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?