62,838 research outputs found
Fast and Exact Spin-s Spherical Harmonic Transforms
We demonstrate a fast spin-s spherical harmonic transform algorithm, which is
flexible and exact for band-limited functions. In contrast to previous work,
where spin transforms are computed independently, our algorithm permits the
computation of several distinct spin transforms simultaneously. Specifically,
only one set of special functions is computed for transforms of quantities with
any spin, namely the Wigner d-matrices evaluated at {\pi}/2, which may be
computed with efficient recursions. For any spin the computation scales as
O(L^3) where L is the band-limit of the function. Our publicly available
numerical implementation permits very high accuracy at modest computational
cost. We discuss applications to the Cosmic Microwave Background (CMB) and
gravitational lensing.Comment: 22 pages, preprint format, 5 figure
CZT-Based Harmonic Analysis in Smart Grid Using Low-Cost Electronic Measurement Boards
This paper validates the use of a harmonic analysis algorithm on a microcontroller to perform measurements of non-stationary signals in the context of smart grids. The increasing presence of electronic devices such as inverters of distributed generators (DG), power converters of charging stations for electric vehicles, etc. can drain non-stationary currents during their operation. A classical fast Fourier transform (FFT) algorithm may not have sufficient spectral resolution for the evaluation of harmonics and inter-harmonics. Thus, in this paper, the implementation of a chirp-Z transform (CZT) algorithm is suggested, which has a spectral resolution independent from the observation window. The CZT is implemented on a low-cost commercial microcontroller, and the absolute error is evaluated with respect to the same algorithm implemented in the LabVIEW environment. The results of the tests show that the CZT implementation on a low-cost microcontroller allows for accurate measurement results, demonstrating the feasibility of reliable harmonic analysis measurements even in non-stationary conditions on smart grids
S2LET: A code to perform fast wavelet analysis on the sphere
We describe S2LET, a fast and robust implementation of the scale-discretised
wavelet transform on the sphere. Wavelets are constructed through a tiling of
the harmonic line and can be used to probe spatially localised, scale-depended
features of signals on the sphere. The scale-discretised wavelet transform was
developed previously and reduces to the needlet transform in the axisymmetric
case. The reconstruction of a signal from its wavelets coefficients is made
exact here through the use of a sampling theorem on the sphere. Moreover, a
multiresolution algorithm is presented to capture all information of each
wavelet scale in the minimal number of samples on the sphere. In addition S2LET
supports the HEALPix pixelisation scheme, in which case the transform is not
exact but nevertheless achieves good numerical accuracy. The core routines of
S2LET are written in C and have interfaces in Matlab, IDL and Java. Real
signals can be written to and read from FITS files and plotted as Mollweide
projections. The S2LET code is made publicly available, is extensively
documented, and ships with several examples in the four languages supported. At
present the code is restricted to axisymmetric wavelets but will be extended to
directional, steerable wavelets in a future release.Comment: 8 pages, 6 figures, version accepted for publication in A&A. Code is
publicly available from http://www.s2let.or
Spherical harmonic transform with GPUs
We describe an algorithm for computing an inverse spherical harmonic
transform suitable for graphic processing units (GPU). We use CUDA and base our
implementation on a Fortran90 routine included in a publicly available parallel
package, S2HAT. We focus our attention on the two major sequential steps
involved in the transforms computation, retaining the efficient parallel
framework of the original code. We detail optimization techniques used to
enhance the performance of the CUDA-based code and contrast them with those
implemented in the Fortran90 version. We also present performance comparisons
of a single CPU plus GPU unit with the S2HAT code running on either a single or
4 processors. In particular we find that use of the latest generation of GPUs,
such as NVIDIA GF100 (Fermi), can accelerate the spherical harmonic transforms
by as much as 18 times with respect to S2HAT executed on one core, and by as
much as 5.5 with respect to S2HAT on 4 cores, with the overall performance
being limited by the Fast Fourier transforms. The work presented here has been
performed in the context of the Cosmic Microwave Background simulations and
analysis. However, we expect that the developed software will be of more
general interest and applicability
Fast directional continuous spherical wavelet transform algorithms
We describe the construction of a spherical wavelet analysis through the
inverse stereographic projection of the Euclidean planar wavelet framework,
introduced originally by Antoine and Vandergheynst and developed further by
Wiaux et al. Fast algorithms for performing the directional continuous wavelet
analysis on the unit sphere are presented. The fast directional algorithm,
based on the fast spherical convolution algorithm developed by Wandelt and
Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature
implementation for Npix pixels on the sphere, and allows one to perform a
directional spherical wavelet analysis of a 10^6 pixel map on a personal
computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE
