Abstract and Applied Analysis

Local Fractional Z-Transforms with Applications to Signals on Cantor Sets

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

Full-text: Open access

Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 638648, 6 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425047791

Digital Object Identifier
doi:10.1155/2014/638648

Mathematical Reviews number (MathSciNet)
MR3182299

Zentralblatt MATH identifier
07022798

Citation

Liu, Kai; Hu, Ren-Jie; Cattani, Carlo; Xie, Gong-Nan; Yang, Xiao-Jun; Zhao, Yang. Local Fractional $Z$ -Transforms with Applications to Signals on Cantor Sets. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 638648, 6 pages. doi:10.1155/2014/638648. https://projecteuclid.org/euclid.aaa/1425047791


Export citation

References

  • B. Davies, Integral Transforms and Their Applications, Texts in Applied Mathematics, Springer, New York, NY, USA, 2002.
  • L. Debnath and D. Bhatta, Integral Transforms and Their Applications, CRC Press, New York, NY, USA, 2010.
  • V. E. Tarasov, Fractional Dynamics, Nonlinear Physical Science, Springer, New York, NY, USA, 2010.
  • R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore, 2011.
  • A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, UK, 2012.
  • M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, Springer, New York, NY, USA, 2011.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010.
  • J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003.
  • R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.
  • L. Miller, “On the controllability of anomalous diffusions generated by the fractional Laplacian,” Mathematics of Control, Signals, and Systems, vol. 18, no. 3, pp. 260–271, 2006.
  • B. J. West, P. Grigolini, R. Metzler, and T. F. Nonnenmacher, “Fractional diffusion and Lévy stable processes,” Physical Review E, vol. 55, no. 1, pp. 99–106, 1997.
  • V. V. Uchaikin, “Anomalous diffusion and fractional stable distributions,” Journal of Experimental and Theoretical Physics, vol. 97, no. 4, pp. 810–825, 2003.
  • Y. Zhang, D. A. Benson, M. M. Meerschaert, E. M. LaBolle, and H. P. Scheffler, “Random walk approximation of fractional-order multiscaling anomalous diffusion,” Physical Review E, vol. 74, no. 2, Article ID 026706, 2006.
  • Q. Yang, I. Turner, F. Liu, and M. Ilic, “Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions,” SIAM Journal on Scientific Computing, vol. 33, no. 3, pp. 1159–1180, 2011.
  • R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance, vol. 190, no. 2, pp. 255–270, 2008.
  • H. Sun, W. Chen, C. Li, and Y. Chen, “Fractional differential models for anomalous diffusion,” Physica A, vol. 389, no. 14, pp. 2719–2724, 2010.
  • M. Xu and W. Tan, “Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion,” Science in China A, vol. 44, no. 11, pp. 1387–1399, 2001.
  • A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, “Fractional diffusion in inhomogeneous media,” Journal of Physics A, vol. 38, no. 42, pp. L679–L684, 2005.
  • G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002.
  • B. Berkowitz, J. Klafter, R. Metzler, and H. Scher, “Physical pictures of transport in heterogeneous media: advection-dispersion, random-walk, and fractional derivative formulations,” Water Resources Research, vol. 38, no. 10, pp. 9-1–9-12, 2002.
  • J. D. Seymour, J. P. Gage, S. L. Codd, and R. Gerlach, “Anomalous fluid transport in porous media induced by biofilm growth,” Physical Review Letters, vol. 93, no. 19, Article ID 198103, 4 pages, 2004.
  • Y. Luchko and A. Punzi, “Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations,” GEM–-International Journal on Geomathematics, vol. 1, no. 2, pp. 257–276, 2011.
  • J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics, Recent Advances, World Scientific, Singapore, 2012.
  • P. Chun and J. Inoue, “Yet another possible mechanism for anomalous transport: theory, numerical method, and experiments,” KSCE Journal of Civil Engineering, vol. 16, no. 1, pp. 45–53, 2012.
  • A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA Journal of Applied Mathematics, vol. 39, no. 2, pp. 159–175, 1987.
  • H. M. Ozaktas and O. Aytür, “Fractional Fourier domains,” Signal Processing, vol. 46, no. 1, pp. 119–124, 1995.
