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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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
Smarajit 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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