Describes how to design a general filter from the Laplace Transform and its pole/zero plots.
Introduction
Analog (Continuous-Time) filters are useful for a wide variety of applications, and are especially useful in that they are very simple to build using standard, passive R,L,C components. Having a grounding in basic filter design theory can assist one in solving a wide variety of signal processing problems.
Estimating frequency response from z-plane
One of the motivating factors for analyzing the pole/zero
plots is due to their relationship to the frequency responseof the system. Based on the position of the poles and zeros,
one can quickly determine the frequency response. This is aresult of the correspondence between the frequency response
and the transfer function evaluated on the unit circle in thepole/zero plots. The frequency response, or DTFT, of the
system is defined as:
Next, by factoring the transfer function into poles and zeros
and multiplying the numerator and denominator by
$e^{iw}$ we arrive at the following equations:
From
[link] we have the
frequency response in a form that can be used to interpretphysical characteristics about the filter's frequency
response. The numerator and denominator contain a product ofterms of the form
$\left|e^{iw}-h\right|$ ,
where
$h$ is either a zero, denoted by
${c}_{k}$ or a pole, denoted by
${d}_{k}$ . Vectors are commonly used to represent
the term and its parts on the complex plane. The pole or zero,
$h$ , is a vector from the origin
to its location anywhere on the complex plane and
$e^{iw}$ is a vector from the origin to its
location on the unit circle. The vector connecting these twopoints,
$\left|e^{iw}-h\right|$ , connects the pole or zero location to a
place on the unit circle dependent on the value of
$w$ . From this, we can begin to
understand how the magnitude of the frequency response is aratio of the distances to the poles and zero present in the
z-plane as
$w$ goes from zero to
pi. These characteristics allow us to interpret
$\left|H(w)\right|$ as follows:
In conclusion, using the distances from the unit circle to the
poles and zeros, we can plot the frequency response of thesystem. As
$w$ goes from
$0$ to
$2\pi $ , the following two properties, taken from the above
equations, specify how one should draw
$\left|H(w)\right|$ .
While moving around the unit circle...
if close to a zero, then the magnitude is small. If a
zero is on the unit circle, then the frequency response iszero at that point.
if close to a pole, then the magnitude is large. If a
pole is on the unit circle, then the frequency responsegoes to infinity at that point.
Drawing frequency response from pole/zero plot
Let us now look at several examples of determining the
magnitude of the frequency response from the pole/zero plot ofa z-transform. If you have forgotten or are unfamiliar with
pole/zero plots, please refer back to the
Pole/Zero Plots module.
In this first example we will take a look at the very simple
z-transform shown below:
$$H(z)=z+1=1+z^{-1}$$$$H(w)=1+e^{-(iw)}$$ For this example, some of the vectors represented by
$\left|e^{iw}-h\right|$ , for random values of
$w$ , are explicitly drawn onto
the complex plane shown in the
figure below. These vectors show how the
amplitude of the frequency response changes as
$w$ goes from
$0$ to
$2\pi $ ,
and also show the physical meaning of the terms in
[link] above. One can see that
when
$w=0$ ,
the vector is the longest and thus the frequency responsewill have its largest amplitude here. As
$w$ approaches
$\pi $ , the length of the vectors decrease
as does the amplitude of
$\left|H(w)\right|$ . Since
there are no poles in the transform, there is only this onevector term rather than a ratio as seen in
[link] .
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
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.
Harper
Do you know which machine is used to that process?
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it