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


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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.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?