elliptic rational functions

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

:$R\_n(\backslash xi,x)=r\_0\backslash ,\backslash frac\{\backslash prod\_\{i=1\}^n\; (x-x\_i)\}\{\backslash prod\_\{i=1\}^n\; (x-x\_\{pi\})\}$      (for n even)

where $x\_i$ are the zeroes and $x\_\{pi\}$ are the poles, and $r\_0$ is a normalizing constant chosen such that $R\_n(\backslash xi,1)=1$. 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:

:$R\_n(\backslash xi,x)=r\_0\backslash ,x\backslash ,\backslash frac\{\backslash prod\_\{i=1\}^\{n-1\}\; (x-x\_i)\}\{\backslash prod\_\{i=1\}^\{n-1\}\; (x-x\_\{pi\})\}$      (for n odd)

File:Rational Elliptic Function (abs, n=3, x=(0,5)).svg

File:Rational Elliptic Function (abs, n=4, x=(0,5)).svg

File:Rational Elliptic Functions (xi varied, x=(-1,1)).svg 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.]]

• $R\_n^2(\backslash xi,x)\backslash le\; 1$ for $|x|\backslash le\; 1\backslash ,$
• $R\_n^2(\backslash xi,x)=\; 1$ at $|x|=\; 1\backslash ,$
• $R\_n^2(\backslash xi,-x)=R\_n^2(\backslash xi,x)$
• $R\_n^2(\backslash xi,x)>1$ for $x>1\backslash ,$
• 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 {{harv|Lutovac|Tošić|Evans|2001|loc=§ 13.2}}. The following properties are derived:

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

:$R\_n(\backslash xi,1)=1\backslash ,$

The nesting property is written:

:$R\_m(R\_n(\backslash xi,\backslash xi),R\_n(\backslash xi,x))=R\_\{m\backslash cdot\; n\}(\backslash xi,x)\backslash ,$

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 $n=2^a3^b$ 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 inversion relationship (see below), the poles can also be found.
• The nesting property implies the nesting property of the discrimination factor:

::$L\_\{m\backslash cdot\; n\}(\backslash xi)=L\_m(L\_n(\backslash xi))$

The elliptic rational functions are related to the Chebyshev polynomials of the first kind $T\_n(x)$ by:

:$\backslash lim\_\{\backslash xi=\backslash rightarrow\backslash ,\backslash infty\}R\_n(\backslash xi,x)=T\_n(x)\backslash ,$

:$R\_n(\backslash xi,-x)=R\_n(\backslash xi,x)\backslash ,$ for n even

:$R\_n(\backslash xi,-x)=-R\_n(\backslash xi,x)\backslash ,$ for n odd

$R\_n(\backslash xi,x)$ has equal ripple of $\backslash pm\; 1$ in the interval $-1\backslash le\; x\backslash le\; 1$. By the inversion relationship (see below), it follows that $1/R\_n(\backslash xi,x)$ has equiripple in $-1/\backslash xi\; \backslash le\; x\backslash le\; 1/\backslash xi$ of $\backslash pm\; 1/L\_n(\backslash xi)$.

The following inversion relationship holds:

:$R\_n(\backslash xi,\backslash xi/x)=\backslash frac\{R\_n(\backslash xi,\backslash xi)\}\{R\_n(\backslash xi,x)\}\backslash ,$

This implies that poles and zeroes come in pairs such that

:$x\_\{pi\}x\_\{zi\}=\backslash xi\backslash ,$

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

The zeroes of the elliptic rational function of order n will be written $x\_\{ni\}(\backslash xi)$ or $x\_\{ni\}$ when $\backslash xi$ 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 {{harv|Lutovac|Tošić|Evans|2001|loc=§ 12.6}}. Using the fact that for any z

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

the defining equation for the elliptic rational functions implies that

:$n\; \backslash frac\{K(1/L\_n)\}\{K(1/\backslash xi)\}\backslash mathrm\{cd\}^\{-1\}(x\_m,1/\backslash xi)=(2m-1)K(1/L\_n)$

so that the zeroes are given by

:$x\_m=\backslash mathrm\{cd\}\backslash left(K(1/\backslash xi)\backslash ,\backslash frac\{2m-1\}\{n\},\backslash frac\{1\}\{\backslash xi\}\backslash right).$

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

From the nesting property, if the zeroes of $R\_m$ and $R\_n$ can be algebraically expressed (i.e. without the need for calculating the Jacobi ellipse functions) then the zeroes of $R\_\{m\backslash cdot\; n\}$ can be algebraically expressed. In particular, the zeroes of elliptic rational functions of order $2^i3^j$ may be algebraically expressed {{harv|Lutovac|Tošić|Evans|2001|loc=§ 12.9, 13.9}}. For example, we can find the zeroes of $R\_8(\backslash xi,x)$ as follows: Define

:$$

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

:$R\_2(\backslash xi,x)=\backslash frac\{(t+1)x^2-1\}\{(t-1)x^2+1\}$

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

:$$

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}

:$$

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:

:$$

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 $R\_8(\backslash xi,x)$ we set $X\_8=0$ in the third equation, calculate the two values of $X\_4$, then use these values of $X\_4$ in the second equation to calculate four values of $X\_2$ and finally, use these values in the first equation to calculate the eight zeroes of $R\_8(\backslash xi,x)$. (The $t\_n$ are calculated by a similar recursion.) Again, using the inversion relationship, these zeroes can be used to calculate the poles.

