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The Winograd Structure can be described in this manner also. Suppose M ( s ) can be factored as M ( s ) = M 1 ( s ) M 2 ( s ) where M 1 and M 2 have no common roots, then C M C M 1 C M 2 where denotes the matrix direct sum. Using this similarity and recalling [link] , the original convolution is decomposed intodisjoint convolutions. This is, in fact, a statement of the Chinese Remainder Theoremfor polynomials expressed in matrix notation. In the case of circular convolution, s n - 1 = d | n Φ d ( s ) , so that S n can be transformed to a block diagonal matrix,

S n C Φ 1 C Φ d C Φ n = d | n C Φ d

where Φ d ( s ) is the d t h cyclotomic polynomial. In this case, each block represents a convolutionwith respect to a cyclotomic polynomial, or a `cyclotomic convolution'.Winograd's approach carries out these cyclotomic convolutions using the Toom-Cook algorithm.Note that for each divisor, d , of n there is a corresponding block on the diagonal of size φ ( d ) , for the degree of Φ d ( s ) is φ ( d ) where φ ( · ) is the Euler totient function. This method is good for short lengths, butas n increases the cyclotomic convolutions become cumbersome,for as the number of distinct prime divisors of d increases, the operation described by k h k C Φ d k becomes more difficult to implement.

The Agarwal-Cooley Algorithm utilizes the fact that

S n = P t S n 1 S n 2 P

where n = n 1 n 2 , ( n 1 , n 2 ) = 1 and P is an appropriate permutation [link] . This converts the one dimensional circular convolutionof length n to a two dimensional one of length n 1 along one dimension and length n 2 along the second.Then an n 1 -point and an n 2 -point circular convolution algorithm can be combined to obtain an n -point algorithm. In polynomial notation, the mapping accomplished bythis permutation P can be informally indicated by

Y ( s ) = X ( s ) H ( s ) s n - 1 P Y ( s , t ) = X ( s , t ) H ( s , t ) s n 1 - 1 , t n 2 - 1 .

It should be noted that [link] implies that a circulant matrix of size n 1 n 2 can be written as a block circulant matrix with circulantblocks.

The Split-Nesting algorithm [link] combines the structures of the Winograd and Agarwal-Cooley methods, so that S n is transformed to a block diagonalmatrix as in [link] ,

S n d | n Ψ ( d ) .

Here Ψ ( d ) = p | d , p P C Φ H d ( p ) where H d ( p ) is the highest power of p dividing d , and P is the set of primes.

S 45 1 C Φ 3 C Φ 9 C Φ 5 C Φ 3 C Φ 5 C Φ 9 C Φ 5

In this structure a multidimensional cyclotomic convolution, represented by Ψ ( d ) , replaces each cyclotomic convolution in Winograd's algorithm (represented by C Φ d in [link] . Indeed, if the product of b 1 , , b k is d and they are pairwise relatively prime, then C Φ d C Φ b 1 C Φ b k . This gives a method for combining cyclotomic convolutionsto compute a longer circular convolution. It is like the Agarwal-Cooley method but requires feweradditions [link] .

Prime factor permutations

One can obtain S n 1 S n 2 from S n 1 n 2 when ( n 1 , n 2 ) = 1 , for in this case, S n is similar to S n 1 S n 2 , n = n 1 n 2 . Moreover, they are related by a permutation.This permutation is that of the prime factor FFT algorithms and is employed in nesting algorithmsfor circular convolution [link] , [link] . The permutation is described by Zalcstein [link] , among others, and it is his description we draw on in the following.

Let n = n 1 n 2 where ( n 1 , n 2 ) = 1 . Define e k , ( 0 k n - 1 ), to be the standard basis vector, ( 0 , , 0 , 1 , 0 , , 0 ) t , where the 1 is in the k t h position. Then, the circular shift matrix, S n , can be described by

Questions & Answers

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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Automatic generation of prime length fft programs. OpenStax CNX. Sep 09, 2009 Download for free at http://cnx.org/content/col10596/1.4
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