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

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Now, suppose we combine length p and q modules with Good's prime factor algorithm (not using twiddles). The following scaling procedure will work:

• Assume the input data has been appropriately loaded into a $pxq$ data array
• Scale the non-DC outputs of the length $p$ module and apply the modified module to all columns of the data array.
• Now all the rows are scaled by $\left(p-1\right)$ except the zeroeth row, corresponding to the DC outputs of the length $p$ modules. Apply a normal length $q$ module to the zeroeth row. Modify the length $q$ module to scale by $1/\left(p-1\right)$ and apply the modified version to all the other rows. The DFT is now complete.

As an example, consider the 3x7 DFT. In the length 3 module scaling the non-DC outputs trades one multiply for one add. When the scaled DFT is constructed, the modified length 3 module is used 7 times. But two rows must be scaled by modified length 7 modules, which brings the total multiply savings to 5 at a cost of 7 adds. This looks like a nice tradeoff. The total number of multiplies in a normal 3x7 PFA is 38.

These ideas can be expanded to multidimensional cases, although it quickly becomes difficult to keep track of which rows and columns need to be counter-scaled.

## Length 11 module: 168 adds / 40 mpys

1. Use the index map $\overline{x}\left(n\right)=x\left(<{8}^{n}{>}_{mod11}\right)$ to convert the DFT into a length 10 convolution, plus a correction term for the DC components.
2. Reduce the length 10 convolution modulo all the irreducible factors of ${z}^{10}-1$
$\begin{array}{ccc}\hfill mod{z}^{5}-1& :& T1,T3,T2,T5,T4\hfill \\ \hfill mod{z}^{5}+1& :& T6,-T8,-T7,-T10,T9\hfill \end{array}$
from ${z}^{5}-1$ data
$\begin{array}{ccc}\hfill modz-1& :& T13\hfill \\ \hfill mod{z}^{5}-1/z-1& :& AM4,AM7,AM3,AM6\left(afterweighting\right)\hfill \end{array}$
from ${z}^{5}+1$ data
$\begin{array}{ccc}\hfill modz+1& :& AM2\left(afterweighting\right)\hfill \\ \hfill mod{z}^{5}+1/z+1& :& S9,S11,S10,S12\left(appearsin\right)\hfill \end{array}$
3. Patch up the DC terms by adding the $z-1$ reduction result to $X\left(I\left(1\right)\right)$ and store the result in AMO.
4. The ${z}^{5}-1$ convolution proceeds in four steps. First, do the irreducible factor reductions, then reduce further with an iterated Toom-Cook procedure, weight all remaining variables, and apply the transpose of the complete reduction stage to the weighted results. The first Toom-Cook reduction uses the factors $z$ , $1/z$ and $z+1$ on the vectors AM4,AM3 and AM7,AM6 which generates the new vector AM4-AM7,AM3-AM6. Each of the original two vectors is then individually reduced using factors of $z$ , $1/z$ and $z+1$ , while the new vector is reduced by $A$ , $1/z$ and $z-1$ . This procedure generates nine variables: AM4,AM3,AM5; AM7,AM6,AM8; S7,S8,AM11. (The expressions for S6 and S8 contain the variables of interest).
5. The nine variables from 4) are weighted along with T13.
6. An exact transpose of the reduction algorithm is applied to the weighted variables (and AMO).
7. The result S16,S15,S18,S17,S19 is the real part of the answer and is mapped back to the output using the map $\overline{x}\left(n\right)=x\left(<{8}^{n+1}>mod11$ . This is an unusual map, but it is perfectly acceptable.
8. A in the length 19 transform the ${z}^{5}+1$ convolution is computed with a variation of the ${z}^{5}-1$ algorithm. First the inputs T6,-T8,-T7,-T1O,T9 are alternately negated, then the ${z}^{5}-1$ algorithm is applied The second stage of the Toom-Cook reductions uses the factors z, liz and z+l for all three length two vectors. Also, the DC patch is not used here. and the outputs alternately negated.
9. The result S21,S20,S23,S22,S24, representing the imaginary part of the answer, is mapped back to the output using the map $\overline{x}\left(n\right)=x\left(<{8}^{n+1}>mod11\right)$ .
10. In both this algorithm and the length 13 DFT plus and minus signs have been freely altered to force all constants to be positive. Also, many shortcut computations were used to save adds, obscuring in some places the logical flow of the algorithm.
11. All coefficients were computed using the author's QR decomposition linear equation solver and are accurate to at least 14 places.

