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We present a simple and general theory of compressive sensing (CS) that relies on elements of the sensing matrix rather than on the number of measurements. We prove the exact recovery using a dual certificate by showing that the sensing matrix satisfies an incoherence property and isotropy property if the sparsity level is kept lower than the reciprocal of the largest element of a matrix created from the sensing matrix. Unlike the CS literature, this unconventional approach does not require a linear relationship between the sparsity and the number of measurements and at the same time, can easily be evaluated. This adaptability captures anisotropic measurements appropriately as with anisotropic measurements, adding more measurements does not really imply that a signal with more nonzero elements will be recovered exactly. As an illustration, we demonstrate the theory’s ability to accurately handle the anisotropic (Green’s function-based sensing matrix) measurements and also its similarity to the existing CS literature for isotropic (Fourier) measurements. Further, we show the usefulness of the theory in comparing different sensing matrices and in generating dielectric images. The dielectric images are perfectly recovered even when there is only a single transmitter.

Discrete complex exponentials are almost periodic signals, not always periodic; when periodic, the frequency determines the period, but not viceversa, the period being a chaotic function of the frequency, expressible in terms of Thomae's function. The absolute value of the frequency is an increasing function of the subadditive functional of average variation. For discrete signals that are either sums or series of periodic complex exponentials, the decomposition into their periodic, additive components allows for their filtering according to period. Likewise, their period-frequency spectrum makes predictable the effects on period of convolution filtering. Ramanujan-Fourier series are a particular case of the signal class of series of periodic complex exponentials, a broad class of signals on which a transform, discrete both in time and in frequency, called the DFDT Transform, is defined.

The waveguide invariant is classically considered as constant in shallow water and can be utilized to estimate the source range. When the environment is variable, the error between the estimated range and the true range fluctuates and becomes unstable. The fluctuation of the range estimation error is large when we use the polar coordinate transform to obtain the slope of the two-dimensional Fourier transform ridge (2DFT-PT). The lines which are parallel in the ridge are not considered and the ridge information is not utilized sufficiently. For this problem, we propose a passive ranging method by using the Radon transform to obtain the slope (2DFT-RT) and achieve a more robust result. Its capability is investigated and compared with the 2DFT-PT method via numerical simulation and the SWellEx-96 experiment.

The classical radial harmonic Fourier moments (RHFMs) and the quaternion radial harmonic Fourier moments (QRHFMs) are gray-scale and color image descriptors. The radial harmonic functions with integer orders are not able to extract fine features from the input images. In this paper, the authors derived novel fractional-order radial harmonic functions in polar coordinates. The obtained functions are used to defined novel multi-channel fractional-order radial harmonic moments (FrMRHFMs) for color image description and analysis. The invariants to geometric transformations for these new moments are derived. A theoretical comparison between FrMRHFMs and QRHFMs is performed from the aspects of kernel function and the spectrum analysis. Numerical simulation is carried out to test these new moments in terms of image reconstruction capabilities, invariance to the similarity transformations, color image recognition and the CPU computational times. The obtained theoretical and numerical results clearly show that the proposed FrMRHFMs is superior to the QRHFMs and the existing fractional-order orthogonal moments.

An efficient parabolic equation solver is proposed to predict the radio wave propagation in forest during the rainy seasons. The method starts with considering the forest model as a mixture of plants and rain particles. Then the equivalent dielectric constant model can be used to simplify the calculation process. At last, the Split-step Fourier transform (SSFT) is used to calculate the loss of radio wave propagation in the forest. Numerical results show that this model can calculate the propagation loss of forests in rainy environment.

In this paper, a half-space wave propagation problem in the vicinity of discontinuities is considered and the cross-sectional Hermite – Gaussian expansion of three dimensional electromagnetic fields at a constant range is investigated. The electromagnetic fields are calculated via a Finite-Difference Time-Domain algorithm and then Fourier transformed. Series expansion coefficients are obtained through an inner product operation with each orthonormal basis function. It is shown that fields can be successfully reconstructed with a limited number of coefficients.

