3.4 Elliptic-function filter properties

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$i=1,3,5,...,N-1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{N}\phantom{\rule{4.pt}{0ex}}\text{even}$

These zeros are purely imaginary and lie on the $\omega$ axis.

Pole locations

The pole locations are somewhat more complicated to find. An approach similar to that used for the Chebyshev filter isused here. $FF\left(s\right)$ becomes infinite when

$1+{ϵ}^{2}{G}^{2}=0$

or

$G=±j\left(1/ϵ\right)$

Using [link] and the periodicity of sn (u,k) , this implies

$G=sn\left(n\phi +2{K}_{1}i,{k}_{1}\right)=±j1/ϵ$

or

$\phi =\left(-2{K}_{1}i+s{n}^{-1}\left(j1/e,{k}_{1}\right)\right)/n$

Define ${\nu }_{0}$ to be the second term in [link] by

$j{\nu }_{0}=\left(s{n}^{-1}\left(j1/e,{k}_{1}\right)\right)/n$

which is similar to the equation for the Chebyshev case. Using properties of $sn$ of an imaginary variable and [link] , ${\nu }_{0}$ becomes

${\nu }_{0}=\left(K/N{K}_{1}\right)s{c}^{-1}\left(1/ϵ,{k}^{\text{'}}\right)$

${s}_{pi}=j\phantom{\rule{4pt}{0ex}}sn\left(Ki/N+j{\nu }_{0},k\right)$

This equation can be more clearly written by using the summation formula [link] for the elliptic sine function to give

${s}_{pi}=\frac{cn\phantom{\rule{4pt}{0ex}}dn\phantom{\rule{4pt}{0ex}}s{n}^{\text{'}}\phantom{\rule{4pt}{0ex}}c{n}^{\text{'}}+jsn\phantom{\rule{4pt}{0ex}}d{n}^{\text{'}}}{1-d{n}^{2}s{n}^{\text{'}2}}$

where

$sn=sn\left(Ki/N,k\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}cn=cn\left(Ki/N,k\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}dn=dn\left(Ki/N,k\right)$
$s{n}^{\text{'}}=sn\left({\nu }_{0},{k}^{\text{'}}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}c{n}^{\text{'}}=cn\left({\nu }_{0},{k}^{\text{'}}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}d{n}^{\text{'}}=dn\left({\nu }_{0},{k}^{\text{'}}\right)$

for

$i=0,2,4,....\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{N}\phantom{\rule{4.pt}{0ex}}\text{odd}$
$i=1,3,5,....\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{N}\phantom{\rule{4.pt}{0ex}}\text{even}$

The theory of Jacobian elliptic functions can be found in [link] and its application to filter design in [link] , [link] , [link] . The best techniques for calculating the elliptic functions seem to use thearithmetic-geometric mean; efficient algorithms are presented in [link] . A design program is given in [link] and a versitile FORTRAN program that is easily related to the theory in this chapter isgiven as Program 8 in the appendix of this book. Matlab has a powerful elliptic function filter design command as well as accurate algorithms forevaluating the Jacobian elliptic functions and integrals.

An alternative to the use of elliptic functions for finding the transfer function $F\left(s\right)$ pole locations is to obtain the zeros from [link] , then find $G\left(\omega \right)$ using the reciprocal relation of the poles and zeros [link] . $F\left(s\right)$ is constructed from $G\left(\omega \right)$ and $ϵ$ from [link] , and the poles are found using a root-finding algorithm. Another possibility is to find the zeros from [link] and the poles from the methods for finding a Chebyshev passband from arbitraryzeros. These approaches avoid calculating ${\nu }_{0}$ by [link] or determining $k$ from $K/{K}^{\text{'}}$ , as is described in [link] . The efficient algorithms for evaluating the elliptic functions and thecommon use of powerful computers make these alternatives less attractive now.

Summary

In this section the basic properties of the Jacobian elliptic functions have been outlined and the necessaryconditions given for an equal-ripple rational function to be defined in terms of them. This rational function was then used toconstruct a filter transfer function with equal-ripple properties. Formulas were derived to calculate the pole and zerolocations for the filter transfer functions and to relate design specifications to the functions. These formulas require theevaluation of elliptic functions and are implemented in Program 8 in the appendix.

Elliptic-function filter design procedures

The equal-ripple rational function $G\left(\omega \right)$ is used to describe an optimal frequency-response function $F\left(j\omega \right)$ and to design the corresponding filter. The squared-magnitude frequency-responsefunction is

${|F\left(j\omega \right)|}^{2}=\frac{1}{1+{ϵ}^{2}G{\left(\omega \right)}^{2}}$

with $G\left(\omega \right)$ defined by Jacobian Elliptic functions, and $ϵ$ being a parameter that controls the passband ripple. The plot of this function for $N=3$ illustrates the relation to the various specification parameters.

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