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The h ( n ) terms are fixed for a digital filter, or they represent the W terms from Multidimensional Index Mapping: Equation 1 if the convolution is being used to calculate a DFT. Because of this, d = B H in [link] can be precalculated and only the A and C operators represent the mathematics done at execution of the algorithm. In order toexploit this feature, it was shown [link] , [link] that the properties of [link] allow the exchange of the more complicated operator C with the simpler operator B . Specifically this is given by

Y = C [ A X o B H ]
Y ' = B T [ A X o C T H ' ]

where H ' has the same elements as H , but in a permuted order, and likewise Y ' and Y . This very important property allows precomputing the more complicated C T H ' in [link] rather than B H as in [link] .

Because B H or C T H ' can be precomputed, the bilinear form of [link] and [link] can be written as a linear form. If an M x M diagonal matrix D is formed from d = C T H , or in the case of [link] , d = B H , assuming a commutative property for o , [link] becomes

Y ' = B T D A X

and [link] becomes

Y = C D A X

In most cases there is no reason not to use the same reduction operations on X and H , therefore, B can be the same as A and [link] then becomes

Y ' = A T D A X

In order to illustrate how the residue reduction is carried out and how the A matrix is obtained, the length-5 DFT algorithmstarted in The DFT as Convolution or Filtering: Matrix 1 will be continued. The DFT is first converted to a length-4 cyclic convolution by theindex permutation from The DFT as Convolution or Filtering: Equation 3 to give the cyclic convolution in The DFT as Convolution or Filtering . To avoid confusion from the permuted order of the data x ( n ) in The DFT as Convolution or Filtering , the cyclic convolution will first be developed without thepermutation, using the polynomial U ( s )

U ( s ) = x ( 1 ) + x ( 3 ) s + x ( 4 ) s 2 + x ( 2 ) s 3
U ( s ) = u ( 0 ) + u ( 1 ) s + u ( 2 ) s 2 + u ( 3 ) s 3

and then the results will be converted back to the permuted x ( n ) . The length-4 cyclic convolution in terms of polynomials is

Y ( s ) = U ( s ) H ( s ) mod ( s 4 - 1 )

and the modulus factors into three cyclotomic polynomials

s 4 - 1 = ( s 2 - 1 ) ( s 2 + 1 )
= ( s - 1 ) ( s + 1 ) ( s 2 + 1 )
= P 1 P 2 P 3

Both U ( s ) and H ( s ) are reduced modulo these three polynomials. The reduction modulo P 1 and P 2 is done in two stages. First it is done modulo ( s 2 - 1 ) , then that residue is further reduced modulo ( s - 1 ) and ( s + 1 ) .

U ( s ) = u 0 + u 1 s + u 2 s 2 + u 3 s 3
U ' ( s ) = ( ( U ( s ) ) ) ( s 2 - 1 ) = ( u 0 + u 2 ) + ( u 1 + u 3 ) s
U 1 ( s ) = ( ( U ' ( s ) ) ) P 1 = ( u 0 + u 1 + u 2 + u 3 )
U 2 ( s ) = ( ( U ' ( s ) ) ) P 2 = ( u 0 - u 1 + u 2 - u 3 )
U 3 ( s ) = ( ( U ( s ) ) ) P 3 = ( u 0 - u 2 ) + ( u 1 - u 3 ) s

The reduction in [link] of the data polynomial [link] can be denoted by a matrix operation on a vector which has the data asentries.

1 0 1 0 0 1 0 1 u 0 u 1 u 2 u 3 = u 0 + u 2 u 1 + u 3

and the reduction in [link] is

1 0 - 1 0 0 1 0 - 1 u 0 u 1 u 2 u 3 = u 0 - u 2 u 1 - u 3

Combining [link] and [link] gives one operator

1 0 1 0 0 1 0 1 1 0 - 1 0 0 1 0 - 1 u 0 + u 2 u 1 + u 3 u 0 - u 2 u 1 - u 3 = u 0 + u 2 u 1 + u 3 u 0 - u 2 u 1 - u 3 = w 0 w 1 v 0 v 1

Further reduction of v 0 + v 1 s is not possible because P 3 = s 2 + 1 cannot be factored over the rationals. However s 2 - 1 can be factored into P 1 P 2 = ( s - 1 ) ( s + 1 ) and, therefore, w 0 + w 1 s can be further reduced as was done in [link] and [link] by

1 1 w 0 w 1 = w 0 + w 1 = u 0 + u 2 + u 1 + u 3
1 - 1 w 0 w 1 = w 0 - w 1 = u 0 + u 2 - u 1 - u 3

Combining [link] , [link] and [link] gives

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

The same reduction is done to H ( s ) and then the convolution of [link] is done by multiplying each residue polynomial of X ( s ) and H ( s ) modulo each corresponding cyclotomic factor of P ( s ) and finally a recombination using the polynomial Chinese RemainderTheorem (CRT) as in Polynomial Description of Signals: Equation 9 and Polynomial Description of Signals: Equation 13 .

Questions & Answers

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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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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