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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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kinnecy Reply
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I'm not sure why it wrote it the other way
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Commplementary angles
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
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rolling four fair dice and getting an even number an all four dice
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Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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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
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I'm interested in Nanotube
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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?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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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