Local fractional \(Z\)-transforms with applications to signals on Cantor sets. (English) Zbl 1468.94171

Liu, Kai; Hu, Ren-Jie; Cattani, Carlo; Xie, Gong-Nan; Yang, Xiao-Jun; Zhao, Yang

Abstr. Appl. Anal. 2014, Article ID 638648, 6 p. (2014).

Summary: The \(Z\)-transform has played an important role in signal processing. In this paper the \(Z\)-transform has been generalized by the coupling of both the \(Z\)-transform and the local fractional complex calculus. In the literature the local fractional \(Z\)-transform is applied to analyze signals, in the following it will be used to analyze signals on Cantor sets. Some examples are also given to show the efficiency and accuracy for handling the signals on Cantor sets.

MSC:

94A12	Signal theory (characterization, reconstruction, filtering, etc.)
28A80	Fractals

References:

[1]	Davies, B., Integral Transforms and Their Applications. Integral Transforms and Their Applications, Texts in Applied Mathematics (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 0996.44001 · doi:10.1007/978-1-4684-9283-5
[2]	Debnath, L.; Bhatta, D., Integral Transforms and Their Applications (2010), New York, NY, USA: CRC Press, New York, NY, USA
[3]	Tarasov, V. E., Fractional Dynamics. Fractional Dynamics, Nonlinear Physical Science (2010), New York, NY, USA: Springer, New York, NY, USA · Zbl 1186.83027 · doi:10.1007/978-3-642-14003-7
[4]	Herrmann, R., Fractional Calculus: An Introduction for Physicists (2011), Singapore: World Scientific, Singapore · Zbl 1232.26006
[5]	Malinowska, A. B.; Torres, D. F. M., Introduction to the Fractional Calculus of Variations (2012), London, UK: Imperial College Press, London, UK · Zbl 1258.49001
[6]	Ortigueira, M. D., Fractional Calculus for Scientists and Engineers. Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1251.26005 · doi:10.1007/978-94-007-0747-4
[7]	Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity (2010), London, UK: Imperial College Press, London, UK · Zbl 1210.26004 · doi:10.1142/9781848163300
[8]	Sabatier, J.; Agrawal, O. P.; Machado, J. A. T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (2007), New York, NY, USA: Springer, New York, NY, USA · Zbl 1116.00014
[9]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[10]	Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[11]	West, B. J.; Bologna, M.; Grigolini, P., Physics of Fractal Operators. Physics of Fractal Operators, Institute for Nonlinear Science (2003), New York, NY, USA: Springer, New York, NY, USA · doi:10.1007/978-0-387-21746-8
[12]	Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[13]	Miller, L., On the controllability of anomalous diffusions generated by the fractional Laplacian, Mathematics of Control, Signals, and Systems, 18, 3, 260-271 (2006) · Zbl 1105.93015 · doi:10.1007/s00498-006-0003-3
[14]	West, B. J.; Grigolini, P.; Metzler, R.; Nonnenmacher, T. F., Fractional diffusion and Lévy stable processes, Physical Review E, 55, 1, 99-106 (1997) · doi:10.1103/PhysRevE.55.99
[15]	Uchaikin, V. V., Anomalous diffusion and fractional stable distributions, Journal of Experimental and Theoretical Physics, 97, 4, 810-825 (2003) · doi:10.1134/1.1625072
[16]	Zhang, Y.; Benson, D. A.; Meerschaert, M. M.; LaBolle, E. M.; Scheffler, H. P., Random walk approximation of fractional-order multiscaling anomalous diffusion, Physical Review E, 74, 2 (2006)
[17]	Yang, Q.; Turner, I.; Liu, F.; Ilic, M., Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM Journal on Scientific Computing, 33, 3, 1159-1180 (2011) · Zbl 1229.35315 · doi:10.1137/100800634
