Double Weighted Fourier Transform Research Articles

Wave field in a layer with a linear background profile and multiscale random irregularities

The paper addresses the problem of determining the field of a reflected wave in a multiscale inhomogeneous medium, which consists of a background deterministic large-scale irregularity and multiscale random irregularities with scales both larger and smaller than the wavelength. A layer with a linear permittivity profile is taken as the background irregularity. The small-scale irregularities are responsible for wide-angle scattering including backscattering. The first approximation of the perturbation theory is used to account for scattering from these irregularities. As the zero approximation and Green’s function in determining the field backscattered by small-scale irregularities, we utilize the integral representation of the field, obtained in our earlier work by combining the method of double weighted Fourier transform (DWFT) and the Fock proper-time method. Asymptotic methods are used to reduce the sevenfold integral representation of a field to lower-order integrals. Conditions for validity of such representations are obtained. Formulas of the frequency coherence functions of waves reflected and backscattered from the turbulent plasma layer are given. The paper presents the results of the simulation of pulse sounding of a randomly inhomogeneous reflecting plasma layer demonstrating the effect of large-scale irregularities on backscattering by small-scale irregularities.

Some Possibilities of Spatial Signal Processing in an Inhomogeneous Medium Based on DWFT

AbstractThe integral representation for the wave field in a multiscale inhomogeneous medium, obtained using the double weighted Fourier transform (DWFT), takes into account different diffraction effects. Spatial radio signal processing based on the DWFT inversion eliminates these effects from received signals. This paper reviews the results of studies on spatial signal processing based on the DWFT inversion. This processing has been shown to reduce amplitude scintillation and improve the resolution in diagnostics of inhomogeneous media. Moreover, elimination of diffraction effects allows us to reduce the residual error in multifrequency global navigation satellite systems. This processing also increases the channel bandwidth and therefore its capacity.

Using spatial radio wave field processing for diagnostics of inhomogeneous plasma

We explore the possibilities of double and single spatial field processings, obtained using the method of double weighted Fourier transform (DWFT), for inhomogeneous plasma diagnostics. We focus on the problem of diagnostics of local small-scale plasma irregularities generating strong radio wave phase variations during small-angle scattering. Numerical calculations of wave phase in DWFT and phase screen approximations have shown that the use of double and single spatial field processings expands Fresnel resolution even in conditions of strong phase variations. However, the single field processing requires knowing coordinates of the virtual screen plane, used for the field processing. For optimal determination of this parameter, we suggest using an algorithm based on the synthesis of Fresnel inversion and DWFT method for remote irregularities. The combined use of the inverse Radon transform and double and single spatial field processings provided images of three Gaussian plasma irregularities with scales less than the Fresnel radius in conditions of strong wave phase variations. We analyze the effect of discretization of elements and size of a transmitting and receiving system on field processing.

Two-Frequency Coherence Function for the Field of a Wave Propagating Through a Multiscale Randomly Inhomogeneous Medium

We study the possibilities of integral representation for the two-frequency mutual coherence function of the wave field in a randomly inhomogeneous ionosphere. The integral representation was obtained using the Double Weighted Fourier Transform (DWFT) method. We point out that the conditions of validity of the geometrical-optics approximation for frequency coherence are weaker than the same condition for individual samples. Examples of calculation of the frequency coherence functions for waves in the ionospheric plasma with the irregularities described by the Gaussian spectrum and Shkarofsky’s model are given. Simulation results show that diffraction effects reduce the width of the frequency coherence function. The capabilities of the methods for spatial processing of the wave field and its two-frequency mutual coherence function, which eliminate these effects through the Fresnel and DWFT inversions, are examined.

Higher Approximations to Study Statistical Characteristics of Waves in Multiscale Inhomogeneous Media

The previously obtained integral field representation in the form of double weighted Fourier transform (DWFT) describes effects of inhomogeneities with different scales. The first DWFT approximation describing the first-order effects does not account for incident wave distortions. However, in inhomogeneous media the multiscale second-order effects can also take place when large-scale inhomogeneities distort the field structure of the wave incident on small-scale inhomogeneities. The paper presents the results of the use of DWFT to derive formulas for wave statistical moments with respect to the first- and second-order effects. It is shown that, for narrow-band signals, the second-order effects do not have a significant influence on the frequency correlation. We can neglect the contribution of the second-order effects to the spatial intensity correlation when thickness of the inhomogeneous layer is small, but these effects become noticeable as the layer thickness increases. Accounting for the second-order effects enabled us to get a spatial intensity correlation function, which at large distances goes to the results obtained earlier by the path integral method. This proves that the incident wave distortion effects act on the intensity fluctuations of a wave propagating in a multiscale randomly inhomogeneous medium.

