Abstract
In the last 40 years the foundations of Gabor analysis, even in the context of locally compact Abelian (LCA) groups, have been widely developed. We know a lot about function spaces, in particular modulation spaces, characterization of these spaces via Gabor expansions, or map** properties of operators between such spaces, even the description of solutions for PDEs can nowadays be given in this context. In contrast, the applied literature gives the impression that the computation of dual Gabor windows in the standard situation, i.e. for the Hilbert space \({{{\varvec{L}}^2}({\mathbb R})}\), and a time–frequency lattice of the form \(\varLambda = a {\mathbb Z} \times {b} {{\mathbb Z}}\) is still the most important problem in (numerical) Gabor analysis. The emphasis of this note is on the value of numerical work, which is much more than just numerical realization of theoretical concepts. It has been in many cases the inspiration for the derivation of theoretical results, based on sometimes surprising observations or systematic numerical simulations. According to our experience, numerical Gabor analysis provides a lot of additional insight about the concrete situation; it may suggest new directions and ask for new theory, but of course efficient algorithms often make use of underlying theory. Overall, we observe that there is an urgent need for a stronger link between computational and theoretical Gabor analysis. The note also contains a number of suggestions and even conjectures which are likely to encourage research in the direction indicated above.
Summer 2017 and 2018 TU Munich, Inst. Theor. Inf. Science
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Recall that at the MATLAB command line the “\(*\)” represents matrix multiplication resp. matrix–vector multiplication!
References
I. Amidror. Mastering the Discrete Fourier Transform in One, Two or Several Dimensions. Pitfalls and Artifacts, volume 43. London: Springer, 2013.
P. Balazs, M. Dörfler, F. Jaillet, N. Holighaus, and G. A. Velasco. Theory, implementation and applications of nonstationary Gabor frames. J. Comput. Appl. Math., 236(6):1481–1496, 2011.
P. Balazs, H. G. Feichtinger, M. Hampejs, and G. Kracher. Double preconditioning for Gabor frames. IEEE Trans. Signal Process., 54(12):4597–4610, December 2006.
S. Bannert, K. Gröchenig, and J. Stöckler. Discretized Gabor frames of totally positive functions. IEEE Trans. Information Theory, 60(1):159–169, 2014.
S. Bannert. Banach-Gelfand Triples and Applications in Time-Frequency Analysis. Master’s thesis, University of Vienna, 2010.
P. Boggiatto and E. Cordero. Anti-Wick quantization with symbols in \({L}^p\) spaces. Proc. Amer. Math. Soc., 130(9):2679–2685, 2002.
P. Boggiatto, E. Cordero, and K. Gröchenig. Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integr. Equ. Oper. Theory, 48(4):427–442, 2004.
A. Borichev, K. Gröchenig, and Y. I. Lyubarskii. Frame constants of Gabor frames near the critical density. J. Math. Pures Appl., 94(2):170–182, August 2010.
P. G. Casazza and O. Christensen. Weyl-Heisenberg frames for subspaces of \({L}^2({R})\). Proc. Amer. Math. Soc., 129(1):145–154, 2001.
O. Christensen, A. Janssen, H. Kim, and R. Kim. Approximately dual Gabor frames and almost perfect reconstruction based on a class of window functions. Adv. Comput. Math., 1–17, 2018.
O. Christensen. An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser Basel, Second edition, 2016.
L. A. Coburn. Berezin-Toeplitz quantization. In Curto, Ral E (ed) et al, Algebraic Methods in Operator Theory Boston, MA: Birkhäuser 101–108. 1994.
E. Cordero and K. Gröchenig. On the product of localization operators. In M. Wong, editor, Modern Trends in Pseudo-differential Operators, volume 172 of Oper. Theory Adv. Appl., 279–295. Birkhäuser, Basel, 2007.
E. Cordero and K. Gröchenig. Symbolic calculus and Fredholm property for localization operators. J. Fourier Anal. Appl., 12(4):371–392, 2006.
E. Cordero and K. Gröchenig. Time-frequency analysis of localization operators. J. Funct. Anal., 205(1):107–131, 2003.
E. Cordero and L. Rodino. Short-time Fourier transform analysis of localization operators. volume 451, pages 47–68. Providence, RI: American Mathematical Society (AMS), 2008.
