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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 n x n z n
Sometimes this equation is referred to as the bilateral z-transform . At times the z-transform is defined as
X z n 0 x n z n
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 ω n x n ω n
Notice that that when the z n is replaced with ω n the z-transform reduces to the Fourier Transform. When the Fourier Transform exists, z ω , 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 r ω , and the angle from the positive, real axis around the plane is denoted by ω . X z is defined everywhere on this plane. X ω on the other hand is defined only where z 1 , which is referred to as the unit circle. So for example, ω 1 at z 1 and ω 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,

n x n z n
must be satisfied for convergence. This is best illustratedby looking at the different ROC's of the z-transforms of α n u n and α n u n 1 .


x n α n u n

x n α n u n where α 0.5 .

X z n x n z n n α n u n z n n 0 α n z n n 0 α z 1 n
This sequence is an example of a right-sided exponential sequence because it is nonzero for n 0 . It only converges when α z 1 . When it converges,
X z 1 1 α z z z α
If α z 1 , then the series, n 0 α z n does not converge. Thus the ROC is the range of values where
α z 1
or, equivalently,
z α

ROC for x n α n u n where α 0.5
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x n α n u n 1

x n α n u n 1 where α 0.5 .

X z n x n z n n α n u -n 1 z n n -1 α n z n n -1 α -1 z n n 1 α -1 z n 1 n 0 α -1 z n
The ROC in this case is the range of values where
α -1 z 1
or, equivalently,
z α
If the ROC is satisfied, then
X z 1 1 1 α -1 z z z α

ROC for x n α n u n 1
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Questions & Answers

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Introduction about quantum dots in nanotechnology
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s. Reply
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are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
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.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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