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A high-radix fast Fourier transformation (FFT) algorithm for computing transforms over GF(sq q), where q is a Mersenne prime, is developed to implement fast circular convolutions. This new algorithm requires substantially fewer multiplications than the conventional FFT.
Operator used in derivation of plasma kinetic equation, expressing integral of pair correlation function for stable and unstable cases via Fourier transform
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A conventional prime factor discrete Fourier transform (DFT) algorithm is used to realize a discrete Fourier-like transform on the finite field, GF(q sub n). A pipeline structure is used to implement this prime factor DFT over GF(q sub n). This algorithm is developed to compute cyclic convolutions of complex numbers and to decode Reed-Solomon codes. Such a pipeline fast prime factor DFT algorithm over GF(q sub n) is regular, simple, expandable, and naturally suitable for VLSI implementation. An example illustrating the pipeline aspect of a 30-point transform over GF(q sub n) is presented.
A conventional prime factor discrete Fourier transform (DFT) algorithm is used to realize a discrete Fourier-like transform on the finite field, GF(q sub n). This algorithm is developed to compute cyclic convolutions of complex numbers and to decode Reed-Solomon codes. Such a pipeline fast prime factor DFT algorithm over GF(q sub n) is regular, simple, expandable, and naturally suitable for VLSI implementation. An example illustrating the pipeline aspect of a 30-point transform over GF(q sub n) is presented.
Impedance of uniform electric current loop in anisotropic magnetoionic plasma, using Maxwell equations and Fourier transform
Accurate synthetic spectra that rely on large Line-By-Line (LBL)-databases are used in a wide range of applications such as high temperature combustion, atmospheric re-entry, planetary surveillance and laboratory plasmas. Conventionally synthetic spectra are calculated by computing a lineshape for every spectral line in the database and adding those together, which may take multiple hours for large databases. In this paper we propose a new approach for spectral synthesis based on an integral transform: the synthetic spectrum is calculated as the integral over the product of a Voigt profile and a newly proposed three-dimensional “lineshape distribution function”, which is a function of spectral position and Gaussian- & Lorentzian width coordinates. A fast discrete version of this transform based on the Fast Fourier Transform (FFT) is proposed, which improves performance compared to the conventional approach by several orders of magnitude while maintaining accuracy. Strategies that minimize the discretization error are discussed. A Python implementation of the method is compared against state-of-the-art spectral code RADIS, and is since adopted as RADIS's default synthesis method. The synthesis of a benchmark CO2 spectrum consisting of 1.8 M spectral lines and 200k spectral points took only 3.1 s using the proposed method (1011 lines × spectral points/s), a factor ~300 improvement over the state-of-the-art, with the relative improvement generally increasing for higher number of lines and/or number of spectral points. Finally, an experimental GPU-implementation of the method was also benchmarked, which demonstrated another 2~3 orders performance increase, achieving up to 5 ∙ 10 14 lines × spectral points/s.
Three methods for performing Hankel transforms with optical or digital processors are described. The first method is applicable when the input data is available in Cartesian (x-y) format and uses the close connection between generalized Hankel transform and the two dimensional Fourier transform in Cartesian coordinates. The second method is useful when the input data is in polar (r - theta) format and uses change of variables to perform the nth order Hankel transform as a correlation integral. The third method utilizes the von Neumann addition theorem for Bessel functions to extract the Hankel coefficients from a correlation between the radial part of the input and a Bessel function. Initial experimental results obtained for optical implementation of the first two methods are presented.
Convection electric field patterns at high latitudes appropriate to periods of low geomagnetic activity are presented. The postulate that the global Joule dissipation rate is a minimum is employed to derive characteristic field distributions for the situation when ring current ohmic losses are negligible and the auroral oval enhancement of height-integrated Hall conductivity is small. The fast Fourier transform is introduced to compute Fourier coefficients of discrete field-aligned current data, and the method is shown to be quite useful in this context as a least squares fitting routine. The model calculations demonstrate the approach of the high-latitude ionosphere toward the perfect shielding configuration predicted by the theory for quiet times.
