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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

This module closely follows the traditional Winograd prime-length approach.

  1. Use the index map x ¯ ( n ) = x ( < 3 n > m o d 17 ) 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).
    m o d z 8 + 1 : r 108 - r 1 1 5 m o d z 8 - 1 : r 100 - r 107
    From z 8 - 1 data
    m o d z 4 + 1 : r 31 - r 34 m o d z 4 - 1 : r 200 - r 203
    From z 4 - 1 data
    m o d z 2 + 1 : r 35 - r 36 m o d z 2 - 1 : r 204 - r 205
    From z 2 - 1 data
    m o d z + 1 : r 38 m o d z - 1 : r 37
  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 ( i ( 1 ) ) .
  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.

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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Source:  OpenStax, Large dft modules: 11, 13, 16, 17, 19, and 25. revised ece technical report 8105. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10569/1.7
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