# 0.6 Winograd's short dft algorithms

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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.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
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
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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