<< Chapter < Page Chapter >> Page >
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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The dft, fft, and practical spectral analysis' conversation and receive update notifications?