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Y ( s ) = H ( s ) X ( s ) mod P ( s )

where P ( s ) = s N - 1 , and if P ( s ) is factored into two relatively prime factors P = P 1 P 2 , using residue reduction of H ( s ) and X ( s ) modulo P 1 and P 2 , the lower degree residue polynomials can be multiplied and the results recombined with theCRT. This is done by

Y ( s ) = ( ( K 1 H 1 X 1 + K 2 H 2 X 2 ) ) P

where

H 1 = ( ( H ) ) P 1 , X 1 = ( ( X ) ) P 1 , H 2 = ( ( H ) ) P 2 , X 2 = ( ( X ) ) P 2

and K 1 and K 2 are the CRT coefficient polynomials from [link] . This allows two shorter convolutions to replace one longer one.

Another property of residue reduction that is useful in DFT calculation is polynomial evaluation. To evaluate F ( s ) at s = x , F ( s ) is reduced modulo s - x .

F ( x ) = ( ( F ( s ) ) ) s - x

This is easily seen from the definition in [link]

F ( s ) = Q ( s ) ( s - x ) + R ( s )

Evaluating s = x gives R ( s ) = F ( x ) which is a constant. For the DFT this becomes

C ( k ) = ( ( X ( s ) ) ) s - W k

Details of the polynomial algebra useful in digital signal processing can be found in [link] , [link] , [link] .

The dft as a polynomial evaluation

The Z-transform of a number sequence x ( n ) is defined as

X ( z ) = n = 0 x ( n ) z - n

which is the same as the polynomial description in [link] but with a negative exponent. For a finite length-N sequence [link] becomes

X ( z ) = n = 0 N - 1 x ( n ) z - n
X ( z ) = x ( 0 ) + x ( 1 ) z - 1 + x ( 2 ) z - 2 + · + x ( N - 1 ) z - N + 1

This N - 1 order polynomial takes on the values of the DFT of x ( n ) when evaluated at

z = e j 2 π k / N

which gives

C ( k ) = X ( z ) | z = e j 2 π k / N = n = 0 N - 1 x ( n ) e - j 2 π n k / N

In terms of the positive exponent polynomial from [link] , the DFT is

C ( k ) = X ( s ) | s = W k

where

W = e - j 2 π / N

is an N t h root of unity (raising W to the N t h power gives one). The N values of the DFT are found from X ( s ) evaluated at the N N t h roots of unity which are equally spaced around the unit circle in the complex s plane.

One method of evaluating X ( z ) is the so-called Horner's rule or nested evaluation. When expressed as a recursivecalculation, Horner's rule becomes the Goertzel algorithm which has some computational advantages especially when only a few values ofthe DFT are needed. The details and programs can be found in [link] , [link] and The DFT as Convolution or Filtering: Goertzel's Algorithm (or A Better DFT Algorithm)

Another method for evaluating X ( s ) is the residue reduction modulo ( s - W k ) as shown in [link] . Each evaluation requires N multiplications and therefore, N 2 multiplications for the N values of C ( k ) .

C ( k ) = ( ( X ( s ) ) ) ( s - W k )

A considerable reduction in required arithmetic can be achieved if some operations can be shared between the reductions for differentvalues of k . This is done by carrying out the residue reduction in stages that can be shared rather than done in one step for each k in [link] .

The N values of the DFT are values of X ( s ) evaluated at s equal to the N roots of the polynomial P ( s ) = s N - 1 which are W k . First, assuming N is even, factor P ( s ) as

P ( s ) = ( s N - 1 ) = P 1 ( s ) P 2 ( s ) = ( s N / 2 - 1 ) ( s N / 2 + 1 )

X ( s ) is reduced modulo these two factors to give two residue polynomials, X 1 ( s ) and X 2 ( s ) . This process is repeated by factoring P 1 and further reducing X 1 then factoring P 2 and reducing X 2 . This is continued until the factors are of first degree which gives the desired DFT values as in [link] . This is illustrated for a length-8 DFT. The polynomial whose roots are W k , factors as

P ( s ) = s 8 - 1
= [ s 4 - 1 ] [ s 4 + 1 ]
= [ ( s 2 - 1 ) ( s 2 + 1 ) ] [ ( s 2 - j ) ( s 2 + j ) ]
= [ ( s - 1 ) ( s + 1 ) ( s - j ) ( s + j ) ] [ ( s - a ) ( s + a ) ( s - j a ) ( s + j a ) ]

where a 2 = j . Reducing X ( s ) by the first factoring gives two third degree polynomials

X ( s ) = x 0 + x 1 s + x 2 s 2 + . . . + x 7 s 7

gives the residue polynomials

X 1 ( s ) = ( ( X ( s ) ) ) ( s 4 - 1 ) = ( x 0 + x 4 ) + ( x 1 + x 5 ) s + ( x 2 + x 6 ) s 2 + ( x 3 + x 7 ) s 3
X 2 ( s ) = ( ( X ( s ) ) ) ( s 4 + 1 ) = ( x 0 - x 4 ) + ( x 1 - x 5 ) s + ( x 2 - x 6 ) s 2 + ( x 3 - x 7 ) s 3

Two more levels of reduction are carried out to finally give the DFT. Close examination shows the resulting algorithm to be thedecimation-in-frequency radix-2 Cooley-Tukey FFT [link] , [link] . Martens [link] has used this approach to derive an efficient DFT algorithm.

Other algorithms and types of FFT can be developed using polynomial representations and some are presented in the generalization in DFT and FFT: An Algebraic View .

Questions & Answers

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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