A novel approach to the design of cloaking devices is presented which takes inspiration from the Fourier diffraction theorem. In particular, we first put on a common basis different linear approximations of the scattering equations. Then, a very simple procedure to get invisibility (under the given approximation at hand) is proposed. The approach is assessed through numerical examples concerning 3-D non-canonical objects.

Fourier-processing of finite discrete two-dimensional signals (FDTD signals) (including images) in informational control systems (IC systems) is the most important method for studying processes and analyzing information. The theoretical basis of Fourier-processing of FDTD signals is two-dimensional discrete Fourier transforms. The practice of using Fourier - processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. In order to solve this problem, the authors have developed new discrete Fourier transforms with variable parameters were developed (2D DFT-VP). The purpose of this work is to study the properties of two-dimensional discrete exponential functions with a variable parameter in the spatial-frequency domain. The introduction of discrete exponential functions with a variable parameter makes it possible to generalize the concept of periodicity of the DEF-VP system. Recall that the periodicity of the DEF system in the classical DFT is understood as a periodic continuation of the DEF system outside the interval of N samples. Moreover, the system of discrete basis functions in the classical DFT does not contain discontinuities. In the case of DFT-VP, for the DEF-VP system to be inseparable, the periodicity should be understood as parametric periodicity. The parametric periodicity of discrete exponential functions with a variable parameter is understood as their periodic continuation with rotation in complex space by a certain angle. Note that the introduced concept of parametric periodicity is valid for 1D and 2D real and complex functions. The theorems of linearity, shift, and correlation are proved for Fourier transforms with variable parameters.

The practice of using Fourier - processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. The purpose of this paper is to develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals. We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. Methods are based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. New discrete Fourier transforms with variable parameters were developed. The proposed discrete Fourier transforms with variable parameters are a generalization of the classical discrete Fourier transform. The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier - processing of two-dimensional signals (including images), padded with zeros.

This paper introduces preliminary results about a novelty formulation to calculate the first reflection over a sloping terrain for a Modified Two-ray model with atmospheric effects. For a case study of a lossy wedge and standard atmosphere conditions, the obtained results through the proposed formulation and the Discrete Mixed Fourier Transform (DMFT) numerical solution are compared, in order to validate the presented approach. The early results showed similar behavior for path loss, in this canonical scenario.

This work examines the characteristic modes and measurement of various circularly distributed array topologies in which element radiators are used independently to deliver both sum and difference beams under lossy conditions. An associated moment generating function is derived, such that analytical patterns use even-odd symmetries for sum difference beam behavior. This approach generalizes the Fourier probabilistic methods by using the Laplace transform to analyze statistical averages catering to the degenerating effects of pattern behavior that is influenced by the environment.

Characterizing the perturbation of scattering coefficients by an additive noise is a key problem to understanding the channel model for a Nonlinear Fourier Transform (NFT) based fibre optical transmission system. In this paper, we show numerically that the perturbation of scattering coefficients is mainly contributed by the noise component (which has significantly lower power than the overall noise) from a dominating noise subspace. An explicit characterization of the subspace is obtained by approximating the derivatives of the scattering coefficients. Refereed/Peer-reviewed

This Ships and offshore structures are exposed to random wave loading in the ocean environment. When predicting the amplitude of the motion response of the ships and offshore structure, it is important to obtain the random wave loading. And the dominant load is due to wind-generated random waves. For wind-generated random waves, the ocean surface elevation is defined by ocean wave energy spectra. There are several ocean wave power spectra in which Pierson-Moskowitz spectrum is the most classical one. Pierson-Moskowitz spectrum describes the energy distribution in a fully developed sea. The power spectral density and the autocorrelation function ACF are Fourier transforms pairs. And the power spectrum is usually obtained by the Fourier transform of the of the signal. But the ACF calculation of the Pierson-Moskowitz spectrum has always been complicated and cumbersome. With the development of mathematical modeling, it is possible to explore the ACF of the wave power spectrum rapidly and efficiently, which can reasonably be considered to be of great significance to the advancement of study on numerical model of ocean wave power spectrum. Additionally, in order to verify the correctness of the result, the discrete Fourier transform is performed based on MATLAB.

