scholarly journals Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral Data

2011 ◽  
Vol 1 (1) ◽  
pp. 9-16 ◽  
Author(s):  
J. Blais
Keyword(s):  
Fourier Transform ◽  
Spectral Data ◽  
Spherical Harmonic ◽  
Fourier Transforms ◽  
Data Matrix ◽  
Geopotential Model ◽  
Double Precision ◽  
Analysis And Synthesis ◽  
Double Precision Arithmetic ◽  
Band Limited

Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral DataSpherical Harmonic Transforms (SHTs) which are non-commutative Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known global strategies for discrete SHTs of band-limited spherical functions are Chebychev quadratures and least squares for equiangular grids. With proper numerical preconditioning, independent of latitude, reliable analysis and synthesis results for degrees and orders over 3800 in double precision arithmetic have been achieved and explicitly demonstrated using white noise simulations. The SHT synthesis and analysis can easily be modified for the ordinary Fourier transform of the data matrix and the mathematical situation is illustrated in a new functional diagram. Numerical analysis has shown very little differences in the numerical conditioning and computational efforts required when working with the two-dimensional (2D) Fourier transform of the data matrix. This can be interpreted as the spectral form of the discrete SHT which can be useful in multiresolution and other applications. Numerical results corresponding to the latest Earth Geopotential Model EGM 2008 of maximum degree and order 2190 are included with some discussion of the implications when working with such spectral sequences of fast decreasing magnitude.

scholarly journals Discrete Spherical Harmonic Transforms of Nearly Equidistributed Global Data

2011 ◽  
Vol 1 (3) ◽  
pp. 251-258 ◽  
Author(s):  
J. Blais
Keyword(s):  
Spherical Harmonic ◽  
Fourier Transforms ◽  
Simulated Data ◽  
Geopotential Model ◽  
Structural Constraints ◽  
Geopotential Models ◽  
Analysis And Synthesis ◽  
Comparative Results ◽  
Efficiency And Reliability ◽  
Global Data

Discrete Spherical Harmonic Transforms of Nearly Equidistributed Global DataDiscrete Spherical Harmonic Transforms (SHTs) are commonly defined for equiangular grids on the sphere. However, when global array data exhibit near equidistributed patterns rather than equiangular grids, discrete SHTs require appropriate adaptations for analysis and synthesis. Computational efficiency and reliability impose structural constraints on possible equidistribution characteristics of data patterns such as for instance with Chebychev quadratures and Fast Fourier Transforms (FFTs). Following some general introduction to discrete SHTs and equidistributions on the sphere, equitriangular (near equiareal) lattices based on the octahedron and the icosahedron are introduced for SHT analysis and synthesis. The developed formulations are described and implemented using simulated data and geopotential models such as the Earth Geopotential Model EGM 2008. Comparative results for analysis and synthesis at different levels of resolution show the potential of the spherical equitriangular approach for geodetic and other applications with nearly equidistributed global data.

A Novel Compression Method of Spectral Data Matrix Based on the Low-Rank Approximation and the Fast Fourier Transform of the Singular Vectors

2021 ◽  
pp. 000370282110447
Author(s):  
Joseph Dubrovkin
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Spectral Data ◽  
Compression Ratio ◽  
Low Rank ◽  
Data Matrix ◽  
Compression Method ◽  
Low Rank Approximation ◽  
Singular Vectors ◽  
Rank Approximation

Storage, processing, and transfer of huge matrices are becoming challenging tasks in the process analytical technology and scientific research. Matrix compression can solve these problems successfully. We developed a novel compression method of spectral data matrix based on its low-rank approximation and the fast Fourier transform of the singular vectors. This method differs from the known ones in that it does not require restoring the low-rank approximated matrix for further Fourier processing. Therefore, the compression ratio increases. A compromise between the losses of the accuracy of the data matrix restoring and the compression ratio was achieved by selecting the processing parameters. The method was applied to multivariate chemometrics analysis of the cow milk for determining fat and protein content using two data matrices (the file sizes were 5.7 and 12.0 MB) restored from their compressed form. The corresponding compression ratios were about 52 and 114, while the loss of accuracy of the analysis was less than 1% compared with processing of the non-compressed matrix. A huge, simulated matrix, compressed from 400 MB to 1.9 MB, was successfully used for multivariate calibration and segment cross-validation. The data set simulated a large matrix of 10 000 low-noise infrared spectra, measured in the range 4000–400 cm−1 with a resolution of 0.5 cm−1. The corresponding file was compressed from 262.8 MB to 19.8 MB. The discrepancies between original and restored spectra were less than the standard deviation of the noise. The method developed in the article clearly demonstrated its potential for future applications to chemometrics-enhanced spectrometric analysis with limited options of memory size and data transfer rate. The algorithm used the standard routines of Matlab software.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lung-Hui Chen
Keyword(s):  
Fourier Transform ◽  
Entire Function ◽  
Convex Hull ◽  
Scattering Theory ◽  
Phase Retrieval ◽  
Indicator Function ◽  
Band Limited ◽  
The Fourier Transform ◽  
Complex Valued ◽  
Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

