# 0.13 Comments: fast fourier transforms

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## Other work and results

This section comes from a note describing results on efficient algorithms to calculate the discrete Fourier transform (DFT) that were collected over years.Perhaps the most interesting is the discovery that the Cooley-Tukey FFT was described by Gauss in 1805 [link] . That gives some indication of the age of research on the topic, and the fact that a1995 compiled bibliography [link] on efficient algorithms contains over 3400 entries indicates its volume. Three IEEE Pressreprint books contain papers on the FFT [link] , [link] , [link] . An excellent general purpose FFT program has been described in [link] , [link] and is used in Matlab and available over the internet.

Efficient FFT algorithms for length- ${2}^{M}$ were described by Gauss and discovered in modern times by Cooley and Tukey [link] . These have been highly developed and good examples of FORTRAN programs canbe found in [link] . Several new algorithms have been published that require the least known amount of total arithmetic [link] , [link] , [link] , [link] , [link] , [link] . Of these, the split-radix FFT [link] , [link] , [link] , [link] seems to have the best structure for programming, and an efficient program has beenwritten [link] to implement it. A mixture of decimation-in-time and decimation-in-frequency with very goodefficiency is given in [link] , [link] and one called the Sine-Cosine FT [link] . Recently a modification to the split-radix algorithm has been described [link] that has a slightly better total arithmetic count. Theoretical bounds on the number ofmultiplications required for the FFT based on Winograd's theories are given in [link] , [link] . Schemes for calculating an in-place, in-order radix-2 FFT are given in [link] , [link] , [link] , [link] . Discussion of various forms of unscramblers is given in [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . A discussion of the relation of the computer architecture, algorithmand compiler can be found in [link] , [link] . A modification to allow lengths of $N=q\phantom{\rule{0.166667em}{0ex}}{2}^{m}$ for $q$ odd is given in [link] .

These efficient algorithms can not only be used on DFT's but on other transforms with a similar structure. They have been applied to thediscrete Hartley transform [link] , [link] and the discrete cosine transform [link] , [link] , [link] .

General length algorithms include [link] , [link] , [link] . For lengths that are not highly composite or prime, the chirp z-transform in agood candidate [link] , [link] for longer lengths and an efficient order- ${N}^{2}$ algorithm called the QFT [link] , [link] , [link] for shorter lengths. A method which automatically generates near-optimal prime lengthWinograd based programs has been given in [link] , [link] , [link] , [link] , [link] . This gives the same efficiency for shorter lengths (i.e. $N\le 19$ ) and new algorithms for much longer lengths and with well-structuredalgorithms. Another approach is given in [link] . Special methods are available for very long lengths [link] , [link] . A very interesting general length FFT systemcalled the FFTW has been developed by Frigo and Johnson at MIT. It uses a library of efficient “codelets" which are composed for a very efficientcalculation of the DFT on a wide variety of computers [link] , [link] , [link] . For most lengths and on most computers, this is the fastest FFT today.Surprisingly, it uses a recursive program structure. The FFTW won the 1999 Wilkinson Prize for Numerical Software.

Various approaches to calculating approximate DFTs have been based on cordic methods, short word lengths, or some form of pruning. A new methodthat uses the characteristics of the signals being transformed has combined the discrete wavelet transform (DWT) combined with the DFT togive an approximate FFT with $O\left(N\right)$ multiplications [link] , [link] , [link] for certain signal classes. A similar approach has been developed using filter banks [link] , [link] .

The study of efficient algorithms not only has a long history and large bibliography, it is still an exciting research field where new resultsare used in practical applications.

More information can be found on the Rice DSP Group's web page

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William
currently
William
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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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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Damian
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Professor
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Damian
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LITNING
scanning tunneling microscope
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Santosh
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Rafiq
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Mahi
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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
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