E. Cordero and L. Rodino. Wick calculus: a time-frequency approach. Osaka J. Math., 42(1):43–63, 2005.
E. Cordero and A. Tabacco. Localization operators via time-frequency analysis. In Advances in Pseudo-differential Operators, volume 155 of Oper. Theory Adv. Appl., pages 131–147. Birkhäuser, 2004.
E. Cordero, H. G. Feichtinger, and F. Luef. Banach Gelfand triples for Gabor analysis. In Pseudo-differential Operators, volume 1949 of Lecture Notes in Mathematics, pages 1–33. Springer, Berlin, 2008.
E. Cordero, K. Gröchenig, and F. Nicola. Approximation of Fourier integral operators by Gabor multipliers. J. Fourier Anal. Appl., 18(4):661–684, 2012.
E. Cordero, K. Gröchenig, F. Nicola, and L. Rodino. Wiener algebras of Fourier integral operators. J. Math. Pures Appl. (9), 99(2):219–233, 2013.
E. Cordero, F. Nicola, and L. Rodino. Time-frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal., 9(1):1–21, 2010.
E. Cordero, L. Rodino, and K. Gröchenig. Localization operators and time-frequency analysis. In N. e. a. Chong, editor, Harmonic, Wavelet and p-adic Analysis, pages 83–110. World Sci. Publ., Hackensack, 2007.
X.-R. Dai and Q. Sun. The \(abc\)-problem for Gabor systems. Mem. Amer. Math. Soc., 244(1152):ix+99, 2016.
X.-R. Dai and Q. Sun. The abc-problem for Gabor systems and uniform sampling in shift-invariant spaces. In Excursions in Harmonic Analysis, Volume 3, pages 177–194. 2015.
I. Daubechies, A. Grossmann, and I. Meyer. Frames in the Bargmann space of entire functions. Comm. Pure Appl. Math., 41:151–164, 1990.
I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271–1283, May 1986.
I. Daubechies, H. J. Landau, and Z. Landau. Gabor time-frequency lattices and the Wexler-Raz identity. J. Fourier Anal. Appl., 1(4):437–478, 1995.
I. Daubechies. Ten Lectures on Wavelets., volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA, 1992.
I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34(4):605–612, July 1988.
J. G. Daugman. Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. IEEE Trans. Acoustics, Speech and Signal Processing, 36(7):1169–1179, 1988.
R. J. Duffin and A. C. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72:341–366, 1952.
M. Englis. Toeplitz operators and localization operators. Trans. Amer. Math. Soc., 361(2):1039–1052, 2009.
M. Faulhuber and S. Steinerberger. Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions. J. Math. Anal. Appl., 445(1):407–422, 2017.
M. Faulhuber. Extremal Bounds of Gaussian Gabor Frames and Properties of Jacobi’s Theta Functions. PhD thesis, 2016.
M. Faulhuber. Gabor frame sets of invariance: a Hamiltonian approach to Gabor frame deformations. J. Pseudo-Differ. Oper. Appl., 7(2):213–235, June 2016.
M. Faulhuber. Geometry and Gabor Frames. Master’s thesis, 2014.
M. Faulhuber. Minimal frame operator norms via minimal theta functions. J. Fourier Anal. Appl., 24(2):545–559, April 2018.
H. G. Feichtinger and K. Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal., 86(2):307–340, 1989.
H. G. Feichtinger and K. Gröchenig. Gabor frames and time-frequency analysis of distributions. J. Funct. Anal., 146(2):464–495, 1997.
H. G. Feichtinger and N. Kaiblinger. 2D-Gabor analysis based on 1D algorithms. In Proc. OEAGM-97 (Hallstatt, Austria), 1997.
H. G. Feichtinger and N. Kaiblinger. Quasi-interpolation in the Fourier algebra. J. Approx. Theory, 144(1):103–118, 2007.
H. G. Feichtinger and N. Kaiblinger. Varying the time-frequency lattice of Gabor frames. Trans. Amer. Math. Soc., 356(5):2001–2023, 2004.
