Sampling signals with finite set of apertures

The purpose of this paper is to research the frequency and time characteristics of the photo-detector using the direct and inverse Fourier transform. The spectral characteristics of the matrix photodetector were selected as the object of research. For the channels of the trans-ducer it was found that the frequency spectrum consists of two elementary bands in the form of Gaussian curves. It was revealed that the spectra have super-broadband properties. The impulse (time) characteristic of the spectra is described by the equation in an analytical form, which is in good agreement with the calculation. The rise time of the transient response is 1.8–2.9 fs. The process of absorption of light is significantly influenced by dielectric relaxa-tion of the charge. The results obtained provide important information on the properties of the photodetector in the frequency and time domain.

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Gap filling and noise reduction of unevenly sampled data by means of the Lomb-Scargle periodogram

Atmospheric Chemistry and Physics Discussions ◽

10.5194/acpd-8-4603-2008 ◽

2008 ◽

Vol 8 (2) ◽

pp. 4603-4623 ◽

Cited By ~ 4

Author(s):

K. Hocke ◽

N. Kämpfer

Keyword(s):

Time Series ◽

Fourier Transform ◽

Noise Reduction ◽

Fourier Spectrum ◽

Inverse Fourier Transform ◽

Reconstruction Method ◽

Sampled Data ◽

Gap Filling ◽

Spectral Components ◽

Mesospheric Ozone

Abstract. The Lomb-Scargle periodogram is widely used for the estimation of the power spectral density of unevenly sampled data. A small extension of the algorithm of the Lomb-Scargle periodogram permits the estimation of the phases of the spectral components. The amplitude and phase information is sufficient for the construction of a complex Fourier spectrum. The inverse Fourier transform can be applied to this Fourier spectrum and provides an evenly sampled series (Scargle, 1989). We are testing the proposed reconstruction method by means of artificial time series and real observations of mesospheric ozone, having data gaps and noise. For data gap filling and noise reduction, it is necessary to modify the Fourier spectrum before the inverse Fourier transform is done. The modification can be easily performed by selection of the relevant spectral components which are above a given confidence limit or within a certain frequency range. Examples with time series of lower mesospheric ozone show that the reconstruction method can reproduce steep ozone gradients around sunrise and sunset and superposed planetary wave-like oscillations observed by a ground-based microwave radiometer at Payerne.

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Gap filling and noise reduction of unevenly sampled data by means of the Lomb-Scargle periodogram

Atmospheric Chemistry and Physics ◽

10.5194/acp-9-4197-2009 ◽

2009 ◽

Vol 9 (12) ◽

pp. 4197-4206 ◽

Cited By ~ 52

Author(s):

K. Hocke ◽

N. Kämpfer

Keyword(s):

Time Series ◽

Fourier Transform ◽

Fourier Spectrum ◽

Inverse Fourier Transform ◽

Reconstruction Method ◽

Sampled Data ◽

Gap Filling ◽

Data Gaps ◽

Spectral Components ◽

Mesospheric Ozone

Abstract. The Lomb-Scargle periodogram is widely used for the estimation of the power spectral density of unevenly sampled data. A small extension of the algorithm of the Lomb-Scargle periodogram permits the estimation of the phases of the spectral components. The amplitude and phase information is sufficient for the construction of a complex Fourier spectrum. The inverse Fourier transform can be applied to this Fourier spectrum and provides an evenly sampled series (Scargle, 1989). We are testing the proposed reconstruction method by means of artificial time series and real observations of mesospheric ozone, having data gaps and noise. For data gap filling and noise reduction, it is necessary to modify the Fourier spectrum before the inverse Fourier transform is done. The modification can be easily performed by selection of the relevant spectral components which are above a given confidence limit or within a certain frequency range. Examples with time series of lower mesospheric ozone show that the reconstruction method can reproduce steep ozone gradients around sunrise and sunset and superposed planetary wave-like oscillations observed by a ground-based microwave radiometer at Payerne. The importance of gap filling methods for climate change studies is demonstrated by means of long-term series of temperature and water vapor pressure at the Jungfraujoch station where data gaps from another instrument have been inserted before the linear trend is calculated. The results are encouraging but the present reconstruction algorithm is far away from being reliable and robust enough for a serious application.

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On the influence of specimen thickness in TEM images of super-conducting vortices

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100166087 ◽

1996 ◽

Vol 54 ◽

pp. 724-725

Author(s):

J. Bonevich ◽

D. Capacci ◽

G. Pozzi ◽

K. Harada ◽

H. Kasai ◽

...

Keyword(s):

Flux Line ◽

Analytical Form ◽

Realistic Model ◽

Electron Holography ◽

Specimen Thickness ◽

Flux Tubes ◽

Analytical Expression ◽

Flux Lines ◽

Normal Core ◽

Two Parameters

The successful observation of superconducting flux lines (fluxons) in thin specimens both in conventional and high Tc superconductors by means of Lorentz and electron holography methods has presented several problems concerning the interpretation of the experimental results. The first approach has been to model the fluxon as a bundle of flux tubes perpendicular to the specimen surface (for which the electron optical phase shift has been found in analytical form) with a magnetic flux distribution given by the London model, which corresponds to a flux line having an infinitely small normal core. In addition to being described by an analytical expression, this model has the advantage that a single parameter, the London penetration depth, completely characterizes the superconducting fluxon. The obtained results have shown that the most relevant features of the experimental data are well interpreted by this model. However, Clem has proposed another more realistic model for the fluxon core that removes the unphysical limitation of the infinitely small normal core and has the advantage of being described by an analytical expression depending on two parameters (the coherence length and the London depth).

