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. 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 ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:

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

where

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

For many cases, in particular for orders of the form n = 2^a3^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: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions.

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.

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

where ${\displaystyle x_{i}}$ are the zeroes and ${\displaystyle x_{pi}}$ are the poles, and ${\displaystyle r_{0}}$ is a normalizing constant chosen such that ${\displaystyle R_{n}(\xi ,1)=1}$. The above form would be true for even 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:

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

Properties

Plot of the absolute value of the third order elliptic rational function with ξ=1.4. There is a zero at x=0 and the pole at infinity. Since the function is antisymmetric, it is seen there are three zeroes and three poles. 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_n

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. 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_n

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 closer the selectivity factor is to unity, the steeper will be the slope in the transition region between x=1 and x=ξ.

The canonical properties

• ${\displaystyle R_{n}^{2}(\xi ,x)\leq 1}$ for ${\displaystyle |x|\leq 1\,}$
• ${\displaystyle R_{n}^{2}(\xi ,x)=1}$ at ${\displaystyle |x|=1\,}$
• ${\displaystyle R_{n}^{2}(\xi ,-x)=R_{n}^{2}(\xi ,x)}$
• ${\displaystyle R_{n}^{2}(\xi ,x)>1}$ for ${\displaystyle x>1\,}$
• The slope at x=1 is as large as possible
• The slope at x=1 is 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 ( Lutovac, Tošić & Evans 2001 , § 13.2). The following properties are derived:

Normalization

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

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

Nesting property

The nesting property is written:

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

This is a very important property:

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

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

Limiting values

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

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

Symmetry

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

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

Equiripple

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

Inversion relationship

The following inversion relationship holds:

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

This implies that poles and zeroes come in pairs such that

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

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

Poles and Zeroes

The zeroes of the elliptic rational function of order n will be written ${\displaystyle x_{ni}(\xi )}$ or ${\displaystyle x_{ni}}$ when ${\displaystyle \xi }$ is implicitly known. The zeroes of the elliptic rational function will be the zeroes of the polynomial in the numerator of the function.

The following derivation of the zeroes of the elliptic rational function is analogous to that of determining the zeroes of the Chebyshev polynomials ( Lutovac, Tošić & Evans 2001 , § 12.6). Using the fact that for any z

${\displaystyle \mathrm {cd} \left((2m-1)K\left(1/z\right),{\frac {1}{z}}\right)=0\,}$

the defining equation for the elliptic rational functions implies that

${\displaystyle n{\frac {K(1/L_{n})}{K(1/\xi )}}\mathrm {cd} ^{-1}(x_{m},1/\xi )=(2m-1)K(1/L_{n})}$

so that the zeroes are given by

${\displaystyle x_{m}=\mathrm {cd} \left(K(1/\xi )\,{\frac {2m-1}{n}},{\frac {1}{\xi }}\right).}$

Using the inversion relationship, the poles may then be calculated.

From the nesting property, if the zeroes of ${\displaystyle R_{m}}$ and ${\displaystyle R_{n}}$ can be algebraically expressed (i.e. without the need for calculating the Jacobi ellipse functions) then the zeroes of ${\displaystyle R_{m\cdot n}}$ can be algebraically expressed. In particular, the zeroes of elliptic rational functions of order ${\displaystyle 2^{i}3^{j}}$ may be algebraically expressed ( Lutovac, Tošić & Evans 2001 , § 12.9, 13.9). For example, we can find the zeroes of ${\displaystyle R_{8}(\xi ,x)}$ as follows: Define

${\displaystyle X_{n}\equiv R_{n}(\xi ,x)\qquad L_{n}\equiv R_{n}(\xi ,\xi )\qquad t_{n}\equiv {\sqrt {1-1/L_{n}^{2}}}.}$

Then, from the nesting property and knowing that

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

where ${\displaystyle t\equiv {\sqrt {1-1/\xi ^{2}}}}$ we have:

${\displaystyle L_{2}={\frac {1+t}{1-t}},\qquad L_{4}={\frac {1+t_{2}}{1-t_{2}}},\qquad L_{8}={\frac {1+t_{4}}{1-t_{4}}}}$

