Russian Microelectronics

, Volume 30, Issue 2, pp 118–124 | Cite as

Digital Simulation of Fractal Measurements

  • P. A. Arutyunov


The principles of digital signal processing are extended for the existing methods of fractal measurements in terms of Hausdorff–Bezikovich and Mandelbrojt dimensions. Fractal dimension as a rational number G1/G2or the ratio of residues GR1/GR2of two equations of measurement (instrument-response equations) is represented in the form of an equivalent digital model of the ratio F1/F2of two sampling rates in some linear digital system. It is shown that such a system is described by generalized convolution, which, in turn, is represented as a cascade connection of interpolators and decimators with integer-valued conversion coefficients. For error-free processing of fractal dimension, it is necessary to convert a rational number to an integer and use Farey fractions in terms of unimodular and multimodular arithmetic. The Farey fractions can then be used as references points in metrological scales.


Signal Processing Convolution Reference Point Fractal Dimension Sampling Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Crochiere, R.E. and Rabiner, L.R., Interpolation and Decimation of Digital Signals—A Tutorial Review, Proc. IEEE, 1981, vol. 69, no. 3, pp. 300–331.Google Scholar
  2. 2.
    Arutyunov, P.A., Digital Processing of Fractal Measurements in Finite Fields, Mikroelektronika, 1999, vol. 28, no. 4, pp. 301–307.Google Scholar
  3. 3.
    Arutyunov, P.A., Algorithmic VLSI Structures for Error-Free Processing of Measurement Signal, Vsesoyuznaya nauchno-tekhnicheskaya konferentsiyaMetrologicheskie problemy mikroelektroniki,” (All-Union Conf. on Metrology Problems in Microelectronics), Moscow: Radio i Svyaz', 1988.Google Scholar
  4. 4.
    Ivanova, V.S., From Dislocations to Fractals, Mezhdistsiplinarnyi seminarFraktaly i prikladnaya sinergetika” (Interdisciplinary Workshop on Fractals and Applied Synergy), Moscow, 1999.Google Scholar
  5. 5.
    Arutyunov, P.A., Residue-Class Systems Applied to the Instrument-Response Law in Digital Signal Processing, Mikroelektronika, 1995, vol. 24, no. 5, pp. 355–359.Google Scholar
  6. 6.
    Arutyunov, P.A., Teoriya i primenenie algoritmicheskikh izmerenii (Algorithmic Measurements: Theory and Applications), Moscow: Energoatomizdat, 1991.Google Scholar
  7. 7.
    Arutyunov, P.A., Indirect Measurements in Finite Fields, Izmer. Tekh., 1999, no. 4, pp. 11–16.Google Scholar
  8. 8.
    Trofilov, V.I., Kartuzov, V.V., and Minakov, N.N., Relation between Fractal Dimension of Fracture Surface and Mechanical Properties, Izmer. Tekh., 1999, no. 4.Google Scholar
  9. 9.
    Ivanova, V.S. and Vstovskii, G.V., How to Determine whether Material Structure is Fractal?, Izmer. Tekh., 1999, no. 4.Google Scholar
  10. 10.
    Ivanova, V.S. and Vstovskii, G.V., Determinate Self-Organization of Fractal Structures in Chaotically Nonlinear Systems, Izmer. Tekh., 1999, no. 4.Google Scholar
  11. 11.
    Malyshev, V.N., Ivanova, V.S., and Vstovskii, G.V., Realization of Synergy Principles and Multifractal Parametrization Conception to Increase Wear Resistance of Oxide Coatings by Optimizing Oxidation Conditions, Izmer. Tekh., 1999, no. 4.Google Scholar
  12. 12.
    Gregory, R.T. and Krishnamurthy, E.V., Methods and Applications of Error-Free Computations, New York: Springer-Verlag, 1984. Translated under the title Bezoshibochnye vychisleniya. Metody i prilozheniya, Moscow: Mir, 1988.Google Scholar
  13. 13.
    Gregory, R.T., A Method for and an Application of Error-Free Computations, Proc. AFCET Symp.Mathematics for Computer Science,” Paris, 1982, pp. 152–158.Google Scholar
  14. 14.
    Gregory, R.T., Exact Computation with Order-N Farey Fractions, Computer Science and Statistics (Proc. 15th Symp. on the Interface), Gentle, J.E., Ed., Amsterdam: North Holland, 1983.Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2001

Authors and Affiliations

  • P. A. Arutyunov
    • 1
  1. 1.Moscow State Institute of Electronics and MathematicsMoscowRussia

Personalised recommendations