Elliptic Rational Function

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)
			(7)
			(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.