Have a personal or library account? Click to login

A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

International Journal of Applied Mathematics and Computer Science
Volume 23 (2013): Issue 3 (September 2013)
By:
Open Access
|Sep 2013

References

  1. Almeida, L. (1994). The fractional Fourier transform and time-frequency representations, IEEE Transactions on SignalProcessing 42(11): 3084-3091.10.1109/78.330368
    Search in Google Scholar
  2. Almeida, L. (1997). Product and convolution theorems for the fractional Fourier transform, IEEE Signal Processing Letters4(1): 15-17.10.1109/97.551689
    Search in Google Scholar
  3. Barshan, B., Ozaktas, H. and Kutey, M. (1997). Optimal filters with linear canonical transformations, Optics Communications 135(1-3): 32-36.10.1016/S0030-4018(96)00598-6
    Search in Google Scholar
  4. Bastiaans, M. (1979). Wigner distribution function and its application to first-order optics, Journal of Optical Societyof America 69(12): 1710-1716.10.1364/JOSA.69.001710
    Search in Google Scholar
  5. Bernardo, L. (1996). ABCD matrix formalism of fractional Fourier optics, Optical Engineering 35(03): 732-740.10.1117/1.600641
    Search in Google Scholar
  6. Bouachache, B. and Rodriguez, F. (1984). Recognition of time-varying signals in the time-frequency domain by means of the Wigner distribution, IEEE International Conferenceon Acoustics, Speech, and Signal Processing, SanDiego, CA, USA, Vol. 9, pp. 239-242.
    Search in Google Scholar
  7. Classen, T.A.C.M. and Mecklenbrauker, W.F.G. (1980). The Wigner distribution: A tool for time-frequency signal analysis, Part I: Continuous time signals, Philips Journalof Research 35(3): 217-250.
    Search in Google Scholar
  8. Cohen, L. (1989). Time-frequency distributions: A review, Proceedingsof the IEEE 77(7): 941-981.10.1109/5.30749
    Search in Google Scholar
  9. Collins, S. (1970). Lens-system diffraction integral written in terms of matrix optics, Journal of Optical Society of America60(9): 1168-1177.10.1364/JOSA.60.001168
    Search in Google Scholar
  10. Deng, B., Tao, R. and Wang, Y. (2006). Convolution theorem for the linear canonical transform and their applications, Science in China, Series F: Information Sciences49(5): 592-603.10.1007/s11432-006-2016-4
    Search in Google Scholar
  11. Gdawiec, K. and Domańska, D. (2011). Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapes, International Journalof Applied Mathematics and Computer Science 21(4): 757-767, DOI: 10.2478/v10006-011-0060-8.10.2478/v10006-011-0060-8
    Search in Google Scholar
  12. Goel, N. and Singh, K. (2011). Analysis of Dirichlet, generalized Hamming and triangular window functions in the linear canonical transform domain, Signal, Image andVideo Processing DOI: 10.1007/s11760-011-0280-2.10.1007/s11760-011-0280-2
    Search in Google Scholar
  13. Healy, J. and Sheridan, J. (2008). Cases where the linear canonical transform of a signal has compact support or is band-limited, Optics Letters 33(3): 228-230.10.1364/OL.33.000228
    Search in Google Scholar
  14. Healy, J. and Sheridan, J.T. (2009). Sampling and discretization of the linear canonical transform, Signal Processing89(4): 641-648.10.1016/j.sigpro.2008.10.011
    Search in Google Scholar
  15. Hennelly, B. and Sheridan, J.T. (2005a). Fast numerical algorithm for the linear canonical transform, Journal ofOptical Society of America A 22(5): 928-937.10.1364/JOSAA.22.00092815898553
    Search in Google Scholar
  16. Hennelly, B. and Sheridan, J.T. (2005b). Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms, Journal of Optical Society of America A22(5): 917-927.10.1364/JOSAA.22.00091715898552
    Search in Google Scholar
  17. Hlawatsch, F. and Boudreaux-Bartels, G.F. (1992). Linear and quadratic time-frequency signal representation, IEEE SignalProcessing Magazine 9(2): 21-67.10.1109/79.127284
    Search in Google Scholar
  18. Hong-yi, F., Ren, H. and Hai-Liang, L. (2008). Convolution theorem of fractional Fourier transformation derived by representation transformation in quantum mechanics, Communication Theoretical Physics 50(3): 611-614.10.1088/0253-6102/50/3/15
    Search in Google Scholar
