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Recall that the impulse response for the capacitor voltage in a series RC circuit is given by

h ( t ) = 1 R C e - t / R C u ( t ) ,

and consider the response to the input voltage

x ( t ) = u ( t ) .

We know that the output for this input voltage is given by the convolution of the impulse response with the input signal

y ( t ) = x ( t ) * h ( t ) .

We would like to compute this operation by beginning in a way that minimizes the algebraic complexity of the expression. Thus, since x ( t ) = u ( t ) is the simpler of the two signals, it is desirable to select it for time reversal and shifting. Thus, we would like to compute

y ( t ) = - 1 R C e - τ / R C u ( τ ) u ( t - τ ) d τ .

The step functions can be used to further simplify this integral by narrowing the region of integration to the nonzero region of the integrand. Therefore,

y ( t ) = 0 max { 0 , t } 1 R C e - τ / R C d τ .

Hence, the output is

y ( t ) = 0 t 0 1 - e - t / R C t > 0

which can also be written as

y ( t ) = 1 - e - t / R C u ( t ) .
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Circular convolution

Continuous time circular convolution is an operation on two finite length or periodic continuous time signals defined by the integral

( f * g ) ( t ) = 0 T f ^ ( τ ) g ^ ( t - τ ) d τ

for all signals f , g defined on R [ 0 , T ] where f ^ , g ^ are periodic extensions of f and g . It is important to note that the operation of circular convolution is commutative, meaning that

f * g = g * f

for all signals f , g defined on R [ 0 , T ] . Thus, the circular convolution operation could have been just as easily stated using the equivalent definition

( f * g ) ( t ) = 0 T f ^ ( t - τ ) g ^ ( τ ) d τ

for all signals f , g defined on R [ 0 , T ] where f ^ , g ^ are periodic extensions of f and g . Circular convolution has several other important properties not listed here but explained and derived in a later module.

Alternatively, continuous time circular convolution can be expressed as the sum of two integrals given by

( f * g ) ( t ) = 0 t f ( τ ) g ( t - τ ) d τ + t T f ( τ ) g ( t - τ + T ) d τ

for all signals f , g defined on R [ 0 , T ] .

Meaningful examples of computing continuous time circular convolutions in the time domain would involve complicated algebraic manipulations dealing with the wrap around behavior, which would ultimately be more confusing than helpful. Thus, none will be provided in this section. However, continuous time circular convolutions are more easily computed using frequency domain tools as will be shown in the continuous time Fourier series section.

Definition motivation

The above operation definition has been chosen to be particularly useful in the study of linear time invariant systems. In order to see this, consider a linear time invariant system H with unit impulse response h . Given a finite or periodic system input signal x we would like to compute the system output signal H ( x ) . First, we note that the input can be expressed as the circular convolution

x ( t ) = 0 T x ^ ( τ ) δ ^ ( t - τ ) d τ

by the sifting property of the unit impulse function. Writing this integral as the limit of a summation,

x ( t ) = lim Δ 0 n x ^ ( n Δ ) δ ^ Δ ( t - n Δ ) Δ

where

δ Δ ( t ) = 1 / Δ 0 t < Δ 0 otherwise

approximates the properties of δ ( t ) . By linearity

H x ( t ) = lim Δ 0 n x ^ ( n Δ ) H δ ^ Δ ( t - n Δ ) Δ

which evaluated as an integral gives

H x ( t ) = 0 T x ^ ( τ ) H δ ^ ( t - τ ) d τ .

Since H δ ( t - τ ) is the shifted unit impulse response h ( t - τ ) , this gives the result

H x ( t ) = 0 T x ^ ( τ ) h ^ ( t - τ ) d τ = ( x * h ) ( t ) .

Hence, circular convolution has been defined such that the output of a linear time invariant system is given by the convolution of the system input with the system unit impulse response.

Graphical intuition

It is often helpful to be able to visualize the computation of a circular convolution in terms of graphical processes. Consider the circular convolution of two finite length functions f , g given by

( f * g ) ( t ) = 0 T f ^ ( τ ) g ^ ( t - τ ) d τ = 0 T f ^ ( t - τ ) g ^ ( τ ) d τ .

The first step in graphically understanding the operation of convolution is to plot each of the periodic extensions of the functions. Next, one of the functions must be selected, and its plot reflected across the τ = 0 axis. For each t R [ 0 , T ] , that same function must be shifted left by t . The product of the two resulting plots is then constructed. Finally, the area under the resulting curve on R [ 0 , T ] is computed.

Convolution demonstration

ConvolutionDemo
Interact (when online) with a Mathematica CDF demonstrating Convolution. To Download, right-click and save target as .cdf.

Convolution summary

Convolution, one of the most important concepts in electrical engineering, can be used to determine the output signal of a linear time invariant system for a given input signal with knowledge of the system's unit impulse response. The operation of continuous time convolution is defined such that it performs this function for infinite length continuous time signals and systems. The operation of continuous time circular convolution is defined such that it performs this function for finite length and periodic continuous time signals. In each case, the output of the system is the convolution or circular convolution of the input signal with the unit impulse response.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
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
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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
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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
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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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 ?
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
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Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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