"JOURNAL OF RADIO ELECTRONICS" (Zhurnal Radioelektroniki ISSN 1684-1719, N 3, 2018

contents of issue       DOI  10.30898/1684-1719.2018.3.11      full text in Russian (pdf)    

Method of searching for optimal gain-frequency characteristics approximations with Chebyshev rational functions


A. V. Smirnov

Moscow Technological University, Vernadsky prospect 78, Moscow 119454, Russia


The paper is received on March 19, 2018


Abstract. Chebyshev rational functions (CRF) have the properties that allow obtaining approximations of gain-frequency characteristics with equiripple attenuation in the pass band and specified attenuation in the stop band. However, the optimal CRF approximation may be found analytically only for some specific tasks. The heuristic algorithm for searching such approximations is proposed and tested in this paper. This problem refers to multiobjective optimization and has three objectives: a nonuniformity of attenuation in the pass band, a departure of attenuation in the stop band and a lower boundary of the stop band. The algorithm searches out the set of Pareto optimal solutions and uses multistart for searching of global extreme for each point of Pareto front. For verification of the algorithm, approximations of elliptical filters were found. A deviation from tabulated parameters turned out not more than 1%. Then some examples of approximations of filters with no analytic solutions were considered. An additional result was received in the experiments: all Pareto optimal CRF approximations with identical order, specified attenuation in the stop band and stop band lower boundary have the same poles of CRF.

Keywords: approximation of gain-frequency characteristics, Chebyshev rational function, Pareto optimality, heuristic algorithm.


1. Beletskii A.F. Teoreticheskie osnovy electroprovodnoi sviazi, Ch.3. Sintez reaktivnyh chetyrehpolusnikov i electricheskih filtrov. [The theoretical foundation of line communication, Vol.3. Synthesis of reactive two-port networks and electrical filters].  Moscow, Sviaz'izdat Publ. 1959. 391 p. (In Russian)

2. Znamenskii A.E., Tepliuk I.N. Aktivnye RC-fil'try. [Active RC filters]. Moscow, Sviaz' Publ. 1970. 280 p. (In Russian)

3. Trifonov I.I. Raschet elektronnyh tsepei s zadannymi chastotnymi haralteristikami. [The design of electronic circuits with specified frequency characteristics]. Moscow, Radio i Sviaz' Publ. 1988. 304 p. (In Russian)  

4. Stankeev M.E. Issledovanie i razrabotka metodov rascheta LC-fil'trov po energeticheskim i massogabaritnym kriteriiam. Avtoreferat dissertatsii na soiskanie uchenoi stepeni kandidata tehnicheskih nauk. [Research and development of LC filters design methods based on power, weight and dimensions measures.]. St-Petersburg. 2014. 162 p. (In Russian) Available at: http://tekhnosfera.com/issledovanie-i-razrabotka-metodov-rascheta-lc-filtrov-po-energeticheskim-i-massogabaritnym-kriteriyam.

5. Eremeev V.P., Mizinenko O.V. [Digital filters based on Chebyshev rational functions. Doklady 3-ei Mezhdunarodnoi konferentsii DSPA -2000. Tom 3. [Proceedings of the 3-rd International Conference DSPA-2000. Vol.3]. Available at: http://www.autex.spb.ru/dspa/dspa2000/part3.htm#57.

6. Gutkin L.S. Optimizatsiia radioelektronnyh ustroistv po sovokupnosti pokazatelei kachestva. [Multiple quality indexes optimization of radio-electronics devices]. Moscow, Sovetskoe Radio Publ. 1975. 368 p. (In Russian)

7. Huelsman L.P., Allen P.E. Introduction to the theory and design of active filters. McGraw-Hill Publ. 1980. 384 p.

8. Handbook of Metaheuristics. Second Edition. Edited by M. Gendreau, J.-Y. Potvin. Springer, 2010. 648 p.

9. Smirnov A.V. The perspectives of heuristic optimization methods employment in a development of radio and telecommunication devices and systems. Rossiiskii tecnologicheskii zhurnal - Russian technological journal, 2012, Vol.5, No.6, pp. 20-33. (In Russian). Available at: https://rtj.mirea.ru/journal-archive/2017/volume-5-number-6/

10. Smirnov A.V. Pareto-optimal approximations of electric filters transfer functions. Aktual'nye problemy gumanitarnyh i estestvennyh nauk - Topical problems of liberal and natural sciences. 2015. No.05 (76). Part 1, pp.74-78. (in Russian)

11. Zhigliavskiy A.A., Zhilinskas A.G. Metody poiska global'nogo ekstremuma. [Methods of global extremum search]. Moscow, Nauka Publ. 1991. 248 p. (in Russian)

12. Ye T., Ran Ch., Xingyi Zh., Yaochu J. PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization.  IEEE Computational Intelligence Magazine, 2017, Vol.12, No.4, pp. 73-87.

13. Lam G. Analog and digital filters. Prentice-Hall, Inc. 1979. 590 p.


For citation:
A. V. Smirnov. Method of searching for optimal gain-frequency characteristics approximations with Chebyshev rational functions. Zhurnal Radioelektroniki - Journal of Radio Electronics. 2018. No. 3. Available at http://jre.cplire.ru/jre/mar18/7/text.pdf.

DOI  10.30898/1684-1719.2018.3.11