# 9.4 Discrete time convolution and the dtft

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This module describes the relationship between discrete convolution and the DTFT.

## Introduction

This module discusses convolution of discrete signals in the time and frequency domains.

## Discrete time fourier transform

The DTFT transforms an infinite-length discrete signal in the time domain into an finite-length (or $2\pi$ -periodic) continuous signal in the frequency domain.

## Dtft

$X(\omega )=\sum_{n=()}$ x n j ω n

## Inverse dtft

$x(n)=\frac{1}{2\pi }\int_{0}^{2\pi } X(\omega )e^{j\omega n}\,d \omega$

## Demonstration Interact (when online) with a Mathematica CDF demonstrating the Discrete Convolution. To Download, right-click and save as .cdf.

## 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 impulse response. The convolution sum is expressed as

$y(n)=\sum_{k=()}$ x k h n k
As with continuous-time, convolution is represented by thesymbol *, and can be written as
$y(n)=(x(n), h(n))$
Convolution is commutative. For more information on the characteristics of convolution,read about the Properties of Convolution .

## Convolution theorem

Let $f$ and $g$ be two functions with convolution $f*g$ .. Let $F$ be the Fourier transform operator. Then

$F\left(f*g\right)=F\left(f\right)·F\left(g\right)$
$F\left(f·g\right)=F\left(f\right)*F\left(g\right)$

By applying the inverse Fourier transform ${F}^{-1}$ , we can write:

$f*g={F}^{-1}\left(F\left(f\right)·F\left(g\right)\right)$

## Conclusion

The Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) corresponds to point-wise multiplication in the other domain (e.g., frequency domain).

#### Questions & Answers

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
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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