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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

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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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