In this paper, a linear transform that is a further generalization of the linear canonical transform (LCT) and the fractional Fourier transform is proposed. Instead of the chirp terms adopted by the LCT, we apply exponent functions with higher order phases before and after applying the scaled Fourier transform. With the proposed higher order phase linear canonical transform (HOPLCT), one can convert the time-frequency distribution with a bent shape, which is usually the case for vocal signals, into a rectangular shape. It can much reduce the bandwidth of a signal and is very helpful for signal sampling and filter design.

The problem of calculating active electric energy losses under the conditions of a non-sinusoidal voltage in the 380 V network is considered. The network voltage is represented as a partially steady signal, with the time dependences of signal component frequencies or harmonic components being piecewise constant functions. This circumstance adds difficulty to the use of classic loss calculation methods, which do not allow the spectrum to be analyzed as a function of time. For solving the problem, the discrete-time short-time Fourier transform is applied. Based on the results of calculating the discrete-time short-time Fourier transform, the steadiness intervals are determined, in which the electric energy losses are subsequently calculated using the well-known formulas. For estimating the validity of the proposed approach, simulation is carried out of the electric network part containing an industrial load connected to a 380 V bus via a cable line in which the electric energy losses are calculated. A conclusion about the applicability of the short-time Fourier transform is drawn.

In this paper, signal components after propagation in a dispersion managed (DM) coherent optical fiber link are analyzed, statistically. In this way, the received signal Gaussianity is investigated using two Gaussian tests, the Anderson-Darling and the Jarque-Bera. Deviation from Gaussian distribution is result of increasing the launch power, that is measured by the tests. Next, we use Johnson $\mathbf{S}_{U}$ distribution for distribution fitting and show its good approximation in DM systems. Bit error rate (BER) is employed for performance evaluation that is calculated in three methods: 1) Gaussian based, 2) Johnson $\mathbf{S}_{U}$ based, and, 3) Monte-Carlo (MC) simulations based on the split-step Fourier method. It is demonstrated that in the nonlinear region, using Johnson $\mathbf{S}_{U}$ distribution will have about 2 dB better power prediction than Gaussian in $\mathbf{BER}=10^{-4}$ , and the performance fits to the MC method with more accuracy.

In this paper, we present a novel method that utilizes amplitude-only data of electric field over two hemispheres for the far-field reconstruction above the perfectly electrically conducting (PEC) ground plane by the iterative Fourier transformation algorithm (IFTA).When the IFTA reaches the maximum number of iterations, then we can calculate the far-field (FF) by hemispherical near-filed to far-field (NF-FF) transformation. The simulated and numerical results are in good agreement with each other, which shows that this method works effectively.

The paper examines the model problem of high-frequency diffraction by a convex surface consisting of two parts. One is soft, the other is hard. The incident wave falls at a small angle to the line which separates soft and hard parts of the surface. The change in the boundary condition provokes the field in the Fock zone to have a quick transverse variation. This causes a special boundary-layer to be formed. The boundary value problem for the three dimensional parabolic equation is reduced to the Riemann problem solved by the factorization in the form of infinite products containing the zeros of the Airy function and zeros of its derivative. the results of this factorization appear under the sign of double Fourier integral in the representation of the field. Both numerical and asymptotic analysis of this representation is carried out and illustrates the effects of high-frequency diffraction caused by the line of the boundary condition discontinuity.

