# 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 .

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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