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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

Do somebody tell me a best nano engineering book for beginners?
s. Reply
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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.
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of graphene you mean?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
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what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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silver nanoparticles could handle the job?
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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