  19. Hong-yi, F. and VanderLinde, J. (1989). Mapping of classical canonical transformations to quantum unitary operators, Physical Review A 39(6): 2987-2993.10.1103/PhysRevA.39.2987
    Search in Google Scholar
  20. Hong-yi, F. and Yue, F. (2003). New eigenmodes of propagation in quadratic graded index media and complex fractional Fourier transform, Communication Theoretical Physics39(1): 97-100.10.1088/0253-6102/39/1/97
    Search in Google Scholar
  21. Hong-yi, F. and Zaidi, H. (1987). New approach for calculating the normally ordered form of squeeze operators, PhysicalReview D 35(6): 1831-1834.10.1103/PhysRevD.35.1831
    Search in Google Scholar
  22. Hua, J., Liu, L. and Li, G. (1997). Extended fractional Fourier transforms, Journal of Optical Society of AmericaA 14(12): 3316-3322.10.1364/JOSAA.14.003316
    Search in Google Scholar
  23. Huang, J. and Pandić, P. (2006). Dirac Operators in RepresentationTheory, Birkhauser/Springer, Boston, MA/ New York, NY.
    Search in Google Scholar
  24. James, D. and Agarwal, G. (1996). The generalized Fresnel transform and its applications to optics, Optics Communications126(4-6): 207-212.10.1016/0030-4018(95)00708-3
    Search in Google Scholar
  25. Kiusalaas, J. (2010). Numerical Methods in Engineering withPython, Cambridge University Press, New York, NY.10.1017/CBO9780511812224
    Search in Google Scholar
  26. Koc, A., Ozaktas, H., Candan, C. and Kutey, M. (2008). Digital computation of linear canonical transforms, IEEE Transactionson Signal Processing 56(6): 2383-2394.10.1109/TSP.2007.912890
    Search in Google Scholar
  27. Li, B., Tao, R. and Wang, Y. (2007). New sampling formulae related to linear canonical transform, Signal Processing87(5): 983-990.10.1016/j.sigpro.2006.09.008
    Search in Google Scholar
  28. Moshinsky, M. and Quesne, C. (1971). Linear canonical transformations and their unitary representations, Journalof Mathematical Physics 12(8): 1772-1783.10.1063/1.1665805
    Search in Google Scholar
  29. Namias, V. (1979). The fractional order Fourier transform and its application to quantum mechanics, IMA Journal of AppliedMathematics 25(3): 241-265.10.1093/imamat/25.3.241
    Search in Google Scholar
  30. Nazarathy, M. and Shamir, J. (1982). First-order optics-a canonical operator representation: Lossless systems, Journalof Optical Society of America A 72(3): 356-364.10.1364/JOSA.72.000356
    Search in Google Scholar
  31. Ogura, A. (2009). Classical and quantum ABCD-transformation and the propagation of coherent and Gaussian beams, Journalof Physics, B: Atomic, Molecular and Optical Physics42(14): 145504, DOI:10.1088/0953-4075/42/14/145504.10.1088/0953-4075/42/14/145504
    Search in Google Scholar
  32. Ogura, A. and Sekiguchi, M. (2007). Algebraic structure of the Feynman propagator and a new correspondence for canonical transformations, Journal of MathematicalPhysics 48(7): 072102, DOI:10.1063/1.2748378.10.1063/1.2748378
    Search in Google Scholar
  33. Oktem, F. and Ozaktas, H. (2010). Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: A generalization of the space-bandwidth product, Journal of Optical Societyof America A 27(8): 1885-1895.10.1364/JOSAA.27.00188520686595
    Search in Google Scholar
  34. Ozaktas, H., Barshan, B., Mendlovic, D. and Onural, L. (1994). Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms, Journal of Optical Society ofAmerica A 11(2): 547-559. 10.1364/JOSAA.11.000547
    Search in Google Scholar
  35. Ozaktas, H., Kutey, M. and Zalevsky, Z. (2000). The FractionalFourier Transform with Applications in Optics and SignalProcessing, John Wiley and Sons, New York, NY.
    Search in Google Scholar
  36. Palma, C. and Bagini, V. (1997). Extension of the Fresnel transform to ABCD systems, Journal of Optical Societyof America A 14(8): 1774-1779.10.1364/JOSAA.14.001774
    Search in Google Scholar
  37. Pei, S. and Ding, J.J. (2001). Relations between fractional operations and time-frequency distributions, and their applications, IEEE Transactions on Signal Processing49(8): 1638-1655.10.1109/78.934134
    Search in Google Scholar
  38. Pei, S. and Ding, J.J. (2002a). Closed-form discrete fractional and affine Fourier transforms, IEEE Transactions on SignalProcessing 48(5): 1338-1353.10.1109/78.839981
