Iterative algorithms to approximate canonical gabor windows: Computational aspects

Journal of Fourier Analysis and Applications

Volumn 13, Issue 2, 2007, Pages 211-241

  Janssen, A J E M ^a   Søndergaard, Peter L ^b  

^a PHILIPS RESEARCH LABORATORIES   (Netherlands)

^b TECHNICAL UNIVERSITY OF DENMARK   (Denmark)

Author keywords

Adjoint orbit; Dual window; Gabor frame; Iterative method; Scaling; Tight window; Zak transform

Indexed keywords

EID: 34147204492     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-006-6069-y     Document Type: Article

References (35)

1. Anderson, E., Bai, Z., Bischof, C., Blackford, L. S., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., and Sorensen, D. (1999). LAPACK Users' Guide, Third Edition, SIAM, Philadelphia.
2. Balazs, P., Feichtinger, H. G., and Kracher, G. (2005). Double preconditioning for Gabor frames, IEEE Trans. Signal Process, submitted for publication.
3. Balazs, P. (2005). Regular and irregular Gabor Multipliers with application to psychoacoustic masking, Ph. D. thesis, Fakultät für Mathematik der Universität Wien, Vienna. June.
4. Bastiaans, M. J. and Geilen, M. C. (1996). On the discrete Gabor transform and the discrete Zak transform, Signal Process. 49(3), 151-166.
5. Björck, Å. and Bowie, C. (1971). An iterative algorithm for computing the best estimate of an orthogonal matrix, SIAM J. Numer. Anal 8(2), 358-364, June.
6. Christensen, O. (2003). An Introduction to Frames and Riesz Bases, Birkhäuser.
7. Christensen, O. and Strohmer, T. (2003). Methods for the approximation of the inverse Gabor frame operator, in Advances in Gabor Analysis, Feichtinger, H. G. and Strohmer, T., Eds., Chapter 8, 171-196. Birkhäuser.
8. Feichtinger, H. G. and Strohmer, T. (1998). Gabor Analysis and Algorithms, Birkhäuser, Boston.
9. Frigo, M. and Johnson, S. G. (2005). The design and implementation of FFTW3., Proceedings of the IEEE, 93(2), 216-231, special issue on Program Generation, Optimization, and Platform Adaptation.
10. Golub, G.H. and van Loan, C.F. (1996). Matrix Computations, Third Edition, John Hopkins University Press.
11. Gröchenig, K. (2001). Foundations of Time-Fmquency Analysis, Birkhäuser.
12. Higham, N. J. (1986). Computing the polar decomposition - with applications, SIAM J. Sci. Statist. Comput. 7(4), 1160-1174, October.
13. Higham, N.J. (1999). Anew sqrtm for MATLAB, Numerical Analysis ReportNo. 336, Manchester Centre for Computational Mathematics, Manchester, England, January.
14. Higham, N. J. and Schreiber, R. S. (1990). Fast polar decomposition of an arbitrary matrix, SIAM J. Sci. Statist. Comput. 11(4), 648-655, July.
15. Janssen, A. J. E. M. Duality and biorthogonality for discrete-time Weyl-Heisenberg frames, Unclassified report, Philips Electronics, 002/94.
16. Janssen, A. J.E.M. (1997). From continuous to discrete Weyl-Heisenberg frames through sampling, J. Fourier Anal. Appl. 3(5), 583-596.
17. Janssen, A. J. E. M. The duality condition for Weyl-Heisenberg frames, in Feichtinger and Strohmer [8], Chapter 1, 33-84.
18. Janssen, A. J.E.M. (2002). Analysis of some fast algorithms to compute canonical windows for Gabor frames, unpublished.
19. Janssen, A. J. E. M. (2003). Some iterative algorithms to compute canonical windows for Gabor frames, in Proceedings of IMS Workshop on Time-Frequency Analysis and Applications, Singapore, September.
20. Janssen, A. J. E. M. and Strohmer, T. (2002). Characterization and computation of canonical tight windows for Gabor frames, J. Fourier Anal. Appl. 8(1), 1-28.
21. Janssen, A. J. E. M. and Strohmer, T. (2002). Hyperbolic secants yield Gabor frames, Appl. Comput. Harmon. Anal. 12(2), 259-267.
22. Kaiblinger, N. (2005). Approximation of the Fourier transform and the dual Gabor window, J. Fourier Anal. Appl. 11(1), 25-42.
23. Kenney, C. and Laub, A. (1990). On scaling Newton's method for polar decomposition and the matrix sign function, Proceedings of the 1990 American Control Conference (IEEE Cat. No.90CH2896-9), 2560-2564, Vol. 3.
24. Kovarik, Z. (1970). Some iterative methods for improving orthonormality, SIAM J. Num. Anal. 7(3), 386-389.
25. Lakic, S. (1995). An iterative method for the computation of a matrix inverse square root, ZAMM Z. Angew. Math. Mech. 75(11), 867-874.
26. Prinz, P. (1996). Calculating the dual Gabor window for general sampling sets, IEEE Trans. Signal Process. 44(8), 2078-2082.
27. Qiu, S. and Feichtinger, H. G. (1995). Discrete gabor structures and optimal representations, IEEE Trans. Signal Process. 43(10), 2258-2268.
28. Schulz, G. (1933). Iterative Berechnung der reziproken Matrix, ZAMMZ. Angew. Math. Mech. 13, 57-59.
29. Sherif, N. (1991). On the computation of a matrix inverse square root, Computing (Vienna/New York), 46(4), 295-305.
30. Søndergaard, P. (2007). Gabor frames by sampling and periodization, Adv. Comput. Math., published online.
31. Strohmer, T. Numerical algorithms for discrete Gabor expansions, in Feichtinger and Strohmer [8], Chapter 8, 267-294.
32. Tolimieri, R. and Orr, R. S. (1995). Poisson summation, the ambiguity function, and the theory of Weyl-Heisenberg systems, J. Fourier Anal. Appl. 1(3), 233-247.
33. Whaley, R. C., Petitet, A., and Dongarra, J. (2000). Automated empirical optimization of software and the ATLAS project, Technical Report UT-CS-00-448, University of Tennessee, Knoxville, TN, September.
34. Zeevi, Y. Y. and Zibulski, M. (1993). Oversampling in the Gabor scheme, IEEE Trans. Signal Process. 41(8), 2679-2687.
35. Zibulski, M. and Zeevi, Y. Y. (1997). Analysis of multiwindow Gabor-type schemes by frame methods, Appl. Comput. Harmon. Anal 4(2), 188-221.