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Corollary: If the modulus polynomial is P ( s ) = s N - 1 , then 2 N - t ( N ) multiplications are necessary to compute x ( s ) h ( s ) modulo P ( s ) , where t ( N ) is the number of positive divisors of N .

Theorem 5 is very general since it allows a general modulus polynomial. The proof of the upper boundinvolves reducing x ( s ) and h ( s ) modulo the k factors of P ( s ) . Each of the k irreducible residue polynomials is then multiplied using the method of Theorem 4 requiring 2 N i - 1 multiplies and the products are combined using the CRT. The total number of multiplies from the k parts is 2 N - k . The theorem also states the structure of these optimal algorithms is essentiallyunique. The special case of P ( s ) = s N - 1 is interesting since it corresponds to cyclic convolution and, as stated in the corollary, k is easily determined. The factors of s N - 1 are called cyclotomic polynomials and have interesting properties [link] , [link] , [link] .

Theorem 6 [link] , [link] Consider calculating the DFT of a prime length real-valued number sequence. If G is chosen as the field of rational numbers, the number of realmultiplications necessary to calculate a length-P DFT is u ( D F T ( N ) ) = 2 P - 3 - t ( P - 1 ) where t ( P - 1 ) is the number of divisors of P - 1 .

This theorem not only gives a lower limit on any practical prime length DFT algorithm, it also gives practicalalgorithms for N = 3 , 5 , and 7. Consider the operation counts in [link] to understand this theorem. In addition to the real multiplications counted by complexity theory, each optimalprime-length algorithm will have one multiplication by a rational constant. That constant corresponds to the residue modulo (s-1)which always exists for the modulus P ( s ) = s N - 1 - 1 . In a practical algorithm, this multiplication must be carried out, andthat accounts for the difference in the prediction of Theorem 6 and count in [link] . In addition, there is another operation that for certain applications must be counted as amultiplication. That is the calculation of the zero frequency term X ( 0 ) in the first row of the example in The DFT as Convolution or Filtering: Matrix 1 . For applications to the WFTA discussed in The Prime Factor and Winograd Fourier Transform Algorithms: The Winograd Fourier Transform Algorithm , that operation must be counted as a multiply. For lengths longer than 7,optimal algorithms require too many additions, so compromise structures are used.

Theorem 7 [link] , [link] If G is chosen as the field of rational numbers, the number of real multiplicationsnecessary to calculate a length-N DFT where N is a prime number raised to an integer power: N = P m , is given by

u ( D F T ( N ) ) = 2 N - ( ( m 2 + m ) / 2 ) t ( P - 1 ) - m - 1

where t ( P - 1 ) is the number of divisors of ( P - 1 ) .

This result seems to be practically achievable only for N = 9 , or perhaps 25. In the case of N = 9 , there are two rational multiplies that must be carried out and arecounted in [link] but are not predicted by Theorem 7 . Experience [link] indicates that even for N = 25 , an algorithm based on a Cooley-Tukey FFT using a type 2 index map givesan over-all more balanced result.

Theorem 8 [link] If G is chosen as the field of rational numbers, the number of real multiplications necessary tocalculate a length-N DFT where N = 2 m is given by

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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