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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
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rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
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J, combine like terms 7x-4y
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how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Ramkumar Reply
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preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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