Shift-invariant and sampling spaces associated with the fractional fourier transform domain

IEEE Transactions on Signal Processing

Volumn 60, Issue 4, 2012, Pages 1627-1637

  Bhandari, Ayush ^a   Zayed, Ahmed I ^b  

^a EPFL   (Switzerland)

^b DEPAUL UNIVERSITY   (United States)

Author keywords

Fractional fourier transform; fractional Zak transform; Poisson summation formula; reproducing kernels; sampling spaces; semi discrete convolution; shift invariant spaces

Indexed keywords

FRACTIONAL FOURIER TRANSFORMS; POISSON SUMMATION FORMULA; REPRODUCING KERNELS; SAMPLING SPACES; SHIFT-INVARIANT SPACE;

CONVOLUTION; DIGITAL SIGNAL PROCESSING; FUNCTIONAL ANALYSIS; IMAGE PROCESSING; ORTHOGONAL FUNCTIONS; SIGNAL SAMPLING;

IMPORTANCE SAMPLING;

EID: 84858328439     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2011.2177260     Document Type: Article

References (41)

1. G. G. Walter, "A sampling theorem for wavelet subspaces," IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 881-884, 1992.
2. A. J. E. M. Janssen, "The Zak transform and sampling theorems for wavelet subspaces," IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3360-3364, 1993.
3. G. Walter and A. I. Zayed, "Multiresolution analysis with sampling subspaces," J. Fractional Calculus Appl. Anal., vol. 1, pp. 109-124, 1998.
4. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. New York: Academic, Sep. 1999.
5. I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992.
6. J. Proakis, Digital Communications. New York: McGraw-Hill Engineering, 2000.
7. H. Nyquist, "Certain topics in telegraph transmission theory," Proc. IEEE, vol. 90, no. 2, pp. 280-305, 1928.
8. X. G. Xia, "A family of pulse-shaping filters with ISI-free matched and unmatched filter properties," IEEE Trans. Commun., vol. 45, no. 10, pp. 1157-1158, Oct. 1997.
9. T. Demeechai, "Pulse-shaping filters with ISI-free matched and unmatched filter properties," IEEE Trans. Commun., vol. 46, no. 8, p. 992, Aug. 1998.
10. A. Aldroubi and M. Unser, "Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory," Numer. Funct. Anal. Opt., vol. 15, no. 1-2, pp. 1-21, 1994.
11. M. Unser, "Sampling-50 years after Shannon," Proc. IEEE, vol. 88, no. 4, pp. 569-587, 2000.
12. M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Process. Mag., vol. 16, no. 6, pp. 22-38, Nov. 1999.
13. E. Hammerich, "Sampling in shift-invariant spaces with Gaussian generator," Sampl. Theory Signal Image Process., vol. 6, pp. 71-86, 2007.
14. T. Blu and M. Unser, "Approximation error for quasi-interpolators and (multi-) wavelet expansions," J. Appl. Comput. Harmon. Anal., vol. 6, no. 2, pp. 219-251, Mar. 1999.
15. Y. C. Eldar and M. Unser, "Nonideal sampling and interpolation from noisy observations in shift-invariant spaces," IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2636-2651, 2006.
16. Y. C. Eldar, "Compressed sensing of analog signals in shift-invariant spaces," IEEE Trans. Signal Process., vol. 57, no. 8, pp. 2986-2997, 2009.
17. Y. Eldar and T. Michaeli, "Beyond bandlimited sampling," IEEE Signal Process. Mag., vol. 26, no. 3, pp. 48-68, 2009.
18. A. Aldroubi and K. Gröchenig, "Nonuniform sampling and reconstruction in shift-invariant spaces," SIAM Rev., vol. 43, no. 4, pp. 585-620, 2001.
19. W. Chen, S. Itoh, and J. Shiki, "On sampling in shift invariant spaces," IEEE Trans. Inf. Theory, vol. 48, no. 10, pp. 2802-2810, 2002.