    Search in Google Scholar
  39. Pei, S. and Ding, J.J. (2002b). Eigenfunctions of linear canonical transform, IEEE Transactions on Signal Processing50(1): 11-26.10.1109/78.972478
    Search in Google Scholar
  40. Portnoff, M. (1980). Time-frequency representation of digital signals and systems based on short-time Fourier analysis, IEEE Transactions on Acoustics, Speech and Signal Processing28(1): 55-69.10.1109/TASSP.1980.1163359
    Search in Google Scholar
  41. Puri, R. (2001). Mathematical Methods of Quantum Optics, Springer-Verlag, Berlin/Heidelberg.10.1007/978-3-540-44953-9
    Search in Google Scholar
  42. Sharma, K. and Joshi, S. (2006). Signal separation using linear canonical and fractional Fourier transforms, Optics Communications265(2): 454-460.10.1016/j.optcom.2006.03.062
    Search in Google Scholar
  43. Sharma, K. and Joshi, S. (2007). Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to super resolution, Optics Communications278(1): 52-59.10.1016/j.optcom.2007.06.022
    Search in Google Scholar
  44. Singh, A. and Saxena, R. (2012). On convolution and product theorems for FRFT, Wireless Personal Communications65(1): 189-201.10.1007/s11277-011-0235-5
    Search in Google Scholar
  45. Stern, A. (2006). Sampling of linear canonical transformed signals, Signal Processing 86(7): 1421-1425. 10.1016/j.sigpro.2005.07.031
    Search in Google Scholar
  46. Świercz, E. (2010). Classification in the Gabor time-frequency domain of non-stationary signals embedded in heavy noise with unknown statistical distribution, International Journalof Applied Mathematics and Computer Science 20(1): 135-147, DOI: 10.2478/v10006-010-0010-x.10.2478/v10006-010-0010-x
    Search in Google Scholar
  47. Shin, Y.J. and Park, C.H. (2011). Analysis of correlation based dimension reduction methods, International Journal of AppliedMathematics and Computer Science 21(3): 549-558, DOI: 10.2478/v10006-011-0043-9.10.2478/v10006-011-0043-9
    Search in Google Scholar
  48. Tao, R., Li, B., Wang, Y. and Aggrey, G. (2008). On sampling of bandlimited signals associated with the linear canonical transform, IEEE Transactions on Signal Processing56(11): 5454-5464.10.1109/TSP.2008.929333
    Search in Google Scholar
  49. Tao, R., Qi, L. and Wang, Y. (2004). Theory and Applicationsof the Fractional Fourier Transform, Tsinghua University Press, Beijing.
    Search in Google Scholar
  50. Wei, D., Ran, Q. and Li, Y. (2012). A convolution and correlation theorem for the linear canonical transform and its application, Circuits, Systems, and Signal Processing31(1): 301-312.10.1007/s00034-011-9319-4
    Search in Google Scholar
  51. Wei, D., Ran, Q., Li, Y., Ma, J. and Tan, L. (2009). A convolution and product theorem for the linear canonical transform, IEEE Signal Processing Letters 16(10): 853-856.10.1109/LSP.2009.2026107
    Search in Google Scholar
  52. Wigner, E. (1932). On the quantum correlation for thermodynamic equilibrium, Physical Review 40(5): 749-759.10.1103/PhysRev.40.749
    Search in Google Scholar
  53. Wolf, K. (1979). Integral Transforms in Science and Engineering, Plenum Press, New York, NY.10.1007/978-1-4757-0872-1
    Search in Google Scholar
  54. Zayad, A. (1998). A product and convolution theorems for the fractional Fourier transform, IEEE Signal Processing Letters5(4): 101-103.10.1109/97.664179
    Search in Google Scholar
  55. Zhang, W., Feng, D. and Gilmore, R. (1990). Coherent states: Theory and some applications, Reviews of Modern Physics62(4): 867-927. 10.1103/RevModPhys.62.867
    Search in Google Scholar
DOI: https://doi.org/10.2478/amcs-2013-0051 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Journal RSS Feed
Language: English
Page range: 685 - 695
Published on: Sep 30, 2013
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 times per year
Keywords:
linear canonical transform,
convolution and product theorem,
quantum mechanical representation
Related subjects:
Mathematics,
Applied mathematics

© 2013 Navdeep Goel, Kulbir Singh, published by University of Zielona Góra
This work is licensed under the Creative Commons License.

Previous articleVolume 23 (2013): Issue 3 (September 2013)