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This integral defines the convolution operator * and provides a way of finding the output y ( t ) of any linear system, given its impulse response h ( t ) and the input x ( t ) .

M atlab has several functions that simplify the numerical evaluation of convolutions. The most obvious of these is conv , which is used in convolex.m to calculate the convolution of an input x (consisting of two delta functions at times t = 1 and t = 3 ) and a system with impulse response h that is an exponential pulse. The convolution gives the output of the system.

Ts=1/100; time=10;             % sampling interval and total time t=0:Ts:time;                   % create time vectorh=exp(-t);                     % define impulse response x=zeros(size(t));              % input is sum of two delta functions...x(1/Ts)=3; x(3/Ts)=2;          % ...at times t=1 and t=3 y=conv(h,x);                   % do convolution and plotsubplot(3,1,1), plot(t,x) subplot(3,1,2), plot(t,h)subplot(3,1,3), plot(t,y(1:length(t)))
convolex.m example of numerical convolution (download file)
The convolution of the input (the top plot) with the impulse response of the system (the middle plot) gives the output in the bottom plot.
The convolution of the input (the top plot) with the impulse response of the system (the middle plot) gives the output in the bottom plot.

[link] shows the input to the system in the top plot,the impulse response in the middle plot, and the output of the system in the bottom plot. Nothing happens before time t = 1 , and the output is zero. When the first spike occurs,the system responds by jumping to 3 and then decaying slowly at a rate dictated by the shape of h ( t ) . The decay continues smoothly until time t = 3 , when the second spike enters. At this point, the output jumps up by 2, andis the sum of the response to the second spike, plus the remainder of the response to the first spike. Since there are no more inputs,the output slowly dies away.

Suppose that the impulse response h ( t ) of a linear system is the exponential pulse

h ( t ) = e - t t 0 0 t < 0 .

Suppose that the input to the system is 3 δ ( t - 1 ) + 2 δ ( t - 3 ) . Use the definition of convolution [link] to show that the output is 3 h ( t - 1 ) + 2 h ( t - 3 ) , where

h ( t - t 0 ) = e - t + t 0 t t 0 0 t < t 0 .

How does your answer compare to [link] ?

Suppose that a system has an impulse response that is an exponential pulse. Mimic the code in convolex.m to find its output when the input is a white noise (recall specnoise.m ).

Mimic the code in convolex.m to find the output of a system when the input is an exponential pulse and theimpulse response is a sum of two delta functions at times t = 1 and t = 3 .

The next two problems show that linear filters commute with differentiation, and with each other.

Use the definition to show that convolution is commutative (i.e., that w 1 ( t ) * w 2 ( t ) = w 2 ( t ) * w 1 ( t ) ). Hint: Apply the change of variables τ = t - λ in [link] .

Suppose a filter has impulse response h ( t ) . When the input is x ( t ) , the output is y ( t ) . If the input is x d ( t ) = x ( t ) t , the output is y d ( t ) . Show that y d ( t ) is the derivative of y ( t ) . Hint: Use [link] and the result of Exercise  [link] .

Let w ( t ) = Π ( t ) be the rectangular pulse of [link] . What is w ( t ) * w ( t ) ? Hint: A pulse shaped like a triangle.

Redo Exercise  [link] numerically by suitably modifying convolex.m . Let T = 1 . 5 seconds.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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.
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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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