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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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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