• Українська
  • English
  • Русский
ISSN 2415-3400 (Online)
ISSN 1028-821X (Print)

Modeling of drone-reflected doppler signals using fractal nondifferentiable functions

Pashchenko, RE, Ivanov, VK, Tsyupak, DO


O. Ya. Usikov Institute for Radiophysics and Electronics of the National Academy of Sciences of Ukraine
12, Proskura st., Kharkov, 61085, Ukraine

E-mail:  r.paschenko@i.ua , ivanov@ire.kharkov.ua , tsyupak87@mail.ru

Language: ukranian


Subject and Purpose. The paper gives analysis to the waveforms of Doppler signals experimentally measured during the sounding of multirotor drones. Various velocities and numbers of the revolving rotors change Doppler signal waveforms. Also, these waveforms are not simple enough to take advantage of general laws of the examined phenomena and write the model equations immediately. In the present work, we seek to check capabilities of fractal nondifferentiable functions as to the empirical modeling of Doppler signals reflected from the revolving rotors of the drone.

Methods and Methodology. In accordance with the standard manner of empirical modeling, we begin with the measurement and analysis of experimental temporal rows of Doppler signals obtained during the drone sounding. On this basis, the model structure (type of the functions) is determined. Then a turn comes of the procedure of the model parameter calculation. Finally, the actual and simulated Doppler signals are analyzed in qualitative and quantitative terms.

Results. The characteristic features of some fractal nondifferentiable functions have been considered for their adoption in the radar signal modeling. The Doppler signal reflected from the revolving rotors of the drone has been represented as a sum of the fractal signal (modelling the low-frequency component of the Doppler signal) and the Weierstrass-Mandelbrot modified function (modelling the high-frequency component of the Doppler signal). The procedure of the model parameter choice has been examined at various speed values and numbers of the revolving rotors of the drone.

Conclusion. The qualitative and quantitative analyses given to the measured and simulated Doppler signals validate the model we have suggested and show that the model provides a good agreement between the waveform features of the measured and simulated signals.

Keywords: Doppler signal, Doppler signal model, fractal nondifferentiable function, fractal signal, multirotor drone

Manuscript submitted  20.10.2019
PASC 02.30.Gp; 05.45.−a; 05.45.Df
Radiofiz. elektron. 2020, 25(3): 16-25
Full text (PDF)




1. In the USA developed the registration system of small drone [online]. Available from: http://www.ato.ru/content/v-ssha-razrabotali-sistemu-registracii-malyh-b...
2. Gosaviasluzhba of Ukraine revised limits on flights drones toward gain in weight, distance and height of flight [online]. Available from: https://interfax.com.ua/news/general/533582.html
3. Lysenko, A.I., Tachinina, E.N., 2015. The Mathematical design of motion quartocopter. Vіsnik AMU. Ser. «Tekhnіka», 2(10), pp. 128-136 (in Russian).
4. Lazarev, V.S., Laschev, A.A., 2018. Development mathematical model of BPLA on a base quartocopter with the frame DJI F-450. The Engineering announcer of Don, Iss. 2 [online]. Available from: ivdon.ru/ru/magazine/archive/ n2y2018/5001
5. Liang, Liu, Popescu, M., Skubic, M., Rantz, M., Yardibi, T., Cuddihy, P., 2011. Automatic fall detection based on Doppler radar motion signature. In: 5th Int. ICST Conf. on Pervasive Computing Technologies for Healthcare. (PervasiveHealth 2011).Proc. Dublin, Ireland, 23-26 May, 2011. DOI:
6. Bezruchko, B.P., Smirnov, D.A., 2005. Mathematical design and chaotic temporal rows. Saratov: GosUNC «College» Publ. (in Russian).
7. Ljung, L., 1991. System Identification: Theory for the User. Translated from English and ed. by Ya.Z. Tsypkin. Moscow: Science Publ. (in Russian).
8. Malineckiy, G.G., Potapov, A.B., 2000. Modern problems of nonlinear dynamics. Moscow: Editorial URSS Publ. (in Russian).
9. Pashchenko, R.E., Ivanov, V.K., Cyupak, D.O., Levadnyy, Yu.V., 2019. Frequency-temporal analysis radar reflections from multirotor drone. Radiofiz. Electron., 24(4), pp. 35-45. DOI:
10. Okorokov, V.A., Sandrakova, E.V., 2009. Fractals are in fundamental physics. Fractal properties of plural formation of particles and topology of selection. Moscow: MIFI Publ. (in Russian).
11. Feder, E., 1991. Fractals. Translated from English by Yu. Danilov and A. Shkurov. Moscow: Mir Publ. (in Russian).
12. Jaggard, D.L., Sun, X., 1990. Fractal surface scattering: A generalized Rayleigh solution. J. Appl. Phys., 68(11), pp. 5456-5462. DOI:
13. Jaggard, D.L., Sun, X., 1989. Scattering from bandlimited fractal fibers. IEEE Trans. Antennas Propag., 37(12), pp. 1591-1597. DOI:
14. Pashchenko, R.E., 2005. Forming of fractal signals. Kharkov: KHOOO "NEO "Ekoperspektiva" Publ. (in Russian).