# 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

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

## 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 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).

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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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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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
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Alexandre
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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
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LITNING
scanning tunneling microscope
Sahil
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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
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
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