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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

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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