## Length 13 module: 188 adds / 40 mpys

1. Use the index map $\overline{x}\left(n\right)=x\left(<{2}^{n}{>}_{mod13}\right)$ to convert the DFT into a length 12 convolution, plus a correction term for the DC components.
2. Reduce the length 12 convolution modulo all the irreducible factors of ${z}^{12}-1$
$\begin{array}{ccc}\hfill mod{z}^{6}+1& :& A7,A8,A9,A10,A11,A12\hfill \\ \hfill mod{z}^{6}-1& :& A1,A2,A3,A4,A5,A6\hfill \end{array}$
from ${z}^{6}-1$ data
$\begin{array}{ccc}\hfill mod{z}^{2}-1& :& A14,A13\hfill \\ \hfill mod{z}^{2}-z+1& :& A23,A22\hfill \\ \hfill mod{z}^{2}+z+1& :& A25,A24\hfill \end{array}$
from ${z}^{2}-1$ data
$\begin{array}{ccc}\hfill modz-1& :& A15\hfill \\ \hfill modz+1& :& implicit\left(A13-A14\right)\hfill \end{array}$
from ${z}^{6}+1$ data
$\begin{array}{ccc}\hfill mod{z}^{2}+1& :& A17,A16\hfill \\ \hfill mod{z}^{4}-{z}^{2}+1& :& A27,A26,-A31,-A30\hfill \end{array}$
3. Patch up the DC terms by adding the $z-1$ reduction result to $X\left(I\left(1\right)\right)$ and store the result in AMO.
4. The ${z}^{2}-z+1$ and ${z}^{2}+z+1$ convolutions are reduced using Toom-cook factors of $z$ , $1/z$ and $z+1$ in one case and $z$ , $1/z$ and $z-1$ in the other case, and then all the reduced quantities are weighted by constants generating new variables: from ${z}^{2}-z+1$
$\begin{array}{ccc}\hfill z& & AM7\hfill \\ \hfill 1/z& & AM6\hfill \\ \hfill z-1& & AM8\hfill \end{array}$
from ${z}^{2}+z+1$
$\begin{array}{ccc}\hfill z& & AM10\hfill \\ \hfill 1/z& & AM9\hfill \\ \hfill z+1& & AM11\hfill \end{array}$
5. The original $modz+1$ reduction quantity is weighted and passed, along with AMO and the above six variables, to a reconstruction procedure which first combines the $z-1$ and ${z}^{2}+z+1$ data to compute the convolution mod ${z}^{3}-1$ (CC4,CC5,CC6), and then combines the $z+1$ and ${z}^{2}-z+1$ data to compute the convolution mod ${z}^{3}+1$ (CC1,CC2,CC3). These two vectors are combined to compute the complete ${z}^{6}-1$ output, which appears in permuted form in CC15 through CC20.
6. The ${z}^{2}+1$ vector is decomposed with Toom-Cook factors of $z$ , $1/z$ and $z+1$ yielding A17,A16 and the implicit term (A16+A17).
7. The ${z}^{4}-{z}^{2}+1$ vector is decomposed with a double iterated Toom-Cook scheme. First the vector is broken into two length two pieces: A27,A26 and A31,A30. Then the vectors are reduced by the factors of $z$ , $1/z$ and $z+1$ operating on whole vectors to produce a set of three length two vectors: Ā27,A26A31,A30 A29,A28 = (A27+A31), (A26+A30)These vectors are not calculated in a straightforward manner. Each length two vector is further reduced, in the second iteration, by the factors $z$ , $1/z$ and $z+1$ to create three new implicit variables $\left(A27+A26\right)$ , $\left(A31+A30\right)$ and $\left(A29+A28\right)$ .
8. The nine variables from [link] and the three variables from [link] are weighted by constants and the $mod{z}^{6}+1$ reconstruction proceeds in an ad-hoc fashion which closely resembles a transposed tensor method, but has some differences. The add count for the reconstruction would have been the same if the transposed tensor method had been applied. The ${z}^{6}+1$ result appears in permuted form in variables CC21 through CC26.
9. The final result is reconstructed from the ${z}^{6}-1$ and ${z}^{6}-1$ vectors. The DC term, $x\left(i\left(1\right)\right)$ is set equal' to AMO.
10. All coefficients were computed using the author's QR decomposition linear equation solver and are accurate to at least 14 places.

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
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