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COMPARATIVE ANALYSIS OF HARMONIC FUNCTION APPROXIMATION METHODS

V. V. Chekushkin¹, S. N. Zhiganov¹, А. A. Bykov¹, К. V. Mikheev^2
1Murom Institute (Branch) of «Vladimir State University named after Alexandr Grigorievich and Nikolay Grigorievich Stoletovs»,
23, Orlovskaya Street, Murom, 602254, Russian Federation
²JSC «Murom Plant of Radio-Measuring Instruments»,
2, Karachaevskoe shosse, Murom, 602267, Russian Federation
E-mail: s_zh_72@mail.ru

ABSTRACT

Introduction. Harmonic signals are widely used in radio engineering systems of information measurement and processing. Measuring instruments are calibrated using these signals and they are the basis for designing frequency synthesizers, which form high-stable radio signals of generators and receiving devices. Recently, with the development of digital methods, harmonic signals are formed algorithmically. The accuracy of measuring instruments and system performance characteristics depend on the quality of their formation. When creating the similar systems, judging by the sine function formation error, speed and computational complexity of algorithms, rational approaches to the choice of sine function approximation methods must be used. The purpose of the article is to conduct the comparative analysis of functional dependence approximation methods, based on the use of continuous piecewise-linear functions and on polynomial representation. As an example, the trigonometric function sin(x) on the interval of values x Î [0, 2p] was considered. The approximation problem can be solved using different methods. The method of continuous piecewise-linear functions, where the function is replaced by line segments on the finite interval of values, is the simplest one. The method, based on the polynomial representation of the approximated function, is also widely used when representing functional dependences. Results. By the example of the trigonometric function sin(x), defined on the interval of values x Î [0, p], two methods of the approximation are compared in the work: piecewise-linear and the one using polynomials of best approximation from the second and fourth degrees. It is shown that the use of polynomial approximation allows improving the fidelity of function reproduction and the increase in the polynomial degree considerably decreases minimum approximation errors but the number of operations, necessary for function reproduction increases. Conclusion. The recommendations for further improvement of harmonic function representation accuracy due to halving the approximation interval and using polynomials of best approximation of odd degrees are given. Polynomials of best approximation of odd degrees with calculated coefficients and corresponding and the corresponding number of necessary operations and maximum approximation errors are presented. In accordance with the strategy of the maximum identity of graphs of reproduced functional dependences, the method of the sinusoid half-time approximation by an «inverted» parabola, has the largest relation of the number of significant figures to the number of required operations of function reproduction at a given error 0,03. The most efficient method of the approximation is the expansion in odd powers in the generally accepted interval of function values x Î [0, p/2] with further use of an argument reduction formula, providing the sequential minimum increment of the complexity of the computational algorithm by 1, 2 operations when the calculation error decreases in the range [0,5; 3,5×10^-9].

KEYWORDS

harmonic signal; piecewise-linear approximation; polynomials of best approximation

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ACKNOWLEDGMENT

The work was carried out with the financial grant support from RFBR №18-37-00077.

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For citation: Chekushkin V. V., Zhiganov S. N., Bykov А. A., Mikheev К. V. Comparative Analysis of Harmonic Function Approximation Methods. Vestnik of Volga State University of Technology. Ser.: Radio Engineering and Infocommunication Systems. 2018. No 3 (39). Pp. 58-66. DOI: 10.15350/2306-2819.2018.3.58