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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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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
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Brian Reply
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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
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LITNING Reply
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is differents between GO and RGO?
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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
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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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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