Representation of operators in the time-frequency domain and generalized gabor multipliers

Journal of Fourier Analysis and Applications

Volumn 16, Issue 2, 2010, Pages 261-293

  Dörfler, Monika ^a   Torrésani, Bruno ^b  

^a UNIVERSITY OF VIENNA   (Austria)

^b AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

Generalized Gabor multipliers; Operator approximation; Spreading function; Twisted convolution

Indexed keywords

EID: 77952099247     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-009-9085-x     Document Type: Article

References (41)

1. Bourouihiya, A.: The tensor product of frames. Sampl. Theory Signal Image Process. (STSIP) 7(1), 65-76 (2008).
2. Balan, R. M., Rickard, S., Poor, H. V., Verdú, S.: Canonical time-frequency, time-scale, and frequency-scale representations of time-varying channels. J. Commun. Inf. Syst. 5(2), 197-226 (2005).
3. Balazs, P.: Hilbert-Schmidt operators and frames-classification, best approximation. Int. J. Wavelets Multiresolut. Inf. Process. 6(2), 315-330 (2008).
4. Balazs, P.: Matrix-representation of operators using frames. Sampl. Theory Signal Image Process. (STSIP) 7(1), 39-54 (2008).
5. Bello, P.: Characterization of randomly time-variant linear channels. IEEE Trans. Commun. 11, 360-393 (1963).
6. Benedetto, J. J., Pfander, G. E.: Frame expansions for Gabor multipliers. Appl. Comput. Harmon. Anal. 20(1), 26-40 (2006).
7. Bölcskei, H., Koetter, R., Mallik, S.: Coding and modulation for underspread fading channels. In: IEEE International Symposium on Information Theory (ISIT) 2002, p. 358. ETH-Zürich, June 2002.
8. Carmona, R., Hwang, W., Torrésani, B.: Practical Time-Frequency Analysis: Continuous Wavelet and Gabor Transforms, with an Implementation in S. Wavelet Analysis and its Applications, vol. 9. Academic Press, San Diego (1998).
9. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992).
10. Dolson, M.: The phase vocoder: a tutorial. Comput. Music. J. 10(4), 11-27 (1986).
11. Dörfler, M., Torrésani, B.: Spreading function representation of operators and Gabor multiplier approximation. In: Sampling Theory and Applications (SAMPTA'07), Thessaloniki, June 2007.
12. Dörfler, M., Torrésani, B.: Representation of operators by sampling in the time-frequency domain. In: Proceedings of SAMPTA09 (2009).
13. Feichtinger, H., Zimmermann, G.: A Banach space of test functions for Gabor analysis. In: Feichtinger, H., Strohmer, T. (eds.) Gabor Analysis and Algorithms: Theory and Applications, pp. 123-170. Birkhäuser, Boston (1998). Chap. 3.
14. Feichtinger, H. G.: On a new Segal algebra. Mon. Math. 92(4), 269-289 (1981).
15. Feichtinger, H. G.: Spline type spaces in Gabor analysis. In: Zhou, D. (ed.) Wavelet Analysis: Twenty Years' Developments. World Scientific, Singapore (2002).
16. Feichtinger, H. G., Hampejs, M., Kracher, G.: Approximation of matrices by Gabor multipliers. IEEE Signal Process. Lett. 11(11), 883-886 (2004).
17. Feichtinger, H. G., Kozek, W.: Quantization of TF lattice-invariant operators on elementary LCA groups. In: Gabor Analysis and Algorithms, pp. 233-266. Birkhäuser, Boston (1998).
18. Feichtinger, H. G., Nowak, K.: A first survey of Gabor multipliers. In: Feichtinger, H. G., Strohmer, T. (eds.) Advances in Gabor Analysis. Birkhäuser, Boston (2002).
19. Feichtinger, H. G., Strohmer, T.: Gabor Analysis and Algorithms. Theory and Applications. Birkhäuser, Boston (1998).
20. Feichtinger, H. G., Strohmer, T.: Advances in Gabor Analysis. Birkhäuser, Boston (2003).
21. Flanagan, J., Golden, R. M.: Phase vocoder. Bell Syst. Tech. 45, 1493-1509 (1966).
22. Folland, G.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989).
23. Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Lecture Notes in Mathematics, vol. 1863. Springer, Berlin (2005).
24. Gel'fand, I. M., Vilenkin, N. Y.: Generalized Functions. Applications of Harmonic Analysis, vol. 4. Academic Press, New York (1964). Translated from the Russian by Amiel Feinstein.
25. Gröchenig, K.: Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston (2001).
26. Gröchenig, K.: Uncertainty principles for time-frequency representations. In: Feichtinger, H., Strohmer, T. (eds.) Advances in Gabor Analysis, pp. 11-30. Birkhäuser, Boston (2003).
27. Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations I: general results. J. Math. Phys. 26, 2473-2479 (1985).
28. Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations II: examples. Ann. Inst. Henri Poincaré 45, 293 (1986).
29. Hlawatsch, F., Matz, G.: Linear time-frequency filters. In: Boashash, B. (ed.) Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, pp. 466-475. Elsevier, Amsterdam (2003).
30. Kailath, T.: Measurements on time-variant communication channels. IEEE Trans. Inf. Theory 8(5), 229-236 (1962).
31. Kennedy, R.: Fading Dispersive Communication Channels. Wiley, New York (1969).
32. Kozek, W.: Matched Weyl-Heisenberg expansions of nonstationary environments. PhD thesis, NuHAG, University of Vienna (1996).
33. Kozek, W.: Adaptation of Weyl-Heisenberg frames to underspread environments. In: Feichtinger, H., Strohmer, T. (eds.) Gabor Analysis and Algorithms: Theory and Applications, pp. 323-352. Birkhäuser, Boston (1997).
34. Kozek, W.: On the transfer function calculus for underspread LTV channels. ISP 45(1), 219-223 (1997).
35. Kozek, W., Pfander, G. E.: Identification of operators with bandlimited symbols. SIAM J. Math. Anal. 37(3), 867-888 (2006).
36. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1998).
37. Matz, G., Hlawatsch, F.: Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory. J. Math. Phys. 39(8), 4041-4070 (1998).
38. Matz, G., Hlawatsch, F.: Time-frequency characterization of random time-varying channels. In: Boashash, B. (ed.) Time Frequency Signal Analysis and Processing: A Comprehensive Reference, pp. 410-419. Elsevier, Amsterdam (2003).
39. Pfander, G. E., Walnut, D. F.: Operator identification and Feichtinger's algebra. Sampl. Theory Signal Image Process. (STSIP) 5(2), 183-200 (2006).
40. Schempp, W.: Harmonic Analysis on the Heisenberg Nilpotent Lie Group. Pitman Series, vol. 147. Wiley, New York (1986).
41. Zadeh, L. A.: Time-varying networks I. In: Proceedings of the IRE, vol. 49, pp. 1488-1502, October 1961.