In this research, we have conducted a study on color image filtering in Bessel-Fourier moments domain. Bessel-Fourier moments of the two testing color images are computed independently from the three color channels (RGB), then lowpass and highpass filters are applied to the data in Bessel-Fourier moments domain for our investigation. For comparison, filters are applied in Fourier Frequency domain as well. The experimental results suggest that Bessel-Fourier moments of the lower orders contain mainly information of smooth varying components of images, while those of the higher orders are more related to details such as sharp transitions in intensity. It is also found that the Gaussian filters would reduce the ringing effect in Bessel-Fourier moments domain as they do in the Fourier Frequency domain.

Fast and high precision numerical simulation under multi-source condition is the key to two and a half dimensional (2.5D) high efficiency and high precision inversion of DC resistivity method. In this paper, a 2.5D multi-source fast and high-precision numerical simulation method based on DC resistivity method is proposed, which can be applied to arbitrary scale and complex geoelectric model. Firstly, the boundary conditions are transferred to make the coefficient matrix independent of the position of the source. On the premise of using the direct solution method to solve the linear equations, the fast back-substitution solution is realized under the multi-source condition. Secondly, a set of wave number selection method that can be adapted to arbitrary scale model is proposed, and the inverse Fourier transform is calculated by using the finite element method. And also, the potential value in the spatial domain of any lateral section is calculated. Finally, an effective solution to the singularity of the field source is proposed. The numerical simulation algorithm shows that the 2.5D numerical simulation of DC resistivity method under multi-source condition proposed in this paper has the characteristics of fast calculation efficiency, high numerical calculation accuracy and high numerical stability.

Methods of determining the envelope (instantaneous amplitude) of a real signal for continuous and discrete cases are investigated. The analysis of advantages and disadvantages of existing methods for extraction the envelope of a real signal for continuous and discrete cases are given. The efficiency of the choice of determining the conjugate signal by Hilbert transform method is substantiated. The problem of determining Hilbert envelope for real, discrete and finite signals by the methods of discrete Fourier transform (DFT) and discrete-time Fourier transform (DTFT) is considered. The reason for the large relative errors in the measurement of Hilbert envelopes by DFT method is revealed. A new effective method and an effective method for measuring the envelope of a real, discrete, finite signal is developed, which reduces the relative error of its measurement by an order of magnitude and reduces by an order of magnitude the computational cost and the necessary memory for implementing the method as well. The theoretical results obtained in the work are confirmed by the results of numerical modeling.

A coordinate-free formulation of four tensor Green’s functions for the general homogeneous uniaxial anisotropic lossless and/or lossy media is developed with the Fourier transformations. In the derivation of these formulas, the product to sum approach is introduced to evaluate the Fourier integrations. For the lossy media, the double-value problem, which is existed in calculating formula of Green’s functions, is effectively avoided by the radiation condition. With a way of finding limit, the pseudo-singularities problem of four Green’s functions, which has to be treated, is discussed deeply and resolved briefly. The robust and efficient computational procedure of Green’s functions is achieved with C++ operator overloading. Modeling results of the triaxial induction well-logging demonstrate efficiency of the computation of the coordinate-free formulas.

Triaxial induction tools can be used to evaluate thinly laminated sand-shale sequences and fractured beds. This type of reservoirs exhibit transversely isotropy (TI) or arbitrarily anisotropy (also called as biaxial anisotropy (BA)). There have been several papers to study the responses in the TI model or the simplified BA model whose conductivity principal coordinate is always consistent with the formation coordinate. However, little work covers the most general biaxial anisotropic model whose conductivity tensor’s orientation is arbitrary. We introduce the Euler angles, then the general biaxial anisotropic conductivity tensor can be determined by three principal components and three ordered Euler angles. To derive the electromagnetic (EM) fields in arbitrarily anisotropic medium, we first convert the Maxwell’s equation of frequency-spatial domain into frequency-wavenumber domain by 2D Fourier transform, and obtain an ordinary differential system about horizontal components of EM fields. Using eigenvalue decomposition of the system matrix, this system can be decomposed into two group of equations associated with upward and downward eigen-waves respectively. We derive the solutions of EM fields in frequency-wavenumber domain by introducing transmission matrix, both local and generalized reflection matrix and propagator matrix After that, we use 2D Gauss-Legendre quadrature to calculate inverse Fourier transformation and obtain Green’s function for simulation of the tri-axial induction responses. The numerical results are compared with 3D numerical method in both vertical and deviated wells and the agreement is satisfactory. Finally, we investigate the response characteristics in several formations with different Euler angles The results show that triaxial induction responses are remarkably influenced by Euler angles even if the values of three principal components of conductivity tensor remain unchanged. Compare to the responses of the simplified BA model, those of general BA model are more complex and contain more nonzero components. The results indicate that using TI model or the simplified BA model in complex environment may cause large errors. Our algorithm are more practical than algorithms based on the simplified model because the real depositional environments are usually complicated.

