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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:

Z-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 .

For

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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For

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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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da
Application of nanotechnology in medicine
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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