  On the discrete Gabor transform and the discrete Zak transform , 1996, Signal Process..

[40]  Hans G. Feichtinger,et al.  Gabor Analysis for Imaging , 2015, Handbook of Mathematical Methods in Imaging.

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

[42]  Michiel Hazewinkel,et al.  METAPLECTIC OPERATORS ON ℂn , 2007 .

[43]  H. Feichtinger,et al.  Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view , 1993 .

[44]  H. Feichtinger,et al.  METAPLECTIC OPERATORS ON C , 2022 .

[45]  Yehoshua Y. Zeevi,et al.  Oversampling in the Gabor scheme , 1993, IEEE Trans. Signal Process..