The problem of E -polarized plane wave diffraction by a strip with fractional boundary conditions (FBC) is analyzed using the Wiener-Hopf technique. Introducing the Fourier transform for the unknown scattered field and applying FBC, the problem is formulated in terms of the Wiener-Hopf equation. The Wiener-Hopf equation is solved via the factorization and decomposition procedure leading to the exact and asymptotic solution. Our final solution is valid for the case where the strip width is large in comparison to the wavelength. The scattered field is evaluated asymptotically based on the saddle point method and a far field expression is derived. Numerical examples on the radar cross section (RCS) are presented for various physical parameters and the scattering characteristics of the strip are discussed in detail.

We considered a problem of reconstruction function in a strip, if we know the integrals of sought function in the family of straight-line segments with a given weight function of a special kind. We obtained an analytical representation of solution in the class of smooth finite functions, the uniqueness theorems for solution of the problem is proved. The stability estimate of solution in Sobolev spaces is presented, which implies its weakly ill-posedness. These theoretical results are used in order to recover the internal structure objects from their integral data. Numerical results and their graphical representation are given.

Jacoby Fourier polynomial shaped phase masks (JFPM) have been shown to be potentially effective for extending the depth of focus in wavefront coding technique. However, some characteristics have not been reported in detail. Here, we will show performance of JFPM for different ${f}$/# and compare with the trefoil phase mask.

In this research, we have analyzed the Chebyshev-Fourier moments computing and found out that the errors are mainly caused by the computational issues of both radial and Fourier polynomials. To increase the accuracy of moments computing, we utilized the k × k numerical scheme in the computations of Chebyshev-Fourier moments and conducted the image reconstructions to verify our solution. The experimental results show that the performances of image reconstructions are highly satisfied. We have also performed image reconstructions from the different radial and Fourier orders of moments, and observed that the radial and Fourier orders of Chebyshev-Fourier moments preserve the information of circular and radial patterns, respectively.

This paper deals with the asymptotic method for describing wave fields in inhomogeneous media, which has the form of the Double Weighted Fourier Transform (DWFT). We show a connection between DWFT and other asymptotic methods for solving the wave equation. We report the results of numerical calculations of phase variations in the wave scattered by irregularities with scales smaller than the Fresnel radius under strong phase variations. As a method able to eliminate diffraction and multipath effects we propose to use spatial field processing obtained by DWFT and and its variants.

Excitation of electromagnetic waves by sources immersed in a homogeneous magnetoplasma is studied using an eigenfunction expansion method. The total field is sought in terms of vector modal solutions of the source-free Maxwell equations. The content of the modal spectrum is determined and the eigenmodes of the homogeneous magnetoplasma are found to form a continuous set. The source-excited field is represented by an integral expansion in terms of the obtained continuous-spectrum modes and the expansion coefficients of such a field representation are derived in explicit form using the mode orthogonality relation. The developed approach is applied to the analysis of radiation from a rotating magnetic-field source in the whistler frequency range.

The problem of representing S-parameters of a periodic structure using the model of an equivalent homogeneous layer is considered. For correct determination of the values of the parameters for such model, it is proposed to use the transition to the time domain and inverse one using the Fourier transform. The use of the proposed approach has been examined for several types of periodic structures. The efficiency of this approach is shown for periodic structures with frequency bands of the anomalous behavior of the phase-frequency characteristic.

