# 3.1 Properties of the fourier transform

 Page 1 / 1
Several of the most important properties of the Fourier transform are derived.

## Properties of the fourier transform

The Fourier Transform (FT) has several important properties which will be useful:

1. Linearity:
$\alpha {x}_{1}\left(t\right)+\beta {x}_{2}\left(t\right)↔\alpha {X}_{1}\left(j\Omega \right)+\beta {X}_{2}\left(j\Omega \right)$
where $\alpha$ and $\beta$ are constants. This property is easy to verify by plugging the left side of [link] into the definition of the FT.
2. Time shift:
$x\left(t-\tau \right)↔{e}^{-j\Omega \tau }X\left(j\Omega \right)$
To derive this property we simply take the FT of $x\left(t-\tau \right)$
${\int }_{-\infty }^{\infty }x\left(t-\tau \right){e}^{-j\Omega t}dt$
using the variable substitution $\gamma =t-\tau$ leads to
$t=\gamma +\tau$
and
$d\gamma =dt$
We also note that if $t=±\infty$ then $\tau =±\infty$ . Substituting [link] , [link] , and the limits of integration into [link] gives
$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{-j\Omega \left(\gamma +\tau \right)}d\gamma & =& {e}^{-j\Omega \tau }{\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{-j\Omega \gamma }d\gamma \hfill \\ & =& {e}^{-j\Omega \tau }X\left(j\Omega \right)\hfill \end{array}$
which is the desired result.
3. Frequency shift:
$x\left(t\right){e}^{j{\Omega }_{0}t}↔X\left(j\left(\Omega -{\Omega }_{0}\right)\right)$
Deriving the frequency shift property is a bit easier than the time shift property. Again, using the definition of FT we get:
$\begin{array}{ccc}\hfill {\int }_{-\infty }^{\infty }x\left(t\right){e}^{j{\Omega }_{0}t}{e}^{-j\Omega t}dt& =& {\int }_{-\infty }^{\infty }x\left(t\right){e}^{-j\left(\Omega -{\Omega }_{0}\right)t}dt\hfill \\ & =& X\left(j\left(\Omega -{\Omega }_{0}\right)\right)\hfill \end{array}$
4. Time reversal :
$x\left(-t\right)↔X\left(-j\Omega \right)$
To derive this property, we again begin with the definition of FT:
${\int }_{-\infty }^{\infty }x\left(-t\right){e}^{-j\Omega t}dt$
and make the substitution $\gamma =-t$ . We observe that $dt=-d\gamma$ and that if the limits of integration for $t$ are $±\infty$ , then the limits of integration for $\gamma$ are $\mp \gamma$ . Making these substitutions into [link] gives
$\begin{array}{ccc}\hfill -{\int }_{\infty }^{-\infty }x\left(\gamma \right){e}^{j\Omega \gamma }d\gamma & =& {\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{j\Omega \gamma }d\gamma \hfill \\ & =& X\left(-j\Omega \right)\hfill \end{array}$
Note that if $x\left(t\right)$ is real, then $X\left(-j\Omega \right)=X{\left(j\Omega \right)}^{*}$ .
5. Time scaling: Suppose we have $y\left(t\right)=x\left(at\right),a>0$ . We have
$Y\left(j\Omega \right)={\int }_{-\infty }^{\infty }x\left(at\right){e}^{-j\Omega t}dt$
Using the substitution $\gamma =at$ leads to
$\begin{array}{cc}\hfill Y\left(j\Omega \right)& =\frac{1}{a}{\int }_{-\infty }^{\infty }x\left(\gamma \right){e}^{-j\Omega \gamma /a}d\gamma \hfill \\ & =\frac{1}{a}X\left(\frac{\Omega }{a}\right)\hfill \end{array}$
6. Convolution: The convolution integral is given by
$y\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right)d\tau$
The convolution property is given by
$Y\left(j\Omega \right)↔X\left(j\Omega \right)H\left(j\Omega \right)$
To derive this important property, we again use the FT definition:
$\begin{array}{ccc}\hfill Y\left(j\Omega \right)& =& {\int }_{-\infty }^{\infty }y\left(t\right){e}^{-j\Omega t}dt\hfill \\ & =& {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right){e}^{-j\Omega t}d\tau dt\hfill \\ & =& {\int }_{-\infty }^{\infty }x\left(\tau \right)\left[{\int }_{-\infty }^{\infty },h,\left(t-\tau \right),{e}^{-j\Omega t},d,t\right]d\tau \hfill \end{array}$
Using the time shift property, the quantity in the brackets is ${e}^{-j\Omega \tau }H\left(j\Omega \right)$ , giving
$\begin{array}{ccc}\hfill Y\left(j\Omega \right)& =& {\int }_{-\infty }^{\infty }x\left(\tau \right){e}^{-j\Omega \tau }H\left(j\Omega \right)d\tau \hfill \\ & =& H\left(j\Omega \right){\int }_{-\infty }^{\infty }x\left(\tau \right){e}^{-j\Omega \tau }d\tau \hfill \\ & =& H\left(j\Omega \right)X\left(j\Omega \right)\hfill \end{array}$
Therefore, convolution in the time domain corresponds to multiplication in the frequency domain.
7. Multiplication (Modulation):
$w\left(t\right)=x\left(t\right)y\left(t\right)↔\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\left(\Omega -\Theta \right)\right)Y\left(j\Theta \right)d\Theta$
Notice that multiplication in the time domain corresponds to convolution in the frequency domain. This property can be understood by applying the inverse Fourier Transform [link] to the right side of [link]
$\begin{array}{ccc}\hfill w\left(t\right)& =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\left(\Omega -\Theta \right)\right)Y\left(j\Theta \right){e}^{j\Omega t}d\Theta d\Omega \hfill \\ & =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right)\left[\frac{1}{2\pi },{\int }_{-\infty }^{\infty },X,\left(j\left(\Omega -\Theta \right)\right),{e}^{j\Omega t},d,\Omega \right]d\Theta \hfill \end{array}$
The quantity inside the brackets is the inverse Fourier Transform of a frequency shifted Fourier Transform,
$\begin{array}{ccc}\hfill w\left(t\right)& =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right)\left[x,\left(t\right),{e}^{j\Theta t}\right]d\Theta \hfill \\ & =& x\left(t\right)\frac{1}{2\pi }{\int }_{-\infty }^{\infty }Y\left(j\Theta \right){e}^{j\Theta t}d\Theta \hfill \\ & =& x\left(t\right)y\left(t\right)\hfill \end{array}$
8. Duality: The duality property allows us to find the Fourier transform of time-domain signals whose functional forms correspond to known Fourier transforms, $X\left(jt\right)$ . To derive the property, we start with the inverse Fourier transform:
$x\left(t\right)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\Omega \right){e}^{j\Omega t}d\Omega$
Changing the sign of $t$ and rearranging,
$2\pi x\left(-t\right)={\int }_{-\infty }^{\infty }X\left(j\Omega \right){e}^{-j\Omega t}d\Omega$
Now if we swap the $t$ and the $\Omega$ in [link] , we arrive at the desired result
$2\pi x\left(-\Omega \right)={\int }_{-\infty }^{\infty }X\left(jt\right){e}^{-j\Omega t}dt$
The right-hand side of [link] is recognized as the FT of $X\left(jt\right)$ , so we have
$X\left(jt\right)↔2\pi x\left(-\Omega \right)$