[18]	Magin, R. L.; Abdullah, O.; Baleanu, D.; Zhou, X. J., Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, Journal of Magnetic Resonance, 190, 2, 255-270 (2008) · doi:10.1016/j.jmr.2007.11.007
[19]	Sun, H.; Chen, W.; Li, C.; Chen, Y., Fractional differential models for anomalous diffusion, Physica A, 389, 14, 2719-2724 (2010) · doi:10.1016/j.physa.2010.02.030
[20]	Xu, M.; Tan, W., Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion, Science in China A, 44, 11, 1387-1399 (2001) · Zbl 1138.76320
[21]	Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M., Fractional diffusion in inhomogeneous media, Journal of Physics A, 38, 42, L679-L684 (2005) · Zbl 1082.76097 · doi:10.1088/0305-4470/38/42/L03
[22]	Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371, 6, 461-580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[23]	Berkowitz, B.; Klafter, J.; Metzler, R.; Scher, H., Physical pictures of transport in heterogeneous media: advection-dispersion, random-walk, and fractional derivative formulations, Water Resources Research, 38, 10, 9-1-9-12 (2002)
[24]	Seymour, J. D.; Gage, J. P.; Codd, S. L.; Gerlach, R., Anomalous fluid transport in porous media induced by biofilm growth, Physical Review Letters, 93, 19 (2004) · doi:10.1103/PhysRevLett.93.198103
[25]	Luchko, Y.; Punzi, A., Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations, GEM—International Journal on Geomathematics, 1, 2, 257-276 (2011) · Zbl 1301.34104 · doi:10.1007/s13137-010-0012-8
[26]	Klafter, J.; Lim, S. C.; Metzler, R., Fractional Dynamics, Recent Advances (2012), Singapore: World Scientific, Singapore · Zbl 1238.93005
[27]	Chun, P.; Inoue, J., Yet another possible mechanism for anomalous transport: theory, numerical method, and experiments, KSCE Journal of Civil Engineering, 16, 1, 45-53 (2012) · doi:10.1007/s12205-012-1269-2
[28]	McBride, A. C.; Kerr, F. H., On Namias’s fractional Fourier transforms, IMA Journal of Applied Mathematics, 39, 2, 159-175 (1987) · Zbl 0644.44005 · doi:10.1093/imamat/39.2.159
[29]	Ozaktas, H. M.; Aytür, O., Fractional Fourier domains, Signal Processing, 46, 1, 119-124 (1995) · Zbl 0875.94037
[30]	Lohmann, A. W.; Mendlovic, D.; Zalevsky, Z., Fractional Hilbert transform, Optics Letters, 21, 4, 281-283 (1996)
[31]	Zhou, N.; Wang, Y.; Gong, L., Novel optical image encryption scheme based on fractional Mellin transform, Optics Communications, 284, 13, 3234-3242 (2011) · doi:10.1016/j.optcom.2011.02.065
[32]	Luchko, Y.; Kiryakova, V., The Mellin integral transform in fractional calculus, Fractional Calculus and Applied Analysis, 16, 2, 405-430 (2013) · Zbl 1312.26016 · doi:10.2478/s13540-013-0025-8
[33]	Mendlovic, D.; Zalevsky, Z.; Mas, D.; García, J.; Ferreira, C., Fractional wavelet transform, Applied Optics, 36, 20, 4801-4806 (1997)
[34]	Shi, J.; Zhang, N.; Liu, X., A novel fractional wavelet transform and its applications, Science China: Information Sciences, 55, 6, 1270-1279 (2012) · Zbl 1245.94047 · doi:10.1007/s11432-011-4320-x
[35]	Dinç, E.; Baleanu, D., Fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two-component mixture, Computers & Mathematics with Applications, 59, 5, 1701-1708 (2010) · Zbl 1189.94028 · doi:10.1016/j.camwa.2009.08.012
[36]	Fan, H. Y., Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation, Optics Letters, 28, 22, 2177-2179 (2003)
[37]	Jury, E. I., Theory and Applications of the Z-Transform Method (1964), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[38]	Machado, J. A. T., Analysis and design of fractional-order digital control systems, Systems Analysis Modelling Simulation, 27, 2-3, 107-122 (1997) · Zbl 0875.93154
[39]	Yang, X. J., Advanced Local Fractional Calculus and Its Applications (2012), New York, NY, USA: World Science, New York, NY, USA
