Convolution Theorem Research Articles

Transmissive coding metasurface with dual-circularly polarized multi-beam.

The Pancharatnam-Berry (PB) phase can be used to control the phase of circularly polarized electromagnetic waves. However, there are few studies on the modulation of dual-circularly polarized multi-beam using the transmissive coding metasurface. A scheme of spin-controlling multi-beam by transmissive coding metasurface is proposed for dual-circular polarization simultaneously. The transmissive coding metasurface (TCMS) can transmit linearly polarized incidence into multi-beam with orthogonally circular polarization. The phase distribution is designed based the convolution theorem, and the elements of metasurface conforming to the PB phase are arranged according to the phase distribution. In order to compensate the emitting spherical waves into plane waves and realize the transmissive waves with dual-circular polarization, an interesting scheme of elements in different regions with different rotating phase are presented based on the principle of phase compensation. TCMS can transmit linearly polarized waves into two left-hand circularly polarized (LHCP) beams and two right-hand circularly polarized (RHCP) beams. The prototype of TCMS is fabricated and measured, and the experimental results agree well with the simulated data. The transmissive metasurface has potential application in holograms and satellite communication.

Characterization and Properties of Bent‐Negabent Functions

A further characterization of the bent-negabent functions is presented. Based on the concept of complete mapping polynomial, we provide a necessary and sufficient condition for a class of quadratic Boolean functions to be bent-negabent. A new characterization of negabent functions can be described by using the parity of Hamming weight. We further generalize the classical convolution theorem and give the nega-Hadamard transform of the composition of a Boolean function and a vectorial Boolean function. The nega-Hadamard transform of a generalized indirect sum is calculated by this composition method.

Analytical solution for torsional vibration of an end-bearing pile in nonhomogeneous unsaturated soil

Incorporation of nonhomogeneous and the three-phased nature of soils (composed of particles, water and gas) can more rationally reflects the dynamic response of pile. Therefore, this work proposes an analytical solution for investigating the torsional vibration response of an end-bearing pile in nonhomogeneous unsaturated soil. Among them, the nonlinear variation of shear modulus of the soil with depth is considered, as well as the inertial coupling effect between pore water and gas components. Utilizing the method of separation of variables, boundary and continuous conditions of the pile-soil system, the analytical solution is derived in the frequency domain. Correspondingly, the quasi-analytical solution for pile head subjected to a semi-sine excitation torque is derived by employing of the inverse Fourier transform and convolution theorem in the time domain. After verifying the derived solution against with the two published analytical solutions for pile in two-phase medium, arithmetical examples are presented to demonstrate the influence of relevant parameters on the torsional vibration of the pile.

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

Clifford-valued linear canonical transform: Convolution and uncertainty principles

Linear canonical transform is a three-parameter class of linear integral transformation, which has found many applications in optics, filter design, speech, image, video, and signal processing due to its flexibility in representing information within the linear canonical domain. In this article, we extend the linear canonical transform to the class of Clifford-valued signals and introduce a novel transform coined as Clifford-valued linear canonical transform. The primary analysis encompasses the derivation of fundamental properties of the proposed transform such as translation and scaling covariance, inner product relation and inversion formula. Subsequently, we investigate the convolution, continuity, and differentiation theorems for the Clifford-valued linear canonical transform. A non-homogeneous partial differential equation has also been solved in the Clifford framework as an application of the differentiation theorem. Finally, we culminate our investigation by deriving several uncertainty principles for the proposed transform.

Solution of Volterra integral equations of the 2nd kind with bulge function using Aboodh transform

A large class of complexities in mathematical physics, applied mathematics, and engineering are expressed as differential equations with few additions and certain conditions. This research article studies the solution of Volterra integral equations of the second kind where bulge functions take as a known function. To obtain an analytical solution, this study uses the Aboodh transform, the Aboodh inverse transform and the convolution theorem whereas it would be required to discover the precise solution of VIEs. We will also compare it with a numerical solution using a modified Simpson method, and finally, we will represent it graphically.

