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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 ( t ) and w 2 ( t ) have Fourier transforms W 1 ( f ) and W 2 ( f ) . Then,

F { w 1 ( t ) * w 2 ( t ) } = W 1 ( f ) W 2 ( f ) .

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

F { w 1 ( t ) * w 2 ( t ) } = t = - w 1 ( t ) * w 2 ( t ) e - j 2 π f t d t = t = - λ = - w 1 ( λ ) w 2 ( t - λ ) d λ e - j 2 π f t d t .

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

F { w 1 ( t ) * w 2 ( t ) } = λ = - w 1 ( λ ) t = - w 2 ( t - λ ) e - j 2 π f t d t d λ = λ = - w 1 ( λ ) W 2 ( f ) e - j 2 π f λ d λ = W 2 ( f ) λ = - w 1 ( λ ) e - j 2 π f λ d λ = W 1 ( f ) W 2 ( f ) .

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,

F { w 1 ( t ) w 2 ( t ) } = W 1 ( f ) W 2 ( f ) = - W 1 ( λ ) W 2 ( f - λ ) d λ .

The usefulness of these convolution properties is apparent when applying them to linear systems.Suppose that H ( f ) is the Fourier transform of the impulse response h ( t ) . Suppose that X ( f ) is the Fourier transform of the input x ( t ) 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,

Y ( f ) = F { y ( t ) } = F { x ( t ) * h ( t ) } = F { h ( t ) } F { x ( t ) } = H ( f ) X ( f ) .

This can be rearranged to solve for

H ( f ) = Y ( f ) X ( f ) ,

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 ( f 1 ) = 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 ( f 2 ) = 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 ( f ) is called the transfer function of the system, since it “transfers” the input x ( t ) (with transform X ( f ) ) into the output y ( t ) (with transform Y ( f ) ).

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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