[40]	Yang, X. J., Local Fractional Functional Analysis and Its Applications (2011), Hong Kong, China: Asian Academic Publisher, Hong Kong, China
[41]	Yang, X. J.; Baleanu, D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 17, 2, 625-628 (2013) · doi:10.2298/TSCI121124216Y
[42]	Yang, X. J.; Baleanu, D.; Zhong, W. P., Approximation solutions for diffusion equation on Cantor time-space, Proceeding of the Romanian Academy A, 14, 2, 127-133 (2013)
[43]	Zhao, Y.; Baleanu, D.; Cattani, C.; Cheng, D. F.; Yang, X. J., Maxwell’s equations on Cantor sets: a local fractional approach, Advances in High Energy Physics, 2013 (2013) · Zbl 1328.82035 · doi:10.1155/2013/686371
[44]	Yang, X. J.; Baleanu, D.; Machado, J. A. T., Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis, Boundary Value Problems, 1, article 131 (20132013) · Zbl 1296.35154 · doi:10.1186/1687-2770-2013-131
[45]	Gao, F.; Zhong, W. P.; Shen, X. M., Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral, Advanced Materials Research, 461, 306-310 (2012) · doi:10.4028/www.scientific.net/AMR.461.306
[46]	Zhao, Y.; Baleanu, D.; Baleanu, M. C.; Cheng, D. F.; Yang, X. J., Mappings for special functions on Cantor sets and special integral transforms via local fractional operators, Abstract and Applied Analysis, 2013 (2013) · Zbl 1295.26008 · doi:10.1155/2013/316978
[47]	He, J.-H., Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012 (2012) · Zbl 1257.35158 · doi:10.1155/2012/916793
[48]	Liu, C. F.; Kong, S. S.; Yuan, S. J., Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem, Thermal Science, 17, 3, 715-721 (2013) · doi:10.2298/TSCI120826075L
[49]	Zhao, C. G.; Yang, A. M.; Jafari, H.; Haghbin, A., The Yang-Laplace transform for solving the IVPs with local fractional derivative, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.34037 · doi:10.1155/2014/386459
[50]	Wang, S. Q.; Yang, Y. J.; Jassim, H. K., Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.35416 · doi:10.1155/2014/176395
[51]	Li, M.; Zhao, W., On bandlimitedness and lag-limitedness of fractional Gaussian noise, Physica A, 392, 9, 1955-1961 (2013) · doi:10.1016/j.physa.2012.12.035
[52]	Cattani, C., Harmonic wavelet approximation of random, fractal and high frequency signals, Telecommunication Systems, 43, 3-4, 207-217 (2010) · doi:10.1007/s11235-009-9208-3
[53]	Nagler, J.; Claussen, J. C., \(1/f^α\) spectra in elementary cellular automata and fractal signals, Physical Review E, 71, 6 (2005) · doi:10.1103/PhysRevE.71.067103
[54]	Gupta, A.; Joshi, S., Variable step-size LMS algorithm for fractal signals, IEEE Transactions on Signal Processing, 56, 4, 1411-1420 (2008) · Zbl 1390.94201 · doi:10.1109/TSP.2007.909374
[55]	West, B. J., Fractal Physiology and Chaos in Medicine (2013), Singapore: World Scientific, Singapore · Zbl 1295.92014
[56]	Li, M., Power spectrum of generalized fractional Gaussian noise, Advances in Mathematical Physics, 2013 (2013) · Zbl 1291.60077 · doi:10.1155/2013/315979
[57]	Li, M., Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space—a further study, Applied Mathematical Modelling, 31, 3, 625-631 (2007) · Zbl 1197.94006 · doi:10.1016/j.apm.2005.11.029
[58]	Li, M.; Lim, S. C., Power spectrum of generalized Cauchy process, Telecommunication Systems, 43, 3-4, 219-222 (2010) · doi:10.1007/s11235-009-9209-2
[59]	Li, M., A class of negatively fractal dimensional Gaussian random functions, Mathematical Problems in Engineering, 2011 (2011) · Zbl 1209.28015 · doi:10.1155/2011/291028
[60]	Yi, G., Local fractional Z transform in fractal space, Advances in Digital Multimedia, 1, 2, 96-102 (2012)