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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

and

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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Kyle
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Adin
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Kyle
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Joe
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Damian Reply
research.net
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Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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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.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
s.
Graphene has a hexagonal structure
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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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