# 0.8 The cooley-tukey fast fourier transform algorithm  (Page 7/8)

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Although there can be significant differences in the efficiencies of the various Cooley-Tukey and Split-Radix FFTs, thenumber of multiplications and additions for all of them is on the order of $NlogN$ . That is fundamental to the class of algorithms.

## The quick fourier transform, an fft based on symmetries

The development of fast algorithms usually consists of using special properties of the algorithm of interest to remove redundant or unnecessary operations of a direct implementation. The discrete Fourier transform(DFT) defined by

$C\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)\phantom{\rule{0.166667em}{0ex}}{W}_{N}^{nk}$

where

${W}_{N}={e}^{-j2\pi /N}$

has enormous capacity for improvement of its arithmetic efficiency. Most fast algorithms use the periodic and symmetric properties of its basisfunctions. The classical Cooley-Tukey FFT and prime factor FFT [link] exploit the periodic properties of the cosine and sine functions. Their use of the periodicities to share and, therefore, reduce arithmeticoperations depends on the factorability of the length of the data to be transformed. For highly composite lengths, the number of floating-pointoperation is of order $N\phantom{\rule{0.166667em}{0ex}}log\left(N\right)$ and for prime lengths it is of order ${N}^{2}$ .

This section will look at an approach using the symmetric properties to remove redundancies. This possibility has long been recognized [link] , [link] , [link] , [link] but has not been developed in any systematic way in the open literature. We will develop an algorithm,called the quick Fourier transform (QFT) [link] , that will reduce the number of floating point operations necessary to compute the DFT by afactor of two to four over direct methods or Goertzel's method for prime lengths. Indeed, it seems the best general algorithm available for primelength DFTs. One can always do better by using Winograd type algorithms but they must be individually designed for each length. The Chirp Z-transform can be used for longer lengths.

## Input and output symmetries

We use the fact that the cosine is an even function and the sine is an odd function. The kernel of the DFT or the basis functions of the expansion isgiven by

${W}_{N}^{nk}={e}^{-j2\pi nk/N}=cos\left(2\pi nk/N\right)+j\phantom{\rule{0.166667em}{0ex}}sin\left(2\pi nk/N\right)$

which has an even real part and odd imaginary part. If the data $x\left(n\right)$ are decomposed into their real and imaginary parts and those into their even andodd parts, we have

$x\left(n\right)=u\left(n\right)+j\phantom{\rule{0.166667em}{0ex}}v\left(n\right)=\left[{u}_{e}\left(n\right)+{u}_{o}\left(n\right)\right]+j\phantom{\rule{0.166667em}{0ex}}\left[{v}_{e}\left(n\right)+{v}_{o}\left(n\right)\right]$

where the even part of the real part of $x\left(n\right)$ is given by

${u}_{e}\left(n\right)=\left(u\left(n\right)+u\left(-n\right)\right)/2$

and the odd part of the real part is

${u}_{o}\left(n\right)=\left(u\left(n\right)-u\left(-n\right)\right)/2$

with corresponding definitions of ${v}_{e}\left(n\right)$ and ${v}_{o}\left(n\right)$ . Using Convolution Algorithms: Equation 32 with a simpler notation, the DFT of Convolution Algorithms: Equation 29 becomes

$C\left(k\right)=\sum _{n=0}^{N-1}\left(u+j\phantom{\rule{0.166667em}{0ex}}v\right)\left(cos-jsin\right).$

The sum over an integral number of periods of an odd function is zero and the sum of an even function over half of the period is one half the sumover the whole period. This causes [link] and [link] to become

$C\left(k\right)=\sum _{n=0}^{N/2-1}\left[{u}_{e}\phantom{\rule{0.166667em}{0ex}}cos+{v}_{o}\phantom{\rule{0.166667em}{0ex}}sin\right]+j\phantom{\rule{0.166667em}{0ex}}\left[{v}_{e}\phantom{\rule{0.166667em}{0ex}}cos-{v}_{o}\phantom{\rule{0.166667em}{0ex}}sin\right].$

for $k=0,1,2,\cdots ,N-1$ .

The evaluation of the DFT using equation [link] requires half as many real multiplication and half as many real additions as evaluating it using [link] or [link] . We have exploited the symmetries of the sine and cosine as functions of the time index $n$ . This is independent of whether the length is composite or not. Another view of this formulation is thatwe have used the property of associatively of multiplication and addition. In other words, rather than multiply two data points by the same value ofa sine or cosine then add the results, one should add the data points first then multiply the sum by the sine or cosine which requires onerather than two multiplications.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
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