<< Chapter < Page Chapter >> Page >
i = 1 , 3 , 5 , . . . , N - 1 N even

These zeros are purely imaginary and lie on the ω axis.

Pole locations

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

1 + ϵ 2 G 2 = 0


G = ± j ( 1 / ϵ )

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

G = s n ( n φ + 2 K 1 i , k 1 ) = ± j 1 / ϵ


φ = ( - 2 K 1 i + s n - 1 ( j 1 / e , k 1 ) ) / n

Define ν 0 to be the second term in [link] by

j ν 0 = ( s n - 1 ( j 1 / e , k 1 ) ) / n

which is similar to the equation for the Chebyshev case. Using properties of s n of an imaginary variable and [link] , ν 0 becomes

ν 0 = ( K / N K 1 ) s c - 1 ( 1 / ϵ , k ' )

The poles are now found from [link] , [link] , [link] , and [link] to be

s p i = j s n ( K i / N + j ν 0 , k )

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

s p i = c n d n s n ' c n ' + j s n d n ' 1 - d n 2 s n ' 2


s n = s n ( K i / N , k ) , c n = c n ( K i / N , k ) , d n = d n ( K i / N , k )
s n ' = s n ( ν 0 , k ' ) , c n ' = c n ( ν 0 , k ' ) , d n ' = d n ( ν 0 , k ' )


i = 0 , 2 , 4 , . . . . N odd
i = 1 , 3 , 5 , . . . . N 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 ( s ) pole locations is to obtain the zeros from [link] , then find G ( ω ) using the reciprocal relation of the poles and zeros [link] . F ( s ) is constructed from G ( ω ) 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 ν 0 by [link] or determining k from K / K ' , 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.


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 ( ω ) is used to describe an optimal frequency-response function F ( j ω ) and to design the corresponding filter. The squared-magnitude frequency-responsefunction is

| F ( j ω ) | 2 = 1 1 + ϵ 2 G ( ω ) 2

with G ( ω ) 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.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing and digital filter design (draft)' conversation and receive update notifications?