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If we are clever enough, we can use these facts to develop a computational algorithm that can compute the Fourier transform of a time series much fasterthan can be obtained using a brute force DFT algorithm. Fortunately, some very clever people have already developed that algorithm. It goes by the name of theFast Fourier Transform, or FFT algorithm.

Steps in the FFT algorithm

In truth, there are several different forms of the FFT algorithm, and the mechanics of each may be slightly different. At least one, and probably many ofthe algorithms operate by performing the following steps:

  • Decompose an N-point complex series into N individual complex series, each consisting of a single complex sample. The order of the decompositionin an FFT algorithm is rather complicated. It is this order of decomposition, and the order of the subsequent recombination of transformresults that causes the FFT algorithm to be so fast. It is also that order that makes the algorithm somewhat difficult to understand. Note that theprogram that I will discuss later does not implement that special order of decomposition and recombination.
  • Calculate the transform of each of the N complex series, each consisting of a single complex sample. This treats each complex sample as if it islocated at the origin of a complex series. This step is trivial. The real part of the transform of a single complex sample located at the origin ofthe series is a complex constant whose values are proportional to the real and imaginary values that make up the complex sample. Since the complexinput series consists of only one complex sample, there is only one complex value in the complex transform.
  • Correct each of the N transform results to reflect the original position of the complex sample in the input series. This involves the application ofsine and cosine curves to the real and imaginary parts of the transform. This step is usually combined with the recombination step that follows.
  • Recombine the N transform results into a single transform result that represents the transform of the original complex series. This is a verycomplicated operation in a real FFT algorithm. It must reverse the order of decomposition in the first step described earlier. As mentioned earlier, itis the order of the decomposition and subsequent recombination that minimizes the arithmetic operations required and gives the FFT itstremendous speed. The program that I will discuss later does not implement the special order of decomposition and recombination used in an actual FFTalgorithm.

A sample program

I want to emphasize at the outset that this program DOES NOT implement an FFT algorithm. Rather, this program illustrates the underlying signal processingconcepts that make the FFT possible in a form that is more easily understood than is normally the case with an actual FFT algorithm.

Separate processes in an FFT algorithm

In summary, a typical FFT algorithm performs the following processes:

  • Decompose an N-point complex series into N individual complex series, each consisting of a single complex sample.
  • Recognize that the complex transform of a single complex sample is equal to the value of the complex sample.
  • Correct the transform for each complex sample to reflect the original position of the complex sample in the input series.
  • Recombine the N transform results into a single transform result that represents the transform of the original complex series.

Questions & Answers

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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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