The properties associated with the Fourier Transform are summarized in [link] .

 Property $y\left(t\right)$ $Y\left(j\Omega \right)$ Linearity $\alpha {x}_{1}\left(t\right)+\beta {x}_{2}\left(t\right)$ $\alpha {X}_{1}\left(j\Omega \right)+\beta {X}_{2}\left(j\Omega \right)$ Time Shift $x\left(t-\tau \right)$ $X\left(j\Omega \right){e}^{-j\Omega \tau }$ Frequency Shift $x\left(t\right){e}^{j{\Omega }_{0}t}$ $X\left(j\left(\Omega -{\Omega }_{0}\right)\right)$ Time Reversal $x\left(-t\right)$ $X\left(-j\Omega \right)$ Time Scaling $x\left(at\right)$ $\frac{1}{a}X\left(\frac{\Omega }{a}\right)$ Convolution $x\left(t\right)*h\left(t\right)$ $X\left(j\Omega \right)H\left(j\Omega \right)$ Modulation $x\left(t\right)w\left(t\right)$ $\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X\left(j\left(\Omega -\Theta \right)\right)W\left(j\Theta \right)d\Theta$ Duality $X\left(jt\right)$ $2\pi x\left(-\Omega \right)$

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
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
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
Got questions? Join the online conversation and get instant answers!