Elliptic rational functions

Plot of elliptic rational functions for x between -1 and 1 for orders 1,2,3 and 4 with discrimination factor ξ=1.1. Note that all are bounded between -1 and 1 and all have the value 1 at x=1.

In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name).

Rational elliptic functions are identified by a positive integer order n and include a parameter $\xi \geq 1$ called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:

$R_{n}(\xi ,x)\equiv \mathrm {cd} \left(n{\frac {K(1/L)}{K(1/\xi )}}\,\mathrm {cd} ^{-1}(x,1/\xi ),1/L\right)$

• cd() is the Jacobi elliptic cosine function.
• K() is a complete elliptic integral of the first kind.
• $L=L_{n}(\xi )=R_{n}(\xi ,\xi )$ is the discrimination factor, equal to the minimum value of the magnitude of $R_{n}(\xi ,x)$ for $|x|\geq \xi$.

For many cases, in particular for orders of the form $n=2^{a}3^{b}$ where a and b are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials.

Expression as a ratio of polynomials

For even orders, the elliptic rational functions may be expressed as a ratio of two polynomials, both of order n.

$R_{n}(\xi ,x)=r_{0}\,{\frac {\prod _{i=1}^{n}(x-x_{zi})}{\prod _{i=1}^{n}(x-x_{pi})}}$      (for n even)

where $x_{zi}$ are the zeroes and $x_{pi}$ are the poles, and $r_{0}$ is a normalizing constant chosen such that $R_{n}(\xi ,1)=1$. The above form would be true for odd orders as well except that for odd orders, there will be a pole at x=∞ and a zero at x=0 so that the above form must be modified to read:

$R_{n}(\xi ,x)=r_{0}\,x\,{\frac {\prod _{i=1}^{n-1}(x-x_{zi})}{\prod _{i=1}^{n-1}(x-x_{pi})}}$      (for n odd)

Properties

Plot of the absolute value of the third order elliptic rational function with ξ=1.4. Note the zero at x=0 and the pole at infinity. Since the function is antisymmetric, it is seen there are three zeroes and three poles. Note also that between the zeroes, the function rises to a value of 1 and between the poles, the function drops to the value of the discrimination factor L

Plot of the absolute value of the fourth order elliptic rational function with ξ=1.4. Since the function is symmetric, it is seen that there are four zeroes and four poles. Note again that between the zeroes, the function rises to a value of 1 and between the poles, the function drops to the value of the discrimination factor L

Plot of the effect of the selectivity factor ξ. The fourth order elliptic rational function is shown with values of ξ varying from nearly unity to infinity. The black curve, corresponding to ξ=∞ is the Chebyshev polynomial of order 4.

The canonical properties

• $R_{n}^{2}(\xi ,x)\leq 1$ for $|x|\leq 1\,$
• $R_{n}^{2}(\xi ,x)=1$ at $|x|=1\,$
• $R_{n}^{2}(\xi ,-x)=R_{n}^{2}(\xi ,x)$
• $R_{n}^{2}(\xi ,x)>1$ for $x>1\,$
• The slope at x=1 should be as large as possible
• The slope at x=1 should be larger than the corresponding slope of the Chebyshev polynomial of the same order.

The only rational function satisfying the above properties is the elliptic rational function. Template:Ref harvard. The following properties are derived.

Normalization

The elliptic rational function is normalized to unity at x=1:

$R_{n}(\xi ,1)=1\,$

Relation between poles and zeroes

The following relationship holds:

$R_{n}(\xi ,x)={\frac {R_{n}(\xi ,\xi )}{R_{n}(\xi ,\xi /x)}}\,$

This implies that poles and zeroes come in pairs such that

$x_{pi}x_{zi}=\xi \,$

Odd order functions will have a zero at x=0 and a corresponding pole at infinity.

Nesting property

The nesting property is written:

$R_{m}(R_{n}(\xi ,\xi ),R_{n}(\xi ,x))=R_{m\cdot n}(\xi ,x)\,$

This is a very important property:

• If $R_{n}$ is known for all prime n, then nesting property gives $R_{n}$ for all n. In particular, since $R_{2}$ and $R_{3}$ can be expressed in closed form without explicit use of the Jacobi elliptic functions, then all $R_{n}$ for n of the form $n=2^{a}3^{b}$ can be so expressed.
• It follows that if the zeroes of $R_{n}$ for prime n are known, the zeros of all $R_{n}$ can be found. Using the pole-zero relationship, the poles can also be found.
• The nesting property implies the nesting property of the discrimination factor:

$L_{m\cdot n}(\xi )=L_{m}(L_{n}(\xi ))$

Limiting values

The elliptic rational functions are related to the Chebyshev polynomials of the first kind $T_{n}(x)$ by:

$\lim _{\xi =\rightarrow \,\infty }R_{n}(\xi ,x)=T_{n}(x)\,$

Symmetry

$R_{n}(\xi ,-x)=R_{n}(\xi ,x)\,$ for n even

$R_{n}(\xi ,-x)=-R_{n}(\xi ,x)\,$ for n odd

Equiripple

$R_{n}(\xi ,x)$ has equal ripple of $\pm 1$ in the interval $-1\leq x\leq 1$. By the pole-zero relationship, it follows that $1/R_{n}(\xi ,x)$ has equiripple in $-1/\xi \leq x\leq 1/\xi$ of $\pm 1/L_{n}(\xi )$.

Particular values

Defining:

$t\equiv {\sqrt {1-{\frac {1}{\xi ^{2}}}}}$

we may write the first few elliptic rational functions as:

$R_{1}(\xi ,x)=x\,$

$R_{2}(\xi ,x)={\frac {(t+1)x^{2}-1}{(t-1)x^{2}+1}}$

$R_{3}(\xi ,x)=x\,{\frac {(1-x_{p}^{2})(x^{2}-x_{z}^{2})}{(1-x_{z}^{2})(x^{2}-x_{p}^{2})}}$

$x_{p}^{2}={\frac {2\xi ^{2}{\sqrt {G}}}{{\sqrt {8\xi ^{2}(\xi ^{2}\!+\!1)+12G\xi ^{2}-G^{3}}}-{\sqrt {G^{3}}}}}$

$G={\sqrt {4\xi ^{2}+(4\xi ^{2}(\xi ^{2}\!-\!1))^{2/3}}}$

$x_{z}^{2}=x_{p}^{2}(3\!-\!2x_{p}^{2})+2x_{p}{\sqrt {(x_{p}^{2}\!-\!1)^{3}}}$

$R_{4}(\xi ,x)=R_{2}(R_{2}(\xi ,\xi ),R_{2}(\xi ,x))={\frac {(1+t)(1+{\sqrt {t}})^{2}x^{4}-2(1+t)(1+{\sqrt {t}})x^{2}+1}{(1+t)(1-{\sqrt {t}})^{2}x^{4}-2(1+t)(1-{\sqrt {t}})x^{2}+1}}$

$R_{6}(\xi ,x)=R_{3}(R_{2}(\xi ,\xi ),R_{2}(\xi ,x))\,$

See Template:Ref harvard for further explicit expressions of order $n=2^{i}3^{j}$.

References

• Science World

• ^{Lutovac 2001} Lutovac, Miroslav D. (2001). Filter Design for Signal Processing using MATLAB© and Mathematica©. New Jersey, USA: Prentice Hall. ISBN 0201361302. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)