The disadvantages of numerical inversion of the Laplace transform via the conventional fast Fourier transform (FFT) are identified and an improved method is presented to remedy them. The improved method is based on introducing a new integration step length Delta(omega) = pi/mT for trapezoidal-rule approximation of the Bromwich integral, in which a new parameter, m, is introduced for controlling the accuracy of the numerical integration. Naturally, this method leads to multiple sets of complex FFT computations. A new inversion formula is derived such that N equally spaced samples of the inverse Laplace transform function can be obtained by (m/2) + 1 sets of N-point complex FFT computations or by m sets of real fast Hartley transform (FHT) computations.
Maximizing stimulated natural and hydraulic fracture network is one of the primary hydraulic fracturing concerns for economic production from a horizontal shale gas well. Geomechanical facies and preexisting fractures in each stage are identified based on similarities in formation characteristics to optimize the locations of perforation clusters. This often requires analyzing large volumes of drilling, Logging While Drilling (LWD) and Measurement While Drilling (MWD) data. In this paper, we develop a methodology that calculates the mechanical specific energy (MSE) using real-time drill string acceleration signals directly from its definition. High resolution vibration signals have been collected using a tri-axial accerlometer, which was an auxiliary tool included in acoustic borehole imager. This technique provides a cost-efficient solution for engineered completion design. Furthermore, we adopt deep Convolutional Neural Network (CNN) with signal processing to build a data pipeline that effectively extracts patterns from dynamic acceleration signals for rock lateral MSE classification. First, we apply discrete wavelet transform and Short-Time Fourier Transform (STFT) for signal denoising and pattern recognition. Then we construct an image dataset using multi-scale image fusion at pixel level from 3 sensor channels, including axial, lateral acceleration spectrograms and zero-padded revolutions per minute (RPM). The resulted RGB image dataset includes 4,000 images of 5 MSE ranges with various rock strength conditions. Our results demonstrate that the proposed deep learning model can achieve more than 90% classification accuracy. The deep learning results, as a reference source, were applied in selected Marcellus Shale Energy and Environmental Lab (MSEEL) wells engineered completion located in the Marcellus shale gas site.
Noise analysis of spectrum obtained by Fourier transformation of interferogram
Dynamic response of simple beam and rectangular plate under traveling shock wave at supersonic speed, using Fourier transforms
This work reports on the computation of the average phase of a beam over an optical element via discrete Fourier transform techniques.
An encoding figure of merit is established for a detector-noise limited Fourier transform spectrometer (FTS) and compared to the comparable figure for a Hadamard transform spectrometer (HTS). The limitation of the Fourier system is partly that it does not truly Fourier analyze the radiation. Instead a cosine squared modulation is imposed on the different spectral frequencies. An additional difficulty is that neither the cosine nor the cosine squared functions form an orthonormal set. This makes the Fellgett's advantage (root-mean-squared figure of merit) for a single detector Michelson interferometer a factor of the square root of (N/8) greater than for a conventional grating instrument - rather than the square root of (N/2). The theoretical limit would be the square root of N.
Maxwell equation solution for source currents in unbounded magnetoionic medium with uniaxial dielectric tensor, using Fourier transform technique
Unitary Fourier transform lies at the core of the multitudinous computational and metrological algorithms. Here we show experimentally how the unitary Fourier transform-based phase estimation protocol, used namely in quantum metrology, can be translated into the classical linear optical framework. The developed setup made of beam splitters, mirrors and phase shifters demonstrates how the classical coherence, similarly to the quantum coherence, poses a resource for obtaining information about the measurable physical quantities. Our study opens route to the reliable implementation of the small-scale unitary algorithms on path-encoded qudits, thus establishing an easily accessible platform for unitary computation.
Source distribution determination from radiation pattern and field amplitudes expressed as Fourier transform pair