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

Navdeep Goel ; Kulbir Singh

International Journal of Applied Mathematics and Computer Science, Tome 23 (2013), p. 685-695 / Harvested from The Polish Digital Mathematics Library

Résumé

The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.

Détail

Publié le : 2013-01-01

Zbl 1317.94026

EUDML-ID : urn:eudml:doc:262405

Bibliographie

[000] Abe, S. and Sheridan, J. (1994a). Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: An operator approach, Journal of Physics, A: Mathematical and General 27(12): 4179-4187. | Zbl 0859.42008

[001] Abe, S. and Sheridan, J. (1994b). Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation, Optics Letters 19(22): 1801-1803.

[002] Alieva, T. and Bastiaans, M. (1999). Powers of transfer matrices determined by means of eigenfunctions, Journal of Optical Society of America A 16(10): 2413-2418.

[003] Almeida, L. (1994). The fractional Fourier transform and time-frequency representations, IEEE Transactions on Signal Processing 42(11): 3084-3091.

[004] Almeida, L. (1997). Product and convolution theorems for the fractional Fourier transform, IEEE Signal Processing Letters 4(1): 15-17.

[005] Barshan, B., Ozaktas, H. and Kutey, M. (1997). Optimal filters with linear canonical transformations, Optics Communications 135(1-3): 32-36.

[006] Bastiaans, M. (1979). Wigner distribution function and its application to first-order optics, Journal of Optical Society of America 69(12): 1710-1716.

[007] Bernardo, L. (1996). ABCD matrix formalism of fractional Fourier optics, Optical Engineering 35(03): 732-740.

[008] Bouachache, B. and Rodriguez, F. (1984). Recognition of time-varying signals in the time-frequency domain by means of the Wigner distribution, IEEE International Conference on Acoustics, Speech, and Signal Processing, San Diego, CA, USA, Vol. 9, pp. 239-242.

[009] 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 Journal of Research 35(3): 217-250.

[010] Cohen, L. (1989). Time-frequency distributions: A review, Proceedings of the IEEE 77(7): 941-981.

[011] Collins, S. (1970). Lens-system diffraction integral written in terms of matrix optics, Journal of Optical Society of America 60(9): 1168-1177.

[012] Deng, B., Tao, R. and Wang, Y. (2006). Convolution theorem for the linear canonical transform and their applications, Science in China, Series F: Information Sciences 49(5): 592-603.

[013] Gdawiec, K. and Domańska, D. (2011). Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapes, International Journal of Applied Mathematics and Computer Science 21(4): 757-767, DOI: 10.2478/v10006-011-0060-8. | Zbl 1283.68314

[014] Goel, N. and Singh, K. (2011). Analysis of Dirichlet, generalized Hamming and triangular window functions in the linear canonical transform domain, Signal, Image and Video Processing DOI: 10.1007/s11760-011-0280-2.

[015] 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.

[016] Healy, J. and Sheridan, J.T. (2009). Sampling and discretization of the linear canonical transform, Signal Processing 89(4): 641-648. | Zbl 1157.94332

[017] Hennelly, B. and Sheridan, J.T. (2005a). Fast numerical algorithm for the linear canonical transform, Journal of Optical Society of America A 22(5): 928-937.

[018] 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 A 22(5): 917-927.

[019] Hlawatsch, F. and Boudreaux-Bartels, G.F. (1992). Linear and quadratic time-frequency signal representation, IEEE Signal Processing Magazine 9(2): 21-67.

[020] 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.

[021] Hong-yi, F. and VanderLinde, J. (1989). Mapping of classical canonical transformations to quantum unitary operators, Physical Review A 39(6): 2987-2993.

[022] Hong-yi, F. and Yue, F. (2003). New eigenmodes of propagation in quadratic graded index media and complex fractional Fourier transform, Communication Theoretical Physics 39(1): 97-100.

[023] Hong-yi, F. and Zaidi, H. (1987). New approach for calculating the normally ordered form of squeeze operators, Physical Review D 35(6): 1831-1834.

[024] Hua, J., Liu, L. and Li, G. (1997). Extended fractional Fourier transforms, Journal of Optical Society of America A 14(12): 3316-3322.

[025] Huang, J. and Pandić, P. (2006). Dirac Operators in Representation Theory, Birkhauser/Springer, Boston, MA/New York, NY.

[026] James, D. and Agarwal, G. (1996). The generalized Fresnel transform and its applications to optics, Optics Communications 126(4-6): 207-212.

[027] Kiusalaas, J. (2010). Numerical Methods in Engineering with Python, Cambridge University Press, New York, NY. | Zbl 1189.65002

[028] Koc, A., Ozaktas, H., Candan, C. and Kutey, M. (2008). Digital computation of linear canonical transforms, IEEE Transactions on Signal Processing 56(6): 2383-2394.

