# 2.8 Stability, causality, and the z-transform

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## Stability, causality, and the $z$ -transform

In going from

$\sum _{k=0}^{N}{a}_{k}y\left[n-k\right]=\sum _{k=0}^{m}{b}_{k}x\left[n-k\right]$

to

$H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}$

we did not specify an ROC. If we factor $H\left(z\right)$ , we can plot the poles and zeros in the $z$ -plane as below.

Several ROCs may be possible. Each ROC corresponds to a different impulse response, so which one should we choose? In general, there is no “right”choice, however, there are some choices that make sense in practice.

In particular, if $h\left[n\right]$ is causal , i.e., if $h\left[n\right]=0$ , $n<0$ , then the ROC extends outward from the outermost pole . This can be seen in the examples up to this point. Moreover, recall that a system is BIBO stable if the impulse response $h\in {\ell }_{1}\left(\mathbb{Z}\right)$ . In this case,

$\begin{array}{cc}\hfill \left|H,\left(,z,\right)\right|& =\left|\sum _{n=-\infty }^{\infty },h,\left[n\right],{z}^{-n}\right|\hfill \\ & \le \sum _{n=-\infty }^{\infty }\left|h,\left[,n,\right]\right|\left|{z}^{-n}\right|\hfill \end{array}$

Consider the unit circle $z={e}^{j\omega }$ . In this case we have $|{z}^{-n}|=|{e}^{-j\omega n}|=1$ , so that

$\left|H,\left(,{e}^{j\omega },\right)\right|\le \sum _{n=-\infty }^{\infty }\left|h,\left[,n,\right]\right|<\infty$

for all $\omega$ . Thus, if a system is BIBO stable, the ROC of $H\left(z\right)$ must include the unit circle. In general, any ROC containing the unit circle will be BIBO stable.

This leads to a key question – are stability and causality always compatible? The answer is no . For example, consider

$H\left(z\right)=\frac{{z}^{2}}{\left(z-2\right)\left(z+\frac{1}{2}\right)}=\frac{\frac{4}{5}z}{z-2}+\frac{\frac{1}{5}z}{z+\frac{1}{2}}$

and its various ROC's and corresponding inverses. If the ROC contains the unit-circle (so that the corresponding system is stable) and is not to contain any poles, then it must extend inward towards the origin, and hence it cannot be causal. Alternatively, if the ROC is to extend outward, it will not contain the unit-circle so that the corresponding system will not be BIBO stable.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
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what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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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?
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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
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what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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