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ω = 2 π f

Care must be taken with the elliptic-function filter where there are two critical frequencies that determine the transitionregion. Both frequencies must be prewarped.

The characteristics of the bilinear transform are the following:

  • The order of the digital filter is the same as the prototype filter.
  • The left-half s-plane is mapped into the unit circle onthe z-plane. This means stability is preserved.
  • Optimal approximations to piecewise constant prototype filters, such as the four cases in Continuous Frequency Definition of Error , transform into optimal digital filters.
  • The cascade of sections designed by the bilinear transform is the same as obtained by transforming the total system.

The bilinear transform is probably the most used method of converting a prototype Laplace transform transfer function into adigital transfer function. It is the one used in most popular filter design programs [link] , because of characteristic 3 above that states optimality is preserved. The maximally flat prototype istransformed into a maximally flat digital filter. This property only holds for approximations to piecewise constant ideal frequencyresponses, because the frequency warping does not change the shape of a constant. If the prototype is an optimal approximation to adifferentiator or to a linear-phase characteristic, the bilinear transform will destroy the optimality. Those approximations have tobe made directly in the digital frequency domain.

The bilinear transformation

To illustrate the bilinear transformation, the third-order Butterworth lowpass filter designed in the Example is converted into adigital filter. The prototype filter transfer function is

F ( s ) = 1 ( s + 1 ) ( s 2 + s + 1 )

The prototype analog filter has a passband edge at u 0 = 1 . A data rate of 1000 samples per second corresponding to T = 0 . 001 seconds is assumed. If the desired digital passband edge is f 0 = 200 Hz, then ω 0 = ( 2 π ) ( 200 ) radians per second, and the total prewarped bilinear transformation from [link] is

s = 1 . 376382 z - 1 z + 1

The digital transfer function in [link] becomes

H ( z ) = 0 . 09853116 ( z + 1 ) 3 ( z - 0 . 158384 ) ( z 2 - 0 . 418856 z + 0 . 355447 )

Note the locations of the poles and zeros in the z-plane. Zeros at infinity in the s-plane always map into the z = -1 point. The exampleillustrate a third-order elliptic-function filter designed using the bilinear transform.

Frequency transformations

For the design of highpass, bandpass, and band reject filters, a particularly powerful combination consists of using thefrequency transformations described in Section elsewhere together with the bilinear transformation. When using this combination, some care must betaken in scaling the specifications properly. This is illustrated by considering the steps in the design of a bandpass filter:

  1. First, the lower and upper digital bandedge frequencies are specified as ω 1 and or ω 1 , ω 2 , ω 3 , and ω 4 if an elliptic-function approximation is used.
  2. These frequencies are prewarped using [link] to give theband edges of the prototype bandpass analogfilter.
  3. These frequencies are converted into a single band- edge ω p or ω s for the Butterworth and Chebyshev and into ω p and ω s for the elliptic-function approximation of the prototype lowpass filter by using Equation 2 from Frequency Transformations and Equation 4 from Frequency Transformations .
  4. The lowpass filter is designed for this ω p and/or ω s by using one of the four approximations in the sections in Continuous Frequency Definition of Error or some other method.
  5. This lowpass analog filter is converted into a bandpass analog filter with the frequency transformation Equation 6 from Frequency Transformations .
  6. The bandpass analog filter is then transformed into the desired bandpass digital filter using the bilineartransformation [link] .

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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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