# 0.8 The cooley-tukey fast fourier transform algorithm  (Page 4/8)

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Recently several papers [link] , [link] , [link] , [link] , [link] have been published on algorithms to calculate a length- ${2}^{M}$ DFT more efficiently than a Cooley-Tukey FFT of any radix. They all havethe same computational complexity and are optimal for lengths up through 16 and until recently was thought to give the best total add-multiply countpossible for any power-of-two length. Yavne published an algorithm with the same computational complexity in 1968 [link] , but it went largely unnoticed. Johnson and Frigo have recently reported the firstimprovement in almost 40 years [link] . The reduction in total operations is only a few percent, but it is a reduction.

The basic idea behind the split-radix FFT (SRFFT) as derived by Duhamel and Hollmann [link] , [link] is the application of a radix-2 index map to the even-indexed terms and a radix-4 map to theodd- indexed terms. The basic definition of the DFT

${C}_{k}=\sum _{n=0}^{N-1}{x}_{n}\phantom{\rule{4pt}{0ex}}{W}^{nk}$

with $W={e}^{-j2\pi /N}$ gives

${C}_{2k}=\sum _{n=0}^{N/2-1}\phantom{\rule{4pt}{0ex}}\left[{x}_{n}+{x}_{n+N/2}\right]\phantom{\rule{4pt}{0ex}}{W}^{2nk}$

for the even index terms, and

${C}_{4k+1}=\sum _{n=0}^{N/4-1}\phantom{\rule{4pt}{0ex}}\left[\left({x}_{n}-{x}_{n+N/2}\right)-j\left({x}_{n+N/4}-{x}_{n+3N/4}\right)\right]\phantom{\rule{4pt}{0ex}}{W}^{n}\phantom{\rule{4pt}{0ex}}{W}^{4nk}$

and

${C}_{4k+3}=\sum _{n=0}^{N/4-1}\phantom{\rule{4pt}{0ex}}\left[\left({x}_{n}-{x}_{n+N/2}\right)+j\left({x}_{n+N/4}-{x}_{n+3N/4}\right)\right]\phantom{\rule{4pt}{0ex}}{W}^{3n}\phantom{\rule{4pt}{0ex}}{W}^{4nk}$

for the odd index terms. This results in an L-shaped “butterfly" shown in [link] which relates a length-N DFT to one length-N/2 DFT and two length-N/4 DFT's with twiddlefactors. Repeating this process for the half and quarter length DFT's until scalars result gives the SRFFT algorithm in much thesame way the decimation-in-frequency radix-2 Cooley-Tukey FFT is derived [link] , [link] , [link] . The resulting flow graph for the algorithm calculated in place looks like a radix-2 FFT except forthe location of the twiddle factors. Indeed, it is the location of the twiddle factors that makes this algorithm use less arithmetic.The L- shaped SRFFT butterfly [link] advances the calculation of the top half by one of the $M$ stages while the lower half, like a radix-4 butterfly, calculates two stages at once. This is illustrated for $N=8$ in [link] .

Unlike the fixed radix, mixed radix or variable radix Cooley-Tukey FFT or even the prime factor algorithm or WinogradFourier transform algorithm , the Split-Radix FFT does not progress completely stage by stage, or, in terms of indices, does notcomplete each nested sum in order. This is perhaps better seen from the polynomial formulation of Martens [link] . Because of this, the indexing is somewhat more complicated than theconventional Cooley-Tukey program.

A FORTRAN program is given below which implements the basic decimation-in-frequency split-radix FFT algorithm. The indexingscheme [link] of this program gives a structure very similar to the Cooley-Tukey programs in [link] and allows the same modifications and improvements such as decimation-in-time, multiplebutterflies, table look-up of sine and cosine values, three real per complex multiply methods, and real data versions [link] , [link] .

SUBROUTINE FFT(X,Y,N,M) N2 = 2*NDO 10 K = 1, M-1 N2 = N2/2N4 = N2/4 E = 6.283185307179586/N2A = 0 DO 20 J = 1, N4A3 = 3*A CC1 = COS(A)SS1 = SIN(A) CC3 = COS(A3)SS3 = SIN(A3) A = J*EIS = J ID = 2*N240 DO 30 I0 = IS, N-1, ID I1 = I0 + N4I2 = I1 + N4 I3 = I2 + N4R1 = X(I0) - X(I2) X(I0) = X(I0) + X(I2)R2 = X(I1) - X(I3) X(I1) = X(I1) + X(I3)S1 = Y(I0) - Y(I2) Y(I0) = Y(I0) + Y(I2)S2 = Y(I1) - Y(I3) Y(I1) = Y(I1) + Y(I3)S3 = R1 - S2 R1 = R1 + S2S2 = R2 - S1 R2 = R2 + S1X(I2) = R1*CC1 - S2*SS1 Y(I2) =-S2*CC1 - R1*SS1X(I3) = S3*CC3 + R2*SS3 Y(I3) = R2*CC3 - S3*SS330 CONTINUE IS = 2*ID - N2 + JID = 4*ID IF (IS.LT.N) GOTO 4020 CONTINUE 10 CONTINUEIS = 1 ID = 450 DO 60 I0 = IS, N, ID I1 = I0 + 1R1 = X(I0) X(I0) = R1 + X(I1)X(I1) = R1 - X(I1) R1 = Y(I0)Y(I0) = R1 + Y(I1) 60 Y(I1) = R1 - Y(I1)IS = 2*ID - 1 ID = 4*IDIF (IS.LT.N) GOTO 50 Split-Radix FFT FORTRAN Subroutine 

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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