Pulse Wave Velocity (PWV) is the optimal index for quantitative evaluation of arteriosclerosis. Owing to the transmission of carotid pulse wave (PW) signals are affected by reflected wave, and the quantization errors occur when moving cross-correlation is used to estimate the pulsation displacement, the accuracy of Time Transit (TT) method in estimating local PWV is decline. For improving the accuracy of PWV detection, a smoothing estimation method based on Fourier-based Filter (FT-Filter) is proposed in this paper. Fourier transform is used to transfer the PW from time domain to frequency domain, and the filter parameters are determined by the spectrum characteristic to filter the high frequency components. Then the time domain denoised PW is obtained by inverse transform. The proposed method is used to process the simulated and clinical PW transmission sequences. The PWV is estimated by TT method, and compared with the estimated results of Savitzky-Golay and Butterworth Filter, the normalized root mean square errors of 30 simulated cases are $0.28\pm 0.05,\ 0.54 \pm 0.07$ and $0.39\pm 0.06$ , respectively. The clinical results of 20 subjects' PWV are 4.51 m/s, 5.86 m/s and 5.19 m/s, respectively. It shows that the proposed FT-Filter improves the accuracy of local PWV estimation based on TT method. There will be helpful to improve the accuracy of clinical diagnosis of local arteriosclerosis.

In this paper, a parabolic equation method for wave propagation in tunnels is raised. To solve parabolic equation, split-step Fourier Transform has been introduced. Then the expressions of refractivity describing the complex meteorological environment are provided. The wave propagation in tunnels can be predicted by amending the refractivity in PE.

The technique for increasing the robustness of the photometric method of ellipsometry according to Malus law is considered. The parameters of the polarization ellipse by the ordinary least squares method and the Fourier method are calculated. The data processing procedure using the bootstrap method is described. The spread of the measured parameters of the polarization ellipses is calculated. The stability and reliability of the obtained results is appreciated. The increase in the robustness of the photometric method is confirmed.

Firstly, based on basic properties of the kernel function of the quaternion Fourier transform we derive in detail relationships among three definitions of the quaternion Fourier transforms. Secondly, based on the quaternion Fourier transform of the quaternion Gaussian function we derive an inversion formula to recovering a quaternion function from the quaternion Fourier transform.

In this paper, we study the recovery of a certain type of shape images modeled as binary-valued images from few Fourier samples. In particular, the boundary of the shapes is assumed to be the zero level-set of a trigonometric curve. This problem was previously considered in [1], and it was shown that such images can be studied withing the framework of signals with finite rate of innovation. In particular, an annihilation filter is introduced to recover the trigonometric boundary curve. It is proved in [1] that 3Λ Fourier samples are sufficient to exactly recover the edges of a binary shape, where Λ is the bandwith of the trigonometric boundary curve. In this paper, we introduce a class of shapes (that include convex shapes as special cases) for which |2Λ + 1| Fourier samples are sufficient for exact recovery using the same annihilation filter. Simulation results support our theoretical results.

We present a novel analytical model for the lossless fiber in the weak-dispersion regime. The model is compared to three other models via simulations using the split-step Fourier method. These simulations explore the validity of the models with respect to parameters that affect the signal after the fiber propagation. In one of the systems under consideration, the proposed model is accurate until 8.7 dBm while the others are accurate only until 2.4 dBm. The impact of the modulation format is analyzed for three of the models. Our results show differences up to 1.82 dB in the maximum power in which the proposed model remains accurate when changing from quadrature phase shifting keying to 8-ary quadrature amplitude modulation.

The quaternion Fourier transformation based on orthogonal planes split is an extension of the two-sided quaternion Fourier transformations using quaternion split. In the present paper we investigate its basic properties such as linearity, frequency-shift and time-frequency shift. We then study the convolution and correlation definitions for the quaternion Fourier transformation based on orthogonal planes split and obtain their convolution and correlation theorems.