Wave reflection from randomly inhomogeneous ionospheric layer: 1. The method of describing the wavefield in a reflecting layer with random irregularities

AbstractIt has been previously proposed to describe wave propagation in inhomogeneous media in a small‐angle approximation with the aid of a double weighted Fourier transform (DWFT) method. This method agrees with the methods of geometrical optics, smooth perturbations, and phase screen in domains of their applicability; therefore, it can be employed to solve direct and inverse problems of radio wave propagation in multiscale inhomogeneous ionospheric plasma. In this paper, for the DWFT wide‐angle generalization a wave equation is preliminary reduced using the Fock proper‐time method to a parabolic equation that then is solved by the DWFT method. The resulting solution is analyzed for the case of wave reflection and scattering by a layer with random irregularities and linear profile of average permittivity. We show the transformation of this solution into strict results in the absence of irregularities and in the single‐scatter approximation, including backscattering, during weak phase fluctuations. Under certain conditions, the solution takes the form of the small‐angle DWFT with respect to refraction in the layer and backscatter effects. Spatial processing in source and observer coordinates brings a beam of received waves into one wave without amplitude fluctuations, which allows an increase in resolution of vertical ionospheric sounding systems.

Numerical Implementation of the Double-Weighted Fourier Transforms in the Problems of Wave Propagation in Inhomogeneous Media

A numerical scheme for calculating direct and inverse double-weighted Fourier transform (DWFT) in the problems of wave propagation in inhomogeneous media on the basis of the fast Fouriertransform algorithm is proposed. The sampling step for which it is possible to recover a continuous field without information loss is estimated. The numerical calculation data are given and the obtained results are discussed. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 56, No. 7, pp. 514–523, July 2013.

About the method of integral field representation as the double weighted Fourier transform in the problems of wave propagation in inhomogeneous media

The limits of applicability of the method of integral field representation as the double weighted Fourier transform (DWFT) are determined on the basis of an exact path representation for the Green function of the parabolic equation. The estimates we have obtained imply that the DWFT method takes into account the effects of diffraction by inhomogeneities with scales equal to or less than the radius of the first Fresnel zone of a plane wave and can be exploited to solve the inverse problems of retrieval of the inhomogeneous-medium parameters from the data measured in the source and receiver planes.

Diffraction tomography of inhomogeneous medium in the presence of strong phase variations

A model of diffraction tomography on the basis of the double weighted Fourier transform (DWFT) is considered. A method for processing the results of tomographic measurements based on the inverse DWFT is proposed. Such processing enables one to obtain super-Fresnel resolution, just as Born’s and Rytov’s models of diffraction tomography but in conditions allowing strong phase variations. The influence of the dimensions and discrete structure of the radiating and receiving antennas on the resolution of the diffraction tomography is studied. It is shown that, using sufficiently large antennas and a small discretization interval, one can obtain super-Fresnel resolution in the presence of phase variations that are significantly greater than those admissible for Born’s and Rytov’s algorithms.

Method for super Fresnel resolution in electromagnetic diagnostics of inhomogeneous plasma

The method of electromagnetic diagnostics is suggested, which promises to perform super Fresnel resolution of plasma inhomogeneities, that is resolution, distinguishing details smaller than Fresnel radius. To realize super Fresnel resolution it is suggested to represent the wave field of the source in the form of double weighted Fourier transform (DWFT), deals with Fourier transform simultaneously in coordinates of the sources and in coordinates of receivers. Important property of DWFT is that DWFT transfers into geometrical optics (GO) approximation for smooth inhomogeneous media and becomes equivalent to the Rytov or to small angle Born approximation in the case of weak inhomogeneities. As a result, inverse DWFT allows obtaining linear integral of plasma density both for large scale inhomogeneities, as in GO approximation, and also for inhomogeneities, whose transverse sizes are small as compared with Fresnel radius. DWFT embraces also the results of the phase screen method and allows to take into account phenomenon of micro-multirayness and to describe strong amplitude fluctuations. Using inverse DWFT algorithm, the authors study resolution of systems consisting of discrete sources and receivers. Both analytical estimates and the results of numerical modeling evidence opportunity to observe small scale plasma inhomogeneities with super Fresnel resolution.

Super-Fresnel resolution of plasma inhomogeneities by electromagnetic sounding

The application of the double weighted Fourier transform (DWFT) method is suggested for extracting the linear integral of the electron density in the inhomogeneous plasma from microwave or IR sounding data. The virtue of DWFT is its ability to localize the measured linear integral in the area, narrow as compared with the radius of the Fresnel zone, that is to realize super-Fresnel resolution. The advantages of the DWFT method are illustrated, firstly, by numerical simulation of a wave propagation through the Gaussian inhomogeneity in conditions of weak scattering, when a small angle Born approximation is applicable, and secondly, by theoretical estimates of resolution in conditions of strong scattering. The method under discussion promises to be helpful for studying the fine structure of turbulent plasma.

Ray Based Diffraction Tomography of the Ionosphere and Laboratory Inhomogeneous Plasma

A new procedure for restoration of the plasma inhomogeneities with improved resolution is suggested. The procedure deals with the double weighted Fourier transform (DWFT) of the observed wavefield in coordinates of both receivers ρ = (x, y) and sources ρ0 = (x0, y0) [1]. Phase increments between the sources and receivers, being found from DWFT representation, can be used for extracting information on small perturbations of the dielectric constant e~(ρ, z) in a way similar to traditional radio tomography. The resulting resolution of the method is close to the diffraction limit Δρ = λh/D in the horizontal direction and Δz = λ(h/D)2 in the vertical direction, where h is the height of inhomogeneities and D is the length of the ground-based receiving system.