<< Chapter < Page Chapter >> Page >

Finally, we should mention that there are many FFTs entirely distinct from Cooley-Tukey. Three notable such algorithms are the prime-factor algorithm for gcd ( n 1 , n 2 ) = 1 [link] , along with Rader's [link] and Bluestein's [link] , [link] , [link] algorithms for prime n . FFTW implements the first two in its codelet generator for hard-coded n "Generating Small FFT Kernels" and the latter two for general prime n (sections "Plans for prime sizes" and "Goals and Background of the FFTW Project" ). There is also the Winograd FFT [link] , [link] , [link] , [link] , which minimizes the number of multiplications at the expense of a largenumber of additions; this trade-off is not beneficial on current processors that have specialized hardware multipliers.

Goals and background of the fftw project

The FFTW project, begun in 1997 as a side project of the authors Frigo and Johnson as graduate students at MIT, has gone through several major revisions, and as of 2008 consists of more than 40,000 lines of code. It is difficult to measure the popularity of a free-software package, but (as of 2008) FFTW has been cited in over 500 academic papers, is used in hundreds of shipping free and proprietary software packages, and the authors have received over 10,000 emails from users of the software. Most of this chapter focuses on performance of FFT implementations, but FFTW would probably not be where it is today if that were the only consideration in its design. One of the key factors in FFTW's success seems to have been its flexibility in addition to its performance. In fact, FFTW is probably the most flexible DFT library available:

  • FFTW is written in portable C and runs well on many architectures and operating systems.
  • FFTW computes DFTs in O ( n log n ) time for any length n . (Most other DFT implementations are either restricted to a subset of sizes or they become Θ ( n 2 ) for certain values of n , for example when n is prime.)
  • FFTW imposes no restrictions on the rank (dimensionality) of multi-dimensional transforms. (Most other implementations arelimited to one-dimensional, or at most two- and three-dimensional data.)
  • FFTW supports multiple and/or strided DFTs; for example, to transform a 3-component vector field or a portion of amulti-dimensional array. (Most implementations support only a single DFT of contiguous data.)
  • FFTW supports DFTs of real data, as well as of real symmetric/anti-symmetric data (also called discrete cosine/sinetransforms).

Our design philosophy has been to first define the most general reasonable functionality, and then to obtain the highest possibleperformance without sacrificing this generality. In this section, we offer a few thoughts about why such flexibility has proved important,and how it came about that FFTW was designed in this way.

FFTW's generality is partly a consequence of the fact the FFTW project was started in response to the needs of a real application forone of the authors (a spectral solver for Maxwell's equations [link] ), which from the beginning had to run on heterogeneous hardware. Our initial application requiredmulti-dimensional DFTs of three-component vector fields (magnetic fields in electromagnetism), and so right away this meant: (i)multi-dimensional FFTs; (ii) user-accessible loops of FFTs of discontiguous data; (iii) efficient support for non-power-of-two sizes(the factor of eight difference between n × n × n and 2 n × 2 n × 2 n was too much to tolerate); and (iv) saving a factor of two for the common real-input case was desirable. That is,the initial requirements already encompassed most of the features above, and nothing about this application is particularly unusual.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
How can I make nanorobot?
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
how can I make nanorobot?
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.
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, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms' conversation and receive update notifications?