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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

Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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RAW Reply
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Brian Reply
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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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
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Introduction about quantum dots in nanotechnology
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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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