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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.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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