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Symmetries in the complex exponential function are again used to expose common computation among each part of the equation; hence

X k = U k + ( ω N k Z k + ω N 3 k Z k ' ) X k + N / 2 = U k - ( ω N k Z k + ω N 3 k Z k ' ) X k + N / 4 = U k + N / 4 - i ( ω N k Z k - ω N 3 k Z k ' ) X k + 3 N / 4 = U k + N / 4 + i ( ω N k Z k - ω N 3 k Z k ' )

which, when recursively applied to the sub-transforms, results in the following recurrence relation for real arithmetic operations:

T ( n ) = T ( n / 2 ) + 2 T ( n / 4 ) + 6 n - 4 for n 2 0 for n = 1

The exact solution T ( n ) = 4 n log 2 n - 6 n + 8 for n 2 was the best arithmetic complexity of all known FFT algorithms for over 30 years, until VanBuskirk was able to break the record in 2004  [link] , as described in "Tangent" .

Van Buskirk's arithmetic complexity breakthrough was based on a variant of the split-radix algorithm known as the “conjugate-pair” algorithm  [link] or the “ - 1 exponent” split-radix algorithm  [link] , [link] . In 1989 the conjugate-pair algorithm was published with the claim that it had broken the record set by Yavne in 1968for the lowest number of arithmetic operations for computing the DFT  [link] . Unfortunately the reduction in the number of arithmetic operations was due to an error in the author's analysis, and thealgorithm was subsequently proven to have an arithmetic count equal to the original split-radixalgorithm  [link] , [link] , [link] . Despite initial claims about the arithmetic savings beingdiscredited, the conjugate-pair algorithm has been used to reduce twiddle factor loads in software implementations of the FFT and fast Hartleytransform (FHT)  [link] , and the algorithm was also recently used as the basis for analgorithm that does reduce the arithmetic operation count, as described in "Tangent" .

The difference between the conjugate-pair algorithm and the split-radix algorithm is in the decomposition of odd elements. In the standardsplit-radix algorithm, the odd elements are decomposed into two parts: x 4 n 4 + 1 and x 4 n 4 + 3 (see [link] ), while in the conjugate-pair algorithm, the last sub-sequence is cyclically shiftedby - 4 , where negative indices wrap around (i.e., x - 1 = x N - 1 ). The result of this cyclic shift is that twiddle factors are nowconjugate pairs. Formally, the conjugate-pair algorithm is defined as:

X k = n 2 = 0 N / 2 - 1 ω N / 2 n 2 k x 2 n 2 + ω N k n 4 = 0 N / 4 - 1 ω N / 4 n 4 k x 4 n 4 + 1 + ω N - k n 4 = 0 N / 4 - 1 ω N / 4 n 4 k x 4 n 4 - 1

As with the ordinary split-radix algorithm, a DIT decomposition of the conjugate-pair algorithm can be expressed as a system of equations:

X k = U k + ( ω N k Z k + ω N - k Z k ' ) X k + N / 2 = U k - ( ω N k Z k + ω N - k Z k ' ) X k + N / 4 = U k + N / 4 - i ( ω N k Z k - ω N - k Z k ' ) X k + 3 N / 4 = U k + N / 4 + i ( ω N k Z k - ω N - k Z k ' )

where k = 0 , , N / 4 - 1 . As can be seen, the trigonometric coefficients are conjugates – a feature that can be exploited to reduce twiddle factorloads.


In 2004, some thirty years after Yavne set the record for the lowest arithmetic operation count, Van Buskirk posted software to Usenet that hadasymptotically reduced the arithmetic operation count by about 6%. Three papers were subsequently published  [link] , [link] , [link] with differing explanations on how to achieve the lowest arithmetic operation count initially demonstrated by Van Buskirk.

Although all three papers describe algorithms that achieve the lowest arithmetic operation count in the same way, and thus can be considered to bedifferent views of the same algorithm, all three papers refer to the algorithms by different names. Lundy and Van Buskirk  [link] refer to their algorithm as “scaled odd tail FFT”, Bernstein  [link] describes an algorithm named “tangent FFT”, while Johnson and Frigo  [link] refer to the algorithm by various names. Many works have cited Johnson and Frigo for the algorithm  [link] . Of these names, “tangent FFT” is used in this work because it is the mostdescriptive; scaling the twiddle factors into tangent form was the linchpin of Van Buskirk's breakthrough in arithmetic complexity.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
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Source:  OpenStax, Computing the fast fourier transform on simd microprocessors. OpenStax CNX. Jul 15, 2012 Download for free at http://cnx.org/content/col11438/1.2
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