<< Chapter < Page Chapter >> Page >
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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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
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
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Information and signal theory' conversation and receive update notifications?