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Stability, causality, and the z -transform

In going from

k = 0 N a k y [ n - k ] = k = 0 m b k x [ n - k ]


H ( z ) = Y ( z ) X ( z )

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

Graph with horizontal axis Re[z] and vertical axis Im[z]. There are three X marks and four circle marks on the graph.

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 [ n ] is causal , i.e., if h [ n ] = 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 1 ( Z ) . In this case,

H ( z ) = n = - h [ n ] z - n n = - h [ n ] z - n

Consider the unit circle z = e j ω . In this case we have | z - n | = | e - j ω n | = 1 , so that

H ( e j ω ) n = - h [ n ] <

for all ω . Thus, if a system is BIBO stable, the ROC of H ( z ) 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 ( z ) = z 2 ( z - 2 ) ( z + 1 2 ) = 4 5 z z - 2 + 1 5 z z + 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.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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