Spatially adaptive blind deconvolution methods for optical coherence tomography

Optical coherence tomography (OCT) is a powerful noninvasive imaging technique for detecting microvascular abnormalities. Following optical imaging principles, an OCT image will be blurred in the out-of-focus domain. Digital deconvolution is a commonly used method for image deblurring. However, the accuracy of traditional digital deconvolution methods, e.g., the Richardson-Lucy method, depends on the prior knowledge of the point spread function (PSF), which varies with the imaging depth and is difficult to determine. In this paper, a spatially adaptive blind deconvolution framework is proposed for recovering clear OCT images from blurred images without a known PSF. First, a depth-dependent PSF is derived from the Gaussian beam model. Second, the blind deconvolution problem is formalized as a regularized energy minimization problem using the least squares method. Third, the clear image and imaging depth are simultaneously recovered from blurry images using an alternating optimization method. To improve the computational efficiency of the proposed method, an accelerated alternating optimization method is proposed based on the convolution theorem and Fourier transform. The proposed method is numerically implemented with various regularization terms, including total variation, Tikhonov, and l1 norm terms. The proposed method is used to deblur synthetic and experimental OCT images. The influence of the regularization term on the deblurring performance is discussed. The results show that the proposed method can accurately deblur OCT images. The proposed acceleration method can significantly improve the computational efficiency of blind demodulation methods.

Fourier Deconvolution Ion Mobility Spectrometry.

Multiplexing the ion packet injection with advanced signal processing is an effective method to improve both the ion throughput and signal-to-noise ratio for ion mobility spectrometry. Generally used multiplexing methods include Hadamard transform ion mobility spectrometry (HT-IMS), Fourier transform ion mobility spectrometry (FT-IMS), and correlation ion mobility spectrometry (C-IMS). However, HT-IMS sometimes suffer from false peaks and further processing is needed, FT-IMS generally requires longer spectra acquisition time than the traditional signal averaging method, and C-IMS also demonstrated drawbacks such as spectra baseline distortions when using traditional on-off binary gating switches. To improve the performance of multiplexing ion mobility spectrometry, this study investigates the Fourier deconvolution to increase the resolving power and signal-to-noise ratio at the same time. This approach modulates the ion gate with a linear square wave chirp sequence and synchronizes the data acquisition and ion gate modulation and then reconstructs the ion mobility spectra based on convolution theorems. The equivalent ion injection period is decreased to microseconds scale with the signal-to-noise ratio improved by up to 13 times on average, and the resolving power is improved by up to 50% compared with traditional signal averaging methods without hardware modifications.

Vertical dynamic response of a floating pile in unsaturated poroelastic media based on the fictitious unsaturated soil pile model

Decades of high-speed railways operation around the world have revealed an increasing number of maintenance issues of pile-supported embankments subjected to long-term dynamic train loads in typical unsaturated soft soil areas. As the physical explanation of vibration characteristics of unsaturated soil piles is a theoretical basis for pile foundation design and pile integrity detection, a developed analytical solution of the vertical dynamic response of a floating pile in homogeneous unsaturated soils is proposed in this work. Based on the fictitious unsaturated soil pile (FUSP) model, the influence of saturation degree on soil shear modulus is considered in governing equations, uncoupled by using potential function and operator decomposition methods. The vertical dynamic response of a floating pile in a frequency domain is derived by employing the pile–soil compatibility conditions and the impedance function transfer method. Time-domain results are obtained by utilizing inverse Fourier transform and the convolution theorem. The correctness of the proposed model is verified by degradation to compare with the two existing solutions. Finally, a parameter study is undertaken to find the most sensitive parameters to the dynamic response of the floating pile in unsaturated soils.

