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A modification of this approach also works for a length which is an odd prime raised to some power: N = P M . This is a bit more complicated [link] , [link] but has been done for lengths of 9 and 25. For longer lengths, the conventional Cooley-Tukey type-two index map algorithm seems to be more efficient. For powers of two, there is no primitive root, and therefore, no simple conversionof the DFT into convolution. It is possible to use two generators [link] , [link] , [link] to make the conversion and there exists a set of length 4, 8, and 16 DFT algorithms of the form in [link] in [link] .

In [link] an operation count of several short DFT algorithms is presented. These are practical algorithms that can beused alone or in conjunction with the index mapping to give longer DFT's as shown in The Prime Factor and Winograd Fourier Transform Algorithms . Most are optimized in having either the theoretical minimum number of multiplications or theminimum number of multiplications without requiring a very large number of additions. Some allow other reasonable trade-offs betweennumbers of multiplications and additions. There are two lists of the number of multiplications. The first is the number of actualfloating point multiplications that must be done for that length DFT. Some of these (one or two in most cases) will be by rationalconstants and the others will be by irrational constants. The second list is the total number of multiplications given in the diagonalmatrix D in [link] . At least one of these will be unity ( theone associated with X ( 0 ) ) and in some cases several will be unity ( for N = 2 M ). The second list is important in programming the WFTA in The Prime Factor and Winograd Fourier Transform Algorithm: The Winograd Fourier Transform Algorithm .

Number of Real Multiplications and Additions for a Length-N DFT of Complex Data
Length N Mult Non-one Mult Total Adds
2 0 4 4
3 4 6 12
4 0 8 16
5 10 12 34
7 16 18 72
8 4 16 52
9 20 22 84
11 40 42 168
13 40 42 188
16 20 36 148
17 70 72 314
19 76 78 372
25 132 134 420
32 68 - 388

Because of the structure of the short DFTs, the number of real multiplications required for the DFT of real data is exactly half that required for complex data. The numberof real additions required is slightly less than half that required for complex data because ( N - 1 ) of the additions needed when N is prime add a real to an imaginary, and that is not actuallyperformed. When N = 2 m , there are ( N - 2 ) of these pseudo additions. The special case for real data is discussed in [link] , [link] , [link] .

The structure of these algorithms are in the form of X ' = C D A X or B T D A X or A T D A X from [link] and [link] . The A and B matrices are generally M by N with M N and have elements that are integers, generally 0 or ± 1 . A pictorial description is given in [link] .

This image consist of 6 vertical lines. The lines are labeled j0.588, j0.362, -j1.539, 0.559, -1.250, and 1.000 from lowest to highest. The lowest line is shorter and from this line there are two diagonal lines that extend upward to the fourth and fifth lines. There are also lines extending up from line two to the top line as well as a line extending from line three to five and then down from that point to line 4. The lines on the figure are symmetrical. So, any line described has an analog line on the opposite side of the figure.
Flow Graph for the Length-5 DFT
This image has a rectangle with an x at its center. There is a trapezoid formed on the right and left sides of the rectangle. Each of these trapezoids contains plus symbols. On the opposite ends of the trapezoid, the sides not formed by the rectangle, there are three lines then three dots and then another line proceeding vertically from top to bottom. These lines are symmetrical on both sides.
Block Diagram of a Winograd Short DFT

The flow graph in [link] should be compared with the matrix description of [link] and [link] , and with the programs in [link] , [link] , [link] , [link] and the appendices. The shape in [link] illustrates the expansion of the data by A . That is to say, A X has more entries than X because M > N . The A operator consists of additions, the D operator gives the M multiplications (some by one) and A T contracts the data back to N values with additions only. M is one half the second list of multiplies in [link] .

Questions & Answers

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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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