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1
Content available Systematic errors of the LIDFT method: analytical form and verification by a Monte Carlo method
Borkowski J.
Metrology and Measurement Systems
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2012
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Vol. 19, nr 4
673-684
EN
This paper derives analytical formulas for the systematic errors of the linear interpolated DFT (LIDFT) method when used to estimating multifrequency signal parameters and verifies this analysis using Monte-Carlo simulations. The analysis is performed on the version of the LIDFT method based on optimal approximation of the unit circle by a polygon using a pair of windows. The analytical formulas derived here take the systematic errors in the estimation of amplitude and frequency of component oscillations in the multifrequency signal as the sum of basic errors and the errors caused by each of the component oscillations. Additional formulas are also included to analyze particular quantities such as a signal consisting of two complex oscillations, and the analyses are verified using Monte-Carlo simulations.
2
Content available remote Optimization of the unit circle approximation by a polygon
Borkowski J.
Przegląd Elektrotechniczny
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2011
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R. 87, nr 7
95-98
EN
The paper presents two optimization criteria of the approximation of the unit circle by a polygon: minimization of maximum approximation errors and minimization of mean square approximation errors. It is shown that application of the unit circle approximation by a polygon requires to compromise between minimization of three types of errors. The most beneficial approximation parameters values range is obtain for optimal application of the presented unit circle approximation by polygon.
PL
Przedstawiono dwa kryteria optymalizacji aproksymacji okręgu jednostkowego przez wielokąt: minimalizacja błędów maksymalnych aproksymacji i minimalizacja błędów średniokwadratowych aproksymacji. Wykazano, że zastosowanie aproksymacji okręgu jednostkowego wielokątem wymaga kompromisu pomiędzy minimalizacją trzech rodzajów błędów. Dla optymalnego stosowania przedstawionej aproksymacji przedstawiono zakres najkorzystniejszych wartości parametrów aproksymacji.
3
Content available Minimization of maximum errors in universal approximation of the unit circle by a polygon
Borkowski J.
Metrology and Measurement Systems
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2011
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Vol. 18, nr 3
391-402
EN
This paper presents a universal approximation of the unit circle by a polygon that can be used in signal processing algorithms. Optimal choice of the values of three parameters of this approximation allows one to obtain a high accuracy of approximation. The approximation described in the paper has a universal character and can be used in many signal processing algorithms, such as DFT, that use the mathematical form of the unit circle. One of the applications of the described approximation is the DFT linear interpolation method (LIDFT). Applying the results of the presented paper to improve the LIDFT method allows one to significantly decrease the errors in estimating the amplitudes and frequencies of multifrequency signal components. The paper presents the derived formulas, an analysis of the approximation accuracy and the region of best values for the approximation parameters.
4
Content available remote On a functional equation related to an automorphism of a unit circle
Kir'ytzkii E.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
119--134
EN
In this article the complete description of the decisions of a functional equation f(w(z)) = f(w(0))f(z) is given, where w(z)-automorphism of a unit circle E and the decisions are searched among analytical in E functions. It is established, that research of a given functional equation is closely connected to property of stationary points of automorphism w(z).
5
Content available remote Asymptotic of Lp extremal polynomials off the unit circle
Khaldi R.
Demonstratio Mathematica
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2005
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Vol. 38, nr 3
623--632
EN
Let sigma be a positive measure whose support is the unit circle gamma plus a denumerable set of mass points, which accumulate at gamma and satisfy Blaschke^s condition. We study the asymptotics of Lp extremal polynomials (0 < p < oo) in the region exterior to r under Szego's condition.
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