# Convolution and linear time-invariant systems

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## Convolution and its numerical approximation

The output $y\left(t\right)$ of a continuous-time linear time-invariant (LTI) system is related to its input $x\left(t\right)$ and the system impulse response $h\left(t\right)$ through the convolution integral expressed as (for details on the theory of convolution and LTI systems, refer to signals and systems textbooks, for example, references [link] - [link] ):

$y\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}h\left(t-\tau \right)x\left(\tau \right)\mathrm{d\tau }$

For a computer program to perform the above continuous-time convolution integral, a numerical approximation of the integral is needed noting that computer programs operate in a discrete – not continuous – fashion. One way to approximate the continuous functions in the Equation (1) integral is to use piecewise constant functions. Define ${\delta }_{\Delta }\left(t\right)$ to be a rectangular pulse of width $\Delta$ and height 1, centered at $t=0$ :

${\delta }_{\Delta }\left(t\right)=\left\{\begin{array}{cc}1& -\Delta /2\le t\le \Delta /2\\ 0& \text{otherwise}\end{array}$

Approximate a continuous function $x\left(t\right)$ with a piecewise constant function ${x}_{\Delta }\left(t\right)$ as a sequence of pulses spaced every $\Delta$ seconds in time with heights $x\left(\mathrm{k\Delta }\right)$ :

${x}_{\Delta }\left(t\right)=\sum _{k=-\infty }^{\infty }x\left(\mathrm{k\Delta }\right){\delta }_{\Delta }\left(t-\mathrm{k\Delta }\right)$

It can be shown in the limit as $\Delta \to 0,{x}_{\Delta }\left(t\right)\to x\left(t\right)$ . As an example, [link] shows the approximation of a decaying exponential $x\left(t\right)=\text{exp}\left(-\frac{t}{2}\right)$ starting from 0 using $\Delta =1$ . Similarly, $h\left(t\right)$ can be approximated by

${h}_{\Delta }\left(t\right)=\sum _{k=-\infty }^{\infty }h\left(\mathrm{k\Delta }\right){\delta }_{\Delta }\left(t-\mathrm{k\Delta }\right)$

One can thus approximate the convolution integral by convolving the two piecewise constant signals as follows:

${y}_{\Delta }\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}{h}_{\Delta }\left(t-\tau \right){x}_{\Delta }\left(\tau \right)\mathrm{d\tau }$

Notice that ${y}_{\Delta }\left(t\right)$ is not necessarily a piecewise constant. For computer representation purposes, discrete output values are needed, which can be obtained by further approximating the convolution integral as indicated below:

${y}_{\Delta }\left(\mathrm{n\Delta }\right)=\Delta \sum _{k=-\infty }^{\infty }x\left(\mathrm{k\Delta }\right)h\left(\left(n-k\right)\Delta \right)$

If one represents the signals ${h}_{\Delta }\left(t\right)$ and ${x}_{\Delta }\left(t\right)$ in a .m file by vectors containing the values of the signals at $t=\mathrm{n\Delta }$ , then Equation (5) can be used to compute an approximation to the convolution of $x\left(t\right)$ and $h\left(t\right)$ . Compute the discrete convolution sum $\sum _{k=-\infty }^{\infty }x\left(\mathrm{k\Delta }\right)h\left(\left(n-k\right)\Delta \right)$ with the built-in LabVIEW MathScript command conv . Then, multiply this sum by $\Delta$ to get an estimate of $y\left(t\right)$ at $t=\mathrm{n\Delta }$ Note that as $\Delta$ is made smaller, one gets a closer approximation to $y\left(t\right)$ .

## Convolution properties

Convolution satisfies the following three properties (see [link] ):

• Commutative property
$x\left(t\right)\ast h\left(t\right)=h\left(t\right)\ast x\left(t\right)$
• Associative property
$x\left(t\right)\ast {h}_{1}\left(t\right)\ast {h}_{2}\left(t\right)=x\left(t\right)\ast \left\{{h}_{1}\left(t\right)\ast {h}_{2}\left(t\right)\right\}$
• Distributive property
$x\left(t\right)\ast \left\{{h}_{1}\left(t\right)+{h}_{2}\left(t\right)\right\}=x\left(t\right)\ast {h}_{1}\left(t\right)+x\left(t\right)\ast {h}_{2}\left(t\right)$

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
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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
many many of nanotubes
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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