Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T09:19:32.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Sampling time-frequency localized functions and constructing localized time-frequency frames

Published online by Cambridge University Press:  19 December 2016

G. A. M. VELASCO  and
M. DÖRFLER
G. A. M. VELASCO
Affiliation:
Institute of Mathematics, University of the Philippines Diliman, 1101 Quezon City, Philippines email: [email protected]
M. DÖRFLER
Affiliation:
Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.

Keywords

framesmultiplierssampling theorysignal theory (filtering, reconstruction)approximation by other special function classes
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 5 , October 2017 , pp. 854 - 876
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abreu, L. D. & Dörfler, M. (2012) An inverse problem for localization operators. Inverse Problems 28 (11), 115001, 16.Google Scholar
[2] Aldroubi, A., Cabrelli, C. A. & Molter, U. (2004) Wavelets on irregular grids with arbitrary dilation matrices, and frame atoms for L 2(ℝ d ). Appl. Comput. Harmon. Anal. (Special Issue: Frames in Harmonic Analysis, Part II) 17 (2), 119–140.Google Scholar
[3] Christensen, O. & Laugesen, R. S. (2010) Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames. Sampl. Theory Signal Image Process. 9 (1–3), 77–89.Google Scholar
[4] Cordero, E. & Gröchenig, K. (2003) Time-frequency analysis of localization operators. J. Funct. Anal. 205 (1), 107–131.Google Scholar
[5] Daubechies, I. (1988) Time-frequency localization operators: A geometric phase space approach. IEEE Trans. Inform. Theory 34 (4), 605–612.Google Scholar
[6] Daubechies, I. (1990) The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 36 (5), 961–1005.Google Scholar
[7] DeMari, F., Feichtinger, H. G. & Nowak, K. (2002) Uniform eigenvalue estimates for time-frequency localization operators. J. London Math. Soc. 65 (3), 720–732.Google Scholar
[8] Dörfler, M. (2011) Quilted Gabor frames - A new concept for adaptive time-frequency representation. Adv. Appl. Math. 47 (4), 668–687.Google Scholar
[9] Dörfler, M. & Matusiak, E. (2014) Nonstationary gabor frames - existence and construction. Int. J. Wavelets Multiresolut. Inf. Process. 12 (3).Google Scholar
[10] Dörfler, M. & Romero, J. L. (2013) Frames of eigenfunctions and localization of signal components. In: Proceedings of the 10th International Conference on Sampling Theory and Applications (SampTA2013), Bremen.Google Scholar
[11] Dörfler, M. & Romero, J. L. (2014) Frames adapted to a phase-space cover. Constr. Approx. 39 (3), 445–484.Google Scholar
[12] Dörfler, M. & Velasco, G. (2014) Adaptive Gabor frames by projection onto time-frequency subspaces. In: Proceedings of the 39th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2014), pp. 3097–3101.Google Scholar
[13] Feichtinger, H. G. (2006) Modulation spaces: Looking back and ahead. Sampl. Theory Signal Image Process. 5 (2), 109–140.CrossRefGoogle Scholar
[14] Feichtinger, H. G. & Kaiblinger, N. (2004) Varying the time-frequency lattice of Gabor frames. Trans. Amer. Math. Soc. 356 (5), 2001–2023.Google Scholar
[15] Feichtinger, H. G. & Nowak, K. (2001) A Szegö-type theorem for Gabor-Toeplitz localization operators. Michigan Math. J. 49 (1), 13–21.CrossRefGoogle Scholar
[16] Feichtinger, H. G. & Werther, T. (2001) Atomic systems for subspaces. In: Proceedings SampTA, Vol. 2001, Orlando, pp. 163–165.Google Scholar
[17] Feichtinger, H. G. & Zimmermann, G. (1998) A Banach space of test functions for Gabor analysis. In: Feichtinger, H. G. & Strohmer, T. (editors), Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Boston, MA, pp. 123–170.Google Scholar
[18] Gröchenig, K. (2001) Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Boston, MA.Google Scholar
[19] Gradshteyn, I. S. & Ryzhik, I. M. (2007) Table of Integrals, Series, and Products, 7th ed., Elsevier/Academic Press.Google Scholar
[20] Gröchenig, K. & Toft, J. (2013) The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces. Trans. Amer. Math. Soc. 365, 4475–4496.Google Scholar
[21] Hogan, J. & Lakey, J. (2015) Frame properties of shifts of prolate spheroidal wave functions. Appl. Comput. Harmon. Anal. 39 (1), 21–32.Google Scholar
[22] Hogan, J. A., Izu, S. & Lakey, J. D. (2010) Sampling approximations for time- and bandlimiting. Sampl. Theory Signal Image Process. 9 (1-3), 91–117.CrossRefGoogle Scholar
[23] Landau, H. J. & Pollak, H. O. (1961) Prolate spheroidal wave functions, Fourier analysis and uncertainty II. Bell Syst. Tech. J. 40, 65–84.CrossRefGoogle Scholar
[24] Landau, H. J. & Pollak, H. O. (1962) Prolate spheroidal wave functions, Fourier analysis and uncertainty, III: The dimension of the space of essentially time- and band-limited signals. Bell Syst. Tech. J. 41, 1295–1336.Google Scholar
[25] Li, S. & Ogawa, H. (2004) Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10 (4), 409–431.Google Scholar
[26] Liuni, M., Robel, A., Matusiak, E., Romito, M. & Rodet, X. (2013) Automatic adaptation of the time-frequency resolution for sound analysis and re-synthesis. IEEE Trans. Audio, Speech, Lang. Process. 21 (5), 959–970.Google Scholar
[27] Matusiak, E. & Eldar, Y. C. (2012) Sub-Nyquist sampling of short pulses. IEEE Trans. Signal Process. 60 (3), 1134–1148.Google Scholar
[28] Ramanathan, J. & Topiwala, P. (1994) Time-frequency localization and the spectrogram. Appl. Comput. Harmon. Anal. 1 (2), 209–215.Google Scholar
[29] Romero, J. L. (2011) Surgery of spline-type and molecular frames. J. Fourier Anal. Appl. 17, 135–174.CrossRefGoogle Scholar
[30] Slepian, D. & Pollak, H. O. (1961) Prolate spheroidal wave functions, fourier analysis and uncertainty –I. Bell Syst. Tech. J. 40 (1), 43–63.Google Scholar
[31] Strohmer, T. (1998) Numerical algorithms for discrete Gabor expansions. In: Feichtinger, H. G. and Strohmer, T. (editors), Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser Boston, Boston, 267–294.Google Scholar
[32] Walter, G. G. & Shen, X. A. (2003) Sampling with prolate spheroidal wave functions. Sampl. Theory Signal Image Process. 2 (1), 25–52.Google Scholar
[33] Wilczok, E. (2000) New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Doc. Math. J. DMV 5, 201–226.Google Scholar
[34] Wong, M.-W. (2002) Wavelet Transforms and Localization Operators. Operator Theory: Advances and Applications, Vol. 136, Birkhäuser, Basel.Google Scholar