Theory, implementation and applications of nonstationary Gabor frames

  • Authors:

  • P. Balazs;M. Dörfler;F. Jaillet;N. Holighaus;G. Velasco

  • Affiliations:

  • Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Wien, Austria;Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Alserbachstraíe 23, 1090 Wien, Austria;Institut de Neurosciences Cognitives de la Méditerranée, UMR 6193 CNRS - Université de la Méditerranée, 31 chemin Joseph Aiguier, 13402 Marseille cedex 20, France;Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Alserbachstraíe 23, 1090 Wien, Austria;Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Alserbachstraíe 23, 1090 Wien, Austria and Institute of Mathematics, University of the Philippines - Diliman, ...

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2011

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Abstract

Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.