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Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window

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Abstract

The Discrete Gabor Transform (DGT) is the most commonly used signal transform for signal analysis and synthesis using a linear frequency scale. The development of the Linear Time-Frequency Analysis Toolbox (LTFAT) has been based on a detailed study of many variants of the relevant algorithms. As a side result of these systematic developments of the subject, two new methods are presented here. Comparisons are made with respect to the computational complexity, and the running time of optimised implementations in the C programming language. The new algorithms have the lowest known computational complexity and running time when a long FIR window is used. 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.

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References

  1. Ahmed, N., Natarajan, T., Rao, K.: Discrete cosine transform. IEEE Trans. Comput. 100(1), 90–93 (1974)

    Article  MathSciNet  Google Scholar 

  2. Allen, J., Rabiner, L.: A unified approach to short-time Fourier analysis and synthesis. Proc. IEEE 65(11), 1558–1564 (1977)

    Article  Google Scholar 

  3. Auslander, L., Gertner, I., Tolimieri, R.: The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals. IEEE Trans. Signal Process. 39(4), 825–835 (1991)

    Article  Google Scholar 

  4. Balazs, P., Dörfler, M., Holighaus, N., Jaillet, F., Velasco, G.: Theory, implementation and application of nonstationary gabor frames. J. Comput. Appl. Math. (2011, accepted)

  5. Bastiaans, M.J., Geilen, M.C.: On the discrete Gabor transform and the discrete Zak transform. Signal Process. 49(3), 151–166 (1996)

    Article  Google Scholar 

  6. Bergland, G.: Numerical analysis: a fast Fourier transform algorithm for real-valued series. Commun. ACM 11(10), 703–710 (1968)

    Article  MATH  Google Scholar 

  7. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Equivalence of DFT filter banks and Gabor expansions. In: SPIE 95, Wavelet Applications in Signal and Image Processing III, vol. 2569, part I, San Diego, July 1995

    Google Scholar 

  8. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46(12), 3256–3268 (2002)

    Article  Google Scholar 

  9. Briggs, W.: The DFT: An Owner’s Manual for the Discrete Fourier Transform. Society for Industrial Mathematics, Philadelphia (1995)

    Book  MATH  Google Scholar 

  10. Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl. 195, 401–414 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  12. Cooley, J., Tukey, J.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19(90), 297–301 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Mesmay, F., Voronenko, Y., Püschel, M.: Offline library adaptation using automatically generated heuristics. In: International Parallel and Distributed Processing Symposium (IPDPS) (2010)

    Google Scholar 

  14. Dongarra, J., Du Croz, J., Hammarling, S., Duff, I.: A set of level 3 basic linear algebra subprograms. ACM Trans. Math. Softw. 16(1), 1–17 (1990)

    Article  MATH  Google Scholar 

  15. Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis and Algorithms. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  16. Feichtinger, H.G., Strohmer, T. (eds.): Advances in Gabor Analysis. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  17. Flanagan, J.L., Meinhart, D., Golden, R., Sondhi, M.: Phase Vocoder. J. Acoust. Soc. Am. 38, 939 (1965)

    Article  Google Scholar 

  18. Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE 93(2), 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”

    Article  Google Scholar 

  19. Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. John Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  20. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    MATH  Google Scholar 

  21. Helms, H.: Fast Fourier transform method of computing difference equations and simulating filters. IEEE Trans. Audio Electroacoust. 15(2), 85–90 (1967)

    Article  Google Scholar 

  22. Janssen, A.J.E.M.: The Zak transform: a signal transform for sampled time-continuous signals. Philips J. Res. 43(1), 23–69 (1988)

    MathSciNet  MATH  Google Scholar 

  23. Janssen, A.J.E.M.: On rationally oversampled Weyl-Heisenberg frames. Signal Process., pp. 239–245 (1995)

  24. Janssen, A.J.E.M.: The duality condition for Weyl-Heisenberg frames. In: Feichtinger, H.G., Strohmer, T. (eds.) Gabor Analysis and Algorithms, pp. 33–84. Birkhäuser, Boston (1998), Chap. 1

