scholarly journals Interpreting the Phase Spectrum in Fourier Analysis of Partial Ranking Data

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Ramakrishna Kakarala
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Phase Spectrum ◽  
Data Sets ◽  
Ranking Data ◽  
Real World Data ◽  
Partial Ranking ◽  
Partial Rankings ◽  
The Fourier Transform ◽  
Processing Properties

Whenever ranking data are collected, such as in elections, surveys, and database searches, it is frequently the case that partial rankings are available instead of, or sometimes in addition to, full rankings. Statistical methods for partial rankings have been discussed in the literature. However, there has been relatively little published on their Fourier analysis, perhaps because the abstract nature of the transforms involved impede insight. This paper provides as its novel contributions an analysis of the Fourier transform for partial rankings, with particular attention to the first three ranks, while emphasizing on basic signal processing properties of transform magnitude and phase. It shows that the transform and its magnitude satisfy a projection invariance and analyzes the reconstruction of data from either magnitude or phase alone. The analysis is motivated by appealing to corresponding properties of the familiar DFT and by application to two real-world data sets.

scholarly journals Spectral Analysis of Traffic Functions in Urban Areas

2015 ◽  
Vol 27 (6) ◽  
pp. 477-484 ◽  
Author(s):  
Florin Nemtanu ◽  
Ilona Madalina Costea ◽  
Catalin Dumitrescu
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Fourier Analysis ◽  
Traffic Flow ◽  
Traffic Control ◽  
Urban Areas ◽  
Urban Traffic ◽  
The Fourier Transform ◽  
The City ◽  
Different Time Scales

The paper is focused on the Fourier transform application in urban traffic analysis and the use of said transform in traffic decomposition. The traffic function is defined as traffic flow generated by different categories of traffic participants. A Fourier analysis was elaborated in terms of identifying the main traffic function components, called traffic sub-functions. This paper presents the results of the method being applied in a real case situation, that is, an intersection in the city of Bucharest where the effect of a bus line was analysed. The analysis was done using different time scales, while three different traffic functions were defined to demonstrate the theoretical effect of the proposed method of analysis. An extension of the method is proposed to be applied in urban areas, especially in the areas covered by predictive traffic control.

scholarly journals Fourier Analysis with Generalized Integration

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1199
Author(s):  
Juan H. Arredondo ◽  
Manuel Bernal ◽  
María Guadalupe Morales
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Integral Theory ◽  
Dense Subspace ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Integrable Functions ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
The Fourier Transform

We generalize the classic Fourier transform operator F p by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the H K -Fourier transform on a dense subspace of L p , 1 < p ≤ 2 . In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function F p ( f ) under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.

Assessing the Influence of Fourier Analysis Parameters on Short-Time Modal Parameter Extraction

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Matthew Lamb ◽  
Vincent Rouillard
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Natural Frequency ◽  
Spectral Resolution ◽  
Parameter Extraction ◽  
Modal Parameter ◽  
Zero Padding ◽  
Spectral Enhancement ◽  
The Fourier Transform ◽  
Modal Parameter Extraction

