A brief definition of the z-transform, explaining its relationship with the Fourier transform and its region of convergence, ROC.
Basic definition of the z-transform
The
z-transform of a sequence is defined as
$X(z)=\sum_{n=()} $∞∞xnzn
Sometimes this equation is referred to as the
bilateral z-transform . At times the z-transform is defined as
$X(z)=\sum_{n=0} $∞xnzn
which is known as the
unilateral z-transform .
There is a close relationship between the z-transform and the
Fourier transform of a discrete time signal,
which is defined as
$X(e^{i\omega})=\sum_{n=()} $∞∞xnωn
Notice that that when the
$z^{-n}$ is replaced with
$e^{-(i\omega n)}$ the z-transform reduces to the Fourier Transform. When the
Fourier Transform exists,
$z()=e^{i\omega}$ , which is to have the magnitude of
$z$ equal to unity.
The complex plane
In order to get further insight into the relationship between
the Fourier Transform and the Z-Transform it is useful to lookat the complex plane or
z-plane . Take a look at
the complex plane:
The Z-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable
$z$ . The position on the complex
plane is given by
$re^{(i\omega )}$ , and the angle from the positive, real axis around the plane is
denoted by
$\omega $ .
$X(z)$ is defined
everywhere on this plane.
$X(e^{i\omega})$ on the other
hand is defined only where
$\left|z\right|=1$ ,
which is referred to as the unit circle. So for example,
$\omega =1$ at
$z=1$ and
$\omega =\pi ()$ at
$z=-1$ .
This is useful because, by representing the Fourier transformas the z-transform on the unit circle, the periodicity of
Fourier transform is easily seen.
Region of convergence
The region of convergence, known as the
ROC , is
important to understand because it defines the region wherethe z-transform exists. The ROC for a given
$x(n)$ , is defined as the range of
$z$ for which the z-transform converges. Since the z-transform is
a
power series , it converges when
$x(n)z^{-n}$ is absolutely summable. Stated differently,
$\sum_{n=()} $∞∞xnzn∞
must be satisfied for convergence. This is best illustratedby looking at the different ROC's of the z-transforms of
$\alpha ^{n}u(n)$ and
$\alpha ^{n}u(n-1)$ .
For
$x(n)=\alpha ^{n}u(n)$
$X(z)=\sum_{n=()} $∞∞xnznn∞∞αnunznn0∞αnznn0∞αz1n
This sequence is an example of a right-sided exponential
sequence because it is nonzero for
$n\ge 0$ .
It only converges when
$\left|\alpha z^{(-1)}\right|< 1$ .
When it converges,
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?