    Chapter  Google Scholar 

  25. Janssen, A.J.E.M., Søndergaard, P.L.: Iterative algorithms to approximate canonical Gabor windows: computational aspects. J. Fourier Anal. Appl. 13(2), 211–241 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Johnson, S., Frigo, M.: A modified split-radix FFT with fewer arithmetic operations. IEEE Trans. Signal Process. 55(1), 111 (2007)

    Article  MathSciNet  Google Scholar 

  27. Moreno-Picot, S., Arevalillo-Herráez, M., Diaz-Villanueva, W.: A linear cost algorithm to compute the discrete Gabor transform. IEEE Trans. Signal Process. 58(5), 2667–2674 (2010)

    Article  MathSciNet  Google Scholar 

  28. Pfander, G., Rauhut, H., Tanner, J.: Identification of matrices having a sparse representation. IEEE Trans. Signal Process. 56(11), 5376–5388 (2008)

    Article  MathSciNet  Google Scholar 

  29. Pfander, G., Rauhut, H., Tropp, J.: The restricted isometry property for time-frequency structured random matrices (2011). Preprint

  30. Portnoff, M.: Implementation of the digital phase vocoder using the fast Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 24(3), 243–248 (1976)

    Article  Google Scholar 

  31. Prinz, P.: Calculating the dual Gabor window for general sampling sets. IEEE Trans. Signal Process. 44(8), 2078–2082 (1996)

    Article  Google Scholar 

  32. Qiu, S.: Discrete Gabor transforms: the Gabor-Gram matrix approach. J. Fourier Anal. Appl. 4(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rader, C.: Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56(6), 1107–1108 (1968)

    Article  Google Scholar 

  34. Rao, K., Yip, P.: Discrete Cosine Transform, Algorithms, Advantages, Applications. Academic Press, San Diego (1990)

    MATH  Google Scholar 

  35. Schafer, R., Rabiner, L.: Design and simulation of a speech analysis-synthesis system based on short-time Fourier analysis. IEEE Trans. Audio Electroacoust. 21(3), 165–174 (1973)

    Article  Google Scholar 

  36. Søndergaard, P.L., Torrésani, B., Balazs, P.: The linear time frequency analysis toolbox. Int. J. Wavelets Multiresol. Anal. Inf. Process. (2011, accepted)

  37. Stockham, T. Jr.: High-speed convolution and correlation. In: Proceedings of the Spring Joint Computer Conference, April 26–28, 1966, pp. 229–233. ACM, New York (1966)

    Google Scholar 

  38. Strohmer, T.: Numerical algorithms for discrete Gabor expansions. In: Feichtinger, H.G., Strohmer, T. (eds.) Gabor Analysis and Algorithms, pp. 267–294. Birkhäuser, Boston (1998). Chap. 8

    Chapter  Google Scholar 

  39. Tao, L., Kwan, H.: Novel DCT-based real-valued discrete Gabor transform and its fast algorithms. IEEE Trans. Signal Process. 57(6), 2151–2164 (2009)

    Article  MathSciNet  Google Scholar 

  40. Walnut, D.F.: Continuity properties of the Gabor frame operator. J. Math. Anal. Appl. 165(2), 479–504 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Whaley, R.C., Petitet, A.: Minimizing development and maintenance costs in supporting persistently optimized BLAS. Softw. Pract. Exp. 35(2), 101–121 (2005)

    Article  Google Scholar 

  42. Zeevi, Y.Y., Zibulski, M.: Oversampling in the Gabor scheme. IEEE Trans. Signal Process. 41(8), 2679–2687 (1993)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Peter L. Søndergaard

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  1. Peter L. Søndergaard
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Correspondence to Peter L. Søndergaard.

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Communicated by Hans G. Feichtinger.

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Søndergaard, P.L. Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window. J Fourier Anal Appl 18, 456–470 (2012). https://doi.org/10.1007/s00041-011-9210-5

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  • DOI: https://doi.org/10.1007/s00041-011-9210-5

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