<< Chapter < Page Chapter >> Page >

It is important in designing filters to choose the particular type that is appropriate. Since in all cases, thefilters are optimal, it is necessary to understand in what sense they are optimal.

The classical Butterworth filter is optimal in the sense that it is the best Taylor's series approximation to the ideallowpass filter magnitude at both ω = 0 and ω = .

The Chebyshev filter gives the smallest maximum magnitude error over the entire passband of any filter that is also a Taylor'sseries approximation at ω = to the ideal magnitude characteristic.

The Inverse-Chebyshev filter is a Taylor's series approximation to the ideal magnitude response at ω = 0 and minimizes the maximum error in the approximation to zero over thestopband. This can also be stated as maximizing the minimum rejection of the filter over the stopband.

The elliptic-function filter (Cauer filter) considers the four parameters of the filter: the passband ripple, thetransition-band width, the stopband ripple, and the order of the filter. For given values of any three of the four, the fourth isminimized.

It should be remembered that all four of these filter designs are magnitude approximations and do not address the phasefrequency response or the time-domain characteristics. For most designs, the Butterworth filter has the smoothest phase curve,followed by the inverse-Chebyshev, then the Chebyshev, and finally the elliptic-function filter having the least smoothphase response.

Recall that in addition to the four filters described in this section, the more general Taylor's series method allows anarbitrary zero locations to be specified but retains the optimal character at ω = 0 . A design similar to this can be obtained by replacing ω by 1 / ω , which allows setting | F ( w ) | 2 equal unity at arbitrary frequencies in the passband and having a Taylor's series approximation to zero at ω = [link] .

These basic normalized lowpass filters can have the passband edge moved from unity to any desired value by a simple change of frequencyvariable, ω replaced with k ω . They can be converted to highpass filters or bandpass or band reject filters by various changessuch as ω with k / ω or ω with a ω + b / ω . In all of these cases the optimality is maintained, because the basic lowpassapproximation is to a piecewise constant ideal. An approximation to a nonpiecewise constant ideal, such as a differentiator, may not be optimalafter a frequency change of variables .

In some cases, especially where time-domain characteristics are important, ripples in the frequency response causeirregularities, such as echoes in the time response. For that reason, the Butterworth and Chebyshev II filters are more desirablethan their frequency response alone might indicate. A fifth approximation has been developed [link] that is similar to the Butterworth. It does not require a Taylor's series approximation at ω = 0 , but only requires that the response monotonically decrease in the passband, thus giving a narrower transition regionthan the Butterworth, but without the ripples of the Cheybshev.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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
Good
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