First we treat filter design for interpolation.
Consider an input signal
$x(n)$ that is
${\omega}_{0}$ -bandlimited in the DTFT domain.
If we upsample by factor
$L$ to get
$v(m)$ , the desired portion of
$V(e^{i\omega})$ is the spectrum in
$\left[\frac{-\pi}{L} , \frac{\pi}{L}\right)$ ,
while the undesired portion is the remainder of
$\left[-\pi , \pi \right)$ .
Noting from
that
$V(e^{i\omega})$ has zero energy in the regions
the anti-imaging filter can be designed with transition bands in
these regions (rather than passbands or stopbands). For a givennumber of taps, the additional degrees of freedom offered by
these transition bands allows for better responses in thepassbands and stopbands. The resulting filter design
specifications are shown in the
bottom subplot below .
Next we treat filter design for decimation. Say that the
desired spectral component of the input
signal is bandlimited to
$\frac{{\omega}_{0}}{M}< \frac{\pi}{M}$ and we have decided to downsample by
$M$ .
The goal is to minimally distort the input spectrum over
$\left[\frac{-{\omega}_{0}}{M} , \frac{{\omega}_{0}}{M}\right)$ ,
i.e. , the post-decimation
spectrum over
$\left[-{\omega}_{0} , {\omega}_{0}\right)$ .
Thus, we must not allow any aliased signals to enter
$\left[-{\omega}_{0} , {\omega}_{0}\right)$ .
To allow for extra degrees of freedom in the filter design,we
do allow aliasing to enter the
post-decimation spectrum outside of
$\left[-{\omega}_{0} , {\omega}_{0}\right)$ within
$\left[-\pi , \pi \right)$ .
Since the input spectral regions which alias outside of
$\left[-{\omega}_{0} , {\omega}_{0}\right)$ are given by
(as shown in
), we can
treat these regions as transition bands in the filter design.The resulting filter design specifications are illustrated in
the middle subplot (
).
Questions & Answers
anyone know any internet site where one can find nanotechnology papers?
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?