[029] Li, B., Tao, R. and Wang, Y. (2007). New sampling formulae related to linear canonical transform, Signal Processing 87(5): 983-990. | Zbl 1186.94201

[030] Moshinsky, M. and Quesne, C. (1971). Linear canonical transformations and their unitary representations, Journal of Mathematical Physics 12(8): 1772-1783. | Zbl 0247.20051

[031] Namias, V. (1979). The fractional order Fourier transform and its application to quantum mechanics, IMA Journal of Applied Mathematics 25(3): 241-265. | Zbl 0434.42014

[032] Nazarathy, M. and Shamir, J. (1982). First-order optics-a canonical operator representation: Lossless systems, Journal of Optical Society of America A 72(3): 356-364.

[033] Ogura, A. (2009). Classical and quantum ABCD-transformation and the propagation of coherent and Gaussian beams, Journal of Physics, B: Atomic, Molecular and Optical Physics 42(14): 145504, DOI:10.1088/0953-4075/42/14/145504.

[034] Ogura, A. and Sekiguchi, M. (2007). Algebraic structure of the Feynman propagator and a new correspondence for canonical transformations, Journal of Mathematical Physics 48(7): 072102, DOI:10.1063/1.2748378. | Zbl 1144.81395

[035] 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 Society of America A 27(8): 1885-1895.

[036] 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 of America A 11(2): 547-559.

[037] Ozaktas, H., Kutey, M. and Zalevsky, Z. (2000). The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley and Sons, New York, NY.

[038] Palma, C. and Bagini, V. (1997). Extension of the Fresnel transform to ABCD systems, Journal of Optical Society of America A 14(8): 1774-1779.

[039] Pei, S. and Ding, J.J. (2001). Relations between fractional operations and time-frequency distributions, and their applications, IEEE Transactions on Signal Processing 49(8): 1638-1655.

[040] Pei, S. and Ding, J.J. (2002a). Closed-form discrete fractional and affine Fourier transforms, IEEE Transactions on Signal Processing 48(5): 1338-1353. | Zbl 1018.94002

[041] Pei, S. and Ding, J.J. (2002b). Eigenfunctions of linear canonical transform, IEEE Transactions on Signal Processing 50(1): 11-26.

[042] Portnoff, M. (1980). Time-frequency representation of digital signals and systems based on short-time Fourier analysis, IEEE Transactions on Acoustics, Speech and Signal Processing 28(1): 55-69. | Zbl 0523.93043

[043] Puri, R. (2001). Mathematical Methods of Quantum Optics, Springer-Verlag, Berlin/Heidelberg. | Zbl 1041.81108

[044] Sharma, K. and Joshi, S. (2006). Signal separation using linear canonical and fractional Fourier transforms, Optics Communications 265(2): 454-460.

[045] Sharma, K. and Joshi, S. (2007). Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to super resolution, Optics Communications 278(1): 52-59.

[046] Singh, A. and Saxena, R. (2012). On convolution and product theorems for FRFT, Wireless Personal Communications 65(1): 189-201.

[047] Stern, A. (2006). Sampling of linear canonical transformed signals, Signal Processing 86(7): 1421-1425. | Zbl 1172.94344

[048] Świercz, E. (2010). Classification in the Gabor time-frequency domain of non-stationary signals embedded in heavy noise with unknown statistical distribution, International Journal of Applied Mathematics and Computer Science 20(1): 135-147, DOI: 10.2478/v10006-010-0010-x. | Zbl 1300.62045

[049] Shin, Y.J. and Park, C.H. (2011). Analysis of correlation based dimension reduction methods, International Journal of Applied Mathematics and Computer Science 21(3): 549-558, DOI: 10.2478/v10006-011-0043-9. | Zbl 1230.68173

[050] 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 Processing 56(11): 5454-5464.

[051] Tao, R., Qi, L. and Wang, Y. (2004). Theory and Applications of the Fractional Fourier Transform, Tsinghua University Press, Beijing.

[052] 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 Processing 31(1): 301-312. | Zbl 1252.94035

[053] 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.

[054] Wigner, E. (1932). On the quantum correlation for thermodynamic equilibrium, Physical Review 40(5): 749-759. | Zbl 58.0948.07

[055] Wolf, K. (1979). Integral Transforms in Science and Engineering, Plenum Press, New York, NY. | Zbl 0409.44001

[056] Zayad, A. (1998). A product and convolution theorems for the fractional Fourier transform, IEEE Signal Processing Letters 5(4): 101-103.

[057] Zhang, W., Feng, D. and Gilmore, R. (1990). Coherent states: Theory and some applications, Reviews of Modern Physics 62(4): 867-927.