Eigenfunctions of fourier and fractional fourier transforms with complex offsets and parameters

IEEE Transactions on Circuits and Systems I: Regular Papers

Volumn 54, Issue 7, 2007, Pages 1599-1611

  Pei, Soo Chang ^a   Ding, Jian Jiun ^a  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Eigenvalue; Eigenvector; Fractional Fourier transform (FRFT); Fractional Laplace transform; Fractional Z transform; Linear canonical transform (LCT); Offset discrete FT (DFT)

Indexed keywords

DISCRETE FOURIER TRANSFORMS; IMAGE PROCESSING; LAPLACE TRANSFORMS; MODULATION; OPTICAL SYSTEMS; Z TRANSFORMS;

FRACTIONAL FOURIER TRANSFORM; HERMITE-GAUSSIAN FUNCTIONS; LINEAR CONICAL TRANSFORM;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 34547107598     PISSN: 10577122     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCSI.2007.900182     Document Type: Article

References (31)

1. R. N. Bracewell, The Fourier Integral and Its Applications. Boston, MA: McGraw Hill, 2000.
2. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform With Applications in Optics and Signal Processing. New York: Wiley, 2000.
3. V. Namias, "The fractional-order Fourier transform and its application to quantum mechanics," J. Inst. Maths. Appl., vol. 25, pp. 241-265, 1980.
4. L. B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3084-3091, Nov. 1994.
5. K. B. Wolf, "Ch. 9: Canonical transforms," in Integral Transforms in Science and Engineering, K. B. Wolf, Ed. New York: Plenum, 1979, pp. 381-416.
6. G. Bongiovanni, P. Corsini, and G. Frosini, "One-dimensional and two-dimensional generalized discrete Fourier transforms," IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-24, no. 1, pp. 97-99, Feb. 1976.
7. O. K. Erosy, "Semisystolic array implementation of circular, skew-circular, and linear convolutions," IEEE Trans. Comput., vol. COM-34, no. 2, pp. 190-194, Feb. 1985.
8. S. Abe and J. T. Sheridan, "Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation," Opt. Lett., vol. 19, pp. 1801-1803, 1994.
9. S. Abe and J. T. Sheridan, "Generalization of the fractional Fourier transformation to an arbitrary linear loseless transformation: An operator approach," J. Phys. A, vol. 27, pp. 4179-4187, 1994.
10. D. F. V. James and G. S. Agarwal, "The generalized Fresnel transform and its applications to optics," Opt. Commun., vol. 126, pp. 207-212, 1996.
11. S. C. Pei and J. J. Ding, "Eigenfunctions of linear canonical transform," IEEE Trans. Signal Process., vol. 50, no. 1, pp. 11-26, Jan. 2002.
12. S. C. Pei and J. J. Ding, "Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms," J. Opt. Soc. Am. A, vol. 20, pp. 522-532, 2003.
13. B. W. Dickinson and K. Steiglitz, "Eigenvectors and functions of the discrete Fourier transform," IEEE Trans. Acoust, Speech, Signal Process., vol. ASSP-30, no. 1, pp. 25-31, Jan. 1982.
14. C. C. Tseng, "Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices," IEEE Trans. Signal Process., vol. 50, no. 4, pp. 866-877, Apr. 2002.
15. S. C. Pei and J. J. Ding, "Generalized eigenvectors and fractionalization of offset DFTs and DCTs," IEEE Trans. Signal Process., vol. 52, no. 7, pp. 2032-2046, Jul. 2004.
16. F. A. Grünbaum, "The eigenvectors of the discrete Fourier transform: A version of the Hermite functions," J. Math. Anal. Appl., vol. 88, pp. 355-363, 1982.
17. S. Clary and D. H. Mugler, "Shifted Fourier matrices and their tridiagonal commutors," SIAM J. Matrix Anal. Appl., vol. 24, pp. 809-821, 2003.
18. S. C. Pei, M. H. Yeh, and C. C. Tseng, "Discrete fractional Fourier transform based on orthogonal projections," IEEE Trans. Signal Process., vol. 47, no. 5, pp. 1335-1348, May 1999.
19. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. New York: McGraw-Hill, 1996.
20. C. Candan, M. A. Kutay, and H. M. Ozaktas, "The discrete fractional Fourier transform," IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1329-1337, May 2000.
21. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, "Graded-index fibers, Wigner distribution and the fractional Fourier transform," Appl. Opt., vol. 33, pp. 6188-6193, 1994.
22. T. Alieva and A. M. Barbe, "Self-fractional Fourier functions and selection of modes," J. Phys. A: Math. Gen., vol. 30, pp. 211-215, 1997.
23. M. Nazarathy and J. Shamir, "First-order optics: Operator representation for systems with loss or gain," J. Opt. Soc. Amer. vol. 72, no. 10, pp. 1398-1408, Oct. 1982.
24. M. Nazarathy, A. Hardy, and J. Shamir, "Generalized mode propagation in first-order optical systems with loss or gain," J. Opt. Soc. Amer. vol. 72, no. 10, pp. 1409-1420, Oct. 1982.
25. K. B. Wolf, "On self-reciprocal functions under a class of integral transforms," J. Math. Phys., vol. 18, pp. 1046-1051, 1977.
26. J. H. McClellan and T. W. Parks, "Eigenvalue and eigenvector decomposition of the discrete Fourier transform," IEEE Trans. Audio Electroacoust., vol. AE-20, no. 1, pp. 66-74, Mar. 1972.
27. N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, "Continuous versus discrete fractional Fourier transforms," J. Comput. Appl. Math., vol. 107, no. 1, pp. 73-95, Jul. 1999.
28. T. Alieva and M. J. Bastiaans, "Self-affinity in phase space," J. Opt. Soc. Amer. A, vol. 17, no. 4, pp. 756-761, Apr. 2000.
29. T. Alieva and M. L. Calvo, "Fractionalization of the linear cyclic transforms," J. Opt. Soc. Amer. A, vol. 17, no. 12, pp. 2330-2338, Dec. 2000.
30. M. T. Hanna, N. P. A. Seif, and W. A. E. M. Ahmed, "Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces," IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2815-2819, Jul. 2006.
31. M. J. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Amer. A, vol. 23, no. 8, pp. 1875-1883, Aug. 2006.