<< Chapter < Page Chapter >> Page >

For mathematical convenience, the four classical IIR filter transfer functions were developed in terms of the Laplace transform ratherthan the z-transform. The prototype Laplace-transform transfer functions are descriptions of analog filters. In this section they are converted toz-transform transfer functions for implementation as IIR digital filters.

There have been several different methods of converting analog systems to digital described over the history of digitalfilters. Two have proven to be useful for most applications. The first is called the impulse-invariant method and results in adigital filter with an impulse response exactly equal to samples of the prototype analog filter. The second method uses afrequency mapping to convert the analog filter to a digital filter. It has the desirable property of preserving theoptimality of the four classical approximations developed in the last section. This section will develop the theory and designformulas to implement both of these conversion approaches.

The impulse-invariant method

Although the transfer functions in Continuous Frequency Definition of Error were designed with criteria in the frequency domain, the impulse-invariant method willconvert them into digital transfer functions using a time-domain constraint [link] , [link] , [link] . The digital filter designed by the impulse-invariant method is required to have an impulse response that isexactly equal to equally spaced samples of the impulse response of the prototype analog filter. If the analog filter has a transfer function F ( s ) with an impulse response f ( t ) , the impulse response of the digital filter h ( n ) is required to match the samples of f ( t ) . For samples at T second intervals, the impulse response is

h ( n ) = F ( T ) | t = T n = F ( T n )

The transfer function of the digital filter is the z-transform of the impulse response of the filter, which is given by

H ( z ) = n = 0 h ( n ) z - n

The transfer function of the prototype analog filter is always a rational function written as

F ( s ) = B ( s ) A ( s )

where B ( s ) is the numerator polynomial with roots that are the zeros of F ( s ) , and A ( s ) is the denominator with roots that are the poles of F ( s ) . If F ( s ) is expanded in terms of partial fractions, it can be written as

F ( s ) = i = 1 N K i s + s i

The impulse response of this filter is the inverse-Laplace transform of [link] , which is

f ( t ) = i = 1 N K e s i t

Sampling this impulse response every T seconds gives

f ( n T ) = i = 1 N K i e - s i n T = i = 1 N K i ( e - s i T ) n

The basic requirement of [link] gives

H ( z ) = n = 0 [ i = 1 N K i ( e - s I T ) n ]
H ( z ) = i = 1 N K i z z - e s I T

which is clearly a rational function of z and is the transfer function of the digital filter, which has samples of the prototype analog filter asits impulse response.

This method has its requirements set in the time domain, but the frequency response is important. In most cases, the prototype analog filter is oneof the classical types, which is optimal in the frequency domain. If the frequency response of the analog filter is denoted by F ( j ω ) and the frequency response of the digital filter designed by the impulse-invariant method is H ( ω ) , it can be shown in a development similar to that used for the sampling theorem

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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?

Ask