scholarly journals On a property of the Reed-Muller-Fourier transform

2018 ◽  
Vol 31 (2) ◽  
pp. 303-311
Author(s):  
Claudio Moraga
Keyword(s):  
Fourier Transform ◽  
Symmetric Function ◽  
Fourier Spectrum ◽  
Additional Result

The Reed-Muller-Fourier is reviewed and a new property is presented: The Reed-Muller-Fourier transform of an n-place p-valued function preserves any permutation of the arguments. This leads to the additional result that the Reed-Muller-Fourier spectrum of an n-place p-valued symmetric function is also symmetric. Furthermore, the Reed-Muller and the Vilenkin-Chrestenson spectra of an n-place p-valued symmetric function are also symmetric.

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

Geophysics ◽  
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.

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

2008 ◽  
Vol 8 (2) ◽  
pp. 4603-4623 ◽  
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.

scholarly journals Experimental Signal Analysis of Robot Impacts in a Fractional Calculus Perspective

2007 ◽  
Vol 11 (9) ◽  
pp. 1079-1085 ◽  
Author(s):  
Miguel F. M. Lima ◽  
◽  
J. A. Tenreiro Machado ◽  
Manuel Crisóstomo ◽  
◽  
...  
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Control System ◽  
Fractional Calculus ◽  
Signal Analysis ◽  
Fourier Spectrum ◽  
Time Windows ◽  
Experimental Setup ◽  
Analysis Package ◽  
Line Analysis

This paper presents a study of the signals captured during impacts and vibrations of a mechanical manipulator. In order to acquire and study the signals an experimental setup is implemented. The system acquires data from the sensors, in real time, and processes it through an off-line analysis package. The study is developed based on a set of signal processing tools, such as the Fast Fourier Transform and the windowed Fourier transform. The signals whose Fourier spectrum presents a non integer behavior are pointed out and analyzed using several time windows. The experimental results provide useful information that can assist in the design of a control system to deal with the unwanted effects of vibrations.

scholarly journals A WAVELET REPRESENTATION OF HVSR TECHNIQUE

2004 ◽  
Vol 36 (3) ◽  
pp. 1269 ◽  
Author(s):  
G. Hloupis ◽  
F. Vallianatos ◽  
J. Stonham
Keyword(s):  
Fourier Transform ◽  
Fourier Spectrum ◽  
Site Amplification ◽  
Wavelet Representation ◽  
Short Period ◽  
Variable Window ◽  
The Fourier Transform ◽  
Fft Algorithms ◽  
Short Time ◽  
Hvsr Technique

In order to evaluate eathquake site amplification characteristics, horizontal to vertical Fourier spectrum of microtremor has been widely used. As long as the Fourier transform (FT) cannot distinguish between stationary and non-stationary coefficients we cannot eliminate the contamination of microtremors signals from short period transients. The wavelet transform (WT), using the property of localization of wavelet bases has been widely used in signal processing. Unlike the Short Time Fourier Transform (STFT) in which the width of window is fixed, the WT localizes signal in a variable window using the dilation parameter. This property, which derived directly from multiresolution analysis provide us the ability to decompose a signal in a well localized set of coefficients and identify the non-stationary portions of it. In the present study we use the WT in order to eliminate the non-stationahties in microtremor signals before we calculate the spectrum of each one using conventional FFT algorithms

scholarly journals Sampling signals with finite set of apertures

2021 ◽  
pp. 55-58
Author(s):  
V. I. Guzhov ◽  
◽  
I. O. Marchenko ◽  
E. E. Trubilina ◽  
A. A. Trubilin ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Fourier Spectrum ◽  
Inverse Fourier Transform ◽  
Analytical Form ◽  
Generalized Functions ◽  
Signal Spectrum ◽  
Original Image ◽  
Analytical Expression ◽  
Image Sampling ◽  
Finite Set

The article discusses the issue of sampling continuous signals using a finite set of apertures. Using the apparatus of generalized functions, an analytical form of sampling is obtained for ideal sampling, for sampling a limited signal and for sampling a signal using a limited set of apertures. It is shown that the signal spectrum is the product of the signal spectrum at ideal sampling by some known factor, the influence of which can be eliminated. The type of this factor can be obtained if the type of aperture is known. The type of analytical expression differs from those known in the literature on image sampling. The use of an analytical expression for sampling can be used to reconstruct the original image from the image obtained with different sets of apertures. For this it is necessary to divide the Fourier spectrum of the sampled image by a factor depending on the selected aperture. Having received the inverse Fourier transform from it, you can get the original one

scholarly journals Determination of Local Strains in a Neighborhood of Cracks in a Welded Seam of Ni-Cr-Fe According to the Power Fourier Spectrum of Kikuchi Patterns

2018 ◽  
Vol 19 (4) ◽  
pp. 307-312 ◽  
Author(s):  
M.D. Borcha ◽  
M.S. Solodkyi ◽  
S.V. Balovsyak ◽  
I.M. Fodchuk ◽  
A.R. Kuzmin ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Gaussian Noise ◽  
Fourier Spectrum ◽  
Electron Backscatter Diffraction ◽  
Two Dimensional ◽  
Electron Backscatter ◽  
Diffraction Patterns ◽  
Welded Seam ◽  
Backscatter Diffraction

A discrete two-dimensional Fourier transform and the power Fourier spectrum are used for determination of average strains near cracks in a welded seam of Ni-Cr-Fe alloy. The alignment of Kikuchi images with the help of genetic algorithms and subtraction of white Gaussian noise made it possible to more fully take into account the influence of instrumental factors on the formation of electron backscatter diffraction patterns.

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

2009 ◽  
Vol 9 (12) ◽  
pp. 4197-4206 ◽  
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.

Electron holography in materials science

1991 ◽  
Vol 49 ◽  
pp. 670-671
Author(s):  
Hannes Lichte ◽  
Edgar Voelkl
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Materials Science ◽  
Fourier Spectrum ◽  
Object Wave ◽  
Electron Holography ◽  
Reconstruction Procedure ◽  
Numerical Reconstruction ◽  
Transmission Electron ◽  
Unique Interpretation

The object wave o(x,y) = a(x,y)exp(iφ(x,y)) at the exit face of the specimen is described by two real functions, i.e. amplitude a(x,y) and phase φ(x,y). In stead of o(x,y), however, in conventional transmission electron microscopy one records only the real intensity I(x,y) of the image wave b(x,y) loosing the image phase. In addition, referred to the object wave, b(x,y) is heavily distorted by the aberrations of the microscope giving rise to loss of resolution. Dealing with strong objects, a unique interpretation of the micrograph in terms of amplitude and phase of the object is not possible. According to Gabor, holography helps in that it records the image wave completely by both amplitude and phase. Subsequently, by means of a numerical reconstruction procedure, b(x,y) is deconvoluted from aberrations to retrieve o(x,y). Likewise, the Fourier spectrum of the object wave is at hand. Without the restrictions sketched above, the investigation of the object can be performed by different reconstruction procedures on one hologram. The holograms were taken by means of a Philips EM420-FEG with an electron biprism at 100 kV.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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