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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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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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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