# 0.6 Results and discussion  (Page 5/5)

 Page 5 / 5

## Applications of this work

This section provides an overview of how the techniques presented in this thesis may be applied to the prime-factor algorithm, sparse Fourier transforms, and multi-threaded transforms.

## Prime-factor algorithm

The techniques presented in this work rely on the fact that FFTs operating on signal lengths that are a power-of-two can be factored into smaller power-of-two length components, which are computed in parallel by being evenly divided into a number of SIMD vector registers that are a power-of-two length.

The prime-factor algorithm factors other lengths of FFTs into components that are co-prime in length, and ultimately small prime components, which do not evenly divide into the power-of-two length SIMD registers, except in the special case where a SIMD register contains only one complex element (such is the case with double-precision on SSE machines).

Because the prime components do not evenly divide into power-of-two length SIMD registers, the algorithm level vectorization techniques presented in this work are not directly applicable. In contrast, the auto-vectorization techniques used in SPIRAL  [link] , [link] , [link] are performed at the instruction level, and are applicable to the prime-factor algorithm, but as the results in [link] show, the downside of SPIRAL's lower level approach is that performance for power-of-two transforms scales poorly with the length of the SIMD register.

## Sparse fourier transforms

The recently published Sparse FFT  [link] , [link] will benefit from the techniques presented in this work because the inner loops use small DFTs (e.g, 512 point for a certain 256k point sparse FFT), which are currently computed with FFTW. Replacing FFTW with SFFT will almost certainly result in improved performance, because SFFT is faster than both FFTW and Intel IPP for the applicable small sizes of transform on an Intel Core i7-2600 (see [link] ).

Version 2.0 of the Sparse FFT code is scalar, and would benefit greatly from explicitly describing the computation with SIMD intrinsics. However, a key difference between the sparse Fourier transform and other FFTs is the use of conditional branches on the input signal data. This has performance implications on all machines, but it is worth noting that some machines will be drastically affected by this, such as the ARM Cortex-A8, where the SIMD pipeline is located behind the main pipeline, resulting in fast transfers from the main CPU unit to the SIMD pipeline, but large penalties when SIMD registers or flags are accessed by the main CPU unit.

MatrixFFT has recently shown that the four-step algorithm  [link] , designed to efficiently use hierarchical or external memory on Cray machines in the 1980's, is useful for computing large multi-threaded transforms on modern machines, providing performance far surpassing that of FFTW's multi-threaded performance  [link] .

The four-step algorithm decomposes a transform of size $N$ into a two-dimensional array of size ${n}_{1}×{n}_{2}$ where $N={n}_{1}{n}_{2}$ , and ${n}_{1}={n}_{2}=\sqrt{N}$ (or close) often obtains the best performance.

The four-steps of the algorithm are:

1. Compute ${n}_{1}$ FFTs of length ${n}_{2}$ along the columns of the array;
2. Multiply each element of the array with ${\omega }_{N}^{ij}$ , where $i$ and $j$ are the array coordinates;
3. Transpose the array;
4. Compute ${n}_{2}$ FFTs of length ${n}_{1}$ along the columns of the array.

Each step can be divided amongst a pool of threads, with a synchronisation barrier between the third and fourth steps. The transforms in steps one and four operate on sequential data, and if they are small enough, they are not subject to bandwidth limitations (and if they are not small enough, they can be further decomposed with the four-step algorithm until they are small enough). The bandwidth bottleneck does not disappear, but it is factored out into the transpose in step three, and because of this, the performance of the small single-threaded 1D transforms used in steps one and four correlate with the overall multi-threaded performance. A simple multi-threaded implementation of the four-step algorithm was benchmarked with SFFT and FFTW transforms, and the results are shown in [link] , which tends to confirm that the performance of single-threaded transforms for steps one and four translates to the overall multi-threaded performance when using the four-step algorithm.

## Similar work

Aside from Bernstein's FFT library, which was designed in the days of scalar microprocessors and has not been updated since 1999, there have been a few other challenges to the automatically adaptive approach of FFTW, but none present concrete results that definitively dismiss the idea. Most recently, Vasilios et al. presented an approach that uses the characteristics of the host machine to choose good FFT parameters at run time  [link] , but their approach has several issues that render it almost irrelevant. First, the approach uses optimizations that only apply to scalar machines, viz. twiddle factor symmetries are exploited to compress the twiddle LUTs, and arithmetic is avoided when twiddle factors contains zeros or ones. The vast majority of microprocessors, even those found in mobile devices such as phones, feature SIMD extensions, and so an approach that is limited to scalar arithmetic is of little consequence. Second, they benchmark the FFTs in a most unusual way. Rather than repeat a large number of iterations of the FFT, they repeat a large number of iterations of a binary that initializes and then executes only one FFT; such an approach is by no means representative of applications where the performance of the FFT is a concern, and is more a measurement of the initialization time rather than the FFT.

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
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
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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