# 2.2 Convolution - analog

 Page 1 / 1
Introduces convolution for analog signals.

In this module we examine convolution for continuous time signals. This willresult in the convolution integral and its properties . These concepts are very important in ElectricalEngineering and will make any engineer's life a lot easier if the time is spent now to truly understand what is going on.

In order to fully understand convolution, you may find it useful to look at the discrete-time convolution as well. Accompanied to this module there is a fully worked out example with mathematics and figures. It will also be helpful to experiment with the Convolution - Continuous time applet available from Johns Hopkins University . These resources offers different approaches to this crucial concept.

## Derivation of the convolution integral

The derivation used here closely follows the one discussed in the motivation section above. To begin this, it is necessary to state theassumptions we will be making. In this instance, the only constraints on our system are that it be linear andtime-invariant.

## Brief overview of derivation steps:

• An impulse input leads to an impulse response output.
• A shifted impulse input leads to a shifted impulse response output. This is due to the time-invariance of the system.
• We now scale the impulse input to get a scaled impulse output. This is using the scalar multiplication property oflinearity.
• We can now "sum up" an infinite number of these scaled impulses to get a sum of an infinite number of scaledimpulse responses. This is using the additivity attribute of linearity.
• Now we recognize that this infinite sum is nothing more than an integral, so we convert both sides into integrals.
• Recognizing that the input is the function $f(t)$ , we also recognize that the output is exactly the convolution integral.

## Convolution integral

As mentioned above, the convolution integral provides an easy mathematical way to express the output of an LTI system basedon an arbitrary signal, $x(t)$ , and the system's impulse response, $h(t)$ . The convolution integral is expressed as

$y(t)=\int_{()} \,d$ x h t
Convolution is such an important tool that it is represented by the symbol *, and can be written as
$y(t)=(x(t), h(t))$
By making a simple change of variables into the convolution integral, $=t-$ , we can easily show that convolution is commutative :
$(x(t), h(t))=(h(t), x(t))$
which gives an equivivalent form of
$y(t)=\int_{()} \,d$ x t h

## Implementation of convolution

Taking a closer look at the convolution integral, we find that we are multiplying the input signal by the time-reversedimpulse response and integrating. This will give us the value of the output at one given value of $t$ . If we then shift the time-reversed impulse response by a small amount, we getthe output for another value of $t$ . Repeating this for every possible value of $t$ , yields the total output function. While we would never actually do thiscomputation by hand in this fashion, it does provide us with some insight into what is actually happening. We find that weare essentially reversing the impulse response function and sliding it across the input function, integrating as we go.This method, referred to as the graphical method , provides us with a much simpler way to solve for the outputfor simple (contrived) signals, while improving our intuition for the more complex cases where we rely on computers. Infact Texas Instruments develops Digital Signal Processors which have special instruction sets for computations such as convolution.

## Summary

Convolution is a truly important concept, which must be well understood.

$y(t)=\int_{()} \,d$ x h t
$y(t)=\int_{()} \,d$ h x t

Go to?

• Introduction
• Convolution - Full example
• Convolution - Discrete time
• Properties of convolution

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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!