# 0.6 Winograd's short dft algorithms  (Page 4/11)

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A modification of this approach also works for a length which is an odd prime raised to some power: $N={P}^{M}$ . This is a bit more complicated [link] , [link] but has been done for lengths of 9 and 25. For longer lengths, the conventional Cooley-Tukey type-two index map algorithm seems to be more efficient. For powers of two, there is no primitive root, and therefore, no simple conversionof the DFT into convolution. It is possible to use two generators [link] , [link] , [link] to make the conversion and there exists a set of length 4, 8, and 16 DFT algorithms of the form in [link] in [link] .

In [link] an operation count of several short DFT algorithms is presented. These are practical algorithms that can beused alone or in conjunction with the index mapping to give longer DFT's as shown in The Prime Factor and Winograd Fourier Transform Algorithms . Most are optimized in having either the theoretical minimum number of multiplications or theminimum number of multiplications without requiring a very large number of additions. Some allow other reasonable trade-offs betweennumbers of multiplications and additions. There are two lists of the number of multiplications. The first is the number of actualfloating point multiplications that must be done for that length DFT. Some of these (one or two in most cases) will be by rationalconstants and the others will be by irrational constants. The second list is the total number of multiplications given in the diagonalmatrix $D$ in [link] . At least one of these will be unity ( theone associated with $X\left(0\right)$ ) and in some cases several will be unity ( for $N={2}^{M}$ ). The second list is important in programming the WFTA in The Prime Factor and Winograd Fourier Transform Algorithm: The Winograd Fourier Transform Algorithm .

 Length N Mult Non-one Mult Total Adds 2 0 4 4 3 4 6 12 4 0 8 16 5 10 12 34 7 16 18 72 8 4 16 52 9 20 22 84 11 40 42 168 13 40 42 188 16 20 36 148 17 70 72 314 19 76 78 372 25 132 134 420 32 68 - 388

Because of the structure of the short DFTs, the number of real multiplications required for the DFT of real data is exactly half that required for complex data. The numberof real additions required is slightly less than half that required for complex data because $\left(N-1\right)$ of the additions needed when $N$ is prime add a real to an imaginary, and that is not actuallyperformed. When $N=2m$ , there are $\left(N-2\right)$ of these pseudo additions. The special case for real data is discussed in [link] , [link] , [link] .

The structure of these algorithms are in the form of ${X}^{\text{'}}=CDAX$ or ${B}^{T}DAX$ or ${A}^{T}DAX$ from [link] and [link] . The $A$ and $B$ matrices are generally $M$ by $N$ with $M\ge N$ and have elements that are integers, generally 0 or $±1$ . A pictorial description is given in [link] .

The flow graph in [link] should be compared with the matrix description of [link] and [link] , and with the programs in [link] , [link] , [link] , [link] and the appendices. The shape in [link] illustrates the expansion of the data by $A$ . That is to say, $AX$ has more entries than $X$ because $M>N$ . The A operator consists of additions, the $D$ operator gives the $M$ multiplications (some by one) and ${A}^{T}$ contracts the data back to $N$ values with additions only. $M$ is one half the second list of multiplies in [link] .

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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
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LITNING
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Sahil
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Santosh
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Rafiq
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Mahi
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Rafiq
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
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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
hi
Loga
what does nano mean?
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Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
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