Smoothed finite element approach for viscoelastic behaviors of general shell structures

This study focuses on the viscoelastic analysis of laminated composite shell structures under long-term creep mechanical loading. An advanced finite element technique named as cell/element-based smoothed discrete shear gap method (CS-DSG3) is employed to obtain the numerical solutions for both elastic and viscoelastic problems owing to its accuracy and rapid convergence. In the CS-DSG3 shell element, each triangular shell element is divided into three DSG3 subtriangles to avoid transverse shear locking. Subsequently, the total element strain is computed from the three partial strains of the subtriangles using a smoothing technique. To overcome the difficulty in the constitutive equations in integral forms for a viscoelastic material, all formulations are transformed into the Laplace domain using the convolution theorem. Finally, time-dependent deformations are obtained and converted back to the real-time domain using inverse Laplace techniques. For validation, various numerical examples of a pinched cylinder, clamped hyperbolic paraboloid, and hemispherical shell formed of isotropic elastic, isotropic viscoelastic, and anisotropic composite materials are selected to investigate creep behavior under mechanical loading. The present study extends the finite element simulation of anisotropic viscoelastic shell structures to achieve high accuracy and efficiency based on the Laplace transform and CS-DSG3.

Convolution and product theorems for the quadratic-phase Fourier transform

Abstract In this paper, we establish the convolution and product theorems for the quadratic-phase Fourier transform (QPFT) and also deduce the convolution theorem associated with generalized translation. Moreover, the Shannon-type reconstruction formula for band-limited signals in the QPFT domain is formulated.

Octonion spectrum of 3D short-time LCT signals

This work is devoted to the development of the octonion linear canonical transform (OLCT) theory proposed by Gao and Li in 2021 that has been designated as an emerging tool in the scenario of signal processing. The purpose of this work is to introduce octonion linear canonical transform of real-valued functions. Further more keeping in mind the varying frequencies, we used the proposed transform to generate a new transform called short-time octonion linear canonical transform (STOLCT). The results of this article focus on the properties like linearity, reconstruction formula and relation with 3D-short-time linear canonical transform (3D-STLCT). The crux of this paper lie in establishing well known uncertainty inequalities and convolution theorem for the proposed transform.

Multifunctional reflection type anisotropic metasurfaces in the terahertz band

Based on the convolution and superposition theorem, we propose a reflective anisotropic metasurface to realize the functions of deflection and superposition of vortex beams, bifocal focusing, and focusing vortex beam. At frequency of 1.04THz, two deflection vortex beams with topological charges of (l=-1 and l=+2) and (l=+1 and l=-2) are generated under x- and y-polarized terahertz wave incidence, respectively. At focal plane, 1200µm from the top layer of the proposed metasurface, one can see that the bifocal focusing along y-axis and x-axis are produced under x- and y-polarized terahertz wave incidence, respectively. Similarly, focusing vortex beams with l=+1 and l=-2 are realized under x- and y-polarized terahertz wave incidence, respectively. The designed metasurface can flexibly manipulate terahertz wave under different polarization waves incidence and has potential application prospects in fields of terahertz communication.

Energy density functional forαclustering and scattering of lightA=4mnuclei

The energy density functional (EDF) is applied to study $\ensuremath{\alpha}$ clustering and $\ensuremath{\alpha}$ scattering in light $A=4m$ nuclei. Our goal is to study the success of the EDF in predicting the ground-state $\ensuremath{\alpha}$ clustering in $4m$-conjugate nuclei. The $\ensuremath{\alpha}$-cluster density is obtained by optimizing the EDF with the help of the convolution theorem. The obtained cluster density reproduced the experimental binding energy of the considered nuclei. The obtained $\ensuremath{\alpha}$-cluster densities are investigated through the elastic scattering of $\ensuremath{\alpha}$ particles from $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{24}\mathrm{Ne}$, $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, $^{32}\mathrm{S}$, and $^{40}\mathrm{Ca}$ in the framework of the optical model. The real part of the optical model potential is calculated using $\ensuremath{\alpha}$-cluster single folding or the conventional double folding models based on the obtained densities. The obtained potentials are used to analyze the elastic scattering of $\ensuremath{\alpha}$-particle from (1) $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{24}\mathrm{Ne}$, $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, $^{32}\mathrm{S}$, and $^{40}\mathrm{Ca}$ at 104 MeV, (2) $^{12}\mathrm{C}$, $^{24}\mathrm{Mg}$, and $^{28}\mathrm{Si}$ at 120 MeV, (3) $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, and $^{40}\mathrm{Ca}$ at 130 MeV. The obtained results are very satisfactory and in agreement with experimental data. This success indicates the validity and applicability of the EDF in $\ensuremath{\alpha}$-cluster calculation for $A=4m$ nuclei.

