The zeros of the transfer function
$H(z)$ of a linear-phase filter lie in specific configurations.
We can write the symmetry condition
$$h(n)=h(N-1-n)$$ in the
$Z$ domain. Taking the
$Z$ -transform of both sides gives
$H(z)=z^{-(N-1)}H(\frac{1}{z})$
Recall that we are assuming that
$h(n)$ is real-valued. If
${z}_{0}$ is a zero of
$H(z)$ ,
$$H({z}_{0})=0$$ then
$$H(\overline{{z}_{0}})=0$$ (Because the roots of a polynomial with real coefficients
exist in complex-conjugate pairs.)
Using the symmetry condition,
, it follows that
$$H({z}_{0})=z^{-(N-1)}H(\frac{1}{{z}_{0}})=0$$ and
$$H(\overline{{z}_{0}})=z^{-(N-1)}H(\frac{1}{\overline{{z}_{0}}})=0$$ or
$$H(\frac{1}{{z}_{0}})=H(\frac{1}{\overline{{z}_{0}}})=0$$
If
${z}_{0}$ is a zero of a (real-valued) linear-phase filter, then so
are
$\overline{{z}_{0}}$ ,
$\frac{1}{{z}_{0}}$ , and
$\frac{1}{\overline{{z}_{0}}}$ .
Zeros locations
It follows that
generic zeros of a linear-phase filter exist in sets of 4.
zeros on the unit circle (
${z}_{0}=e^{i{}_{0}}$ ) exist in sets of 2. (
${z}_{0}\neq (1)$ )
zeros on the real line (
${z}_{0}=a$ ) exist in sets of 2. (
${z}_{0}\neq (1)$ )
zeros at 1 and -1 do not imply the existence of zeros at
other specific points.
Zero locations: automatic zeros
The frequency response
${H}^{f}()$ of a Type II FIR filter always has a zero at
$=\pi $ :
$$h(n)$$
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