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Y ( s ) = K 1 ( s ) U 1 ( s ) H 1 ( s ) + K 2 ( s ) U 2 ( s ) H 2 ( s ) + K 3 ( s ) U 3 ( s ) H 3 ( s )

mod ( s 4 - 1 )

where U 1 ( s ) = r 1 and U 2 ( s ) = r 2 are constants and U 3 ( s ) = v 0 + v 1 s is a first degree polynomial. U 1 times H 1 and U 2 times H 2 are easy, but multiplying U 3 time H 3 modulo ( s 2 + 1 ) is more difficult.

The multiplication of U 3 ( s ) times H 3 ( s ) can be done by the Toom-Cook algorithm [link] , [link] , [link] which can be viewed as Lagrange interpolation or polynomial multiplication modulo a specialpolynomial with three arbitrary coefficients. To simplify the arithmetic, the constants are chosen to be plus and minus one andzero. The details of this can be found in [link] , [link] , [link] . For this example it can be verified that

( ( v 0 + v 1 s ) ( h 0 + h 1 s ) ) ) s 2 + 1 = ( v 0 h 0 - v 1 h 1 ) + ( v 0 h 1 + v 1 h 0 ) s

which by the Toom-Cook algorithm or inspection is

1 - 1 0 - 1 - 1 1 1 0 0 1 1 1 v 0 v 1 o 1 0 0 1 1 1 h 0 h 1 = y 0 y 1

where o signifies point-by-point multiplication. The total A matrix in [link] is a combination of [link] and [link] giving

A X = A 1 A 2 A 3 X
= 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 - 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 - 1 0 0 1 0 - 1 u 0 u 1 u 2 u 3 = r 0 r 1 v 0 v 1

where the matrix A 3 gives the residue reduction s 2 - 1 and s 2 + 1 , the upper left-hand part of A 2 gives the reduction modulo s - 1 and s + 1 , and the lower right-hand part of A1 carries out the Toom-Cook algorithm modulo s 2 + 1 with the multiplication in [link] . Notice that by calculating [link] in the three stages, seven additions are required. Also notice that A 1 is not square. It is this “expansion" that causes more than N multiplications to be required in o in [link] or D in [link] . This staged reduction will derive the A operator for [link]

The method described above is very straight-forward for the shorter DFT lengths. For N = 3 , both of the residue polynomials are constants and the multiplication given by o in [link] is trivial. For N = 5 , which is the example used here, there is one first degree polynomial multiplication required but the Toom-Cookalgorithm uses simple constants and, therefore, works well as indicated in [link] . For N = 7 , there are two first degree residue polynomials which can each be multiplied by the sametechniques used in the N = 5 example. Unfortunately, for any longer lengths, the residue polynomials have an order of three orgreater which causes the Toom-Cook algorithm to require constants of plus and minus two and worse. For that reason, the Toom-Cook methodis not used, and other techniques such as index mapping are used that require more than the minimum number of multiplications, but donot require an excessive number of additions. The resulting algorithms still have the structure of [link] . Blahut [link] and Nussbaumer [link] have a good collection of algorithms for polynomial multiplication that can be used with thetechniques discussed here to construct a wide variety of DFT algorithms.

The constants in the diagonal matrix D can be found from the CRT matrix C in [link] using d = C T H ' for the diagonal terms in D . As mentioned above, for the smaller prime lengths of 3, 5, and 7 this works well but for longer lengths the CRT becomesvery complicated. An alternate method for finding D uses the fact that since the linear form [link] or [link] calculates the DFT, it is possible to calculate a known DFT of a given x ( n ) from the definition of the DFT in Multidimensional Index Mapping: Equation 1 and, given the A matrix in [link] , solve for D by solving a set of simultaneous equations. The details of this procedure are described in [link] .

Questions & Answers

how can chip be made from sand
Eke Reply
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
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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