An interplay between quadratic-phase Fourier and Zak transforms

The advent of the quadratic-phase Fourier transform (QPFT) is one of the contemporary developments in the context of integral transforms. This article focuses on an interplay between the well-known Zak transform and the QPFT. To begin with, we formulate the Zak transform in quadratic-phase Fourier domain, braced with an example. In addition, we establish a novel convolution structure in the context of quadratic-phase Zak transform and obtain the corresponding convolution theorem. Finally, we study the Weyl–Heisenberg frames associated with the Quadratic-phase Zak Transform.

Uniqueness of inverse source problems for general evolution equations

In this paper, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at [Formula: see text], which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from [Formula: see text], which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.

Wideband RCS Reduction of Microstrip Antenna Array Using Coding Metasurface With LowQResonators and Fast Optimization Method

In this letter, a novel coding metasurface (CM) based on low <i>Q</i> resonators and fast optimization method is proposed to achieve wideband radar cross section (RCS) reduction of the microstrip antenna array (MAA) while maintaining its radiation properties. Theoretical analysis reveals the relationship between the <i>Q</i> value and the phase shift of the resonator, which indicates that the wideband control of the reflected wave can be achieved with the low <i>Q</i> resonator. Moreover, the fast optimization method based on the convolution theorem is proposed, which enhances the optimization efficiency of the coding matrix. Finally, the CM with two kinds of low <i>Q</i> resonators is constructed and applied to the MAA for wideband RCS reduction. The measured results indicate that the proposed low RCS MAA (LRMAA) can realize more than 10 dB RCS reduction in 5.8&#x2013;21.5 GHz, which also demonstrates superior specular scattering suppression. In parallel, the radiation properties of the LRMAA are consistent with those of MAA. The proposed strategy achieves the high integration of radiation and scattering properties, which has potential applications in antenna systems of stealth platforms.

Generalized Bloch type periodicity and applications to semi-linear differential equations in banach spaces

AbstractIn this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

Reduced Biquaternion Convolutional Neural Network for Color Image Processing

Reduced biquaternion is a four dimensional hyper–complex number which is commutative algebra and an extension of complex. Due to this property, the corresponding fast algorithm is designed in time frequency analysis which can better fit the convolution theorem than the non-commutative quaternion. In addition, the reduced biquaternion can be interpreted as a single point in 2-dimensional hyperbolic space with its algebraic structure which has more capacity and variable ability than the Euclidean space. However, the algebra properties of the reduced biquaternion have not yet been well explored in the deep convolutional neural network. In this paper, we develop a new deep network structure, namely reduced biquaternion valued convolutional neural network (RQV-CNN). The proposed RQV-CNN includes basic modules of reduced biquaternion convolution layer and fully connection layer. Extensive experiments on color image classification and color image denoising are conducted to evaluate the promising performance of the RQV-CNN framework. The results show that RQV-CNN outperforms the real-valued CNN (RV-CNN), complex-valued CNN (CV-CNN), and quaternion-valued CNN (QV-CNN) with same structures. The source code can be found at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/tasteimage/RQVCNN</uri> .

Multifunctional Coding Metasurface With Left and Right Circularly Polarized and Multiple Beams

In this paper, a multifunctional coding metasurface (MCMS) has been proposed to realize dual-circularly polarized beams and beam focusing with transmission and reflection. The phase of transmissive wave is controlled by rotating the elements, and the corresponding element, which consists of two quadrate voids etched on a single layer substrate, is designed for the metasurface with Pancharatnam-Berry (PB) phase. The phase distribution of the circularly polarized four-beam is determined according to the convolution theorem of patterns and the phase compensation principle. In order to validate the proposed metasurface, the multifunctional meta-device is fabricated and measured to illustrate the four-beam with left circular polarization in transmissive space and the right circularly polarized four-beam in reflective space by MCMS with x-polarized incidence. The experimental results heavily agree with the simulated data. The MCMS has potential applications in wireless communications due to its low profile, compact, and lightweight features.