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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 N log N . 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 ( k ) = n = 0 N - 1 x ( n ) W N n k


W N = e - j 2 π / 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 log ( N ) 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 n k = e - j 2 π n k / N = cos ( 2 π n k / N ) + j sin ( 2 π n k / N )

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

x ( n ) = u ( n ) + j v ( n ) = [ u e ( n ) + u o ( n ) ] + j [ v e ( n ) + v o ( n ) ]

where the even part of the real part of x ( n ) is given by

u e ( n ) = ( u ( n ) + u ( - n ) ) / 2

and the odd part of the real part is

u o ( n ) = ( u ( n ) - u ( - n ) ) / 2

with corresponding definitions of v e ( n ) and v o ( n ) . Using Convolution Algorithms: Equation 32 with a simpler notation, the DFT of Convolution Algorithms: Equation 29 becomes

C ( k ) = n = 0 N - 1 ( u + j v ) ( cos - j sin ) .

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 ( k ) = n = 0 N / 2 - 1 [ u e cos + v o sin ] + j [ v e cos - v o sin ] .

for k = 0 , 1 , 2 , , 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.

Questions & Answers

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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