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Next we take advantage of the symmetries of the sine and cosine as functions of the frequency index k . Using these symmetries on [link] gives

C ( k ) = n = 0 N / 2 - 1 [ u e cos + v o sin ] + j [ v e cos - v o sin ]
C ( N - 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 / 2 - 1 . This again reduces the number of operations by a factor of two, this time because it calculates two outputvalues at a time. The first reduction by a factor of two is always available. The second is possible only if both DFT values are needed. Itis not available if you are calculating only one DFT value. The above development has not dealt with the details that arise with the differencebetween an even and an odd length. That is straightforward.

Further reductions if the length is even

If the length of the sequence to be transformed is even, there are further symmetries that can be exploited. There will be four data values that areall multiplied by plus or minus the same sine or cosine value. This means a more complicated pre-addition process which is a generalization of thesimple calculation of the even and odd parts in [link] and [link] will reduce the size of the order N 2 part of the algorithm by still another factor of two or four. It the length is divisible by 4, theprocess can be repeated. Indeed, it the length is a power of 2, one can show this process is equivalent to calculating the DFT in terms ofdiscrete cosine and sine transforms [link] , [link] with a resulting arithmetic complexity of order N log ( N ) and with a structure that is well suited to real data calculations and pruning.

If the flow-graph of the Cooley-Tukey FFT is compared to the flow-graph of the QFT, one notices both similarities and differences. Both progress instages as the length is continually divided by two. The Cooley-Tukey algorithm uses the periodic properties of the sine and cosine to give thefamiliar horizontal tree of butterflies. The parallel diagonal lines in this graph represent the parallel stepping through the data in synchronismwith the periodic basis functions. The QFT has diagonal lines that connect the first data point with the last, then the second with the nextto last, and so on to give a “star" like picture. This is interesting in that one can look at the flow graph of an algorithm developed by somecompletely different strategy and often find section with the parallel structures and other parts with the star structure. These must be usingsome underlying periodic and symmetric properties of the basis functions.

Arithmetic complexity and timings

A careful analysis of the QFT shows that 2 N additions are necessary to compute the even and odd parts of the input data. This is followed by thelength N / 2 inner product that requires 4 ( N / 2 ) 2 = N 2 real multiplications and an equal number of additions. This is followed by thecalculations necessary for the simultaneous calculations of the first half and last half of C ( k ) which requires 4 ( N / 2 ) = 2 N real additions. This means the total QFT algorithm requires M 2 real multiplications and N 2 + 4 N real additions. These numbers along with those for the Goertzel algorithm [link] , [link] , [link] and the direct calculation of the DFT are included in the following table. Of the various order- N 2 DFT algorithms, the QFT seems to be the most efficient general method for anarbitrary length N .

Algorithm Real Mults. Real Adds Trig Eval.
Direct DFT 4 N 2 4 N 2 2 N 2
Mod. 2nd Order Goertzel N 2 + N 2 N 2 + N N
QFT N 2 N 2 + 4 N 2 N

Timings of the algorithms on a PC in milliseconds are given in the following table.

Algorithm N = 125 N = 256
Direct DFT 4.90 19.83
Mod. 2O. Goertzel 1.32 5.55
QFT 1.09 4.50
Chirp + FFT 1.70 3.52

These timings track the floating point operation counts fairly well.

Conclusions

The QFT is a straight-forward DFT algorithm that uses all of the possible symmetries of the DFT basis function with no requirements on the lengthbeing composite. These ideas have been proposed before, but have not been published or clearly developed by [link] , [link] , [link] , [link] . It seems that the basic QFT is practical and useful as a general algorithmfor lengths up to a hundred or so. Above that, the chirp z-transform [link] or other filter based methods will be superior. For special cases and shorter lengths, methods based on Winograd's theories willalways be superior. Nevertheless, the QFT has a definite place in the array of DFT algorithms and is not well known. A Fortran program isincluded in the appendix.

It is possible, but unlikely, that further arithmetic reduction could be achieved using the fact that W N has unity magnitude as was done in second-order Goertzel algorithm. It is also possible that some way ofcombining the Goertzel and QFT algorithm would have some advantages. A development of a complete QFT decomposition of a DFT of length- 2 M shows interesting structure [link] , [link] and arithmetic complexity comparable to average Cooley-Tukey FFTs. It does seem better suited toreal data calculations with pruning.

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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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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