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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

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, 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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