${\displaystyle X_{2}={\frac {(t+1)x^{2}-1}{(t-1)x^{2}+1}},\qquad X_{4}={\frac {(t_{2}+1)X_{2}^{2}-1}{(t_{2}-1)X_{2}^{2}+1}},\qquad X_{8}={\frac {(t_{4}+1)X_{4}^{2}-1}{(t_{4}-1)X_{4}^{2}+1}}.}$

These last three equations may be inverted:

${\displaystyle x={\frac {1}{\pm {\sqrt {1+t\,\left({\frac {1-X_{2}}{1+X_{2}}}\right)}}}},\qquad X_{2}={\frac {1}{\pm {\sqrt {1+t_{2}\,\left({\frac {1-X_{4}}{1+X_{4}}}\right)}}}},\qquad X_{4}={\frac {1}{\pm {\sqrt {1+t_{4}\,\left({\frac {1-X_{8}}{1+X_{8}}}\right)}}}}.\qquad }$

To calculate the zeroes of ${\displaystyle R_{8}(\xi ,x)}$ we set ${\displaystyle X_{8}=0}$ in the third equation, calculate the two values of ${\displaystyle X_{4}}$, then use these values of ${\displaystyle X_{4}}$ in the second equation to calculate four values of ${\displaystyle X_{2}}$ and finally, use these values in the first equation to calculate the eight zeroes of ${\displaystyle R_{8}(\xi ,x)}$. (The ${\displaystyle t_{n}}$ are calculated by a similar recursion.) Again, using the inversion relationship, these zeroes can be used to calculate the poles.

Particular values

We may write the first few elliptic rational functions as:

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

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

where

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

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

where

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

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

${\displaystyle x_{z}^{2}=\xi ^{2}/x_{p}^{2}}$

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

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

See Lutovac, Tošić & Evans (2001 , § 13) for further explicit expressions of order n=5 and ${\displaystyle n=2^{i}\,3^{j}}$.

The corresponding discrimination factors are:

${\displaystyle L_{1}(\xi )=\xi \,}$

${\displaystyle L_{2}(\xi )={\frac {1+t}{1-t}}=\left(\xi +{\sqrt {\xi ^{2}-1}}\right)^{2}}$

${\displaystyle L_{3}(\xi )=\xi ^{3}\left({\frac {1-x_{p}^{2}}{\xi ^{2}-x_{p}^{2}}}\right)^{2}}$

${\displaystyle L_{4}(\xi )=\left({\sqrt {\xi }}+(\xi ^{2}-1)^{1/4}\right)^{4}\left(\xi +{\sqrt {\xi ^{2}-1}}\right)^{2}}$

${\displaystyle L_{6}(\xi )=L_{3}(L_{2}(\xi ))\,}$ etc.

The corresponding zeroes are ${\displaystyle x_{nj}}$ where n is the order and j is the number of the zero. There will be a total of n zeroes for each order.

${\displaystyle x_{11}=0\,}$

${\displaystyle x_{21}=\xi {\sqrt {1-t}}\,}$

${\displaystyle x_{22}=-x_{21}\,}$

${\displaystyle x_{31}=x_{z}\,}$

${\displaystyle x_{32}=0\,}$

${\displaystyle x_{33}=-x_{31}\,}$

${\displaystyle x_{41}=\xi {\sqrt {\left(1-{\sqrt {t}}\right)\left(1+t-{\sqrt {t(t+1)}}\right)}}\,}$

${\displaystyle x_{42}=\xi {\sqrt {\left(1-{\sqrt {t}}\right)\left(1+t+{\sqrt {t(t+1)}}\right)}}\,}$

${\displaystyle x_{43}=-x_{42}\,}$

${\displaystyle x_{44}=-x_{41}\,}$

From the inversion relationship, the corresponding poles ${\displaystyle x_{p,ni}}$ may be found by ${\displaystyle x_{p,ni}=\xi /(x_{ni})}$

References

• MathWorld
• Daniels, Richard W. (1974). Approximation Methods for Electronic Filter Design. New York: McGraw-Hill. ISBN   0-07-015308-6.
• Lutovac, Miroslav D.; Tošić, Dejan V.; Evans, Brian L. (2001). Filter Design for Signal Processing using MATLAB© and Mathematica©. New Jersey, USA: Prentice Hall. ISBN   0-201-36130-2.