Electromagnetic Signal Processing and Noncommutative Tomography

A generic electromagnetic signal described by Maxwell equations both in vacuum and media is considered in the tomographic representation. The Ville–Wigner phase-space representation of the electromagnetic field is also discussed. Relations between different representations of the electromagnetic signal are elucidated. The connection of the Fourier analysis of the electromagnetic signal and other mathematical approaches like the Radon transform of the analytic signal is presented. The distinguishing property of the tomogram to coincide with the probability density of a random variable considered in a reference frame in the signal's phase space is pointed out. The entropy of the signal related to the probability density is studied.     

Application of the Wavelet Transform for Detecting Signals of Nuclear Quadrupole Resonance

In the present study, the wavelet transform is used to increase the signal-to-noise ratio of nuclear quadrupole resonance (NQR) signals in the direct pulse method. The efficiency of the wavelet and Fourier transforms used for detecting the NQR signal component is compared. The signal consists of noise, magnetoacoustic and piezoelectric signals, noise from external sources, and NQR signal. Signals from ¹⁴ N nuclei in hexahydro-1,3,5-trinitro-s-triazine C₃H₆N₆O₆ (RDX) are investigated at a temperature of 298 K.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 47–50, January, 2005.     

负二进制编码小波变换

提出了种新的数学编码方法来计算小波变换，并利用非相干光学相关器进行实验，结果与理论分析相符。这种算法原理简单。，操作方便且精度可望达到极高。     

On the Radon transformation of Wigner function altered with various optical processes

We discuss what happens to the Radon transformation of signal's Wigner functions (i.e., signal's Wigner transformation (WT)) if the signal function undergoes various optical processes, such as Fraunhofer diffraction, lens transformation and Fresnel diffraction, etc. Because the usual Wigner transforms can be studied via their corresponding transforms of the Wigner operator, we use the Weyl ordered form of the Wigner operator and the Weyl ordering invariance under similar transformations to derive the result, we find that the alteration of Radon transformation of signal's Wigner function (or named the variation of tomogram function), through these optical processes, can be ascribed to the variation of Radon transformation parameters once the parameter of WT is given.     

MEI, SOI and mid-range correlations in the onset of El Niño–Southern Oscillation

Entropy analysis and wavelet transforms are used to study the Southern Oscillation index (SOI) and multivariate ENSO index (MEI) indexes for the El Niño–Southern Oscillation. Trends to large decreasing symbolic chains responsible for La Niña and interrupted increasing chains for El Niño are found in the MEI persistence series. These are all located in regions where the wavelet transforms of both indexes reveal the existence of mid-range correlations. The SOI and MEI indexes are mutually correlated in a non-trivial manner in the time.     

Generalized analytic signal associated with linear canonical transform

Analytic signal is tightly associated with Hilbert transform and Fourier transform. The linear canonical transform is the generalization of many famous linear integral transforms, such as Fourier transform, fractional Fourier transform and Fresnel transform. Based on the parameter (a, b)-Hilbert transform and the linear canonical transform, in this paper, we develop some issues on generalized analytic signal. The generalized analytic signal can suppress the negative frequency components in the linear canonical transform domain. Furthermore, we prove that the kernel function of the inverse linear canonical transform satisfies the generalized analytic condition and get the generalized analytic pairs. We show the generalized Bedrosian theorem is valid in the linear canonical transform domain.     

Upon the utility of spherical harmonic expansions in the evaluation of cluster integrals

A novel procedure for the analytic evaluation of cluster integrals is given. By means of a result of Silverstone and Moats which transforms the spherical harmonic expansion of a function around a given point into a new spherical harmonic expansion around a displaced point, a 3N-dimensional cluster integral forN point particles (N > 2) may be reduced to 2N+1 trivial integrals andN– 1 interesting integrals, an improvement over the usual reduction to six trivial integrals and3N–6 nontrivial integrals. For hard spheres, theN–1 integrals involve only a series of simple polynomials taken between linear algebraic bounds.This work was supported in part by the National Science Foundation under Grant No. CHE79-20389.     