  • A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional Hilbert transform,” Optics Letters, vol. 21, no. 4, pp. 281–283, 1996.
  • N. Zhou, Y. Wang, and L. Gong, “Novel optical image encryption scheme based on fractional Mellin transform,” Optics Communications, vol. 284, no. 13, pp. 3234–3242, 2011.
  • Y. Luchko and V. Kiryakova, “The Mellin integral transform in fractional calculus,” Fractional Calculus and Applied Analysis, vol. 16, no. 2, pp. 405–430, 2013.
  • D. Mendlovic, Z. Zalevsky, D. Mas, J. García, and C. Ferreira, “Fractional wavelet transform,” Applied Optics, vol. 36, no. 20, pp. 4801–4806, 1997.
  • J. Shi, N. Zhang, and X. Liu, “A novel fractional wavelet transform and its applications,” Science China: Information Sciences, vol. 55, no. 6, pp. 1270–1279, 2012.
  • E. Dinç and D. Baleanu, “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, vol. 59, no. 5, pp. 1701–1708, 2010.
  • H. Y. Fan, “Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation,” Optics Letters, vol. 28, no. 22, pp. 2177–2179, 2003.
  • E. I. Jury, Theory and Applications of the Z-Transform Method, John Wiley & Sons, New York, NY, USA, 1964.
  • J. A. T. Machado, “Analysis and design of fractional-order digital control systems,” Systems Analysis Modelling Simulation, vol. 27, no. 2-3, pp. 107–122, 1997.
  • X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  • X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, China, 2011.
  • X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 2, pp. 625–628, 2013.
  • X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solutions for diffusion equation on Cantor time-space,” Proceeding of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013.
  • Y. Zhao, D. Baleanu, C. Cattani, D. F. Cheng, and X. J. Yang, “Maxwell's equations on Cantor sets: a local fractional approach,” Advances in High Energy Physics, vol. 2013, Article ID 686371, 6 pages, 2013.
  • X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis,” Boundary Value Problems, no. 1, article 131, 20132013.
  • F. Gao, W. P. Zhong, and X. M. Shen, “Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral,” Advanced Materials Research, vol. 461, pp. 306–310, 2012.
  • Y. Zhao, D. Baleanu, M. C. Baleanu, D. F. Cheng, and X. J. Yang, “Mappings for special functions on Cantor sets and special integral transforms via local fractional operators,” Abstract and Applied Analysis, vol. 2013, Article ID 316978, 6 pages, 2013.
  • J.-H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
  • C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,” Thermal Science, vol. 17, no. 3, pp. 715–721, 2013.
  • C. G. Zhao, A. M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 386459, 5 pages, 2014.
  • S. Q. Wang, Y. J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 176395, 7 pages, 2014.
  • M. Li and W. Zhao, “On bandlimitedness and lag-limitedness of fractional Gaussian noise,” Physica A, vol. 392, no. 9, pp. 1955–1961, 2013.
  • C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207–217, 2010.
  • J. Nagler and J. C. Claussen, “1/f$^{\alpha }$ spectra in elementary cellular automata and fractal signals,” Physical Review E, vol. 71, no. 6, Article ID 067103, 4 pages, 2005.
  • A. Gupta and S. Joshi, “Variable step-size LMS algorithm for fractal signals,” IEEE Transactions on Signal Processing, vol. 56, no. 4, pp. 1411–1420, 2008.
  • B. J. West, Fractal Physiology and Chaos in Medicine, World Scientific, Singapore, 2013.
  • M. Li, “Power spectrum of generalized fractional Gaussian noise,” Advances in Mathematical Physics, vol. 2013, Article ID 315979, 3 pages, 2013.
  • M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space–-a further study,” Applied Mathematical Modelling, vol. 31, no. 3, pp. 625–631, 2007.
  • M. Li and S. C. Lim, “Power spectrum of generalized Cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219–222, 2010.
  • M. Li, “A class of negatively fractal dimensional Gaussian random functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 291028, 18 pages, 2011.
  • G. Yi, “Local fractional Z transform in fractal space,” Advances in Digital Multimedia, vol. 1, no. 2, pp. 96–102, 2012. \endinput