scholarly journals Preferred Orientation, PDF and Debye Equation

2014 ◽  
Vol 70 (a1) ◽  
pp. C1077-C1077
Author(s):  
Reinhard Neder
Keyword(s):  
Fourier Transform ◽  
Diffraction Pattern ◽  
Complex Function ◽  
Good Alternative ◽  
Preferred Orientation ◽  
Spherical Average ◽  
Area Detector ◽  
Orientation Axis ◽  
The Fourier Transform ◽  
Debye Equation

The effect of preferred orientation is currently neglected in the Debye Equation and PDF calculations. This is to a large extend justified, especially for the PDF, as the scattering by large sample volumes is detected by an area detector. The integration of powder rings reduces the effects of preferred orientation. As more laboratory PDF measurements become available that use linear position sensitive detectors or single counter detectors, preferred orientation needs to be reconsidered. A Rietveld calculation treats preferred orientation by multiplying the Bragg intensity by a factor that depends on the angle between the reciprocal space vector and the preferred orientation axis. The powder intensity I(Q) is thus multiplied by a complex function that depends at each Q on the degree of preferred orientation, the lattice parameters, reflection multiplicity etc. The effect on the PDF is therefore the convolution by the Fourier transform of this complex function. The Debye equation is derived from a spherical average of the scattering intensity of a finite object. Thus completely random orientation of the powder grains is implicitly assumed. Both, the Debye algorithm and the PDF algorithm calculate the powder pattern, respectively the PDF from a histogram of interatomic distances, which correspond to a spherical average of all interatomic distance vectors. This histogram does not allow for a Rietveld preferred orientation correction. The effects of preferred orientation on the PDF will be presented on the basis of simulated diffraction pattern. An algorithm to describe the changes in the PDF and the inverse sine Fourier transform as a convenient tool to calculate the powder diffraction pattern will be introduced. This is a good alternative to the Debye equation and allows to take preferred orientation into account both for the PDF and the calculated powder.

scholarly journals Generalized Parton Distributions of proton for nonzero skewness in transverse and longitudinal position spaces

2015 ◽  
Vol 30 (02) ◽  
pp. 1550010 ◽  
Author(s):  
Narinder Kumar ◽  
Harleen Dahiya
Keyword(s):  
Fourier Transform ◽  
Transverse Momentum ◽  
Diffraction Pattern ◽  
Wave Functions ◽  
Position Space ◽  
Parton Distributions ◽  
Generalized Parton Distributions ◽  
The Fourier Transform ◽  
Longitudinal Position ◽  
Transverse Position

We investigate the Generalized Parton Distributions (GPDs) of proton by expressing them in terms of overlaps of light front wave functions (LFWFs) using a simulated model which is able to qualitatively improve the convergence near the end points of x. We study the spin nonflip H(x, ζ, t) and spin flip E(x, ζ, t) parts of GPDs for the particle conserving n → n overlap in the DGLAP region (ζ < x < 1). The Fourier transform (FT) of the GPDs with respect to the transverse momentum transfer as well as the FT of the GPDs with respect to ζ has also been obtained by giving the distribution of partons in the transverse position space and the distribution in the longitudinal position space, respectively. Diffraction pattern is obtained for both [Formula: see text] and [Formula: see text] in the longitudinal position space.

Some properties of windowed linear canonical transform and its logarithmic uncertainty principle

2016 ◽  
Vol 14 (03) ◽  
pp. 1650015 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Complex Function ◽  
Young Inequality ◽  
Orthogonality Relation ◽  
Linear Canonical Transform ◽  
Heisenberg Uncertainty Principle ◽  
Heisenberg Uncertainty ◽  
The Fourier Transform

Based on the relationship between the Fourier transform (FT) and linear canonical transform (LCT), a logarithmic uncertainty principle and Hausdorff–Young inequality in the LCT domains are derived. In order to construct the windowed linear canonical transform (WLCT), Gabor filters associated with the LCT is introduced. Using the basic connection between the classical windowed Fourier transform (WFT) and the WLCT, a new proof of inversion formula for the WLCT is provided. This relation allows us to derive Lieb’s uncertainty principle associated with the WLCT. Some useful properties of the WLCT such as bounded, shift, modulation, switching, orthogonality relation, and characterization of range are also investigated in detail. By the Heisenberg uncertainty principle for the LCT and the orthogonality relation property for the WLCT, the Heisenberg uncertainty principle for the WLCT is established. This uncertainty principle gives information how a complex function and its WLCT relate. Lastly, the logarithmic uncertainty principle associated with the WLCT is obtained.

A Remark on Fourier Transforms

1936 ◽  
Vol 32 (2) ◽  
pp. 321-327 ◽  
Author(s):  
A. Zygmund
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Complex Function ◽  
The Fourier Transform ◽  
Image Position ◽  
The Right

1. Let f(x) be a complex function belonging to LP (−∞, ∞); i.e. let f(x) be measurable, and |f(x)|p integrable, over (−∞, ∞). The functionis called the Fourier transform of f(x), if the integral on the right exists, in some sense, for almost every value of y. It is well known that, if 1 ≤ p ≤ 2, the integral (1) converges in mean, with index p′ = p/(p – l)† i.e. thatwhere

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

Determination of starch crystallinity with the Fourier-transform terahertz spectrometer

2021 ◽  
Vol 262 ◽  
pp. 117928
Author(s):  
Shusaku Nakajima ◽  
Shuhei Horiuchi ◽  
Akifumi Ikehata ◽  
Yuichi Ogawa
Keyword(s):  
Fourier Transform ◽  
The Fourier Transform ◽  
Starch Crystallinity

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.

scholarly journals Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Angela A. Albanese ◽  
Claudio Mele
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Strong Operator ◽  
Dual Spaces ◽  
Structure Theorems ◽  
The Fourier Transform ◽  
Decreasing Functions

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

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