20. W. Chen, B. Han, and R.-Q. Jia, "On simple oversampled A/D conversion in shift-invariant spaces," IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 648-657, 2005.
21. A. G. García,M. A. Hernandez-Medina, and G. Perez-Villalon, "Oversampling in shift-invariant spaces with a rational sampling period," IEEE Trans. Signal Process., vol. 57, no. 9, pp. 3442-3449, 2009.
22. C. D. Boor, R. A. Devore, and A. Ron, "Approximation from shiftinvariant subspaces of ," Trans. Amer. Math. Soc., vol. 341, no. 2, pp. 787-806, 1994.
23. W. Sun and X. Zhou, "Reconstruction of band-limited signals from local averages," IEEE Trans. Inf. Theory, vol. 48, no. 11, pp. 2955-2963, Nov. 2002.
24. A. Bhandari and P. Marziliano, "Fractional delay filters based on generalized cardinal exponential splines," IEEE Signal Process. Lett., vol. 17, no. 3, pp. 225-228, 2010.
25. T. Laakso, V. Valimaki, M. Karjalainen, and U. Laine, "Splitting the unit delay," IEEE Signal Process. Mag., vol. 13, no. 1, pp. 30-60, Jan. 1996.
26. C. E. Shannon, "Communications in the presence of noise," Proc. IRE, vol. 37, pp. 10-21, January 1949.
27. J. Zak, "Finite translations in solid-state physics," Phys. Rev. Lett., vol. 19, pp. 1385-1387, Dec. 1967.
28. L. B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3084-3091, Nov. 1994.
29. X. G. Xia, "On bandlimited signals with fractional Fourier transform," IEEE Signal Process. Lett., vol. 3, no. 3, pp. 72-74, Mar. 1996.
30. A. I. Zayed, "A class of fractional integral transform: A generalization of the fractional Fourier transform," IEEE Trans. Signal Process., vol. 50, pp. 619-627, Mar. 2002.
31. A. I. Zayed, "On the relationship between the Fourier and fractional Fourier transforms," IEEE Signal Process. Lett., vol. 3, no. 12, pp. 310-311, Dec. 1996.
32. T. Erseghe, P. Kraniauskas, and G. Carioraro, "Unified fractional Fourier transform and sampling theorem," IEEE Trans. Signal Process., vol. 47, no. 12, pp. 3419-3423, Dec. 1999.
33. A. I. Zayed and A. G. García, "New sampling formulae for the fractional Fourier transform," Signal Process., vol. 77, no. 1, pp. 111-114, 1999.
34. A. G. García, "Orthogonal sampling formulas: A unified approach," SIAM Rev., vol. 42, no. 3, pp. 499-512, 2000.
35. C. Candan and H. M. Ozaktas, "Sampling and series expansion theorems for fractional Fourier and other transforms," Signal Process., vol. 83, no. 11, pp. 2455-2457, 2003.
36. R. Torres, P. F. Pellat, and Y. Torres, "Sampling theorem for fractional bandlimited signals: A self-contained proof. application to digital holography," IEEE Signal Process. Lett., vol. 13, no. 11, pp. 676-679, Nov. 2006.
37. R. Tao, B.Deng, Z.-Q.Wei, andY.Wang, "Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain," IEEE Trans. Signal Process., vol. 56, no. 1, pp. 158-171, Jan. 2008.
38. H. M. Ozaktas and M. A. Kutay, Introduction to the Fractional Fourier Transform and Its Applications. New York: Academic, 1999.
39. A. I. Zayed, "A convolution and product theorem for the fractional Fourier transform," IEEE Signal Process. Lett., vol. 5, no. 4, pp. 101-103, Apr. 1998.
40. B. Z. Li, R. Tao, T.-Z. Xu, and Y. Wang, "The Poisson sum formulae associated with the fractional Fourier transform," Signal Process., vol. 89, no. 5, pp. 851-856, 2009.
41. A. Bhandari and P.Marziliano, "Sampling and reconstruction of sparse signals in fractional Fourier domain," IEEE Signal Process. Lett., vol. 17, no. 3, pp. 221-224, 2010.