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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

where we get a research paper on Nano chemistry....?
Maira Reply
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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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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