Identification of Static Unbalance Wheel of Passenger Car Carried out on a Road

2014 ◽  
Vol 214 ◽  
pp. 48-57 ◽  
Author(s):  
Krzysztof Prażnowski ◽  
Sebastian Brol ◽  
Andrzej Augustynowicz
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Constant Velocity ◽  
Fourier Transforms ◽  
Static Condition ◽  
Rotational Frequency ◽  
Acceleration Sensor ◽  
Short Time Fourier Transform ◽  
Road Roughness ◽  
Short Time

This paper presents a method of identification of non-homogeneity or static unbalance of the structure of a car wheel based on a simple road test. In particular a method the detection of single wheel unbalance is proposed which applies an acceleration sensor fixed on windscreen. It measures accelerations cause by wheel unbalance among other parameters. The location of the sensor is convenient for handling an autonomous device used for diagnostic purposes. Unfortunately, its mounting point is located away from wheels. Moreover, the unbalance forces created by wheels spin are dumped by suspension elements as well as the chassis itself. It indicates that unbalance acceleration will be weak in comparison to other signals coming from engine vibrations, road roughness and environmental effects. Therefore, the static unbalance detection in the standard way is considered problematic and difficult. The goal of the undertaken research is to select appropriate transformations and procedures in order to determine wheel unbalance in these conditions. In this investigation regular and short time Fourier transform were used as well as wavelet transform. It was found that the use of Fourier transforms is appropriate for static condition (constant velocity) but the results proves that the wavelet transform is more suitable for diagnostic purposes because of its ability of producing clearer output even if car is in the state of acceleration or deceleration. Moreover it was proved that in the acceleration spectrum of acceleration measured on the windscreen a significant peak can be found when car runs with an unbalanced wheel. Moreover its frequency depends on wheel rotational frequency. For that reason the diagnostic of single wheel unbalance can be made by applying this method.

Generalized signal-tuned Gabor approach for signal representation and analysis

2017 ◽  
Vol 28 (01) ◽  
pp. 1750001 ◽  
Author(s):  
José R. A. Torreão
Keyword(s):  
Fourier Transform ◽  
Visual Cortex ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Receptive Fields ◽  
Gabor Functions ◽  
Plausible Model ◽  
Spectral Signal ◽  
Potential Applications ◽  
Space Frequency

The signal-tuned Gabor approach is based on spatial or spectral Gabor functions whose parameters are determined, respectively, by the Fourier and inverse Fourier transforms of a given “tuning” signal. The sets of spatial and spectral signal-tuned functions, for all possible frequencies and positions, yield exact representations of the tuning signal. Moreover, such functions can be used as kernels for space-frequency transforms which are tuned to the specific features of their inputs, thus allowing analysis with high conjoint spatio-spectral resolution. Based on the signal-tuned Gabor functions and the associated transforms, a plausible model for the receptive fields and responses of cells in the primary visual cortex has been proposed. Here, we present a generalization of the signal-tuned Gabor approach which extends it to the representation and analysis of the tuning signal’s fractional Fourier transform of any order. This significantly broadens the scope and the potential applications of the approach.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

scholarly journals Synthesis and antimicrobial activities of some (E)-N'-1-(substituted benzylidene)benzohydrazides

2017 ◽  
Vol 5 (1) ◽  
pp. 17 ◽  
Author(s):  
V. Manikandan ◽  
S. Balaji ◽  
R. Senbagam ◽  
R. Vijayakumar ◽  
M. Rajarajan ◽  
...  
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Fourier Transform ◽  
Magnetic Resonance ◽  
Spectral Data ◽  
Ultra Violet ◽  
Antimicrobial Activities ◽  
Antibacterial And Antifungal Activities ◽  
Ft Ir ◽  
Fourier Transform Ir ◽  
Antibacterial And Antifungal

About ten substituted (E)-N'-1-(substituted benzylidene) benzohydrazides have been synthesized. They are analyzed by their analytical, ultra violet (UV), Fourier transform-IR (FT-IR) and nuclear magnetic resonance (NMR) spectral data and evaluated by antimicrobial activities such antibacterial and antifungal activities.

IMPLEMENTING FAST FOURIER TRANSFORMS ON DISTRIBUTED-MEMORY MULTIPROCESSSORS USING DATA REDISTRIBUTIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 477-488 ◽  
Author(s):  
S.K.S. GUPTA ◽  
C.-H. HUANG ◽  
P. SADAYAPPAN ◽  
R.W. JOHNSON
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Distributed Memory ◽  
Fourier Transforms ◽  
Experimental Performance ◽  
Minimum Number ◽  
Data Redistribution ◽  
Using Data ◽  
Performance Results ◽  
Fft Algorithms

Implementations of various fast Fourier transform (FFT) algorithms are presented for distributed-memory multiprocessors. These algorithms use data redistribution to localize the computation. The goal is to optimize communication cost by using a minimum number of redistribution steps. Both analytical and experimental performance results on the Intel iPSC/860 system are presented.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

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