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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

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
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
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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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