Limited projections laser speckle tomography of complex flows

Different approaches are discussed applied to reconstruction of local flow parameters using line-of-sight laser probing. The first one is based on single projection measurement and detailed numerical simulation of flow pattern. The second one is based on multi-projectional line-of-sight measurement and reconstructing local flow parameters by solving inverse integral transforms. Holographic interferometry (HI) and speckle photography (SP) are used for laser probing of complex 3D media under study. Computer-assisted tomography (CAT) approach based on Radon integral transform is described and examples of reconstructed flowfields are given including compressible flows with shock waves. Statistical information about flow studied is extracted as well, and maps of local statistical turbulence parameters are quantitatively determined by using obtained SP-data and a new Erbeck–Merzkirch integral transform.     

A computational method for wave propagation from a point load in an anisotropic material

One approach which is employed to solve dynamic point load problems in plates and laminates is to take integral transforms to reduce the governing equations to a system of ordinary differential equations with respect to the depth variable. The solution of this system leads to expressions for the transforms of the displacement and stress components at any level in the plate and the transient response at any location may then be recovered by inversion of the multiple transforms. The formal transform inversion involves a double infinite integral but by making a change of variable this may be replaced by an infinite integral associated with a line source and a finite integral with respect to the orientation of the line. A first attempt at applying this approach to obtain the point load response of quasi-isotropic fibre composite laminate led to a non-causal predicted signal. This paper deals with an investigation of this proposed method applied to the simpler model problem of wave propagation in a two-dimensional anisotropic medium. Results are obtained for two different time histories of point loads, namely: a delta function; and a single period of a sine function. In the case of the delta function source a comparison is made with the analytic solution and the errors arising from the numerical approach are discussed. Graphs are also presented showing the non-causal contributions to the overall response which arise at individual angles of orientation of the line source.     

Baryon-baryon spin-orbit and tensor interactions from quark-exchange kernels

Dibaryon quark-exchange kernels are constructed in explicit analytic form for the tensor and spin-orbit terms of the one-gluon-exchange quark-quark Breit interaction and for spin-orbit terms generated by quark-confinement mechanisms. The spin operators needed are defined through their spin-reduced matrix elements including those needed for interactions coupling NN, NΔ, and ΔΔ channels. Effective baryon-baryon spin-orbit potentials, generated through the Wigner transforms of the quark-exchange kernels with the use of a local momentum approximation, show that the NN spin-orbit interaction derived from the symmetric spin-orbit term of the one-gluon-exchange quark-quark interaction is in general agreement with the short-range part of phenomenological potentials derived from NN scattering. With the inclusion of the antisymmetric spin-orbit one-gluon-exchange terms and spin-orbit terms generated by confining potentials the full triplet-odd NN spin-orbit potential is greatly reduced in the 0.5–1 fm range. The uncertainties associated with spin-orbit terms generated by quark-confinement mechanisms are emphasized. The relative importance of various possible quark-gluon exchange terms is studied and shows that models which neglect some types of exchange terms are open to question. An SU(3)-flavor symmetric model for N-hyperon spin-orbit potentials leads to an NΛ spin-orbit potential only slightly weaker than the NN spin-orbit potential.     

Precise determination of Mössbauer lineshape parameters including interference

Using 100 Ci¹⁸³Ta and 5 Ci¹⁸²Ta sources, with LiF and NaCl crystal monochromating filters, we have measured the lineshape parameters for the 46.5 keV and 99.1 keV Mossbauer effect (ME) transitions of¹⁸³W and the 100.1 keV transition of¹⁸²W. Using an analytic representation of the convolution integral and utilizing asymptotic analyses of the lineshape, we find, for both transmission and microfoil internal conversion (MICE) experiments, accurate values of all ME parameters including width, position, cross section, and interference. This new approach allows deconvolution of source and absorber spectra and gives a simple analytic expression for both as well as their Fourier transforms. The line widths for the 46.5, 99.1, and 100.1 keV transitions are 3.10(10), 0.369(18), and 0.195(12) cm/s, respectively. The interference parameters are –0.00257(9), –0.0093(12), and –0.0107(12) in the same respective order. The agreement between transmission and MICE /1,2/ measurements for the above lineshape parameters is within the experimental errors. We believe these measurements are the first having sufficient precision to allow a quantitative comparison with dispersion theory /3,4/ and they indicate interference parameters 10 to 20% smaller than predicted. Our measured line widths are less than earlier reported values. This is because our analysis of the true lineshape and the study of line asymptotics permits a quantitative determination of the isomer lifetimes rather than the usual lower bound found in earlier ME experiments.This material was prepared with the support of the U.S. Department of Energy, Grants No. DE-FG02-85 ER 45199 A00 and De-FG02-85 ER 45200.     

