# 8.4 Continuous time convolution and the ctft

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This module describes the relationship between continuous convolution and the fourier transform.

## Introduction

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

## Continuous time fourier transform

The CTFT transforms a infinite-length continuous signal in the time domain into an infinite-length continuous signal in the frequency domain.

## Ctft

$ℱ(\Omega )=\int_{()} \,d t$ f t Ω t

## Inverse ctft

$f(t)=\frac{1}{2\pi }\int_{()} \,d \Omega$ Ω Ω t

## Convolution integral

The convolution integral expresses the output of an LTI system based on an input signal, $x(t)$ , and the system's impulse response, $h(t)$ . The convolution integral is expressed as

$y(t)=\int_{()} \,d \tau$ x τ h t τ
Convolution is such an important tool that it is representedby the symbol $()$ , and can be written as
$y(t)=(x(t), h(t))$

## Convolution theorem

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

$F\left(f*g\right)=F\left(f\right)·F\left(g\right)$
$F\left(f·g\right)=\frac{1}{2\pi }F\left(f\right)*F\left(g\right)$

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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Joseph
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no can't
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William
currently
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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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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narayan
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ya I also want to know the raman spectra
Bhagvanji
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Damian
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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
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Damian
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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Mahi
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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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
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