# Limit cycles

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Overflow can cause large-scale limit cycles in IIR filters. Saturation arithmetic prevents zero-input limit cycles. Small-scale limit cycles can arise from quantization in recursive filters.

## Large-scale limit cycles

When overflow occurs, even otherwise stable filters may get stuck in a large-scale limit cycle , which is a short-period, almost full-scale persistent filter output caused by overflow.

Consider the second-order system $H(z)=\frac{1}{1-z^{(-1)}+\frac{1}{2}z^{-2}}$

with zero input and initial state values ${z}_{0}(0)=0.8$ , ${z}_{1}(0)=-0.8$ . Note $y(n)={z}_{0}(n+1)$ .

The filter is obviously stable, since the magnitude of the poles is $\frac{1}{\sqrt{2}}=0.707$ , which is well inside the unit circle. However, with wraparound overflow, note that $y(0)={z}_{0}(1)=\frac{4}{5}-\frac{1}{2}-\left(\frac{4}{5}\right)=\frac{6}{5}=-\left(\frac{4}{5}\right)$ , and that ${z}_{0}(2)=y(1)=-\left(\frac{4}{5}\right)-\frac{1}{2}\frac{4}{5}=-\left(\frac{6}{5}\right)=\frac{4}{5}$ , so $y(n)=-\left(\frac{4}{5}\right),\frac{4}{5},-\left(\frac{4}{5}\right),\frac{4}{5},$ even with zero input.

Clearly, such behavior is intolerable and must be prevented. Saturation arithmetic has been proved to prevent zero-input limit cycles , which is one reason why all DSP microprocessors support this feature. In many applications,this is considered sufficient protection. Scaling to prevent overflow is another solution, if as well the inital statevalues are never initialized to limit-cycle-producing values. The normal-form structure also reduces the chance ofoverflow.

## Small-scale limit cycles

Small-scale limit cycles are caused by quantization. Consider the system

Note that when ${z}_{0}> {z}_{0}-\frac{{}_{B}}{2}$ , rounding will quantize the output to the current level (with zero input), so the output will remain at thislevel forever. Note that the maximum amplitude of this "small-scale limit cycle" is achieved when ${z}_{0}={z}_{0}-\frac{{}_{B}}{2}=={z}_{\mathrm{max}}=\frac{{}_{B}}{2(1-)}$ In a higher-order system, the small-scale limit cycles are oscillatory in nature. Anyquantization scheme that never increases the magnitude of any quantized value prevents small-scale limit cycles.
Two's-complement truncation does not do this; it increases the magnitude of negative numbers.
However, this introduces greater errorand bias. Since the level of the limit cycles is proportional to ${}_{B}$ , they can be reduced by increasing the number of bits. Poles close to the unit circle increase themagnitude and likelihood of small-scale limit cycles.

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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
Abigail
Do somebody tell me a best nano engineering book for beginners?
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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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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
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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
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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