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In 1990, it was estimated that Cray Research's installed base of approximately 200 machines spent 40% of all CPU cycles computing the fast Fourier transform (FFT)  [link] . With each machine worth about USD$25 million, the performance of the FFT was of prime importance.

Today, use of the FFT is even more pervasive, and it is counted among the 10 algorithms that have had the greatest influence on the development andpractice of science and engineering in the 20 th century  [link] . Huge numbers of mobile smartphones, tablets and PCs  [link] , [link] , most of which are equipped with single-instruction, multiple-data (SIMD)  [link] , [link] microprocessors, compute the FFT on a large scale for a plethora of sound, video and imageprocessing applications. In the space of a few years, mobile applications have become a part of many people's everyday lives  [link] .

This thesis shows that the key to optimizing the performance of the split-radix FFT algorithms on SIMD microprocessors is latency and spatiallocality optimizations, and in some cases, a variant of the split-radix FFT called the conjugate-pairalgorithm  [link] , [link] , [link] , [link] . It is also shown that extensive machine specific calibration may besuperfluous.

Hypotheses

FFTW  [link] , [link] , [link] , SPIRAL  [link] , [link] , [link] and UHFFT  [link] , [link] , [link] , [link] , [link] are state of the art FFT libraries that employ automatic empirical optimization. SPIRAL automatically performs machine-specific optimizations atcompile time, and FFTW and UHFFT automatically adapt to a machine at run-time. Aside from the use of automatic optimization, a common denominator among theselibraries is the use of large straight line blocks of code and optimized memory locality.

The hypotheses outlined below test whether good heuristics and model-based optimization can be used in the place of automatic empirical optimization.

Hypothesis 1: accessing memory in sequential “streams” is critical for best performance

Large FFT exhibit poor temporal locality, and when computing these transforms on microprocessor based systems that feature a cache, bestperformance is typically achieved when “streaming” sequential data through the CPU. Hypothesis 1 is tested in Implementation Details with replicated coefficient lookup tables that trade-off increased memory size forbetter spatial locality, and in Streaming FFT by topologically sorting a directed acyclic graph (DAG) of sub-transforms to again improve spatial locality.

Hypothesis 2: the conjugate-pair algorithm is faster than the ordinary split-radix algorithm

Hypothesis 2 is based on the idea that memory bandwidth is a bottleneck, and on the fact that the conjugate-pair algorithm requires only half the number oftwiddle factor loads. This hypothesis is tested in Split-radix vs. conjugate-pair , where a highly optimized implementation of the conjugate-pair algorithm is benchmarked against anequally highly optimized implementation of the ordinary split-radix algorithm.

Hypothesis 3: the performance of an fft can be predicted based on characteristics of the underlying machine and the compiler

Exploratory experiments suggest that good results can be obtained without empirical techniques, and that certain parameters can be predicted based on thecharacteristics of the underlying machine and the compiler used. Hypothesis 3 is tested in Results and Discussion by building a model that predicts performance, and by benchmarking FFTW against an implementation that does notrequire extensive calibration, on 18 different machines.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket 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
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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