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A major application of the FFT is fast convolution or fast filtering where the DFT of the signal is multiplied term-by-term by the DFT ofthe impulse (helps to be doing finite impulse response (FIR) filtering) and the time-domain output is obtained by taking the inverse DFT ofthat product. What is less well-known is the DFT can be calculated by convolution. There are several different approaches to this,each with different application.

Rader's conversion of the dft into convolution

In this section a method quite different from the index mapping or polynomial evaluation is developed. Rather than dealingwith the DFT directly, it is converted into a cyclic convolution which must then be carried out by some efficient means. Those meanswill be covered later, but here the conversion will be explained. This method requires use of some number theory, which can befound in an accessible form in [link] or [link] and is easy enough to verify on one's own. A good general reference on numbertheory is [link] .

The DFT and cyclic convolution are defined by

C ( k ) = n = 0 N - 1 x ( n ) W n k
y ( k ) = n = 0 N - 1 x ( n ) h ( k - n )

For both, the indices are evaluated modulo N . In order to convert the DFT in [link] into the cyclic convolution of [link] , the n k product must be changed to the k - n difference. With real numbers, this can be done with logarithms, but it is more complicated when working in a finite set of integersmodulo N . From number theory [link] , [link] , [link] , [link] , it can be shown that if the modulus is a prime number, a base (called aprimitive root) exists such that a form of integer logarithm can be defined. This is stated in the following way. If N is a prime number, a number r called a primitive roots exists such that the integer equation

n = ( ( r m ) ) N

creates a unique, one-to-one map of the N - 1 member set m = { 0 , . . . , N - 2 } and the N - 1 member set n = { 1 , . . . , N - 1 } . This is because the multiplicative group of integers modulo a prime, p , is isomorphic to the additive group of integers modulo ( p - 1 ) and is illustrated for N = 5 below.

Table of Integers n = ( ( r m ) ) modulo 5, [* not defined]
r m= 0 1 2 3 4 5 6 7
1 1 1 1 1 1 1 1 1
2 1 2 4 3 1 2 4 3
3 1 3 4 2 1 3 4 2
4 1 4 1 4 1 4 1 4
5 * 0 0 0 * 0 0 0
6 1 1 1 1 1 1 1 1

[link] is an array of values of r m modulo N and it is easy to see that there are two primitiveroots, 2 and 3, and [link] defines a permutation of the integers n from the integers m (except for zero). [link] and a primitive root (usually chosen to be the smallest of those that exist) can be used to convert the DFT in [link] to the convolution in [link] . Since [link] cannot give a zero, a new length-(N-1) data sequence is defined from x ( n ) by removing the term with index zero. Let

n = r - m


k = r s

where the term with the negative exponent (the inverse) is defined as the integer that satisfies

( ( r - m r m ) ) N = 1

If N is a prime number, r - m always exists. For example, ( ( 2 - 1 ) ) 5 = 3 . [link] now becomes

C ( r s ) = m = 0 N - 2 x ( r - m ) W r - m r s + x ( 0 ) ,

for s = 0 , 1 , . . , N - 2 , and

C ( 0 ) = n = 0 N - 1 x ( n )

New functions are defined, which are simply a permutation in the order of the original functions, as

x ' ( m ) = x ( r - m ) , C ' ( s ) = C ( r s ) , W ' ( n ) = W r n

[link] then becomes

C ' ( s ) = m = 0 N - 2 x ' ( m ) W ' ( s - m ) + x ( 0 )

which is cyclic convolution of length N-1 (plus x ( 0 ) ) and is denoted as

C ' ( k ) = x ' ( k ) * W ' ( k ) + x ( 0 )

Applying this change of variables (use of logarithms) to the DFT can best be illustrated from the matrix formulation of the DFT. [link] is written for a length-5 DFT as

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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
ya I also want to know the raman spectra
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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