# 0.5 Convolution - discrete time

 Page 1 / 1
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.

$y(n)=(x(n))=(\sum )$ 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.

## 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

$y(n)=\sum_{k=()}$ 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 :
$y(n)=\sum_{k=()}$ h k x n k

## 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,

$y(n)=\sum_{l=0}^{m-1} x(l)h(n-l)$
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:

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.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
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
learn
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
nanocopper obvius
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
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!