Signal reconstruction is one of the most important problems in signal processing and sampling theorems are one of the main tools used for such reconstructions. There is a vast literature on sampling in one and higher dimensions of bandlimited signals. Because the sampling formulas and points depend on the geometry of the domain on which the signals are confined, explicit representations of the reconstruction formulas exist mainly for domains that are geometrically simple, such as intervals or parallelepiped symmetric about the origin.In this talk we derive sampling theorem for the reconstruction of signals that are bandlimited to a disc centered at the origin. This will be done for a more general class of signals than those that are bandlimited in the Fourier transform domain. The sampling points are related to the zeros of the Bessel function.

This paper presents the numerical stability analysis of multiple one-dimensional locally one-dimensional finite-difference time-domain (M1-D LOD-FDTD) method for inhomogeneous coupled transmission lines. The stability analysis is first performed using the von Neumann method based on Fourier amplification matrix for homogeneous coupled transmission lines. The transition matrix for whole computational domain is next analyzed, which takes into consideration inhomogeneous media. While the M1-D LOD-FDTD method for homogeneous coupled transmission lines is stable based on the von Neumann method, the eigenvalues of transition matrix for inhomogeneous coupled transmission lines may indicate instability. Results show that the stability analysis based on von Neumann Fourier modes alone may not be sufficient to prove stability as it generally assumes homogeneous media.

This paper studies the dual-phase-lag heat conduction system, the model of which improves the classical Fourier equation by adding two positive delay variables. Assume that these two delay parameters are vanished in a subregion of the spatial domain [-1,1], which yields a mixed heat conduction system. By the resolvent estimate, we obtain two different energy decay rate of the system under certain conditions respect to the relationship between these two delay parameters. Especially, a faster energy decay rate is proved compared to the one with the globally positive delay parameters conditions are fulfilled.

A unified time- and frequency- domain equation in terms of the active reflection coefficient (ARC) in four-dimensional (4D) antenna arrays including mutual coupling effect is presented. The "instantaneous" active reflection coefficient (ARC) as a function of time and the "averaged" active reflection coefficient as a function of frequency are introduced to evaluate the port matching and the mutual coupling in 4D arrays, respectively. The two analysis approaches from different viewpoints are demonstrated to be unified and complementary through Fourier transform. The results of a 16-element 4D linear array with tightly coupled elements are used to validate the unified equation.

The report proposes a method for calculating the electromagnetic field on rough terrain using a parabolic equation based on elevation data obtained from electronic terrain maps. The presence of inhomogeneous relief significantly affects the propagation of radio waves above the Earth's surface. Obstacles serve as a barrier, as well as with a special arrangement and geometry can be a natural repeater. The calculation of the electromagnetic field using the parabolic equation method with the step-by-step Fourier transform allows for a three-dimensional model of space to take into account the point-built inhomogeneity of the relief, if you specify the desired step.

The paper presents an imaging algorithm of millimeter wave radar. The radar scans the space by changing the antenna positions and acquires the signals at different frequencies in each point. The stolt interpolation algorithm converses $k_{x}^{\prime}-f$ domain to $k_{x}^{\prime}-k_{z}^{\prime}$ domain. Then by inverse Fourier transform we obtain the image of the object in the space plane. Finally, the effectiveness of the algorithm is verified by MATLAB simulation.

The paper studies a method of forming a mathematical model of a synchronous machine when exposed to voltage drops on the exciter. A local integral transform was used as a partial local Fourier transform for the transition from the continuous variables to the equations in finite differences of constant structure. After that, the reverse transition to the mathematical model of the constant structure in continuous variables.

The Casimir energy has been calculated using the path integral method [2], [3] for a geometry of cylindrical perfect electric conductors. A similar derivation is proposed by making use of a decomposition rule for the calculation of functional determinants in terms of the Fourier components of Greens functions. This method yields a set of different impedance matrices that are validated analytically for the parallel plate geometry.