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Computational experiment

Experiment design

We wanted to determine the most computationally efficient demodulation method for the digital multitone scheme. In our experiment, we compared the operation counts of an optimized FFT meta algorithm, a partial DFT computation (the DFT computed only for the nonzero coefficients), and FFAST in demodulating various message signals. We chose these demodulation methods because they are currently the most efficient methods for DMT demodulation. The experiments were run using MATLAB 2014a. We chose to run our experiment in MATLAB for its rapid prototyping environment.

While we were interested in comparing the computational efficiency of these algorithms, we chose to record operation counts rather than run times. Run times are unreliable metrics on machines with multitasking operating systems, especially when using highly optimized programs like MATLAB. We chose to count complex additions and multiplications as one operation each. We did not count conditional statements as operations because in most general processors, they only require a single cycle. We counted complex exponentials and trigonometric functions as one operation because they may be implemented using lookup tables.

Meta-fft algorithm implementation

The two main goals for our implementation of the optimized FFT meta algorithm were to i) create an algorithm that performs without the need to zero-pad the signal and ii) allows us to count operations. Well-behaved signal lengths in our implementation of the algorithm have the form of N = n 0 n 1 n 2 , where n i are coprimes. To exploit this structure, the optimized FFT meta algorithm, implemented in meta_fft.m , uses a self-sorting mixed-radix complex FFT [link] . For sub-transforms of length 2 N 6 , short-length Winograd transforms are applied to conserve operation count. All other transforms are computed using Rader's FFT algorithm [link] .

Ffast implementation

The FFAST algorithm implementation requires the signal itself, the length of the signal, and a vector of the downsampling coefficients.

The FFAST algorithm was implemented in two files. The first file, ffast_front_end.m , downsamples the signal by each of the coprimes and feeds shifted and unshifted versions of the downsampling to the meta_fft.m file, to get operation counts and the relevant FFTs. Once the relevant DFT pairs are generated, ffast_front_end.m calls the back end of the algorithm.

The second file, peeling_decoder.m , implements the peeling module of FFAST to backsolve the bipartite graph. The program will return a flag if the algorithm encounters no singletons at a stage where it has not been fully solved.

Numerical results

In our first experiment, we varied the signal length N and observed the operation counts required for the optimized FFT meta algorithm, the partial DFT, and FFAST. We constructed each signal in the Fourier domain by randomly selecting k values from the set of integers { 1 , , N } and setting the corresponding DFT coefficients to k 2 . For each signal, we put k = N 1 / 3 , which is the greatest allowed sparsity in our scheme. The choice of k and values of the k nonzero DFT coefficients are consistent with the DMT scheme. We used MATLAB's library function ifft() to compute the corresponding signal and counted the number of operations it took each algorithm to compute the DFT. We observed that the operation count required for FFAST was usually an order of magnitude less that the operation counts required for both the meta FFT algorithm and the partial DFT. See Fig  [link] for the results of the first experiment.

Experiment 1

In our second experiment, we varied the signal sparsity k and observed the operation count required for the optimized FFT meta algorithm, the partial DFT, and FFAST. For this experiment, signal length N = 8740 and the signals were constructed in the Fourier domain, as before. We observed an 80 % computational decrease from the optimized FFT to FFAST for all k < N 1 3 . However, for k > N 1 3 , we observed signals for which FFAST did not converge. See Fig  [link] for the results of the second experiment.

Experiment 2

Note that the operation count for the partial DFT is less than the operation count for FFAST for k < 8 ; this is somewhat misleading because the way that the partial DFT is computed is slightly optimistic. The partial DFT computes only the nonzero coefficients, which are known a priori in this experiment. In the general framework of DMT, one would need to compute all possibly nonzero DFT coefficients, resulting in an operation count higher than that of FFAST.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
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Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Using ffast to decrease computation time in digital multitone communication. OpenStax CNX. Dec 17, 2014 Download for free at http://legacy.cnx.org/content/col11731/1.1
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