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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 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
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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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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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