<< Chapter < Page Chapter >> Page >
Number of Real Multiplications and Additions for Complex Single Radix FFTs
N M1 M2 M3 M5 A1 A2 A3 A5
2 4 0 0 0 6 4 4 4
4 16 4 0 0 24 18 16 16
8 48 20 8 4 72 58 52 52
16 128 68 40 28 192 162 148 148
32 320 196 136 108 480 418 388 388
64 768 516 392 332 1152 1026 964 964
128 1792 1284 1032 908 2688 2434 2308 2308
256 4096 3076 2568 2316 6144 5634 5380 5380
512 9216 7172 6152 5644 13824 12802 12292 12292
1024 20480 16388 14344 13324 30720 28674 27652 27652
2048 45056 36868 32776 30732 67584 63490 61444 61444
4096 98304 81924 73736 69644 147456 139266 135172 135172
4 12 0 0 0 22 16 16 16
16 96 36 28 24 176 146 144 144
64 576 324 284 264 1056 930 920 920
256 3072 2052 1884 1800 5632 5122 5080 5080
1024 15360 11268 10588 10248 28160 26114 25944 25944
4096 73728 57348 54620 53256 135168 126978 126296 126296
8 32 4 4 4 66 52 52 52
64 512 260 252 248 1056 930 928 928
512 6144 4100 4028 3992 12672 11650 11632 11632
4096 65536 49156 48572 48280 135168 126978 126832 126832
16 80 20 20 20 178 148 148 148
256 2560 1540 1532 1528 5696 5186 5184 5184
4096 61440 45060 44924 44856 136704 128514 128480 128480
2 0 0 0 0 4 4 4 4
4 8 0 0 0 20 16 16 16
8 24 8 4 4 60 52 52 52
16 72 32 28 24 164 144 144 144
32 184 104 92 84 412 372 372 372
64 456 288 268 248 996 912 912 912
128 1080 744 700 660 2332 2164 2164 2164
256 2504 1824 1740 1656 5348 5008 5008 5008
512 5688 4328 4156 3988 12060 11380 11380 11380
1024 12744 10016 9676 9336 26852 25488 25488 25488
2048 28216 22760 22076 21396 59164 56436 56436 56436
4096 61896 50976 49612 48248 129252 123792 123792 123792

In [link] Mi and Ai refer to the number of real multiplications and real additions used by an FFT with i separately written butterflies.The first block has the counts for Radix-2, the second for Radix-4, the third for Radix-8, thefourth for Radix-16, and the last for the Split-Radix FFT. For the split-radix FFT, M3 and A3 refer to the two- butterfly-plus programand M5 and A5 refer to the three-butterfly program.

The first evaluations of FFT algorithms were in terms of the number of real multiplications required as that was the slowestoperation on the computer and, therefore, controlled the execution speed. Later with hardware arithmetic both the number ofmultiplications and additions became important. Modern systems have arithmetic speeds such that indexing and data transfer times becomeimportant factors. Morris [link] has looked at some of these problems and has developed a procedure called autogen to writepartially straight-line program code to significantly reduce overhead and speed up FFT run times. Some hardware, such as theTMS320 signal processing chip, has the multiply and add operations combined. Some machines have vector instructions or have parallelprocessors. Because the execution speed of an FFT depends not only on the algorithm, but also on the hardware architecture andcompiler, experiments must be run on the system to be used.

In many cases the unscrambler or bit-reverse-counter requires 10% of the execution time, therefore, if possible, it should beeliminated. In high-speed convolution where the convolution is done by multiplication of DFT's, a decimation-in-frequency FFT can becombined with a decimation-in-time inverse FFT to require no unscrambler. It is also possible for a radix-2 FFT to do theunscrambling inside the FFT but the structure is not very regular [link] , [link] . Special structures can be found in [link] and programs for data that are real or have special symmetries are in [link] , [link] , [link] .

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
Maciej
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.
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
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
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
Good
Berger describes sociologists as concerned with
Mueller Reply
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, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms' conversation and receive update notifications?

Ask