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Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

Abstract and Applied Analysis ◽

10.1155/2014/890456 ◽

2014 ◽

Vol 2014 ◽

pp. 1-24 ◽

Cited By ~ 2

Author(s):

David W. Pravica ◽

Njinasoa Randriampiry ◽

Michael J. Spurr

Keyword(s):

Fourier Transform ◽

Theta Function ◽

Legendre Polynomial ◽

Legendre Polynomials ◽

Inverse Fourier Transform ◽

Recursion Relations ◽

Jacobi Theta Function ◽

Convergence Results ◽

The Family ◽

Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

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Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

Volume 2E: Turbomachinery ◽

10.1115/gt2020-14754 ◽

2020 ◽

Author(s):

Sen Zhang ◽

Dingxi Wang ◽

Yi Li ◽

Hangkong Wu ◽

Xiuquan Huang

Keyword(s):

Fourier Transform ◽

Stability Analysis ◽

Condition Number ◽

Inverse Fourier Transform ◽

Real Life ◽

Equation System ◽

Time Instant ◽

Transform Matrix ◽

Spectral Equation ◽

The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

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An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽

10.1190/1.1442017 ◽

1985 ◽

Vol 50 (9) ◽

pp. 1500-1501

Author(s):

B. N. P. Agarwal ◽

D. Sita Ramaiah

Keyword(s):

Fourier Transform ◽

Gravity Anomaly ◽

Fourier Spectrum ◽

Fourier Transforms ◽

Weighted Average ◽

Window Function ◽

Average Spectrum ◽

Amplitude Spectra ◽

Distance Data ◽

The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

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MULTIDIMENSIONAL REPRESENTATION OF AUTOMATIC DOSING DEVICE SIGNALS

VESTNIK OF ASTRAKHAN STATE TECHNICAL UNIVERSITY SERIES MANAGEMENT COMPUTER SCIENCE AND INFORMATICS ◽

10.24143/2072-9502-2018-4-26-32 ◽

2018 ◽

pp. 26-32

Author(s):

Denis Borisovich Fedosenkov ◽

Anna Alekseevna Simikova ◽

Boris Andreevich Fedosenkov ◽

Stanislav Matveevich Kulakov

Keyword(s):

Fourier Transform ◽

Wigner Distribution ◽

Low Frequency ◽

Stationary Flow ◽

Complex Model ◽

Signal Spectrum ◽

One Dimensional ◽

Time Frequency ◽

Cohen's Class ◽

Discrete Values

The article describes the development of a special approach based on using multidimensional wavelet distributions principle to monitor and control the feed dozing processes in the mix preparation unit. As a key component, this approach uses the multidimensional time-frequency Wigner-Ville distribution, which is the part of Cohen's class distributions. The research focuses on signals characterizing mass transfer processes in the form of material flow measuring signals in relevant points of the unit. Wigner-Ville distribution has been shown in time terms as Fourier transform of products of multiplied parts of the signal under consideration for past and future time moments; corresponding distribution for the frequency spectrum is shown as Fourier transform of the products of signal parts for high-frequency and low-frequency fragments of the signal spectrum. It has been noted that when using a complex model of a dozing signal, discrete values (samples) of the latter are considered as its real values. The description of the signal parameters (amplitude, phase, frequency) has been carried out with the help of Hilbert transform. In Cohen's class distributions which represent one-dimensional non-stationary flow signals, the concept of ‘instantaneous frequency’ has been introduced. A graphical explanation for the transformation of a process flow signal from a one-dimensional time domain to a time-frequency 2 D/ 3 D -space is presented. The technology of developing a multidimensional image in the form of Wigner distribution for one-dimensional signals of continuous spiral or screw-type feeders has been examined in detail. There have been considered the features to support Wigner distribution, which allow to guess the presence or absence of time-frequency distribution elements in the interval of signal recording. There has been demonstrated how Wigner distribution can be obtained for a continuous-intermittent feeding signal. It has been concluded that for a certain types of the signal for zero fragments of the latter, non-zero time-frequency elements (i.e. virtual, anomalous ones) appear on the distribution. In addition to Wigner distribution, two other distributions - of Rihachek and Page - are considered. They display the same signal and also contain virtual elements, but in different domains of the time-frequency space. A generalized multidimensional compound signal distribution with a so-called distribution kernel available in it is presented, which includes a correction parameter that allows controlling the intensity of the virtual signal energy.

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The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

International Mathematics Research Notices ◽

10.1093/imrn/rnab161 ◽

2021 ◽

Author(s):

Yeansu Kim ◽

Loren Spice ◽

Sandeep Varma

Keyword(s):

Fourier Transform ◽

Characteristic Function ◽

Lie Algebra ◽

Equivalence Relation ◽

Characteristic Zero ◽

Closed Subset ◽

Inverse Fourier Transform ◽

Reductive Group ◽

Associate Class ◽

Weak Associate

Abstract Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

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On the Inverse Fourier Transform of the Planck-Einstein law

10.20944/preprints202109.0369.v1 ◽

2021 ◽

Author(s):

Alireza Jamali

Keyword(s):

Fourier Transform ◽

Inverse Fourier Transform ◽

Maximum Frequency ◽

Principle Of Maximum

After proposing a natural metric for the space in which particles spin which implements the principle of maximum frequency, E=hf is generalised and its inverse Fourier transform is calculated.

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