This form allows for easy inversions of each term of the sum
using the
inspection
method and the
transform table . If the numerator is
a polynomial, however, then it becomes necessary to use
partial-fraction
expansion to put
$H(s)$ in the above form. If
$M\ge N$ then
$H(s)$ can be expressed as
Find the inverse z-transform of
$$H(s)=\frac{1+2s^{(-1)}+s^{-2}}{1-3s^{(-1)}+2s^{-2}}$$ where the ROC is
$\left|s\right|> 2$ .
In this case
$M=N=2$ ,
so we have to use long division to get
$$H(s)=\frac{1}{2}+\frac{\frac{1}{2}+\frac{7}{2}s^{(-1)}}{1-3s^{(-1)}+2s^{-2}}$$ Next factor the denominator.
$$H(s)=2+\frac{-1+5s^{(-1)}}{(1-2s^{(-1)})(1-s^{(-1)})}$$ Now do partial-fraction expansion.
$$H(s)=\frac{1}{2}+\frac{{A}_{1}}{1-2s^{(-1)}}+\frac{{A}_{2}}{1-s^{(-1)}}=\frac{1}{2}+\frac{\frac{9}{2}}{1-2s^{(-1)}}+\frac{-4}{1-s^{(-1)}}$$ Now each term can be inverted using the inspection method
and the Laplace-transform table. Thus, since the ROC is
$\left|s\right|> 2$ ,
$$h(t)=\frac{1}{2}\delta (t)+\frac{9}{2}2^{t}u(t)-4u(t)$$
In this case, since there were no poles, we multiplied thefactors of
$H(s)$ .
Now, by inspection, it is clear that
$$h(t)=\delta (t+2)+\frac{5}{2}\delta (t+1)+\frac{1}{2}\delta (t)+-\delta (t-1)$$ .
One of the advantages of the power series expansion method is
that many functions encountered in engineering problems havetheir power series' tabulated. Thus functions such as log,
sin, exponent, sinh, etc, can be easily inverted.
where
$r$ is a counter-clockwise contour in the ROC of
$H(s)$ encircling the origin of the s-plane. To further expand on
this method of finding the inverse requires the knowledge ofcomplex variable theory and thus will not be addressed in this
module.
Demonstration of contour integration
Conclusion
The Inverse Laplace-transform is very useful to know for the purposes of designing a filter, and there are many ways in which to calculate it, drawing from many disparate areas of mathematics. All nevertheless assist the user in reaching the desired time-domain signal that can then be synthesized in hardware(or software) for implementation in a real-world filter.
Questions & Answers
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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
can nanotechnology change the direction of the face of the world