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The number of additions depends on the order of the pre- and postweave operators. For example in the length-15 WFTA in [link] , if the length-5 had been done first and last, there would have beensix row addition preweaves in the preweave operator rather than the five shown. It is difficult to illustrate the algorithm for three ormore factors of N, but the ideas apply to any number of factors. Each length has an optimal ordering of the pre- and postweaveoperators that will minimize the number of additions.

A program for the WFTA is not as simple as for the FFT or PFA because of the very characteristic that reduces the number ofmultiplications, the nesting. A simple two-factor example program is given in [link] and a general program can be found in [link] , [link] . The same lengths are possible with the PFA and WFTA and the same short DFT modules can be used, however, themultiplies in the modules must occur in one place for use in the WFTA.

Modifications of the pfa and wfta type algorithms

In the previous section it was seen how using the permutation property of the elementary operators in the PFA allowedthe nesting of the multiplications to reduce their number. It was also seen that a proper ordering of the operators could minimize thenumber of additions. These ideas have been extended in formulating a more general algorithm optimizing problem. If the DFT operator F in [link] is expressed in a still more factored form obtained from Winograd’s Short DFT Algorithms: Equation 30 , a greater variety of ordering can be optimized. For example if the A operators have two factors

F 1 = A 1 T A 1 ' T D 1 A 1 ' A 1

The DFT in [link] becomes

X = A 2 T A ' 2 T D 2 A ' 2 A 2 A 1 T A ' 1 T D 1 A ' 1 A 1 x

The operator notation is very helpful in understanding the central ideas, but may hide some important facts. It has been shown [link] , [link] that operators in different F i commute with each other, but the order of the operators within an F i cannot be changed. They represent the matrix multiplications in Winograd’s Short DFT Algorithms: Equation 30 or Winograd’s Short DFT Algorithms: Equation 8 which do not commute.

This formulation allows a very large set of possible orderings, in fact, the number is so large that some automatictechnique must be used to find the “best". It is possible to set up a criterion of optimality that not only includes the number ofmultiplications but the number of additions as well. The effects of relative multiply-add times, data transfer times, CPU register andmemory sizes, and other hardware characteristics can be included in the criterion. Dynamic programming can then be applied to derive anoptimal algorithm for a particular application [link] . This is a very interesting idea as there is no longer a single algorithm, buta class and an optimizing procedure. The challenge is to generate a broad enough class to result in a solution that is close to a globaloptimum and to have a practical scheme for finding the solution.

Results obtained applying the dynamic programming method to the design of fairly long DFT algorithms gave algorithms that hadfewer multiplications and additions than either a pure PFA or WFTA [link] . It seems that some nesting is desirable but not total nesting for four or more factors. There are also some interestingpossibilities in mixing the Cooley-Tukey with this formulation. Unfortunately, the twiddle factors are not the same for all rows andcolumns, therefore, operations cannot commute past a twiddle factor operator. There are ways of breaking the total algorithm intohorizontal paths and using different orderings along the different paths [link] , [link] . In a sense, this is what the split-radix FFT does with its twiddle factors when compared to a conventionalCooley-Tukey FFT.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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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