H. G. Feichtinger and W. Kozek. Quantization of TF lattice-invariant operators on elementary LCA groups. In H. G. Feichtinger and T. Strohmer, editors, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., pages 233–266. Birkhäuser, Boston, MA, 1998.
H. G. Feichtinger and F. Luef. Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis. Collect. Math., 57(Extra Volume (2006)):233–253, 2006.
H. G. Feichtinger and K. Nowak. A first survey of Gabor multipliers. In H. G. Feichtinger and T. Strohmer, editors, Advances in Gabor Analysis, Appl. Numer. Harmon. Anal., pages 99–128. Birkhäuser, 2003.
H. G. Feichtinger and K. Nowak. A Szegö-type theorem for Gabor-Toeplitz localization operators. Michigan Math. J., 49(1):13–21, 2001.
H. G. Feichtinger and D. Onchis. Constructive realization of dual systems for generators of multi-window spline-type spaces. J. Comput. Appl. Math., 234(12):3467–3479, 2010.
H. G. Feichtinger and T. Strohmer. Advances in Gabor Analysis. Birkhäuser, Basel, 2003.
H. G. Feichtinger and T. Strohmer. Gabor Analysis and Algorithms. Theory and Applications. Birkhäuser, Boston, 1998.
H. G. Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis. In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, pages 123–170. Birkhäuser Boston, 1998.
H. G. Feichtinger, W. Kozek, and F. Luef. Gabor Analysis over finite Abelian groups. Appl. Comput. Harmon. Anal., 26(2):230–248, 2009.
H. G. Feichtinger. A compactness criterion for translation invariant Banach spaces of functions. Analysis Mathematica, 8:165–172, 1982.
H. G. Feichtinger. Banach Gelfand triples for applications in physics and engineering. volume 1146 of AIP Conf. Proc., pages 189–228. Amer. Inst. Phys., 2009.
H. G. Feichtinger. Compactness in translation invariant Banach spaces of distributions and compact multipliers. J. Math. Anal. Appl., 102:289–327, 1984.
H. G. Feichtinger. Modulation spaces of locally compact Abelian groups. In R. Radha, M. Krishna, and S. Thangavelu, editors, Proc. Internat. Conf. on Wavelets and Applications, pages 1–56, Chennai, January 2002, 2003. New Delhi Allied Publishers.
H. G. Feichtinger. Modulation Spaces: Looking Back and Ahead. Sampl. Theory Signal Image Process., 5(2):109–140, 2006.
H. G. Feichtinger. Spline-type spaces in Gabor analysis. In D. X. Zhou, editor, Wavelet Analysis: Twenty Years Developments Proceedings of the International Conference of Computational Harmonic Analysis, Hong Kong, China, June 4–8, 2001, volume 1 of Ser. Anal., pages 100–122. World Sci.Pub., River Edge, NJ, 2002.
D. Gabor. Theory of communication. J. IEE, 93(26):429–457, 1946.
K. Gröchenig and S. Koppensteiner. Gabor Frames: characterizations and coarse structure. Ar**v e-prints, mar 2018.
K. Gröchenig and M. Leinert. Wiener’s lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc., 17:1–18, 2004.
K. Gröchenig and J. Stöckler. Gabor frames and totally positive functions. Duke Math. J., 162(6):1003–1031, 2013.
K. Gröchenig and G. Zimmermann. Spaces of test functions via the STFT. J. Funct. Spaces Appl., 2(1):25–53, 2004.
K. Gröchenig. Acceleration of the frame algorithm. IEEE Trans. SSP, 41/12:3331–3340, 1993.
K. Gröchenig. An uncertainty principle related to the Poisson summation formula. Studia Math., 121(1):87–104, 1996.
K. Gröchenig. Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston, MA, 2001.
M. Hampejs and L. Tóth. On the subgroups of finite Abelian groups of rank three. Annales Univ. Sci. Budapest., Sect. Comp., 39:111–124, 2013.
X. He and H. Li. On the \(abc\)-problem in Weyl-Heisenberg frames. Czechoslovak Math. J., 64(139)(2):447–458, 2014.
F. Hlawatsch, G. Matz, H. Kirchauer, and W. Kozek. Time–frequency formulation and design of time–varying optimal filters. IEEE Trans. Signal Process., 1997.
