# 2.8 Stability, causality, and the z-transform

 Page 1 / 1

## 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.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Got questions? Join the online conversation and get instant answers!