# 0.3 Modelling corruption  (Page 5/11)

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## The delta “function”

One way to see how a system behaves is to kick it and see how it responds.Some systems react sluggishly, barely moving away from their resting state, while othersrespond quickly and vigorously. Defining exactly what is meant mathematicallyby a “kick” is trickier than it seems because the kick must occur over a very short amount of time,yet must be energetic in order to have any effect. This section defines the impulse (or delta) function $\delta \left(t\right)$ , which is a useful “kick” for the study of linear systems.

The criterion that the impulse be energetic is translated to the mathematical statement that itsintegral over all time must be nonzero, and it is typically scaled to unity, that is,

${\int }_{-\infty }^{\infty }\delta \left(t\right)dt=1.$

The criterion that it occur over a very short time span is translated to the statement that, for every positive $ϵ$ ,

$\delta \left(t\right)=\left\{\begin{array}{cc}0\hfill & t<-ϵ\hfill \\ 0\hfill & t>ϵ\hfill \end{array}\right).$

Thus, the impulse $\delta \left(t\right)$ is explicitly defined to be equal to zero for all $t\ne 0$ . On the other hand, $\delta \left(t\right)$ is implicitly defined when $t=0$ by the requirement that its integral be unity. Together, these guarantee that $\delta \left(t\right)$ is no ordinary function. The impulse is called a distribution and is the subject of considerable mathematical investigation.

The most important consequence of the definitions [link] and [link] is the sifting property

${\int }_{-\infty }^{\infty }w\left(t\right)\delta \left(t-{t}_{0}\right)dt=w\left(t\right){|}_{t={t}_{0}}=w\left({t}_{0}\right),$

which says that the delta function picks out the value of the function $w\left(t\right)$ from under the integral at exactly the time whenthe argument of the $\delta$ function is zero, that is, when $t={t}_{0}$ . To see this, observe that $\delta \left(t-{t}_{0}\right)$ is zero whenever $t\ne {t}_{0}$ , and hence $w\left(t\right)\delta \left(t-{t}_{0}\right)$ is zero whenever $t\ne {t}_{0}$ . Thus,

$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }w\left(t\right)\delta \left(t-{t}_{0}\right)dt& =& {\int }_{-\infty }^{\infty }w\left({t}_{0}\right)\delta \left(t-{t}_{0}\right)dt\hfill \\ & =& w\left({t}_{0}\right){\int }_{-\infty }^{\infty }\delta \left(t-{t}_{0}\right)dt\hfill \\ & =& w\left({t}_{0}\right)·1=w\left({t}_{0}\right).\hfill \end{array}$

Sometimes it is helpful to think of the impulse as a limit. For instance, define the rectangular pulseof width $1/n$ and height $n$ by

${\delta }_{n}\left(t\right)=\left\{\begin{array}{cc}0,\hfill & t<-1/2n\hfill \\ n,\hfill & -1/2n\le t\le 1/2n\hfill \\ 0,\hfill & t>1/2n\hfill \end{array}\right).$

Then $\delta \left(t\right)={lim}_{n\to \infty }{\delta }_{n}\left(t\right)$ fulfills both criteria [link] and [link] . Informally, it is not unreasonable to think of $\delta \left(t\right)$ as being zero everywhere except at $t=0$ , where it is infinite.While it is not really possible to “plot” the delta function $\delta \left(t-{t}_{0}\right)$ , it can be represented in graphical form as zero everywhereexcept for an up-pointing arrow at ${t}_{0}$ . When the $\delta$ function is scaled by a constant, the value of the constant is often placedin parenthesis near the arrowhead. Sometimes, when the constant is negative, the arrow is drawn pointing down.For instance, [link] shows a graphical representation of the function $w\left(t\right)=\delta \left(t+10\right)-2\delta \left(t+1\right)+3\delta \left(t-5\right)$ .

What is the spectrum (Fourier transform) of $\delta \left(t\right)$ ? This can be calculated directly from the definitionby replacing $w\left(t\right)$ in [link] with $\delta \left(t\right)$ :

$\mathcal{F}\left\{\delta \left(t\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(t\right){e}^{-j2\pi ft}dt.$

Apply the sifting property [link] with $w\left(t\right)={e}^{-j2\pi ft}$ and ${t}_{0}=0$ . Thus $\mathcal{F}\left\{\delta \left(t\right)\right\}={e}^{-j2\pi ft}{|}_{t=0}=1$ .

Alternatively, suppose that $\delta$ is a function of frequency, that is, a spike at zero frequency.The corresponding time domain function can be calculated analogously using the definitionof the inverse Fourier transform, that is, by substituting $\delta \left(f\right)$ for $W\left(f\right)$ in [link] and integrating:

${\mathcal{F}}^{-1}\left\{\delta \left(f\right)\right\}={\int }_{-\infty }^{\infty }\delta \left(f\right){e}^{j2\pi ft}df={e}^{j2\pi ft}{|}_{f=0}=1.$

Thus a spike at frequency zero is a “DC signal” (a constant) in time.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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