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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.

Tangent

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

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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good afternoon madam
AMJAD
what is system testing
AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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