It is sometimes necessary to determine the manner in which materials and structures deteriorate with respect to time when subjected to sustained random dynamic loads. In such cases a system’s fatigue characteristics can be obtained by continuously monitoring its modal parameters. This allows for any structural deterioration, often manifested as a loss in stiffness, to be detected. Many common structural integrity assessment techniques make use of Fourier analysis for modal parameter extraction. For continual modal parameter extraction, the Fourier transform requires that a compromise be made between the accuracy of the estimates and how frequently they can be obtained. The limitations brought forth by this compromise can be significantly reduced by selecting suitable values for the analysis parameters, mainly subrecord length and number of averages. Further improvements may also be possible by making use of spectral enhancement techniques, specifically overlapped averaging and zero padding. This paper uses the statistical analysis of results obtained from numerous physical and numerical experiments to evaluate the influence of the analysis parameters and spectral enhancement techniques on modal estimates obtained from limited data sets. This evaluation will assist analysts in selecting the most suitable inputs for parameter extraction purposes. The results presented in this paper show that when using the Fourier transform to extract modal characteristics, any variation in the parameters used for analysis can have a significant influence on the extraction of natural frequency estimates from systems subjected to random excitation. It was found that for records containing up to 10% noise, subrecord length; hence spectral resolution, has a more pronounced influence on the accuracy of modal estimates than the level of spectral averaging; therefore spectral uncertainty. It was also found that while zero padding may not increase the actual spectral resolution, it does allow for improved natural frequency estimates with the introduction of interpolated estimates at the nondescribed frequencies. Finally, it was found that for modal parameter extraction purposes (in this case natural frequency), increased amounts of overlapped averaging can significantly reduce the variance of the estimates obtained. This is particularly useful as it allows for increased precision without compromising temporal resolution.

scholarly journals Bayesian 14C-rationality, Heisenberg Uncertainty, and Fourier Transform

2020 ◽  
Vol 47 ◽  
pp. 536-559
Author(s):  
Bernhard Weninger ◽  
Kevan Edinborough
Keyword(s):  
Fourier Transform ◽  
Quantum Physics ◽  
Domain Switching ◽  
Data Sets ◽  
14C Dating ◽  
Calendar Time ◽  
Time Space ◽  
Wave Particle Duality ◽  
Heisenberg Uncertainty ◽  
The Fourier Transform

Following some 30 years of radiocarbon research during which the mathematical principles of 14C-calibration have been on loan to Bayesian statistics, here they are returned to quantum physics. The return is based on recognition that 14C-calibration can be described as a Fourier transform. Following its introduction as such, there is need to reconceptualize the probabilistic 14C-analysis. The main change will be to replace the traditional (one-dimensional) concept of 14C-dating probability by a two-dimensional probability. This is entirely analogous to the definition of probability in quantum physics, where the squared amplitude of a wave function defined in Hilbert space provides a measurable probability of finding the corresponding particle at a certain point in time/space, the so-called Born rule. When adapted to the characteristics of 14C-calibration, as it turns out, the Fourier transform immediately accounts for practically all known so-called quantization properties of archaeological 14C-ages, such as clustering, age-shifting, and amplitude-distortion. This also applies to the frequently observed chronological lock-in properties of larger data sets, when analysed by Gaussian wiggle matching (on the 14C-scale) just as by Bayesian sequencing (on the calendar time-scale). Such domain-switching effects are typical for a Fourier transform. They can now be understood, and taken into account, by the application of concepts and interpretations that are central to quantum physics (e.g. wave diffraction, wave-particle duality, Heisenberg uncertainty, and the correspondence principle). What may sound complicated, at first glance, simplifies the construction of 14C-based chronologies. The new Fourier-based 14C-analysis supports chronological studies on previously unachievable geographic (continental) and temporal (Glacial-Holocene) scales; for example, by temporal sequencing of hundreds of archaeological sites, simultaneously, with minimal need for development of archaeological prior hypotheses, other than those based on the geo-archaeological law of stratigraphic superposition. As demonstrated in a variety of archaeological case studies, just one number, defined as a gauge-probability on a scale 0–100%, can be used to replace a stacked set of subjective Bayesian priors.

