# 0.14 Conclusions: fast fourier transforms

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This book has developed a class of efficient algorithms based on index mapping and polynomial algebra. This provides aframework from which the Cooley-Tukey FFT, the split-radix FFT, the PFA, and WFTA can be derived. Even the programs implementing thesealgorithms can have a similar structure. Winograd's theorems were presented and shown to be very powerful in both deriving algorithmsand in evaluating them. The simple radix-2 FFT provides a compact, elegant means for efficiently calculating the DFT. If someelaboration is allowed, significant improvement can be had from the split-radix FFT, the radix-4 FFT or the PFA. If multiplications areexpensive, the WFTA requires the least of all.

Several method for transforming real data were described that are more efficient than directly using a complex FFT. Acomplex FFT can be used for real data by artificially creating a complex input from two sections of real input. An alternative andslightly more efficient method is to construct a special FFT that utilizes the symmetries at each stage.

As computers move to multiprocessors and multicore, writing and maintaining efficient programs becomes more and more difficult.The highly structured form of FFTs allows automatic generation of very efficient programs that are tailored specifically to aparticular DSP or computer architecture.

For high-speed convolution, the traditional use of the FFT or PFA with blocking is probably the fastest method although rectangular transforms,distributed arithmetic, or number theoretic transforms may have a future with special VLSI hardware.

The ideas presented in these notes can also be applied to the calculation of the discrete Hartley transform [link] , [link] , the discrete cosine transform [link] , [link] , and to number theoretic transforms [link] , [link] , [link] .

There are many areas for future research. The relationship of hardware to algorithms, the proper use of multiple processors,the proper design and use of array processors and vector processors are all open. There are still many unanswered questions inmulti-dimensional algorithms where a simple extension of one-dimensional methods will not suffice.

how can chip be made from sand
is this allso about nanoscale material
Almas
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 is the latest information on a no technology how can I find it
William
currently
William
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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