# Large dft modules: 11, 13, 16, 17, 19, and 25  (Page 2/5)

 Page 2 / 5

The length 25 module does not follow the traditional Winograd approach. This module is an in-line code version of a common-factor 5x5 DFT. Each length 5 DFT is a prime-length convolutional module. The output unscrambling is included in the assignment statements at the end of the program. Some of the length 5 modules used in this program are implemented as scaled versions of conventional length 5 modules in order to save some multiplies by 1/4. The scaling factors are then compensated for by adjusting the twiddle factors. This module has three multiply sections, one for the row DFT's with a data expansion factor of 6/5, one for the twiddle factors (expansion=33/25) and on for the column DFT's (expansion=6/5).

Modules for lengths 11 and 13 are very similar in spirit to the length 19 and 17 modules. Derivations are presented for both the 11 and 13 length modules which are consistent with the listings, although these interpretations may not agree with the original intentions of the designer [link] they are correct in the sense that the algorithms could have been derived in the stated manner. Both the modules are of prime length and they are implemented in Winograd's convolutional style.

FORTRAN listings for all five modules are included with this report in a subroutine form suitable for use in Burrus' PFA program [link] . Addition and multiplication counts given are for complex input data.

## 17 module: 314 adds / 70 mpys

1. Use the index map $\overline{x}\left(n\right)=x\left(<{3}^{n}{>}_{mod17}\right)$ to convert the DFT into a length 16 convolution, plus a correction term for the DC component.
2. Reduce the length 16 convolution modulo all the irreducible factors of ${z}^{16}-1$ . (Irreducible over the rationals).
$\begin{array}{ccc}\hfill mod{z}^{8}+1& :& r108-r115\hfill \\ \hfill mod{z}^{8}-1& :& r100-r107\hfill \end{array}$
From ${z}^{8}-1$ data
$\begin{array}{ccc}\hfill mod{z}^{4}+1& :& r31-r34\hfill \\ \hfill mod{z}^{4}-1& :& r200-r203\hfill \end{array}$
From ${z}^{4}-1$ data
$\begin{array}{ccc}\hfill mod{z}^{2}+1& :& r35-r36\hfill \\ \hfill mod{z}^{2}-1& :& r204-r205\hfill \end{array}$
From ${z}^{2}-1$ data
$\begin{array}{ccc}\hfill modz+1& :& r38\hfill \\ \hfill modz-1& :& r37\hfill \end{array}$
3. Reduce the convolution modulo ${z}^{2}+1$ using Toom-Cook factors of $z$ , $1/z$ and $z+1$ . This creates variables r35, r36, and r314.
4. Reduce the modulo ${z}^{4}+1$ convolution with an iterated Toom-Cook reduction using the factors $z$ , $1/z$ and $z-1$ for the first step, and the factors $z$ , $1/z$ and $z+1$ for the second step. The first step produces r310 and r39, and the second step computes r313, r312 and r311. This is exactly the reduction procedure used in Nussbaumer's ${z}^{4}+1$ convolution algorithm.
5. Patch up the DC term by adding the $z-1$ reduction result to $x\left(i\left(1\right)\right)$ .
6. Use Nussbaumer's ${z}^{8}+1$ convolution algorithm [link] on r108-r115. This is the only exception to the strict use of transposing the tensor, as his algorithm saves two additions by computing the transposed reconstruction procedure in an obscure fashion. The result, however, is an exact calculation of the transpose. This reduction computes twenty-one values, r315-r335, which must be weighted by coefficients to produce the reconstructed ${z}^{8}+1$ output, t115-t135.
7. Weight the variables r31-r39, r310-r314 by coefficients to produce t11-t19, t110-t114.
8. The reconstruction procedure for the ${z}^{8}-1$ terms is a straightforward transpose of the reduction procedure.
9. The ${z}^{16}-1$ convolution result is reconstructed from the ${z}^{8}-1$ (real) and ${z}^{8}+1$ (imaginary) vectors and mapped back to the outputs using the reverse of the input map.
10. All coefficients were computed using the author's QR decompositionlinear equation solver and are accurate to at least 14 places.

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
Do somebody tell me a best nano engineering book for beginners?
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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
I'm interested in nanotube
Uday
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!