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 

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
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
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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