Trans. Sig. Pro
Modern Methods of Time-Frequency Warping of Sound Signals
Tato prĂĄce se zabĂœvĂĄ reprezentacĂ nestacionĂĄrnĂch harmonickĂœch signĂĄlĆŻ s ÄasovÄ promÄnnĂœmi komponentami. PrimĂĄrnÄ je zamÄĆena na Harmonickou transformaci a jeji variantu se subkvadratickou vĂœpoÄetnĂ sloĆŸitostĂ, Rychlou harmonickou transformaci. V tĂ©to prĂĄci jsou prezentovĂĄny dva algoritmy vyuĆŸĂvajĂcĂ Rychlou harmonickou transformaci. Prvni pouĆŸĂvĂĄ jako metodu odhadu zmÄny zĂĄkladnĂho kmitoÄtu sbĂranĂ© logaritmickĂ© spektrum a druhĂĄ pouĆŸĂvĂĄ metodu analĂœzy syntĂ©zou. Oba algoritmy jsou pouĆŸity k analĂœze ĆeÄovĂ©ho segmentu pro porovnĂĄnĂ vystupĆŻ. Nakonec je algoritmus vyuĆŸĂvajĂcĂ metody analĂœzy syntĂ©zou pouĆŸit na reĂĄlnĂ© zvukovĂ© signĂĄly, aby bylo moĆŸnĂ© zmÄĆit zlepĆĄenĂ reprezentace kmitoÄtovÄ modulovanĂœch signĂĄlĆŻ za pouĆŸitĂ HarmonickĂ© transformace.This thesis deals with representation of non-stationary harmonic signals with time-varying components. Its main focus is aimed at Harmonic Transform and its variant with subquadratic computational complexity, the Fast Harmonic Transform. Two algorithms using the Fast Harmonic Transform are presented. The first uses the gathered log-spectrum as fundamental frequency change estimation method, the second uses analysis-by-synthesis approach. Both algorithms are used on a speech segment to compare its output. Further the analysis-by-synthesis algorithm is applied on several real sound signals to measure the increase in the ability to represent real frequency-modulated signals using the Harmonic Transform.
Efficient Spherical Harmonic Transforms aimed at pseudo-spectral numerical simulations
In this paper, we report on very efficient algorithms for the spherical
harmonic transform (SHT). Explicitly vectorized variations of the algorithm
based on the Gauss-Legendre quadrature are discussed and implemented in the
SHTns library which includes scalar and vector transforms. The main
breakthrough is to achieve very efficient on-the-fly computations of the
Legendre associated functions, even for very high resolutions, by taking
advantage of the specific properties of the SHT and the advanced capabilities
of current and future computers. This allows us to simultaneously and
significantly reduce memory usage and computation time of the SHT. We measure
the performance and accuracy of our algorithms. Even though the complexity of
the algorithms implemented in SHTns are in (where N is the maximum
harmonic degree of the transform), they perform much better than any third
party implementation, including lower complexity algorithms, even for
truncations as high as N=1023. SHTns is available at
https://bitbucket.org/nschaeff/shtns as open source software.Comment: 8 page
Wavemoth -- Fast spherical harmonic transforms by butterfly matrix compression
We present Wavemoth, an experimental open source code for computing scalar
spherical harmonic transforms (SHTs). Such transforms are ubiquitous in
astronomical data analysis. Our code performs substantially better than
existing publicly available codes due to improvements on two fronts. First, the
computational core is made more efficient by using small amounts of precomputed
data, as well as paying attention to CPU instruction pipelining and cache
usage. Second, Wavemoth makes use of a fast and numerically stable algorithm
based on compressing a set of linear operators in a precomputation step. The
resulting SHT scales as O(L^2 (log L)^2) for the resolution range of practical
interest, where L denotes the spherical harmonic truncation degree. For low and
medium-range resolutions, Wavemoth tends to be twice as fast as libpsht, which
is the current state of the art implementation for the HEALPix grid. At the
resolution of the Planck experiment, L ~ 4000, Wavemoth is between three and
six times faster than libpsht, depending on the computer architecture and the
required precision. Due to the experimental nature of the project, only
spherical harmonic synthesis is currently supported, although adding support or
spherical harmonic analysis should be trivial.Comment: 13 pages, 6 figures, accepted by ApJ
A pseudospectral matrix method for time-dependent tensor fields on a spherical shell
We construct a pseudospectral method for the solution of time-dependent,
non-linear partial differential equations on a three-dimensional spherical
shell. The problem we address is the treatment of tensor fields on the sphere.
As a test case we consider the evolution of a single black hole in numerical
general relativity. A natural strategy would be the expansion in tensor
spherical harmonics in spherical coordinates. Instead, we consider the simpler
and potentially more efficient possibility of a double Fourier expansion on the
sphere for tensors in Cartesian coordinates. As usual for the double Fourier
method, we employ a filter to address time-step limitations and certain
stability issues. We find that a tensor filter based on spin-weighted spherical
harmonics is successful, while two simplified, non-spin-weighted filters do not
lead to stable evolutions. The derivatives and the filter are implemented by
matrix multiplication for efficiency. A key technical point is the construction
of a matrix multiplication method for the spin-weighted spherical harmonic
filter. As example for the efficient parallelization of the double Fourier,
spin-weighted filter method we discuss an implementation on a GPU, which
achieves a speed-up of up to a factor of 20 compared to a single core CPU
implementation.Comment: 33 pages, 9 figure
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