F. Hlawatsch, G. Matz, H. Kirchauer, and W. Kozek. Time-frequency formulation, design, and implementation of time-varying optimal filters for signal estimation. IEEE Trans. Signal Process., 48(5):1417–1432, 2000.
N. Holighaus, M. Dörfler, G. A. Velasco, and T. Grill. A framework for invertible, real-time constant-Q transforms. IEEE Trans. Audio Speech Lang. Process., 21(4):775 –785, 2013.
N. Holighaus, M. Hampejs, C. Wiesmeyr, and L. Tóth. Representing and counting the subgroups of the group \({Z}_m \times {Z}_n\). J. Number Theory, 2014:6, 2014.
N. Holighaus. Theory and Implementation of Adaptive Time-Frequency Transforms. PhD thesis, University of Vienna, 2013.
O. Hutnik. On Toeplitz localization operators. Arch. Math. (Basel), 97:333–344, 2011.
M. S. Jakobsen and H. G. Feichtinger. The inner kernel theorem for a certain Segal algebra. Ar**v e-prints, June 2018.
A. J. E. M. Janssen. Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl., 1(4):403–436, 1995.
A. J. E. M. Janssen. Gabor representation of generalized functions. J. Math. Anal. Appl., 83:377–394, October 1981.
N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J. Fourier Anal. Appl., 11(1):25–42, 2005.
T. Kloos and J. Stöckler. Zak transforms and Gabor frames of totally positive functions and exponential B-splines. J. Approx. Theory, 184:209–237, 2014.
T. Kloos, J. Stöckler, and K. Gröchenig. Implementation of discretized Gabor frames and their duals. IEEE Trans. Inform. Theory, 62(5):2759–2771, 2016.
W. Kozek and K. Riedel. Quadratic time-varying spectral estimation for underspread processes. In Proc. IEEE-SP Internat. Symp. on Time-Frequency and Time-scale Analysis, pages 417–420, Philadelphia/PA, October 1994.
W. Kozek. Adaptation of Weyl-Heisenberg frames to underspread environments. In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, pages 323–352. Birkhäuser Boston, Boston, 1997.
W. Kozek. On the underspread/overspread classification of nonstationary random processes. In K. Kirchgaessner, Mahrenholtz, and R. Mennicken, editors, Proc. ICIAM 95, Hamburg, July 1995, volume 3 of Mathematical Research, pages 63–66, Berlin, 1996. Akademieverlag.
J. Lemvig. On some Hermite series identities and their applications to Gabor analysis. Monatsh. Math., 182(4):899–912, April 2017.
Y. Lyubarskii. Frames in the Bargmann space of entire functions. In Entire and Subharmonic Functions, Advances in Soviet Mathematics, 11:167–180, Amer. Math. Soc., 1992.
A. Missbauer. Gabor Frames and the Fractional Fourier Transform. Master’s thesis, University of Vienna, 2012.
M. Porat and Y. Y. Zeevi. The generalized Gabor scheme of image representation in biological and machine vision. IEEE Trans. Patt. Anal. Mach. Intell., 10(4):452–468, July 1988.
Z. Prusa, P. Sondergaard, P. Balazs, and N. Holighaus. LTFAT: A Matlab/Octave toolbox for sound processing. In Proc. 10th International Symposium on Computer Music Multidisciplinary Research (CMMR), pages 299–314, 2013.
Z. Prusa. The Phase Retrieval Toolbox. In Audio Engineering Society Conference: 2017 AES International Conference on Semantic Audio, 2017.
S. Qiu and H. G. Feichtinger. Discrete Gabor structures and optimal representation. IEEE Trans. Signal Process., 43(10):2258–2268, October 1995.
S. Qiu and H. G. Feichtinger. The structure of the Gabor matrix and efficient numerical algorithms for discrete Gabor expansion. In Proc. SPIE VCIP94, SPIE 2308, pages 1146–1157, Chicago, 1994.
S. Qiu. Characterizations of Gabor-Gram matrices and efficient computations of discrete Gabor coefficients. In A. F. Laine, M. A. Unser, and V. Wickershauser, editors, Proc. SPIE Wavelet Applications in Signal and Image Processing III, volume 2569, pages 678–688, 1995.
