# 0.6 Winograd's short dft algorithms  (Page 2/11)

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The $h\left(n\right)$ terms are fixed for a digital filter, or they represent the $W$ terms from Multidimensional Index Mapping: Equation 1 if the convolution is being used to calculate a DFT. Because of this, $d=BH$ in [link] can be precalculated and only the $A$ and $C$ operators represent the mathematics done at execution of the algorithm. In order toexploit this feature, it was shown [link] , [link] that the properties of [link] allow the exchange of the more complicated operator $C$ with the simpler operator $B$ . Specifically this is given by

$Y=C\left[AXoBH\right]$
${Y}^{\text{'}}={B}^{T}\left[AXo{C}^{T}{H}^{\text{'}}\right]$

where $H$ ' has the same elements as $H$ , but in a permuted order, and likewise $Y$ ' and $Y$ . This very important property allows precomputing the more complicated ${C}^{T}H$ ' in [link] rather than $BH$ as in [link] .

Because $BH$ or ${C}^{T}H$ ' can be precomputed, the bilinear form of [link] and [link] can be written as a linear form. If an $M$ x $M$ diagonal matrix $D$ is formed from $d={C}^{T}H$ , or in the case of [link] , $d=BH$ , assuming a commutative property for $o$ , [link] becomes

${Y}^{\text{'}}={B}^{T}DAX$

$Y=CDAX$

In most cases there is no reason not to use the same reduction operations on $X$ and $H$ , therefore, $B$ can be the same as $A$ and [link] then becomes

${Y}^{\text{'}}={A}^{T}DAX$

In order to illustrate how the residue reduction is carried out and how the A matrix is obtained, the length-5 DFT algorithmstarted in The DFT as Convolution or Filtering: Matrix 1 will be continued. The DFT is first converted to a length-4 cyclic convolution by theindex permutation from The DFT as Convolution or Filtering: Equation 3 to give the cyclic convolution in The DFT as Convolution or Filtering . To avoid confusion from the permuted order of the data $x\left(n\right)$ in The DFT as Convolution or Filtering , the cyclic convolution will first be developed without thepermutation, using the polynomial $U\left(s\right)$

$U\left(s\right)=x\left(1\right)+x\left(3\right)s+x\left(4\right){s}^{2}+x\left(2\right){s}^{3}$
$U\left(s\right)=u\left(0\right)+u\left(1\right)s+u\left(2\right){s}^{2}+u\left(3\right){s}^{3}$

and then the results will be converted back to the permuted $x\left(n\right)$ . The length-4 cyclic convolution in terms of polynomials is

$Y\left(s\right)=U\left(s\right)\phantom{\rule{4pt}{0ex}}H\left(s\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{mod}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left({s}^{4}-1\right)$

and the modulus factors into three cyclotomic polynomials

$s4-1=\left({s}^{2}-1\right)\left({s}^{2}+1\right)$
$=\left(s-1\right)\left(s+1\right)\left({s}^{2}+1\right)$
$={P}_{1}\phantom{\rule{4pt}{0ex}}{P}_{2}\phantom{\rule{4pt}{0ex}}{P}_{3}$

Both $U\left(s\right)$ and $H\left(s\right)$ are reduced modulo these three polynomials. The reduction modulo ${P}_{1}$ and ${P}_{2}$ is done in two stages. First it is done modulo $\left({s}^{2}-1\right)$ , then that residue is further reduced modulo $\left(s-1\right)$ and $\left(s+1\right)$ .

