0.6 Winograd's short dft algorithms  (Page 12/11)

 Page 1 / 11

The operation of discrete convolution defined by

$y\left(n\right)=\sum _{k}h\left(n-k\right)\phantom{\rule{4pt}{0ex}}x\left(k\right)$

is called a bilinear operation because, for a fixed $h\left(n\right)$ , $y\left(n\right)$ is a linear function of $x\left(n\right)$ and for a fixed $x\left(n\right)$ it is a linear function of $h\left(n\right)$ . The operation of cyclic convolution is the same but with all indices evaluated modulo $N$ .

Recall from Polynomial Description of Signals: Equation 3 that length-N cyclic convolution of $x\left(n\right)$ and $h\left(n\right)$ can be represented by polynomial multiplication

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

This bilinear operation of [link] and [link] can also be expressed in terms of linear matrix operators and a simpler bilinearoperator denoted by $o$ which may be only a simple element-by-element multiplication of the two vectors [link] , [link] , [link] . This matrix formulation is

$Y=C\left[AXoBH\right]$

where $X$ , $H$ and $Y$ are length-N vectors with elements of $x\left(n\right)$ , $h\left(n\right)$ and $y\left(n\right)$ respectively. The matrices $A$ and $B$ have dimension $M$ x $N$ , and $C$ is $N$ x $M$ with $M\ge N$ . The elements of $A$ , $B$ , and $C$ are constrained to be simple; typically small integers or rational numbers. It will be thesematrix operators that do the equivalent of the residue reduction on the polynomials in [link] .

In order to derive a useful algorithm of the form [link] to calculate [link] , consider the polynomial formulation [link] again. To use the residue reduction scheme, the modulus is factored into relatively prime factors. Fortunately the factoringof this particular polynomial, ${s}^{N}-1$ , has been extensively studied and it has considerable structure. When factored over the rationals,which means that the only coefficients allowed are rational numbers, the factors are called cyclotomic polynomials [link] , [link] , [link] . The most interesting property for our purposes is that most of the coefficients of cyclotomic polynomialsare zero and the others are plus or minus unity for degrees up to over one hundred. This means the residue reduction will generallyrequire no multiplications.

The operations of reducing $X\left(s\right)$ and $H\left(s\right)$ in [link] are carried out by the matrices $A$ and $B$ in [link] . The convolution of the residue polynomials is carried out by the $o$ operator and the recombination by the CRT is done by the $C$ matrix. More details are in [link] , [link] , [link] , [link] , [link] but the important fact is the $A$ and $B$ matrices usually contain only zero and plus or minus unity entries and the $C$ matrix only contains rational numbers. The only general multiplications are those represented by $o$ . Indeed, in the theoretical results from computational complexity theory,these real or complex multiplications are usually the only ones counted. In practical algorithms, the rational multiplicationsrepresented by $C$ could be a limiting factor.

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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
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.
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!