S. Qiu. Gabor-type matrix algebra and fast computations of dual and tight Gabor wavelets. Opt. Eng., 36(1):276–282, 1997.
S. Qiu. Generalized dual Gabor atoms and best approximations by Gabor family. Signal Process., 49:167–186, 1996.
S. Qiu. Matrix approaches to discrete Gabor transform. PhD thesis, Dept. Mathematics, Univ. Vienna, June 2000.
S. Qiu. Structural properties of the Gabor transform and numerical algorithms. Proc. SPIE, vol. 2491, Wavelet Applications for Dual Use, pages 980–991, 1995.
S. Qiu. Super-fast computations of dual and tight Gabor atoms. In F. Laine, M. A. Unser, and V. Wickershauser, editors, Proc. SPIE Wavelet Applications in Signal and Image Processing III, volume 2569, pages 464–475, 1995.
A. Ron and Z. Shen. Frames and stable bases for subspaces of \({L}^2({R}^d)\): the duality principle of Weyl-Heisenberg sets. In M. Chu, R. Plemmons, D. Brown, and D. Ellison, editors, Proceedings of the Lanczos Centenary Conference Raleigh, NC., pages 422–425. SIAM, 1993.
A. Ron and Z. Shen. Weyl-Heisenberg frames and Riesz bases in \({L}_2({\mathbb{R}}^{d})\). Duke Math. J., 89(2):237–282, 1997.
C. Schoerkhuber, A. Klapuri, N. Holighaus, and M. Dörfler. A MATLAB toolbox for efficient perfect reconstruction time-frequency transforms with log-frequency resolution. volume 53. Proceedings of the 53rd AES international conference on semantic audio. London, UK, 2014.
K. Seip, and R. Wallsten. Density theorems for sampling and interpolation in the Bargmann-Fock space II. J. Reine Angew. Math., 429:107–113, 1992.
K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space I. J. Reine Angew. Math., 429:91–106, 1992.
P. Sondergaard, B. Torrésani, and P. Balazs. The Linear Time Frequency Analysis Toolbox. International Journal of Wavelets, Multiresolution and Information Processing, 10(4):1250032, 2012.
T. Strohmer and S. Beaver. Optimal OFDM system design for time-frequency dispersive channels. IEEE Trans. Comm., 51(7):1111–1122, July 2003.
J. Toft and P. Boggiatto. Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces. Adv. Math., 217(1):305–333, 2008.
L. Toth. The number of subgroups of the group \({Z}_m \times {Z}_n \times {Z}_r \times {Z}_s\). Ar**v e-prints, November 2016.
T. Tschurtschenthaler. The Gabor Frame Operator (its Structure and Numerical Consequences). Master’s thesis, University of Vienna, 2000.
G. A. Velasco. Time-Frequency Localization and Sampling in Gabor Analysis. PhD thesis, Universität Wien, 2015.
D. G. Voelz. Computational Fourier Optics. A MATLAB Tutorial. Tutorial text. Vol. 89. SPIE, 2011.
J. von Neumann. Mathematical Foundations of Quantum Mechanics., Robert T. Beyer, 1955.
D. F. Walnut. Continuity properties of the Gabor frame operator. J. Math. Anal. Appl., 165(2):479–504, 1992.
J. Wexler and S. Raz. Discrete Gabor expansions. Signal Process., 21:207–220, 1990.
C. Wiesmeyr. Construction of frames by discretization of phase space. PhD thesis, Universität Wien, 2014.
M. Zibulski and Y. Y. Zeevi. Analysis of multiwindow Gabor-type schemes by frame methods. Appl. Comput. Harmon. Anal., 4(2):188–221, 1997.
M. Zibulski and Y. Y. Zeevi. Discrete multiwindow Gabor-type transforms. IEEE Trans. Signal Process., 45(6):1428–1442, 1997.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Feichtinger, H.G. (2019). Gabor Expansions of Signals: Computational Aspects and Open Questions. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05210-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-05210-2_7
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-05209-6
Online ISBN: 978-3-030-05210-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)