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This chapter describes the benchmarking methods used to evaluate the performance and accuracy of various FFT implementations throughout this thesis.

The two architectures of interest are the Intel x86 architecture and the ARM architecture. A comprehensive set of results collected from a wide range of machines implementing these architectures is presented in Results and discussion , but throughout the rest of the thesis, benchmarks are performed on an Apple Macbook Air 4,2; a widely available and currently state-of-the-art machine that is equipped with an Intel Core i5-2557M. [link] summarizes the specifications of the machine.

For the x86 benchmarks, an existing framework called BenchFFT   [link] was used. For the ARM benchmarks, which were performed on iOS devices, there was no existing FFT benchmark software, and so an application was written for this purpose, which is described in "ARM architecture" .

Specifications of the primary test machine
Macbook Air 4,2
CPU Dual-core Intel Core i5 (i5-2557M)
CPU clock 1.7 GHz (turbo to 2.7GHz with one core)
L1 cache 32KB I-cache&32KB D-cache
L2 cache 256KB
L3 cache 3MB shared
Memory 4 GB of 1333 MHz DDR3 SDRAM
OS OS X 10.7.2
SIMD extensions SSE and AVX

X86 architecture

The x86 benchmarks were performed with BenchFFT, a collection of FFT libraries and benchmarking software assembled by Frigo and Johnson, the authors of FFTW  [link] . The benchmarks in BenchFFT use timing and calibration code from lmbench , a performance analysis tool written by Larry McVoy and Carl Staelin  [link] .


BenchFFT measures the initialization time and runtime of an FFT separately. The initialization time is measured only once, and thus outliersdue to effects from external factors such as OS scheduling are occasionally observed. Routines from lmbench are then used to calibrate the minimum number of FFT iterations required for accurate measurementusing the gettimeofday function. Finally, the time taken to run the minimum number of iterations is measured eight times, from which the minimumtime divided by the number of iterations is used, in order to factor out effects from external factors.

The minimum time for a transform is then used to determine a scaled inverse time measurement, sometimes known as CTGs. CTG are defined as:

C T G s = 5 N log 2 ( N ) 10 9 t

for complex transforms and

C T G s = 2 . 5 N log 2 ( N ) 10 9 t

for real transforms, where t is the time taken to run one transform (in seconds). Unless the Cooley-Tukey radix-2 algorithm is used, a measurement expressed in CTGs is not an actual FLOP count – it is a roughmeasure of an algorithm's efficiency relative to the radix-2 algorithm and the clock speed of the machine.

When a transform has several variants (such as direction or radix), BenchFFT reports the speed of the FFT as being the fastest of the possible options.


To measure the accuracy of a transform, BenchFFT compares an FFT with an arbitrary-precision FFT computed on the same inputs, and reports therelative RMS error. The inputs are pseudo-random in the range [ 0 . 5 , 0 . 5 ) and the arbitrary-precision FFT has over 40 decimal places of accuracy.

When a transform has several variants (such as direction or radix), BenchFFT reports the accuracy as being worst of the results.


Except where otherwise noted, ICC version 12.1.0 for OS X was used to compile 64-bit code. For OS X builds, the compiler flags used were “-O3”,while “-O3 -msse2” (or equivalent) was used for Linux builds. In the cases where the FFT uses AVX, the code is compiled with “-xAVX” or“-mavx” (depending on compiler).

Some libraries included in the BenchFFT software have their own compilation scripts which override the defaults, and in the case of commercial libraries(such as Intel IPP and Apple vDSP), the compiler flags are of little consequence because the libraries are distributed in binary form.

Data format

FFT libraries use interleaved format and/or complex format to store the data. In the case of interleaved format, the real and imaginary parts ofcomplex numbers are stored adjacently in memory, while in the case of split format, the real and imaginary parts are stored in separate arrays.

The majority of FFT libraries use interleaved format to store data. In the case where the library supports interleaved or split format, BenchFFT uses interleaved format. However there are a few libraries that only supportsplit format, and in theses cases it should be noted the results are not strictly comparable (Apple vDSP is one such case).

Arm architecture

There was no existing FFT benchmarking software for iOS on ARM devices, and so a benchmarking tool was written. The tool runs the benchmarking in athread of normal priority.


The code was compiled with Apple clang compiler 3.0 for ARMv7 targets running iOS 5.0. The compiler flags used were “-O3 -mfpu=neon”.


The Apple A4 and A5 SoCs are built around the ARM Cortex-A8 and Cortex-A9 cores, which have hardware cycle counters that can be used for precise timing.The cycle counter control registers can only be accessed in kernel mode, and so the high resolution timer available through the mach_absolute_time function was used instead.

For a given size of transform, a calibration routine determines the number of iterations that must be run such that the total runtime is approximately onesecond. After calibration, each FFT to be evaluated is run for the pre-determined number of iterations – this loop is run eight times, and thefastest time divided by the number of iterations is taken to be the FFTs runtime. By running each FFT for approximately one second, and repeatingthe measurement eight times to find the best time, the effects from external factors such as OS scheduling are minimized. As with BenchFFT, the time isexpressed in CTGs.

Questions & Answers

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
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
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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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