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Shows a full example of convolution including math and figures.

Basic example

Let us look at a basic continuous-time convolution example to help express some of the important ideas. We will convolve together two square pulses, x t and h t , as shown in

Two basic signals that we will convolve together.

Reflect and shift

Now we will take one of the functions and reflect it around the y-axis. Then we must shift the function, such that theorigin, the point of the function that was originally on the origin, is labeled as point t . This step is shown in , h t τ .

Reflected square pulse.
Reflected and shifted square pulse.
h τ and h t τ .
Note that in τ is the 1st axis variable while t is a constant (in this figure).Since convolution is commutative it will never matter which function is reflected and shifted; however, asthe functions become more complicated reflecting and shifting the "right one" will often make the problem much easier.

Regions of integration

We start out with the convolution integral, y t τ x τ h t τ . The value of the function y at time t is given by the amount of overlap(to be precise the integral of theoverlapping region) between h t τ and x τ .

Next, we want to look at the functions and divide the span of the functions into different limits of integration.These different regions can be understood by thinking about how we slide h t τ over x τ , see .

No overlap.
h t τ on its way "into" x τ
h t τ on its way "out of" x τ
No overlap.
Figures to help understand the regions of intergration
In this case we will have the following four regions. Compare these limits of integration to thefour illustrations of h t τ and x τ in .

    Four limits of integration

  • t 0
  • 0 t 1
  • 1 t 2
  • t 2

Using the convolution integral

Finally we are ready for a little math. Using the convolution integral, let us integrate the product of x τ h t τ . For our first and fourth region this will be trivial as it will always be 0 . The second region, 0 t 1 , will require the following math:

y t τ 0 t 1 t
The third region, 1 t 2 , is solved in much the same manner. Take note of the changes in our integration though. As we move h t τ across our other function, the left-hand edge of the function, t 1 , becomes our lowlimit for the integral. This is shown through our convolution integral as
y t τ t 1 1 1 1 t 1 2 t
The above formulas show the method for calculating convolution; however, do not let the simplicity of thisexample confuse you when you work on other problems. The method will be the same, you will just have to deal withmore math in more complicated integrals.

Note that the value of y t at all time is given by the integral of the overlapping functions. In this example y for a given t equals the gray area in the plots in .

Convolution results

Thus, we have the following results for our four regions:

y t 0 t 0 t 0 t 1 2 t 1 t 2 0 t 2
Now that we have found the resulting function for each of the four regions, we can combine them together and graph theconvolution of x t h t .

Shows the system's output in response to the input, x t .

Common sense approach

By looking at we can obtain the system output, y t , by "common" sense.For t 0 there is no overlap, so y t is 0. As t goes from 0 to 1 the overlap will linearly increase with a maximum for t 1 , the maximum corresponds to the peak value in the triangular pulse.As t goes from 1 to 2 the overlap will linearly decrease. For t 2 there will be no overlap and hence no output.

We see readily from the "common" sense approach that the output function y t is the same as obtained above with calculations. When convolving to squarepulses the result will always be a triangular pulse. Its origin, peak value and strech will, of course, vary.

  • Introduction
  • Convolution - Discrete time
  • Convolution - Analog
  • Properties of convolution

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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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