$U\left(s\right)=u0+u1s+{u}_{2}{s}^{2}+{u}_{3}{s}^{3}$
${U}^{\text{'}}\left(s\right)={\left(\left(U\left(s\right)\right)\right)}_{\left({s}^{2}-1\right)}=\left({u}_{0}+{u}_{2}\right)+\left({u}_{1}+{u}_{3}\right)s$
$U1\left(s\right)={\left(\left({U}^{\text{'}}\left(s\right)\right)\right)}_{{P}_{1}}=\left({u}_{0}+{u}_{1}+{u}_{2}+{u}_{3}\right)$
$U2\left(s\right)={\left(\left({U}^{\text{'}}\left(s\right)\right)\right)}_{{P}_{2}}=\left({u}_{0}-{u}_{1}+{u}_{2}-{u}_{3}\right)$
$U3\left(s\right)={\left(\left(U\left(s\right)\right)\right)}_{{P}_{3}}=\left({u}_{0}-{u}_{2}\right)+\left({u}_{1}-{u}_{3}\right)s$

The reduction in [link] of the data polynomial [link] can be denoted by a matrix operation on a vector which has the data asentries.

$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{0}\\ {u}_{1}\\ {u}_{2}\\ {u}_{3}\end{array}\right]=\left[\begin{array}{c}{u}_{0}+{u}_{2}\\ {u}_{1}+{u}_{3}\end{array}\right]$

and the reduction in [link] is

$\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 1& 0& -1\end{array}\right]\left[\begin{array}{c}{u}_{0}\\ {u}_{1}\\ {u}_{2}\\ {u}_{3}\end{array}\right]=\left[\begin{array}{c}{u}_{0}-{u}_{2}\\ {u}_{1}-{u}_{3}\end{array}\right]$

$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& -1& 0\\ 0& 1& 0& -1\end{array}\right]\left[\begin{array}{c}{u}_{0}+{u}_{2}\\ {u}_{1}+{u}_{3}\\ {u}_{0}-{u}_{2}\\ {u}_{1}-{u}_{3}\end{array}\right]=\left[\begin{array}{c}{u}_{0}+{u}_{2}\\ {u}_{1}+{u}_{3}\\ {u}_{0}-{u}_{2}\\ {u}_{1}-{u}_{3}\end{array}\right]=\left[\begin{array}{c}{w}_{0}\\ {w}_{1}\\ {v}_{0}\\ {v}_{1}\end{array}\right]$

Further reduction of ${v}_{0}+{v}_{1}s$ is not possible because ${P}_{3}={s}^{2}+1$ cannot be factored over the rationals. However ${s}^{2}-1$ can be factored into ${P}_{1}{P}_{2}=\left(s-1\right)\left(s+1\right)$ and, therefore, ${w}_{0}+{w}_{1}s$ can be further reduced as was done in [link] and [link] by

$\left[\begin{array}{cc}1& 1\end{array}\right]\left[\begin{array}{c}{w}_{0}\\ {w}_{1}\end{array}\right]={w}_{0}+{w}_{1}={u}_{0}+{u}_{2}+{u}_{1}+{u}_{3}$
$\left[\begin{array}{cc}1& -1\end{array}\right]\left[\begin{array}{c}{w}_{0}\\ {w}_{1}\end{array}\right]={w}_{0}-{w}_{1}={u}_{0}+{u}_{2}-{u}_{1}-{u}_{3}$

$\left[\begin{array}{cccc}1& 1& 0& 0\\ 1& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& -1& 0\\ 0& 1& 0& -1\end{array}\right]\left[\begin{array}{c}{u}_{0}\\ {u}_{1}\\ {u}_{2}\\ {u}_{3}\end{array}\right]=\left[\begin{array}{c}{r}_{0}\\ {r}_{1}\\ {v}_{0}\\ {v}_{1}\end{array}\right]$

The same reduction is done to $H\left(s\right)$ and then the convolution of [link] is done by multiplying each residue polynomial of $X\left(s\right)$ and $H\left(s\right)$ modulo each corresponding cyclotomic factor of $P\left(s\right)$ and finally a recombination using the polynomial Chinese RemainderTheorem (CRT) as in Polynomial Description of Signals: Equation 9 and Polynomial Description of Signals: Equation 13 .

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
Is there any normative that regulates the use of silver nanoparticles?
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Kyle
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Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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Daniel
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it is a goid question and i want to know the answer as well
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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.
Tarell
what is the actual application of fullerenes nowadays?
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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