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Fast Fourier transform (FFT) algorithms efficiently compute the discrete Fourier transform (DFT).There are different types of FFT algorithms for different DFT lengths; lengths equal to a power of two are the simplest and by far the most commonly used.The prime-factor algorithm yields fast algorithms for some other lengths, and along with the chirp z-transform and Rader's conversion allow fast algorithms for DFTs of any length.

A fast Fourier transform , or FFT , is not a new transform, but is a computationally efficient algorithm for the computingthe DFT . The length- N DFT, defined as

X k n N 1 0 x n 2 n k N
where X k and x n are in general complex-valued and 0 k , n N 1 , requires N complex multiplies to compute each X k . Direct computation of all N frequency samples thus requires N 2 complex multiplies and N N 1 complex additions. (This assumes precomputation of the DFT coefficients W N n k 2 n k N ; otherwise, the cost is even higher.) For the large DFT lengths used in many applications, N 2 operations may be prohibitive. (For example, digital terrestrial television broadcastin Europe uses N = 2048 or 8192 OFDM channels, and the SETI project uses up to length-4194304 DFTs.)DFTs are thus almost always computed in practice by an FFT algorithm . FFTs are very widely used in signal processing, for applicationssuch as spectrum analysis and digital filtering via fast convolution .

History of the fft

It is now known that C.F. Gauss invented an FFT in 1805 or so to assist the computation of planetary orbits via discrete Fourier series . Various FFT algorithms were independently invented over the next twocenturies, but FFTs achieved widespread awareness and impact only with the Cooley and Tukey algorithm published in 1965, which cameat a time of increasing use of digital computers and when the vast range of applications of numerical Fourier techniques was becoming apparent.Cooley and Tukey's algorithm spawned a surge of research in FFTs and was also partly responsible for the emergence of Digital Signal Processing (DSP) as adistinct, recognized discipline. Since then, many different algorithms have been rediscovered or developed,and efficient FFTs now exist for all DFT lengths.

Summary of fft algorithms

The main strategy behind most FFT algorithms is to factor a length- N DFT into a number of shorter-length DFTs, the outputs of which are reused multipletimes (usually in additional short-length DFTs!) to compute the final results.The lengths of the short DFTs correspond to integer factors of the DFT length, N , leading to different algorithms for different lengths and factors.By far the most commonly used FFTs select N 2 M to be a power of two, leading to the very efficient power-of-two FFT algorithms , including the decimation-in-time radix-2 FFT and the decimation-in-frequency radix-2 FFT algorithms, the radix-4 FFT ( N 4 M ), and the split-radix FFT . Power-of-two algorithms gain their high efficiencyfrom extensive reuse of intermediate results and from the low complexity of length-2 and length-4DFTs, which require no multiplications. Algorithms for lengths with repeated common factors (such as 2 or 4 in the radix-2 and radix-4 algorithms, respectively) require extra twiddle factor multiplications between the short-length DFTs, which together leadto a computational complexity of O N N , a very considerable savings over direct computation of the DFT.

The other major class of algorithms is the Prime-Factor Algorithms (PFA) . In PFAs, the short-length DFTs must be of relatively prime lengths.These algorithms gain efficiency by reuse of intermediate computations and by eliminating twiddle-factor multiplies,but require more operations than the power-of-two algorithms to compute the short DFTs of various prime lengths. In the end, the computational costs of the prime-factorand the power-of-two algorithms are comparable for similar lengths, as illustrated in Choosing the Best FFT Algorithm . Prime-length DFTs cannot be factored into shorter DFTs,but in different ways both Rader's conversion and the chirp z-transform convert prime-length DFTs into convolutions of other lengths that can be computed efficiently using FFTsvia fast convolution .

Some applications require only a few DFT frequency samples, in which case Goertzel's algorithm halves the number of computations relative to the DFT sum. Other applications involve successive DFTs of overlappedblocks of samples, for which the running FFT can be more efficient than separate FFTs of each block.

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
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The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
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kinnecy Reply
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I'm not sure why it wrote it the other way
I got X =-6
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Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
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Differences Between Laspeyres and Paasche Indices
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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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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I'm interested in Nanotube
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can nanotechnology change the direction of the face of the world
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
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