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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
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
how do they get the third part x = (32)5/4
kinnecy Reply
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
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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?
bamidele 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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