# 0.6 Winograd's short dft algorithms  (Page 9/11)

 Page 9 / 11

Here we use three methods to facilitate the construction of prime length FFT modules. First, the matrix exchange property [link] , [link] , [link] is used so that the transpose of the reduction operator can be used rather than the morecomplicated CRT reconstruction operator. This is then combined with the numerical method [link] for obtaining the multiplication coefficients rather than the direct use of the CRT. We also deviate from the Toom-Cook algorithm,because it requires too many additions for the lengths in which we are interested. Instead we use an iterated polynomial multiplication algorithm [link] . We have incorporated these three ideas into a single structural procedure that automatesthe design of prime length FFTs.

## Matrix description

It is important that each step in the Winograd FFT can be described using matrices. By expressing cyclic convolution as a bilinear form, a compact form of prime length DFTs can be obtained.

If $y$ is the cyclic convolution of $h$ and $x$ , then $y$ can be expressed as

$y=C\left[Ax.*Bh\right]$

where, using the Matlab convention, $.*$ represents point by point multiplication. When $A$ , $B$ , and $C$ are allowed to be complex, $A$ and $B$ are seen to be the DFT operator and $C$ , the inverse DFT. When only real numbers are allowed, $A$ , $B$ , and $C$ will be rectangular. This form of convolution is presented with many examples in [link] . Using the matrix exchange property explained in [link] and [link] this form can be written as

$y=R{B}^{T}\left[{C}^{T}Rh.*Ax\right]$

where $R$ is the permutation matrix that reverses order.

When $h$ is fixed, as it is when considering prime length DFTs, the term ${C}^{T}Rh$ can be precomputed and a diagonal matrix $D$ formed by $D=diag\left\{{C}^{T}Rh\right\}$ . This is advantageous because in general, $C$ is more complicated than $B$ , so the ability to “hide" $C$ saves computation. Now $y=R{B}^{T}DAx$ or $y=R{A}^{T}DAx$ since $A$ and $B$ can be the same; they implement a polynomial reduction. The form $y={R}^{T}DAxT$ can also be used for the prime length DFTs, it is only necessary to permute the entries of x and to ensure that theDC term is computed correctly. The computation of the DC term is simple, for the residue of a polynomial modulo $a-1$ is always the sum of the coefficients. After adding the ${x}_{0}$ term of the original input sequence, to the $s-l$ residue, the DC term is obtained. Now $DFT\left\{x\right\}=R{A}^{T}DAx$ . In [link] Johnson observes that by permuting the elements on the diagonal of $D$ , the output can be permuted, so that the $R$ matrix can be hidden in $D$ , and $DFT\left\{x\right\}={A}^{T}DAx$ . From the knowledge of this form, once $A$ is found, $D$ can be found numerically [link] .

## Programming the design procedure

Because each of the above steps can be described by matrices, the development of a prime length FFTs is made convenient with the use of a matrix oriented programminglanguage such as Matlab. After specifying the appropriate matrices that describe the desired FFT algorithm, generating code involves compiling the matrices into the desiredcode for execution.

Each matrix is a section of one stage of the flow graph that corresponds to the DFT program. The four stages are:

1. Permutation Stage: Permutes input and output sequence.
2. Reduction Stage: Reduces the cyclic convolution to smaller polynomial products.
3. Polynomial Product Stage: Performs the polynomial multiplications.
4. Multiplication Stage: Implements the point-by-point multiplication in the bilinear form.

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
how did you get the value of 2000N.What calculations are needed to arrive at it
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