# 0.3 Modelling corruption  (Page 9/11)

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Suppose that a system has an impulse response that is a sinc function, as shown in [link] , and that the input to the system is a white noise (as in specnoise.m ).

1. Mimic convolex.m to numerically find the output.
2. Plot the spectrum of the input and the spectrum of the output (using plotspec.m ). What kind of filter would you call this?

## Convolution $⇔$ Multiplication

While the convolution operator [link] describes mathematically how a linear system acts on a given input, time domain approaches are often notparticularly revealing about the general behavior of the system.Who would guess, for instance in Exercises  [link] and [link] , that convolution with exponentials and sinc functions would act like lowpass filters?By working in the frequency domain, however, the convolution operator is transformed into a simpler point-by-pointmultiplication, and the generic behavior of the system becomes clearer.

The first step is to understand the relationship between convolution intime, and multiplication in frequency. Suppose that the two time signals ${w}_{1}\left(t\right)$ and ${w}_{2}\left(t\right)$ have Fourier transforms ${W}_{1}\left(f\right)$ and ${W}_{2}\left(f\right)$ . Then,

$\mathcal{F}\left\{{w}_{1}\left(t\right)*{w}_{2}\left(t\right)\right\}={W}_{1}\left(f\right){W}_{2}\left(f\right).$

To justify this property, begin with the definition of the Fourier transform [link] and apply the definition of convolution [link] to obtain

$\begin{array}{ccc}\hfill \mathcal{F}\left\{{w}_{1}\left(t\right)*{w}_{2}\left(t\right)\right\}& =& {\int }_{t=-\infty }^{\infty }{w}_{1}\left(t\right)*{w}_{2}\left(t\right){e}^{-j2\pi ft}dt\hfill \\ & =& {\int }_{t=-\infty }^{\infty }\left[{\int }_{\lambda =-\infty }^{\infty },{w}_{1},\left(\lambda \right),{w}_{2},\left(t-\lambda \right),d,\lambda \right]{e}^{-j2\pi ft}dt.\hfill \end{array}$

Reversing the order of integration and using the time shift property [link] produces

$\begin{array}{ccc}\hfill \mathcal{F}\left\{{w}_{1}\left(t\right)*{w}_{2}\left(t\right)\right\}& =& {\int }_{\lambda =-\infty }^{\infty }{w}_{1}\left(\lambda \right)\left[{\int }_{t=-\infty }^{\infty },{w}_{2},\left(t-\lambda \right),{e}^{-j2\pi ft},d,t\right]d\lambda \hfill \\ & =& {\int }_{\lambda =-\infty }^{\infty }{w}_{1}\left(\lambda \right)\left[{W}_{2},\left(f\right),{e}^{-j2\pi f\lambda }\right]d\lambda \hfill \\ & =& {W}_{2}\left(f\right){\int }_{\lambda =-\infty }^{\infty }{w}_{1}\left(\lambda \right){e}^{-j2\pi f\lambda }d\lambda \hfill \\ & =& {W}_{1}\left(f\right){W}_{2}\left(f\right).\hfill \end{array}$

Thus, convolution in the time domain is the same as multiplication in the frequency domain. See [link] .

The companion to the convolution property is the multiplication property, which says that multiplication in the time domainis equivalent to convolution in the frequency domain (see [link] ); that is,

$\begin{array}{ccc}\hfill \mathcal{F}\left\{{w}_{1}\left(t\right){w}_{2}\left(t\right)\right\}& =& {W}_{1}\left(f\right)☆{W}_{2}\left(f\right)\hfill \\ & =& {\int }_{-\infty }^{\infty }{W}_{1}\left(\lambda \right){W}_{2}\left(f-\lambda \right)d\lambda .\hfill \end{array}$

The usefulness of these convolution properties is apparent when applying them to linear systems.Suppose that $H\left(f\right)$ is the Fourier transform of the impulse response $h\left(t\right)$ . Suppose that $X\left(f\right)$ is the Fourier transform of the input $x\left(t\right)$ that is applied to the system. Then [link] and [link] show that the Fourier transform of the output is exactly equal to the product of the transforms of theinput and the impulse response, that is,

$\begin{array}{ccc}\hfill Y\left(f\right)=\mathcal{F}\left\{y\left(t\right)\right\}& =& \mathcal{F}\left\{x\left(t\right)*h\left(t\right)\right\}\hfill \\ & =& \mathcal{F}\left\{h\left(t\right)\right\}\mathcal{F}\left\{x\left(t\right)\right\}=H\left(f\right)X\left(f\right).\hfill \end{array}$

This can be rearranged to solve for

$H\left(f\right)=\frac{Y\left(f\right)}{X\left(f\right)},$

which is called the frequency response of the system because it shows, for each frequency $f$ , how the system responds. For instance, suppose that $H\left({f}_{1}\right)=3$ at some frequency ${f}_{1}$ . Then whenever a sinusoid of frequency ${f}_{1}$ is input into the system, it will be amplified by a factor of 3.Alternatively, suppose that $H\left({f}_{2}\right)=0$ at some frequency ${f}_{2}$ . Then whenever a sinusoid of frequency ${f}_{2}$ is input into the system, it is removed from the output(because it has been multiplied by a factor of 0).

The frequency response shows how the system treats inputs containing various frequencies. In fact, this propertywas already used repeatedly in [link] when drawing curves that describe the behavior of lowpass and bandpassfilters. For example, the filters of Figures  [link] , [link] , and  [link] are used to remove unwanted frequencies from the communications system. In each of these cases, the plotof the frequency response describes concretely and concisely how the system (or filter) affects the input, and how thefrequency content of the output relates to that of the input. Sometimes, the frequency response $H\left(f\right)$ is called the transfer function of the system, since it “transfers” the input $x\left(t\right)$ (with transform $X\left(f\right)$ ) into the output $y\left(t\right)$ (with transform $Y\left(f\right)$ ).

#### Questions & Answers

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
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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
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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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