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The results presented in this thesis show that vectorization at the algorithm level of abstraction produces good performance results, theconjugate-pair algorithm is in many cases faster than the ordinary split-radix algorithm, and that there are good heuristics for predicting the performance ofthe FFT on SIMD microprocessors (i.e., the need for empirical optimization may be overstated).

This work concludes with a review of the hypotheses, a summary of the contributions, some ideas for directions that future work might take, and a fewfinal remarks.

Revisiting the hypotheses

This section discusses the hypotheses of Introduction with reference to the experiments in Implementation details and Streaming FFT and the results in Results and discussion .

Hypothesis 1: accessing memory in sequential “streams” is critical for best performance

The simple implementation in Simple programs used a LUT to store precomputed coefficients, but for every size of sub-transformthat composes a particular transform, the LUT is accessed non-contiguously, with vector gather operations of varying strides. In Vectorized loops , vector intrinsics and a sequentially accessed LUT for each size of sub-transform are shown to improveperformance. Although the set of LUTs increases the memory footprint, the speed improves markedly, by over 30% in many cases.

In Improving memory locality in the leaves , a DAG representing the computation was topologically sorted so that accesses to the input data, which are effectivelypseudo-random for a decimation-in-time decomposition, are ordered into sequential streams. Benchmark results in Results and discussion show that this technique, in tandem with several others, achieves good results, being faster than FFTW in manycases.

The results from the above two cases confirm the idea that accessing data in sequential streams provides big performance gains, even in the somewhatcounter-intuitive case where data is duplicated and more memory is required.

Hypothesis 2: the conjugate-pair algorithm is faster than the ordinary split-radix algorithm

Hypothesis 2 is based on the idea that memory bandwidth is a bottleneck, and on the fact that the conjugate-pair algorithm requires only half the number oftwiddle factor loads.

In Results and discussion , a highly optimized implementation of the conjugate-pair algorithm is benchmarked against an equally highlyoptimized implementation of the ordinary split-radix algorithm. For smaller sizes of transform, the ordinary split-radix algorithm is faster, but above acertain size (4096 in this case), the conjugate-pair algorithm is faster.

Thus, Hypothesis 2 is confirmed with the proviso that the transform is larger than a particular size.

Hypothesis 3: the performance of an fft can be predicted based on characteristics of the underlying machine and the compiler

In Results and discussion , SFFT and FFTW were benchmarked on sixteen x86 machines and two ARM NEON machines, and SFFT was found to be as fast as,or faster than FFTW, suggesting that the performance of an FFT running on a certain machine can be predicted and reasoned about, and that extensivemachine calibration might not be required.

In Predicting performance , a model was evaluated with 10-fold cross-validation to have 74.8% precision when using characteristics of theunderlying machine and the compiler to predict performance, further supporting the idea that the performance of the FFT on SIMD microprocessors canbe predicted and reasoned about.

Contributions

The contributions of this work are summarized as follows:

  1. Three methods of computing the conjugate-pair algorithm on SIMD microprocessors are presented in Streaming FFT . The three techniques are suited for different sizes of transform, but in general, all techniquesare amenable to algorithm level vectorization, and latency and memory locality optimizations. These techniques are shown to produce results thatare, in many cases, faster than state of the art libraries such as FFTW and SPIRAL, but without extensive machine calibration;
  2. The source code for the library developed in this thesis, SFFT, is publicly available under a permissive open source license on github . A permissive open source license will hopefully ensure that SFFT is developed further.

Future work

This section presents some ideas for future work that can be divided into four categories: measurement, modelling, systems and applications.

Measurement

FFTW could be instrumented to collect data on the huge space of transforms it evaluates, which could then be used to build more accurate models.

The existing FFT benchmarking infrastructure could be improved by detecting interruption by other system processes and re-running the affectedresults. Benchmarks could then be run on a much wider range of machines, under more controlled conditions, which would increase the accuracy of models builtfrom the data.

Modelling

It might be possible to build a classifier that predicts whether a transform is likely, given some threshold, to be the fastest. The fastest is then selectedfrom a subset of those that are likely to be the fastest, and thus the number of transforms that must be evaluated during calibration is reduced, whilesacrificing little or no performance.

Systems

SFFT could be extended to multi-dimensional, multi-threaded, real, large (megapoint and above) and arbitrary sized transforms. Additionally, support forother architectures such as POWER and Cell B.E. could be added. Code could be optimized between transforms in a library, which would reduce binary size, butthere may be other effects.

Algorithms

So far, there have been no known attempts to seriously optimize the tangent FFT, and the results of optimizing the tangent FFT to the samedegree as the conjugate-pair FFT in this thesis would be very interesting.

SFFT could be utilized in the sparse FFT algorithms which have recently been published, perhaps improving their performance even further.

Applications

Applications such as the SETI@home client could be patched to support SFFT. The results of benchmarks between SFFT, FFTW and other libraries, when used in realworld applications such as SETI@home, would be of great interest.

Final remarks

This thesis showed that high-performance computation of the FFT is by no means a solved problem, and it is hoped that this work will serve as a catalystor foundation for future efforts that push efficiency and performance even further.

Questions & Answers

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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Computing the fast fourier transform on simd microprocessors. OpenStax CNX. Jul 15, 2012 Download for free at http://cnx.org/content/col11438/1.2
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