We may write the first few elliptic rational functions as:

:$R\_1(\backslash xi,x)=x\backslash ,$

:$R\_2(\backslash xi,x)=\backslash frac\{(t+1)x^2-1\}\{(t-1)x^2+1\}$

::: where

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

:$R\_3(\backslash xi,x)=x\backslash ,\backslash frac\{(1-x\_p^2)(x^2-x\_z^2)\}\{(1-x\_z^2)(x^2-x\_p^2)\}$

::: where

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

:::$x\_p^2\backslash equiv\backslash frac\{2\backslash xi^2\backslash sqrt\{G\}\}\{\backslash sqrt\{8\backslash xi^2(\backslash xi^2\backslash !+\backslash !1)+12G\backslash xi^2-G^3\}-\backslash sqrt\{G^3\}\}$

:::$x\_z^2=\backslash xi^2/x\_p^2$

:$R\_4(\backslash xi,x)=R\_2(R\_2(\backslash xi,\backslash xi),R\_2(\backslash xi,x))=\backslash frac$

{(1+t)(1+\sqrt{t})^2x^4-2(1+t)(1+\sqrt{t})x^2+1}

{(1+t)(1-\sqrt{t})^2x^4-2(1+t)(1-\sqrt{t})x^2+1}

:$R\_6(\backslash xi,x)=R\_3(R\_2(\backslash xi,\backslash xi),R\_2(\backslash xi,x))\backslash ,$ etc.

See {{harvtxt|Lutovac|Tošić|Evans|2001|loc=§ 13}} for further explicit expressions of order n=5 and $n=2^i\backslash ,3^j$.

The corresponding discrimination factors are:

:$L\_1(\backslash xi)=\backslash xi\backslash ,$

:$L\_2(\backslash xi)=\backslash frac\{1+t\}\{1-t\}=\backslash left(\backslash xi+\backslash sqrt\{\backslash xi^2-1\}\backslash right)^2$

:$L\_3(\backslash xi)=\backslash xi^3\backslash left(\backslash frac\{1-x\_p^2\}\{\backslash xi^2-x\_p^2\}\backslash right)^2$

:$L\_4(\backslash xi)=\backslash left(\backslash sqrt\{\backslash xi\}+(\backslash xi^2-1)^\{1/4\}\backslash right)^4\backslash left(\backslash xi+\backslash sqrt\{\backslash xi^2-1\}\backslash right)^2$

:$L\_6(\backslash xi)=L\_3(L\_2(\backslash xi))\backslash ,$ etc.

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

:$x\_\{11\}=0\backslash ,$

:$x\_\{21\}=\backslash xi\backslash sqrt\{1-t\}\backslash ,$

:$x\_\{22\}=-x\_\{21\}\backslash ,$

:$x\_\{31\}=x\_z\backslash ,$

:$x\_\{32\}=0\backslash ,$

:$x\_\{33\}=-x\_\{31\}\backslash ,$

:$x\_\{41\}=\backslash xi\backslash sqrt\{\backslash left(1-\backslash sqrt\{t\}\backslash right)\backslash left(1+t-\backslash sqrt\{t(t+1)\}\backslash right)\}\backslash ,$

:$x\_\{42\}=\backslash xi\backslash sqrt\{\backslash left(1-\backslash sqrt\{t\}\backslash right)\backslash left(1+t+\backslash sqrt\{t(t+1)\}\backslash right)\}\backslash ,$

:$x\_\{43\}=-x\_\{42\}\backslash ,$

:$x\_\{44\}=-x\_\{41\}\backslash ,$

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

• [http://mathworld.wolfram.com/EllipticRationalFunction.html MathWorld]
• {{cite book |last=Daniels |first=Richard W. |title=Approximation Methods for Electronic Filter Design |year=1974 |publisher=McGraw-Hill |location=New York |isbn=0-07-015308-6 }}
• {{cite book |last1=Lutovac |first1=Miroslav D. | last2= Tošić | first2=Dejan V.|last3=Evans| first3=Brian L. |title=Filter Design for Signal Processing using MATLAB© and Mathematica© |year=2001 |publisher=Prentice Hall |location=New Jersey, USA |isbn=0-201-36130-2}}

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Category:Rational functions

Category:Elliptic functions