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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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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