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The fourier transform

Many practical problems in signal analysis involve either infinitely long or very long signals where the Fourier series is not appropriate. For thesecases, the Fourier transform (FT) and its inverse (IFT) have been developed. This transform has been used with great success in virtually all quantitativeareas of science and technology where the concept of frequency is important. While the Fourier series was used before Fourier worked on it, the Fouriertransform seems to be his original idea. It can be derived as an extension of the Fourier series by letting the length increase to infinity or the Fouriertransform can be independently defined and then the Fourier series shown to be a special case of it. The latter approach is the more general of the two, but the former is more intuitive .

Definition of the fourier transform

The Fourier transform (FT) of a real-valued (or complex) function of the real-variable t is defined by

X ( ω ) = x ( t ) e j ω t t
giving a complex valued function of the real variable ω representing frequency. The inverse Fourier transform (IFT) is given by
x ( t ) = 1 2 π X ( ω ) e j ω t ω .
Because of the infinite limits on both integrals, the question of convergence is important. There are useful practical signals that do not have Fouriertransforms if only classical functions are allowed because of problems with convergence. The use of delta functions (distributions) in both the time andfrequency domains allows a much larger class of signals to be represented .

Examples of the fourier transform

Deriving a few basic transforms and using the properties allows a large class of signals to be easily studied. Examples of modulation, sampling, and otherswill be given.

  • If x ( t ) = δ ( t ) then X ( ω ) = 1
  • If x ( t ) = 1 then X ( ω ) = 2 π δ ( ω )
  • If x ( t ) is an infinite sequence of delta functions spaced T apart, x ( t ) = n = δ ( t n T ) , its transform is also an infinite sequence of delta functions of weight 2 π / T spaced 2 π / T apart, X ( ω ) = 2 π k = δ ( ω 2 π k / T ) .
  • Other interesting and illustrative examples can be found in .

Note the Fourier transform takes a function of continuous time into a function of continuous frequency, neither function being periodic. If "distribution"or "delta functions" are allowed, the Fourier transform of a periodic function will be a infinitely long string of delta functions with weights thatare the Fourier series coefficients.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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