A convolution integral for Fourier transforms on the groupSL(2,C)

The Fourier transform of a product of two functions onSL(2,C) is expressed as a convolution integral of the Fourier transforms of its factors. With the help of this convolution integral we present the Fourier transform of a polynomially bounded function as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.     

Feature separation using ICA for a one-dimensional time series and its application in fault detection

Principal component analysis (PCA) is a method that transforms multiple data series into uncorrelated data series. Independent component analysis (ICA) is a method that separates multiple data series into independent data series. Both methods have been used in fault detection. However, both require signals from at least two separate sensors. To overcome this requirement and utilize the fault detection capability of ICA and PCA, we propose to use wavelet transform to pre-process the data collected from a single sensor and then use the coefficients of the wavelet transforms at different scales as input to ICA and PCA. The effectiveness of this method is demonstrated by applying it to both a simulated signal series and a vibration signal series collected from a gearbox. The results show that the method of combining wavelet transform and ICA works better than the method of combining wavelet transform and PCA for impulse detection based on a one-dimensional vibration data series.     

The elliptical elegant Laguerre–Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new kind of laser beam called the elliptical elegant Laguerre–Gaussian beam (EELGB) is defined by using tensor method. By using the generalized diffraction integral formulas for light beam passing through paraxial optical system, the analytical propagation formulas for EELGB passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, the propagation properties of EELGBs in free space propagation and through a misaligned thin lens are studied.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.     

Exact broken-symmetry states and Hartree–Fock solutions for quantum dots at high magnetic fields

Wigner molecules formed at high magnetic fields in circular and elliptic quantum dots are studied by exact diagonalization (ED) and unrestricted Hartree–Fock (UHF) methods with multicenter basis of displaced lowest Landau level wave functions. The broken symmetry states with semi-classical charge density constructed from superpositions of the ED solutions are compared to the UHF results. UHF overlooks the dependence of the few-electron wave functions on the actual relative positions of electrons localized in different charge puddles and partially compensates for this neglect by an exaggerated separation of charge islands which are more strongly localized than in the exact broken-symmetry states.     

Analysis of the Spectral Composition of Fluorescence from Laser-Excited Cold Atomic Clouds

A consistent quantum approach is used to calculate the signal of fluorescence from a laser-excited cold atomic ensemble. The possibility of analyzing the spectral composition of this nonstationary signal is investigated. Application of wavelet transforms is shown to be a very efficient method that allows achieving a good spectral or temporal resolution, depending on specific problem. On the basis of this approach, the spectral composition of fluorescence and its dynamics after excitation are studied. The spectral properties of the secondary emission of the ensemble are compared for different directions and polarizations.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for derivingmiscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can alsobe easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transformsand the squeezing transforms in quantum optics is investigated.     

Wavelet analysis of extremely short femtosecond pulse propagating in dispersion-shifted single-mode fiber

Results of the wavelet analysis of an extremely short femtosecond pulse propagating in a single-mode optical fiber with a shifted dispersion characteristic are described. The advantages of wavelet analysis as compared with Wigner analysis are shown. In particular, wavelet analysis allows improving the spectral resolution of the studied pulse in the high-frequency range.     

On the formal expansion of the possibilities of the method of integral transformations in investigating linear distributed systems with constant parameters

The Green's functions for a series of boundary-value problems are formulated in transforms with respect to t by means of matrix functions that are introduced and allow the scalar dependence on the weighted differences of the coordinate argument to be isolated in them. It was possible in general form to carry out the procedure of the method of integral transformations in solving the integral equations for the function that generates periodic motion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 323–331, March, 1972.