Introduction

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Fourier Transform ◽  
Magnetic Resonance ◽  
Fourier Analysis ◽  
Fourier Integral ◽  
Sampling Theorem ◽  
Two Dimensional ◽  
The Fourier Transform ◽  
Array Imaging ◽  
Nuclear Magnetic

Jean Baptiste Joseph Fourier’s powerful idea of decomposition of a signal into sinusoidal components has found application in almost every engineering and science field. An incomplete list includes acoustics [1497], array imaging [1304], audio [1290], biology [826], biomedical engineering [1109], chemistry [438, 925], chromatography [1481], communications engineering [968], control theory [764], crystallography [316, 498, 499, 716], electromagnetics [250], imaging [151], image processing [1239] including segmentation [1448], nuclear magnetic resonance (NMR) [436, 1009], optics [492, 514, 517, 1344], polymer characterization [647], physics [262], radar [154, 1510], remote sensing [84], signal processing [41, 154], structural analysis [384], spectroscopy [84, 267, 724, 1220, 1293, 1481, 1496], time series [124], velocity measurement [1448], tomography [93, 1241, 1242, 1327, 1330, 1325, 1331], weather analysis [456], and X-ray diffraction [1378], Jean Baptiste Joseph Fourier’s last name has become an adjective in the terms like Fourier series [395], Fourier transform [41, 51, 149, 154, 160, 437, 447, 926, 968, 1009, 1496], Fourier analysis [151, 379, 606, 796, 1472, 1591], Fourier theory [1485], the Fourier integral [395, 187, 1399], Fourier inversion [1325], Fourier descriptors [826], Fourier coefficients [134], Fourier spectra [624, 625] Fourier reconstruction [1330], Fourier spectrometry [84, 355], Fourier spectroscopy [1220, 1293, 1438], Fourier array imaging [1304], Fourier transform nuclear magnetic resonance (NMR) [429, 1004], Fourier vision [1448], Fourier optics [419, 517, 1343], and Fourier acoustics [1496]. Applied Fourier analysis is ubiquitous simply because of the utility of its descriptive power. It is second only to the differential equation in the modelling of physical phenomena. In contrast with other linear transforms, the Fourier transform has a number of physical manifestations. Here is a short list of everyday occurrences as seen through the lens of the Fourier paradigm. • Diffracting coherent waves in sonar and optics in the far field are given by the two dimensional Fourier transform of the diffracting aperture. Remarkably, in free space, the physics of spreading light naturally forms a two dimensional Fourier transform. • The sampling theorem, born of Fourier analysis, tells us how fast to sample an audio waveform to make a discrete time CD or an image to make a DVD.

Advances in Analysis

2014 ◽  
Author(s):  
Charles Fefferman ◽  
Alexandru D. Ionescu ◽  
D.H. Phong ◽  
Stephen Wainger
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Fourier Analysis ◽  
Interpolation Theorem ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Restriction Theorem ◽  
Several Variables ◽  
Hp Spaces ◽  
The Fourier Transform

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.

Multicomponent Analysis Using Fourier Transform Infrared and UV Spectra

1988 ◽  
Vol 42 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Steven M. Donahue ◽  
Chris W. Brown ◽  
Robert J. Obremski
Keyword(s):  
Factor Analysis ◽  
Fourier Transform ◽  
Fourier Transforms ◽  
Uv Spectra ◽  
Data Sets ◽  
Transform Method ◽  
Multicomponent Analysis ◽  
P Matrix ◽  
The Fourier Transform ◽  
Selection Of

Two- and three-component mixtures of methylated benzenes were analyzed with the use of both infrared and UV spectra. The spectra of known mixtures were Fourier transformed and coefficients from the transforms selected to form coordinates of vectors. The resulting vectors were subjected to factor analysis to obtain representations for multicomponent analysis. A total of eight data sets were analyzed by factor analysis after preprocessing by taking the Fourier transforms of the spectra. The eight data sets were also analyzed by the P-matrix method (inverse Beer's law) in the spectral domain after preprocessing of the data to allow selection of the optimum analytical wavenumbers. This spectral method was compared to the Fourier transform method using cross-validation, in which one sample at a time was left out of the standards and treated as an unknown. The Standard Error of Prediction (SEP) was calculated for the two methods for all possible numbers of vectors and numbers of wavenumbers, starting with the number equal to the number of components and increasing up to a total number of standards or some reasonable cut-off value. Processing in the Fourier domain clearly produced the best results for seven of the data sets and equal results for the other set.

