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This chapter describes SFFT: a high-performance FFT library for SIMD microprocessors that is, in many cases, faster than the state of the art FFT libraries reviewed in Existing libraries .

Implementation details described some simple implementations of the FFT and concluded with an analysis of the performance bottlenecks. The implementations presented in this chapter are designed to improve spatial locality, and utilize larger straight line blocks of code at the leaves, corresponding to sub-transforms of sizes 8 through to 64, in order to reduce latency and stack overheads.

In distinct contrast to the simple FFT programs of Chapter 3 , this chapter employs meta-programming. Rather than describe FFT programs, we describe programs that statically elaborate the FFT into a DAG of nodes representing the computation, apply some optimizing transformations to the graph, and then generate code. Many other auto-vectorization techniques, such as those employed by SPIRAL, operate at the instruction level  [link] , but the techniques presented in this chapter vectorize blocks of computation at the algorithm level of abstraction, thus enabling some of the algorithms structure to be utilized.

Three types of implementation are described in this chapter, and the performance of each depends on the parameters of the transform to be computed and the characteristics of the underlying machine.For a given machine and FFT to be computed (which has parameters such as length and precision), the fastest configuration is selected from among a small set of up to eight possible FFT configurations – a much smaller space compared to FFTW's exhaustive search of all possible FFTs. The fastest configuration is easily selected by timing each of the possible options, but it is shown in Results and discussion that it is also possible to use machine learning to build a classifier that will predict the fastest based on attributes such as the size of the cache.

SFFT comprises three types of conjugate-pair implementation, which are:

  1. Fully hard-coded FFTs;
  2. Four-step FFTs with hard-coded sub-transforms;
  3. FFTs with hard-coded leaves.

Fully hard-coded

Statically elaborating a DAG that represents a depth-first recursive FFT is much like computing a depth-first recursive FFT: instead of performing computation at the leaves of the recursion and where smaller DFTs are combined into one, a node representing the computation is appended to the end of a list, and the list of nodes, i.e., a topological ordering of the DAG, is later translated into a program that can be compiled and executed.

Emitting code with a vector length of 1 (i.e., scalar code or vector code where only one complex element fits in a vector register) is relatively simple and is described in "Vector length 1" . For vector lengths above 1, vectorizing the topological ordering of nodes poses some subtle challenges, and these details are described in "Other vector lengths" . The fully hard-coded FFTs described in this section are generally only practical for smaller sizes of transforms, typically where N 128 , however these techniques are expanded in later sections to scale the performance to larger sizes.

Questions & Answers

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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Computing the fast fourier transform on simd microprocessors. OpenStax CNX. Jul 15, 2012 Download for free at http://cnx.org/content/col11438/1.2
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