# 2.1 System classifications and properties

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## Introduction

In this module some of the basic classifications of systems will be briefly introduced and the most important propertiesof these systems are explained. As can be seen, the properties of a system provide an easy way to separate onesystem from another. Understanding these basic difference's between systems, and their properties, will be a fundamentalconcept used in all signal and system courses, such as digital signal processing (DSP). Once a set of systems can beidentified as sharing particular properties, one no longer has to deal with proving a certain characteristic of a system eachtime, but it can simply be accepted do the systems classification. Also remember that this classificationpresented here is neither exclusive (systems can belong to several different classifications) nor is it unique.

## Classification of systems

Along with the classification of systems below, it is also important to understand other Classification of Signals .

## Continuous vs. discrete

This may be the simplest classification to understand as the idea of discrete-time and continuous-time is one of the mostfundamental properties to all of signals and system. A system where the input and output signals are continuous is a continuous system , and one where the input and output signals are discrete is a discrete system .

## Linear vs. nonlinear

A linear system is any system that obeys the properties of scaling (homogeneity) and superposition(additivity), while a nonlinear system is any system that does not obey at least one of these.

To show that a system $H$ obeys the scaling property is to show that

$H(kf(t))=kH(f(t))$

To demonstrate that a system $H$ obeys the superposition property of linearity is to show that

$H({f}_{1}(t)+{f}_{2}(t))=H({f}_{1}(t))+H({f}_{2}(t))$

It is possible to check a system for linearity in a single (though larger) step. To do this, simply combine the firsttwo steps to get

$H({k}_{1}(){f}_{1}(t)+{k}_{2}(){f}_{2}(t))={k}_{2}()H({f}_{1}(t))+{k}_{2}()H({f}_{2}(t))$

## Time invariant vs. time variant

A time invariant system is one that does not depend on when it occurs: the shape of the output does notchange with a delay of the input. That is to say that for a system $H$ where $H(f(t))=y(t)$ , $H$ is time invariant if for all $T$

$H(f(t-T))=y(t-T)$

When this property does not hold for a system, then it is said to be time variant , or time-varying.

## Causal vs. noncausal

A causal system is one that is nonanticipative ; that is, the output may depend on current and past inputs, but not future inputs. All"realtime" systems must be causal, since they can not have future inputs available to them.

One may think the idea of future inputs does not seem to make much physical sense; however, we have only beendealing with time as our dependent variable so far, which is not always the case. Imagine rather that we wanted to doimage processing. Then the dependent variable might represent pixels to the left and right (the "future") of the currentposition on the image, and we would have a noncausal system.

## Stable vs. unstable

A stable system is one where the output does not diverge as long as the input does not diverge. Thereare many ways to say that a signal "diverges"; for example it could have infinite energy. One particularly usefuldefinition of divergence relates to whether the signal is bounded or not. Then a system is referred to as bounded input-bounded output (BIBO) stable if every possible bounded input produces a bounded output.

Representing this in a mathematical way, a stable system must have the following property, where $x(t)$ is the input and $y(t)$ is the output. The output must satisfy the condition

$\left|y(t)\right|\le {M}_{y}()$
when we have an input to the system that can be described as
$\left|x(t)\right|\le {M}_{x}()$
${M}_{x}$ and ${M}_{y}$ both represent a set of finite positive numbers and these relationships hold for all of $t$ .

If these conditions are not met, i.e. a system's output grows without limit (diverges) from abounded input, then the system is unstable . Note that the BIBO stability of a linear time-invariantsystem (LTI) is neatly described in terms of whether or notits impulse response is absolutely integrable .

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
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
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
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.
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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