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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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
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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
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