Search results for "Fourier Analysis"
showing 10 items of 113 documents
Optical illustration of a varied fractional Fourier-transform order and the Radon-Wigner display.
2010
Based on an all-optical system, a display of a fractional Fourier transform with many fractional orders is proposed. Because digital image-processing terminology is used, this display is known as the Radon–Wigner transform. It enables new aspects for signal analysis that are related to time- and spatial-frequency analyses. The given approach for producing this display starts with a one-dimensional input signal although the output signal contains two dimensions. The optical setup for obtaining the fractional Fourier transform was adapted to include only fixed free-space propagation distances and variable lenses. With a set of two multifacet composite holograms, the Radon–Wigner display has b…
Ultrarelativistic (Cauchy) spectral problem in the infinite well
2016
We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|\Delta |^{1/2}$, provided its action is constrained exclusively to the interior of the interval $[-1,1] \subset R$. To this end both analytic and numerical methods are employed. New high-accuracy spectral data are obtained. A direct analytic proof is given that trigonometric functions $\cos(n\pi x/2)$ and $\sin(n\pi x)$, for integer $n$ are {\it not} the eigenfunctions of $|\Delta |_D^{1/2}$, $D=(-1,1)$. This clearly demonstrates that the traditional Fourier multiplier representation of $|\Delta |^{1/2}$ becomes defective, while passing from $R$ to a bounded spatial domain $D\subset R$.
Impact of a temporal sinusoidal phase modulation on the optical spectrum
2018
International audience; We discuss the effects of imparting a temporal sinusoidal phase modulation to a continuous wave on the frequency spectrum. While a practical analytical solution to this problem already exists, we present here a physical interpretation based on interference processes. This simple model will help the students better understand the origin of the oscillatory structure that can be observed in the resulting spectrum and that is characteristic of Bessel functions of the first kind. We illustrate our approach with an example from the field of optics.
Teaching Fourier optics through ray matrices
2005
In this work we examine the use of ray-transfer matrices for teaching and for deriving some topics in a Fourier optics course, exploiting the mathematical simplicity of ray matrices compared to diffraction integrals. A simple analysis of the physical meaning of the elements of the ray matrix provides a fast derivation of the conditions to obtain the optical Fourier transform. We extend this derivation to fractional Fourier transform optical systems, and derive the order of the transform from the ray matrix. Some examples are provided to stress this point of view, both with classical and with graded index lenses. This formulation cannot replace the complete explanation of Fourier optics prov…
White-light optical implementation of the fractional fourier transform with adjustable order control.
2000
An optical implementation of the fractional Fourier transform (FRT) with broadband illumination is proposed by use of a single imaging element, namely, a blazed diffractive lens. The setup displays an achromatized version of the FRT of order P of any two-dimensional input function. This fractional order can be tuned continuously by shifting of the input along the optical axis. Our compact and flexible configuration is tested with a chirplike input signal, and the good experimental results obtained support the theory.
Statistics of Galaxy Clustering
2006
In this introductory talk we will establish connections between the statistical analysis of galaxy clustering in cosmology and recent work in mainstream spatial statistics. The lecture will review the methods of spatial statistics used by both sets of scholars, having in mind the cross-fertilizing purpose of the meeting series. Special topics will be: description of the galaxy samples, selection effects and biases, correlation functions, nearest neighbor distances, void probability functions, Fourier analysis, and structure statistics.
Removal of streaking artefact in the images of the Pierre Auger Observatory infra-red cameras
2011
In this paper the problem of removing concentric semi-circular stripe artefacts induced by the operating hardware of the infra red cameras of the Pierre Auger Observatory, is tackled. The method builds on top of a recent algorithm for the removal of artefacts, which presents a robust filter, obtained as a combination of Wavelet and Fourier analysis, capable of eliminating horizontal and vertical stripes in images, while trying to preserve structural features and quantitative values of the image. The method requires several parameters which have been tuned by an exhaustive test on a large set of images. The results show that the method is capable to satisfactorily remove the stripe artefacts.
On the Minimal Solution of the Problem of Primitives
2000
Abstract We characterize the primitives of the minimal extension of the Lebesgue integral which also integrates the derivatives of differentiable functions (called the C -integral). Then we prove that each BV function is a multiplier for the C -integral and that the product of a derivative and a BV function is a derivative modulo a Lebesgue integrable function having arbitrarily small L 1 -norm.
On some dual frames multipliers with at most countable spectra
2021
A dual frames multiplier is an operator consisting of analysis, multiplication and synthesis processes, where the analysis and the synthesis are made by two dual frames in a Hilbert space, respectively. In this paper we investigate the spectra of some dual frames multipliers giving, in particular, conditions to be at most countable. The contribution extends the results available in literature about the spectra of Bessel multipliers with symbol decaying to zero and of multipliers of dual Riesz bases.
Fourier analysis of periodic Radon transforms
2019
We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on $H^s$ Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data.