An Improved Fourier-Mellin Algorithm Based on the Image Amplitude Spectrum

2014 ◽  
Vol 687-691 ◽  
pp. 3773-3776
Author(s):  
Xiao Qiang Ji ◽  
Jie Zhang Cheng ◽  
Lei Jiang ◽  
Ting Ting Zhang ◽  
Mei Jiao Wang
Keyword(s):  
Fourier Transform ◽  
Motion Vector ◽  
Amplitude Spectrum ◽  
Phase Spectrum ◽  
Video Image ◽  
Experimental Results ◽  
Image Motion ◽  
Motion Model ◽  
Image Translation ◽  
The Fourier Transform

This paper proposes an improved amplitude spectrum based Fourier-Mellin algorithm Fourier by studying the nature of the Fourier transform image of the amplitude spectrum and the application of phase spectrum in estimation for image motion vector according to the shortcomings of traditional Fourier-Mellin algorithm when the video image translation, rotation and scaling of the situation exist. Experimental results show that the algorithm can estimate the rotation, translation and other vector parameters of a complicated motion model .The complexity of the algorithm has greatly improved comparing to the traditional Fourier-Mellin algorithm.

Comparison of Numerical and Experimental Strain Measurements of a Composite Panel Using Image Decomposition

2011 ◽  
Vol 70 ◽  
pp. 63-68 ◽  
Author(s):  
Christopher M Sebastian ◽  
Eann A Patterson ◽  
Donald Ostberg
Keyword(s):  
Fourier Transform ◽  
Image Decomposition ◽  
Image Correlation ◽  
Composite Panel ◽  
Data Sets ◽  
Full Field ◽  
Strain Measurements ◽  
Data Points ◽  
Optical Strain ◽  
The Fourier Transform

Image decomposition is used to address the problem of accurately and concisely describing the strain in an inhomogeneous composite panel that is bolted to a vehicle structure. In-service, the composite panel is subject to structural loads from the vehicle which can cause unintended damage to the panel. Finite element simulations have been performed with the plan to establish their fidelity using full-field optical strain measurements obtained using digital image correlation. A methodology is presented based on using orthogonal shape descriptors to decompose the data-rich maps of strain into information-preserved data sets of reduced dimensionality that facilitate a quantitative comparison of the computational and experimental results. The decomposition is achieved employing the Fourier transform followed by fitting Tchebichef moments to the maps of the magnitude of the Fourier transform. The results show that this approach is fast and reliably describes the strain fields using less than fifty moments as compared to the thousands of data points in each strain map.

Fourier Analysis of Digital STEM Images in the Study of Transparency in Human Cornea And Sclera

1988 ◽  
Vol 46 ◽  
pp. 178-179
Author(s):  
S . Vaezy
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Visible Light ◽  
Light Wave ◽  
Incident Light ◽  
Human Cornea ◽  
Quantitative Measurements ◽  
Spatial Fluctuations ◽  
The Fourier Transform ◽  
Incident Light Wave

The structure of transparent human cornea is similar to that of the opaque human sclera, and they both consist of collagen fibers embedded in a mucopolysaccharide matrix.[1] This similarity in structure and composition, in contrast to a large difference in turbidity, presents an important question regarding the basis of transparency in the cornea.In general, a transparent structure permits most of the incident light to be transmitted, whereas an opaque structure scatters it. Scattering of an incident light wave occurs when the spatial fluctuations in the index of the refraction of the medium have dimensions close to the wavelength of visible light.[2] To analyze spatial fluctuations in cornea and sclera we have developed techniques based on Fourier analysis of digital STEM images.Fourier analysis allows direct quantitative measurements of the fluctuations in any signal. The signal is resolved into its constituent sinusoids or Fourier components. Amplitude, and frequency of each component are quantitatively obtained using the Fourier Transform (FT).

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