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The key to evaluating a convolution integral such as

x ( t ) * h ( t ) = - x ( τ ) h ( t - τ ) d τ

is to realize that as far as the integral is concerned, the variable t is a constant and the integral is over the variable τ . Therefore, for each t , we are finding the area of the product x ( τ ) h ( t - τ ) . Let's look at an example that illustrates how this works.

Example 3.1 Find the convolution of x ( t ) = u ( t ) and h ( t ) = e - t u ( t ) . The convolution integral is given by

h ( t ) * x ( t ) = - e - τ u ( τ ) u ( t - τ ) d τ

[link] shows the graph of e - τ u ( τ ) , e - t u ( t ) , and their product. From the graph of the product, it is easy to see the the convolution integral becomes

0 t e - τ d τ = 1 - e - t , t 0 0 , t < 0
Graphs of signals used in Example [link] .

Signals which can be expressed in functional form should be convolved as in the above example. Other signals may not have an easy functional representation but rather may be piece-wise linear. In order to convolve such signals, one must evaluate the convolution integral over different intervals on the t -axis so that each distinct interval corresponds to a different expression for x ( t ) * h ( t ) . The following example illustrates this:

Example 3.2 Suppose we attempt to convolve the unit step function x ( t ) = u ( t ) with the trapezoidal function

h ( t ) = t , 0 t < 1 1 , 1 t < 2 0 , elsewhere

From [link] , it can be seen that on the interval 0 t < 1 , the product x ( t - τ ) h ( τ ) is an equilateral triangle with area t 2 / 2 . On the interval 1 t < 2 , the area of x ( t - τ ) h ( τ ) is t - 1 / 2 . This latter area results by adding the area of an equilateral triangle having a base of 1, and the area of a rectangle having a base of t - 1 and a height of 1. For all values of t greater than 2, the convolution is 1.5 since x ( t - τ ) h ( τ ) = h ( τ ) and h ( τ ) is a trapezoid having an area of 1.5. Finally, for t < 0 , the convolution is zero since x ( t - τ ) h ( τ ) = 0 .

Graphs of signals used in Example [link] .

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals, systems, and society. OpenStax CNX. Oct 07, 2012 Download for free at http://cnx.org/content/col10965/1.15
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