<< Chapter < Page Chapter >> Page >

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

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
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 ?
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?
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
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
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
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Computing the fast fourier transform on simd microprocessors' conversation and receive update notifications?