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Reference [link] should be consulted for the details of these conditions and examples. Two classes of index maps are definedfrom these conditions.

Type-one index map:

The map of [link] is called a type-one map when integers a and b exist such that

K 1 = a N 2 and K 2 = b N 1

Type-two index map:

The map of [link] is called a type-two map when when integers a and b exist such that

K 1 = a N 2 or K 2 = b N 1 , but not both.

The type-one can be used only if the factors of N are relatively prime, but the type-two can be used whether they are relatively prime ornot. Good [link] , Thomas, and Winograd [link] all used the type-one map in their DFT algorithms. Cooley and Tukey [link] used the type-two in their algorithms, both for a fixed radix ( N = R M ) and a mixed radix [link] .

The frequency index is defined by a map similar to [link] as

k = ( ( K 3 k 1 + K 4 k 2 ) ) N

where the same conditions, [link] and [link] , are used for determining the uniqueness of this map in terms of the integers K 3 and K 4 .

Two-dimensional arrays for the input data and its DFT are defined using these index maps to give

x ^ ( n 1 , n 2 ) = x ( ( K 1 n 1 + K 2 n 2 ) ) N
X ^ ( k 1 , k 2 ) = X ( ( K 3 k 1 + K 4 k 2 ) ) N

In some of the following equations, the residue reduction notation will be omitted for clarity. These changes of variablesapplied to the definition of the DFT given in [link] give

C ( k ) = n 2 = 0 N 2 - 1 n 1 = 0 N 1 - 1 x ( n ) W N K 1 K 3 n 1 k 1 W N K 1 K 4 n 1 k 2 W N K 2 K 3 n 2 k 1 W N K 2 K 4 n 2 k 2

where all of the exponents are evaluated modulo N .

The amount of arithmetic required to calculate [link] is the same asin the direct calculation of [link] . However, because of the special nature of the DFT, the integer constants K i can be chosen in such a way that the calculations are “uncoupled" andthe arithmetic is reduced. The requirements for this are

( ( K 1 K 4 ) ) N = 0 and/or ( ( K 2 K 3 ) ) N = 0

When this condition and those for uniqueness in [link] are applied, it is found that the K i may always be chosen such that one of the terms in [link] is zero. If the N i are relatively prime, it is always possible to make both terms zero. If the N i are not relatively prime, only one of the terms can be set to zero. When they are relatively prime, there is a choice, itis possible to either set one or both to zero. This in turn causes one or both of the center two W terms in [link] to become unity.

An example of the Cooley-Tukey radix-4 FFT for a length-16 DFT uses the type-two map with K 1 = 4 , K 2 = 1 , K 3 = 1 , K 4 = 4 giving

n = 4 n 1 + n 2
k = k 1 + 4 k 2

The residue reduction in [link] is not needed here since n does not exceed N as n 1 and n 2 take on their values. Since, in this example, the factors of N have a common factor, only one of the conditions in [link] can hold and, therefore, [link] becomes

C ^ ( k 1 , k 2 ) = C ( k ) = n 2 = 0 3 n 1 = 0 3 x ( n ) W 4 n 1 k 1 W 16 n 2 k 1 W 4 n 2 k 2

Note the definition of W N in [link] allows the simple form of W 16 K 1 K 3 = W 4

This has the form of a two-dimensional DFT with an extra term W 16 , called a “twiddle factor". The inner sum over n 1 represents four length-4 DFTs, the W 16 term represents 16 complex multiplications, and the outer sum over n 2 represents another four length-4 DFTs. This choice of the K i “uncouples" the calculations since the first sum over n 1 for n 2 = 0 calculates the DFT of the first row of the data array x ^ ( n 1 , n 2 ) , and those data values are never needed in the succeeding row calculations. The row calculations are independent,and examination of the outer sum shows that the column calculations are likewise independent. This is illustrated in [link] .

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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