Elliptic rational functions explained

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:

Rn(\xi,x)\equivcd\left(n

K(1/Ln(\xi))
K(1/\xi)

cd-1(x,1/\xi),1/Ln(\xi)\right)

where

Ln(\xi)=Rn(\xi,\xi)

is the discrimination factor, equal to the minimum value of the magnitude of

Rn(\xi,x)

for

|x|\ge\xi

.

For many cases, in particular for orders of the form n = 2a3b 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.

Rn(\xi,x)=r

0
n
\prod(x-xi)
i=1
n
\prod(x-xpi)
i=1
     (for n even)

where

xi

are the zeroes and

xpi

are the poles, and

r0

is a normalizing constant chosen such that

Rn(\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:

Rn(\xi,x)=r

0x
n-1
\prod(x-xi)
i=1
n-1
\prod(x-xpi)
i=1
     (for n odd)

Properties

The canonical properties

2(\xi,x)\le
R
n

1

for

|x|\le1

2(\xi,x)=
R
n

1

at

|x|=1

2(\xi,x)
R
n
2(\xi,x)>1
R
n
for

x>1

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

Normalization

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

Rn(\xi,1)=1

Nesting property

The nesting property is written:

Rm(Rn(\xi,\xi),Rn(\xi,x))=Rm(\xi,x)

This is a very important property:

Rn

is known for all prime n, then nesting property gives

Rn

for all n. In particular, since

R2

and

R3

can be expressed in closed form without explicit use of the Jacobi elliptic functions, then all

Rn

for n of the form

n=2a3b

can be so expressed.

Rn

for prime n are known, the zeros of all

Rn

can be found. Using the inversion relationship (see below), the poles can also be found.

Lm(\xi)=Lm(Ln(\xi))

Limiting values

The elliptic rational functions are related to the Chebyshev polynomials of the first kind

Tn(x)

by:

\lim\xi= → inftyRn(\xi,x)=Tn(x)

Symmetry

Rn(\xi,-x)=Rn(\xi,x)

for n even

Rn(\xi,-x)=-Rn(\xi,x)

for n odd

Equiripple

Rn(\xi,x)

has equal ripple of

\pm1

in the interval

-1\lex\le1

. By the inversion relationship (see below), it follows that

1/Rn(\xi,x)

has equiripple in

-1/\xi\lex\le1/\xi

of

\pm1/Ln(\xi)

.

Inversion relationship

The following inversion relationship holds:

R
n(\xi,\xi/x)=Rn(\xi,\xi)
Rn(\xi,x)

This implies that poles and zeroes come in pairs such that

xpixzi=\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

xni(\xi)

or

xni

when

\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 . Using the fact that for any z

cd\left((2m-1)K\left(1/z\right),1
z

\right)=0

the defining equation for the elliptic rational functions implies that

n

K(1/Ln)
K(1/\xi)

cd-1(xm,1/\xi)=(2m-1)K(1/Ln)

so that the zeroes are given by

x,
m=cd\left(K(1/\xi)2m-1
n
1
\xi

\right).

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

From the nesting property, if the zeroes of

Rm

and

Rn

can be algebraically expressed (i.e. without the need for calculating the Jacobi ellipse functions) then the zeroes of

Rm

can be algebraically expressed. In particular, the zeroes of elliptic rational functions of order

2i3j

may be algebraically expressed . For example, we can find the zeroes of

R8(\xi,x)

as follows: Define

Xn\equivRn(\xi,x)    Ln\equivRn(\xi,\xi)    tn\equiv

2}.
\sqrt{1-1/L
n

Then, from the nesting property and knowing that

R
2(\xi,x)=(t+1)x2-1
(t-1)x2+1

where

t\equiv\sqrt{1-1/\xi2}

we have:
L
2=1+t
1-t

,   

L
4=1+t2
1-t2

,   

L
8=1+t4
1-t4

X
2=(t+1)x2-1
(t-1)x2+1

,   

X
4=
(t
2-1
2
2+1)X
(t
2+1
2
2-1)X

,   

X
8=
(t
2-1
4
4+1)X
(t
2+1
4
4-1)X

.

These last three equations may be inverted:

x=

1
\pm\sqrt{1+t
\left(1-X2
1+X2
\right)
},\qquadX_2=\frac,\qquadX_4=\frac.\qquad

To calculate the zeroes of

R8(\xi,x)

we set

X8=0

in the third equation, calculate the two values of

X4

, then use these values of

X4

in the second equation to calculate four values of

X2

and finally, use these values in the first equation to calculate the eight zeroes of

R8(\xi,x)

. (The

tn

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:

R1(\xi,x)=x

R
2(\xi,x)=(t+1)x2-1
(t-1)x2+1

where

t\equiv\sqrt{1-

1
\xi2
}
R
3(\xi,x)=x
2)(x
(1-x
2)
z
p
2)(x
(1-x
2)
p
z

where

G\equiv\sqrt{4\xi2+(4\xi2(\xi2-1))2/3

}
2\equiv2\xi2\sqrt{G
x
p
}
2=\xi
x
z
2
p

R4(\xi,x)=R2(R2(\xi,\xi),R

2x
2(\xi,x))=(1+t)(1+\sqrt{t
)

4-2(1+t)(1+\sqrt{t})x2+1} {(1+t)(1-\sqrt{t})2x4-2(1+t)(1-\sqrt{t})x2+1}

R6(\xi,x)=R3(R2(\xi,\xi),R2(\xi,x))

etc.

See for further explicit expressions of order n=5 and

n=2i3j

.

The corresponding discrimination factors are:

L1(\xi)=\xi

L
2(\xi)=1+t
1-t

=\left(\xi+\sqrt{\xi2-1}\right)2

3\left(
2
1-x
p
2
\xi
p
L
3(\xi)=\xi

\right)2

2-1)
L
4(\xi)=\left(\sqrt{\xi}+(\xi

1/4\right)4\left(\xi+\sqrt{\xi2-1}\right)2

L6(\xi)=L3(L2(\xi))

etc.

The corresponding zeroes are

xnj

where n is the order and j is the number of the zero. There will be a total of n zeroes for each order.

x11=0

x21=\xi\sqrt{1-t}

x22=-x21

x31=xz

x32=0

x33=-x31

x41=\xi\sqrt{\left(1-\sqrt{t}\right)\left(1+t-\sqrt{t(t+1)}\right)}

x42=\xi\sqrt{\left(1-\sqrt{t}\right)\left(1+t+\sqrt{t(t+1)}\right)}

x43=-x42

x44=-x41

From the inversion relationship, the corresponding poles

xp,ni

may be found by

xp,ni=\xi/(xni)

References