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
can someone help me with some logarithmic and exponential equations.
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
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.