# 0.8 The cooley-tukey fast fourier transform algorithm  (Page 5/8)

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As was done for the other decimation-in-frequency algorithms, the input index map is used and the calculations aredone in place resulting in the output being in bit-reversed order. It is the three statements following label 30 that do the specialindexing required by the SRFFT. The last stage is length- 2 and, therefore, inappropriate for the standard L-shaped butterfly, so itis calculated separately in the DO 60 loop. This program is considered a one-butterfly version. A second butterfly can be addedjust before statement 40 to remove the unnecessary multiplications by unity. A third butterfly can be added to reduce the number ofreal multiplications from four to two for the complex multiplication when W has equal real and imaginary parts. It is also possible toreduce the arithmetic for the two- butterfly case and to reduce the data transfers by directly programming a length-4 and length-8butterfly to replace the last three stages. This is called a two-butterfly-plus version. Operation counts for the one, two,two-plus and three butterfly SRFFT programs are given in the next section. Some details can be found in [link] .

The special case of a SRFFT for real data and symmetric data is discussed in [link] . An application of the decimation-in-time SRFFT to real data is given in [link] . Application to convolution is made in [link] , to the discrete Hartley transform in [link] , [link] , to calculating the discrete cosine transform in [link] , and could be made to calculating number theoretic transforms.

An improvement in operation count has been reported by Johnson and Frigo [link] which involves a scaling of multiplying factors. The improvement is small but until this result, it wasgenerally thought the Split-Radix FFT was optimal for total floating point operation count.

## Evaluation of the cooley-tukey fft algorithms

The evaluation of any FFT algorithm starts with a count of the real (or floating point) arithmetic. [link] gives the number of real multiplications and additions required to calculate a length-NFFT of complex data. Results of programs with one, two, three and five butterflies are given to show the improvement that can beexpected from removing unnecessary multiplications and additions. Results of radices two, four, eight and sixteen for the Cooley-TukeyFFT as well as of the split-radix FFT are given to show the relative merits of the various structures. Comparisons of these data shouldbe made with the table of counts for the PFA and WFTA programs in The Prime Factor and Winograd Fourier Transform Algorithms: Evaluation of the PFA and WFTA . All programs use the four-multiply-two-add complex multiply algorithm. A similar table can be developed for thethree-multiply-three-add algorithm, but the relative results are the same.

From the table it is seen that a greater improvement is obtained going from radix-2 to 4 than from 4 to 8 or 16. This ispartly because length 2 and 4 butterflies have no multiplications while length 8, 16 and higher do. It is also seenthat going from one to two butterflies gives more improvement than going from two tohigher values. From an operation count point of view and from practical experience, a three butterfly radix-4 or a two butterflyradix-8 FFT is a good compromise. The radix-8 and 16 programs become long, especially with multiple butterflies, and they give a limitedchoice of transform length unless combined with some length 2 and 4 butterflies.

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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