<< Chapter < Page Chapter >> Page >
Time discrete convolution

Introduction

The idea of discrete-time convolution is exactly the same as that of continuous-time convolution . For this reason, it may be useful to look at both versions to help yourunderstanding of this extremely important concept. Convolution is a very powerful tool in determining asystem's output from knowledge of an arbitrary input and the system's impulse response.

It also helpful to see convolution graphically, i.e. by using transparencies or Java Applets. Johns Hopkins University has an excellent Discrete time convolution applet. Using this resource will help understanding this crucial concept.

Derivation of the convolution sum

We know that any discrete-time signal can be represented by a summation of scaled and shifted discrete-time impulses, see . Since we are assuming the system to be linear and time-invariant, itwould seem to reason that an input signal comprised of the sum of scaled and shifted impulses would give rise to an outputcomprised of a sum of scaled and shifted impulse responses. This is exactly what occurs in convolution . Below we present a more rigorous and mathematical look at thederivation:

Letting be a discrete time LTI system, we start with the folowing equation and work our waydown the the convoluation sum.

y n x n k x k n k k x k n k k x k n k k x k h n k
Let us take a quick look at the steps taken in the above derivation. After our initial equation we rewrite the function x n as a sum of the function times the unit impulse. Next, we can move around theoperator and the summation because is a linear, DT system. Because of this linearity and the fact that x k is a constant, we pull the constant out and simply multiply it by . Finally, we use the fact that is time invariant in order to reach our final state - the convolution sum!

Above the summation is taken over all integers. Howerer, in many practical cases either x n or h n or both are finite, for which case the summations will be limited.The convolution equations are simple tools which, in principle, can be used for all input signals. Following is an example to demonstrate convolution;how it is calculated and how it is interpreted.

Graphical illustration of convolution properties

A quick graphical example may help in demonstrating why convolution works.

A single impulse input yields the system's impulse response.
A scaled impulse input yields a scaled response, due to the scaling property of the system's linearity.
We now use the time-invariance property of the system to show that a delayed input results in an output of the sameshape, only delayed by the same amount as the input.
We now use the additivity portion of the linearity property of the system to complete the picture. Since anydiscrete-time signal is just a sum of scaled and shifteddiscrete-time impulses, we can find the output from knowing the input and the impulse response.

Convolution sum

As mentioned above, the convolution sum provides a concise, mathematical way to express the output of an LTI system basedon an arbitrary discrete-time input signal and the system's response. The convolution sum is expressed as

y n k x k h n k
As with continuous-time, convolution is represented by the symbol *, and can be written as
y n x n h n
By making a simple change of variables into the convolution sum, k n k , we can easily show that convolution is commutative :
y n x n h n h n x n
From we get a convolution sum that is equivivalent to the sum in :
y n k h k x n k
For more information on the characteristics of convolution, read about the Properties of Convolution .

Convolution through time (a graphical approach)

In this section we will develop a second graphical interpretation of discrete-time convolution. We will beginthis by writing the convolution sum allowing x to be a causal, length-m signal and h to be a causal, length-k, LTI system. This gives us the finite summation,

y n l 0 m 1 x l h n l
Notice that for any given n we have a sum of the m products of x l and a time-delayed h n l . This is to say that we multiply the terms of x by the terms of a time-reversed h and add them up.

Going back to the previous example:

This is the end result that we are looking to find.
Here we reverse the impulse response, h , and begin its traverse at time 0 .
We continue the traverse. See that at time 1 , we are multiplying two elements of the input signal bytwo elements of the impulse respone.
If we follow this through to one more step, n 4 , then we can see that we produce the same output as we saw in the intial example.

What we are doing in the above demonstration is reversing the impulse response in time and "walking it across" the inputsignal. Clearly, this yields the same result as scaling, shifting and summing impulse responses.

This approach of time-reversing, and sliding across is a common approach to presenting convolution, since itdemonstrates how convolution builds up an output through time.

Questions & Answers

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
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit 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
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Media processing in processing. OpenStax CNX. Nov 10, 2010 Download for free at http://cnx.org/content/col10268/1.14
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Media processing in processing' conversation and receive update notifications?

Ask