# 0.8 The cooley-tukey fast fourier transform algorithm  (Page 2/8)

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$TF:\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{W}_{8}^{{n}_{2}{k}_{1}}=\left[\begin{array}{cc}{W}^{0}& {W}^{0}\\ {W}^{0}& {W}^{1}\\ {W}^{0}& {W}^{2}\\ {W}^{0}& {W}^{3}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& W\\ 1& -j\\ 1& -jW\end{array}\right]$

The twiddle factor array will always have unity in the first row and first column.

To complete [link] at this point, after the row DFT's are multiplied by the TF array, the ${N}_{1}$ length- ${N}_{2}$ DFT's of the columns are calculated. However, since the columns DFT's are oflength ${R}^{M-1}$ , they can be posed as a ${R}^{M-2}$ by $R$ array and the process repeated, again using length- $R$ DFT's. After $M$ stages of length- $R$ DFT's with TF multiplications interleaved, the DFT is complete. The flow graph of a length-2 DFT is given in Figure 1 and is called a butterfly because of its shape. The flow graph of thecomplete length-8 radix-2 FFT is shown in Figure 2 .

This flow-graph, the twiddle factor map of [link] , and the basic equation [link] should be completely understood before going further.

A very efficient indexing scheme has evolved over the years that results in a compact and efficient computer program. A FORTRANprogram is given below that implements the radix-2 FFT. It should be studied [link] to see how it implements [link] and the flow-graph representation.

N2 = N DO 10 K = 1, MN1 = N2 N2 = N2/2E = 6.28318/N1 A = 0DO 20 J = 1, N2 C = COS (A)S =-SIN (A) A = J*EDO 30 I = J, N, N1 L = I + N2XT = X(I) - X(L) X(I) = X(I) + X(L)YT = Y(I) - Y(L) Y(I) = Y(I) + Y(L)X(L) = XT*C - YT*S Y(L) = XT*S + YT*C30 CONTINUE 20 CONTINUE10 CONTINUE A Radix-2 Cooley-Tukey FFT Program 

This discussion, the flow graph of Winograd’s Short DFT Algorithms: Figure 2 and the program of [link] are all based on the input index map of Multidimensional Index Mapping: Equation 6 and [link] and the calculations are performed in-place. According to Multidimensional Index Mapping: In-Place Calculation of the DFT and Scrambling , this means the output is scrambled in bit-reversed order and should be followed by anunscrambler to give the DFT in proper order. This formulation is called a decimation-in-frequency FFT [link] , [link] , [link] . A very similar algorithm based on the output index map can be derived whichis called a decimation-in-time FFT. Examples of FFT programs are found in [link] and in the Appendix of this book.

## Modifications to the basic cooley-tukey fft

Soon after the paper by Cooley and Tukey, there were improvements and extensions made. One very important discovery wasthe improvement in efficiency by using a larger radix of 4, 8 or even 16. For example, just as for the radix-2 butterfly, there areno multiplications required for a length-4 DFT, and therefore, a radix-4 FFT would have only twiddle factor multiplications. Becausethere are half as many stages in a radix-4 FFT, there would be half as many multiplications as in a radix-2 FFT. In practice, becausesome of the multiplications are by unity, the improvement is not by a factor of two, but it is significant. A radix-4 FFT is easilydeveloped from the basic radix-2 structure by replacing the length-2 butterfly by a length-4 butterfly and making a few othermodifications. Programs can be found in [link] and operation counts will be given in "Evaluation of the Cooley-Tukey FFT Algorithms" .

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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.
Bharti
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Damian Reply
absolutely yes
Daniel
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Akash Reply
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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
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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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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
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or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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China
Cied
types of nano material
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many many of nanotubes
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Source:  OpenStax, Fast fourier transforms (6x9 version). OpenStax CNX. Aug 24, 2009 Download for free at http://cnx.org/content/col10683/1.5
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