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Date: 15-5-2018
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Date: 15-9-2019
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Date: 23-8-2019
904
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Elliptic rational functions are a special class of rational functions that have nice properties for approximating other functions over the interval . In particular, they are equiripple, satisfy over , are minimax approximations over , exhibit monotonic increase on , and have minimal order . Additional properties include symmetry
(1) |
normalization
(2) |
the property
(3) |
and the nesting property
(4) |
(Lutovac et al. 2001).
Letting the discrimination factor be the largest value of for , the elliptic rational functions can be defined by
(5) |
where is a complete elliptic integral of the first kind, is a Jacobi elliptic function, and is an inverse Jacobi elliptic function. For , 2, and 3, the functions are given by
(6) |
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(7) |
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(8) |
where . can be expressed in closed form without using elliptic functions for of the form .
The elliptic rational functions are related to the Chebyshev polynomials of the first kind by
(9) |
REFERENCES:
Antoniou, A. Digital Filters: Analysis and Design. New York: McGraw-Hill, 1979.
Daniels, R. W. Approximation Methods for Electronic Filter Design. New York: McGraw-Hill, 1974.
Lutovac, M. D.; Tosic, D. V.; and Evans, B. L. Filter Design for Signal Processing Using MATLAB and Mathematica. Upper Saddle River, NJ: Prentice-Hall, 2001.
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