# 3.9 Evaluation of convolution integrals

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The key to evaluating a convolution integral such as

$x\left(t\right)*h\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right)d\tau$

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

Example 3.1 Find the convolution of $x\left(t\right)=u\left(t\right)$ and $h\left(t\right)={e}^{-t}u\left(t\right)$ . The convolution integral is given by

$h\left(t\right)*x\left(t\right)={\int }_{-\infty }^{\infty }{e}^{-\tau }u\left(\tau \right)u\left(t-\tau \right)d\tau$

[link] shows the graph of ${e}^{-\tau }u\left(\tau \right)$ , ${e}^{-t}u\left(t\right)$ , and their product. From the graph of the product, it is easy to see the the convolution integral becomes

${\int }_{0}^{t}{e}^{-\tau }d\tau =\left\{\begin{array}{cc}1-{e}^{-t},& t\ge 0\\ 0,& t<0\end{array}\right)$

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 $x\left(t\right)*h\left(t\right)$ . The following example illustrates this:

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

$h\left(t\right)=\left\{\begin{array}{cc}t,& 0\le t<1\\ 1,& 1\le t<2\\ 0,& \text{elsewhere}\end{array}\right)$

From [link] , it can be seen that on the interval $0\le t<1$ , the product $x\left(t-\tau \right)h\left(\tau \right)$ is an equilateral triangle with area ${t}^{2}/2$ . On the interval $1\le t<2$ , the area of $x\left(t-\tau \right)h\left(\tau \right)$ 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 $x\left(t-\tau \right)h\left(\tau \right)=h\left(\tau \right)$ and $h\left(\tau \right)$ is a trapezoid having an area of 1.5. Finally, for $t<0$ , the convolution is zero since $x\left(t-\tau \right)h\left(\tau \right)=0$ .

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yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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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
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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