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In 1976, S. Winograd [link] presented a new DFT algorithm which had significantly fewer multiplications than the Cooley-TukeyFFT which had been published eleven years earlier. This new Winograd Fourier Transform Algorithm (WFTA) is based on the type- one indexmap from Multidimensional Index Mapping with each of the relatively prime length short DFT's calculated by very efficient special algorithms. It isthese short algorithms that this section will develop. They use the index permutation of Rader described in the another module toconvert the prime length short DFT's into cyclic convolutions. Winograd developed a method for calculating digital convolution withthe minimum number of multiplications. These optimal algorithms are based on the polynomial residue reduction techniques of Polynomial Description of Signals: Equation 1 to break the convolution into multiple small ones [link] , [link] , [link] , [link] , [link] , [link] .

The operation of discrete convolution defined by

y ( n ) = k h ( n - k ) x ( k )

is called a bilinear operation because, for a fixed h ( n ) , y ( n ) is a linear function of x ( n ) and for a fixed x ( n ) it is a linear function of h ( n ) . The operation of cyclic convolution is the same but with all indices evaluated modulo N .

Recall from Polynomial Description of Signals: Equation 3 that length-N cyclic convolution of x ( n ) and h ( n ) can be represented by polynomial multiplication

Y ( s ) = X ( s ) H ( s ) mod ( s N - 1 )

This bilinear operation of [link] and [link] can also be expressed in terms of linear matrix operators and a simpler bilinearoperator denoted by o which may be only a simple element-by-element multiplication of the two vectors [link] , [link] , [link] . This matrix formulation is

Y = C [ A X o B H ]

where X , H and Y are length-N vectors with elements of x ( n ) , h ( n ) and y ( n ) respectively. The matrices A and B have dimension M x N , and C is N x M with M N . The elements of A , B , and C are constrained to be simple; typically small integers or rational numbers. It will be thesematrix operators that do the equivalent of the residue reduction on the polynomials in [link] .

In order to derive a useful algorithm of the form [link] to calculate [link] , consider the polynomial formulation [link] again. To use the residue reduction scheme, the modulus is factored into relatively prime factors. Fortunately the factoringof this particular polynomial, s N - 1 , has been extensively studied and it has considerable structure. When factored over the rationals,which means that the only coefficients allowed are rational numbers, the factors are called cyclotomic polynomials [link] , [link] , [link] . The most interesting property for our purposes is that most of the coefficients of cyclotomic polynomialsare zero and the others are plus or minus unity for degrees up to over one hundred. This means the residue reduction will generallyrequire no multiplications.

The operations of reducing X ( s ) and H ( s ) in [link] are carried out by the matrices A and B in [link] . The convolution of the residue polynomials is carried out by the o operator and the recombination by the CRT is done by the C matrix. More details are in [link] , [link] , [link] , [link] , [link] but the important fact is the A and B matrices usually contain only zero and plus or minus unity entries and the C matrix only contains rational numbers. The only general multiplications are those represented by o . Indeed, in the theoretical results from computational complexity theory,these real or complex multiplications are usually the only ones counted. In practical algorithms, the rational multiplicationsrepresented by C could be a limiting factor.

Questions & Answers

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
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
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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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