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Chained-Function Filter Synthesis Based on the Legendre Polynomials

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Abstract

A new kind of filter called Legendre chained-function (LCF) filter with characteristic function given by the product of low-degree Legendre orthogonal polynomials (seed functions) is studied in this paper. LCF filter magnitude response exhibits unequal ripple level in the passband compared to Chebyshev chained-function filter with identical edge ripple factor at the passband edge. A proper combination of seed functions, i.e., a product of them, is used to control the maximum ripple in the passband, which affects the return loss, selectivity and a group delay characteristics of a filter. By selecting one of the possible combinations of seed functions (thus obtaining various degrees of freedom in filter design), filters with improved performances compared to traditional approximation techniques can be obtained. The degrees of freedom increase if the degree of filter increases. Compared to existing Chebyshev chained-function (CCF) filters, whose performances are also presented, and Butterworth function filters as a special case of both LCF and CCF filters, the new family of LCF filters has many advantages. A table summarizing properties of CCF and LCF filters is given for design purpose.

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Notes

  1. The loaded quality factor \(Q_\mathrm{L}\), defined as the ratio of the magnitude of the resonator response at resonance frequency to the total circuit resistance, can be calculated for each resonator using [19]:

    $$\begin{aligned} 1/Q_\mathrm{L}=1/Q_\mathrm{u}+1/Q_\mathrm{e} \end{aligned}$$

    where \(Q_\mathrm{e}\) is external Q-factor.

  2. Approximately 99.7% of the data values lie within three standard deviations from the mean.

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Acknowledgements

The authors wish to thank Professor V. S. Stojanović of the University of Niš, Niš, Serbia for his valuable comments and suggestions. The work presented here was partly supported by the Serbian Ministry of Education and Science in the frame of the Projects TR 32009.

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Authors and Affiliations

  1. Faculty of Electronic Engineering, University of Niš, A. Medvedeva 14, 18000, Niš, Serbia

    Nikola Stojanović

  2. Faculty of Natural Science and Mathematics, University of Priština, 38220, Kosovska Mitrovica, Serbia

    Negovan Stamenković

  3. Faculty of Technical Science, University of Priština, Knjaza Miloša 7, 38220, Kosovska Mitrovica, Serbia

    Ivan Krstić

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  1. Nikola Stojanović
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  2. Negovan Stamenković
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  3. Ivan Krstić
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Correspondence to Nikola Stojanović.

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Stojanović, N., Stamenković, N. & Krstić, I. Chained-Function Filter Synthesis Based on the Legendre Polynomials. Circuits Syst Signal Process 37, 2001–2020 (2018). https://doi.org/10.1007/s00034-017-0651-1

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