Convolution Theorem Research Articles

The Laplace Transform Shortcut Solution to a One-Dimensional Heat Conduction Model with Dirichlet Boundary Conditions

When using the Laplace transform to solve a one-dimensional heat conduction model with Dirichlet boundary conditions, the integration and transformation processes become complex and cumbersome due to the varying properties of the boundary function f(t). Meanwhile, if f(t) has a complex functional form, e.g., an exponential decay function, the product of the image function of the Laplace transform and the general solution to the model cannot be obtained directly due to the difficulty in solving the inverse. To address this issue, operators are introduced to replace f(t) in the transformation process. Based on the properties of the Laplace transform and the convolution theorem, without the direct involvement of f(t) in the transformation, a general theoretical solution incorporating f(t) is derived, which consists of the product of erfc(t) and f(0), as well as the convolution of erfc(t) and the derivative of f(t). Then, by substituting f(t) into the general theoretical solution, the corresponding analytical solution is formulated. Based on the general theoretical solution, analytical solutions are given for f(t) as a commonly used function. Finally, combined with an exemplifying application demonstration based on the test data of temperature T(x, t) at point x away from the boundary and the characteristics of curve T(x, t) − t and curve 𝜕T(x, t)/𝜕t − t, the inflection point and curve fitting methods are established for the inversion of model parameters.

One-dimensional quaternion Laplace transform: Properties and its application to quaternion-valued differential equations

This paper is devoted to the study of the quaternion Laplace transform, which is a natural generalization of the classical Laplace transform using the quaternion algebra. We find that some of its properties such as derivative, convolution and correlation theorems are quite different from the corresponding properties of the classical Laplace transform. Other properties like linearity, shifting, scaling and derivative are also studied in detail. Finally, the utility of the quaternion Laplace transform for solving quaternion-valued differential equations is presented.

Wigner–Ville Distribution Associated with Clifford Geometric Algebra Cln,0, n=3(mod 4) Based on Clifford–Fourier Transform

In this study, the Wigner–Ville distribution is associated with the one sided Clifford–Fourier transform over Rn, n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of τ∈Rn. Moreover, powerful results on the WVD-CFT have been derived such as Parseval’s theorem, convolution theorem, Moyal’s formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

Generalized Function Projective Synchronization of Two Different Chaotic Systems with Uncertain Parameters

This study proposes a new approach to realize generalized function projective synchronization (GFPS) between two different chaotic systems with uncertain parameters. The GFPS condition is derived by converting the differential equations describing the synchronization error systems into a series of Volterra integral equations on the basis of the Laplace transform method and convolution theorem, which are solved by applying the successive approximation method in the theory of integral equations. Compared with the results obtained by constructing Lyapunov functions or calculating the conditional Lyapunov exponents, the uncertain parameters and the scaling function factors considered in this paper have fewer restrictions, and the parameter update laws designed for the estimation of the uncertain parameters are simpler and easier to realize physically.

Convolution, Correlation and Uncertainty Principle in the One-Dimensional Quaternion Quadratic-Phase Fourier Transform Domain

In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on R. Later on, we show that 1D-QQPFT satisfies all the respective properties such as inversion formula, linearity, Moyal’s formula, convolution theorem, correlation theorem and uncertainty principle. Moreover, we use the proposed transform to obtain an inversion formula for two-dimensional quaternion quadratic-phase Fourier transform. Finally, we highlight our paper with some possible applications.

Reconfigurable Terahertz Spatial Deflection Varifocal Metamirror.

A traditional optical lens usually has a fixed focus, and its focus controlling relies on a bulky lens component, which makes integration difficult. In this study, we propose a kind of terahertz spatial varifocal metamirror with a consistent metal-graphene unit structure whose focus can be flexibly adjusted. The focus deflection angle can be theoretically defined by superimposing certain encoded sequence on it according to Fourier convolution theorem. The configurable metamirror allows for the deflection of the focus position in space. The proposed configuration approach presents a design concept and offers potential advancements in the field of developing novel terahertz devices.

Evidence for a short-lived resonance state in enzyme catalysis via rate-equationconvolution.

At the cellular level, all biological function relies on enzymes to provide catalytic acceleration of essential biochemical processes driving cellular metabolism. The enzyme is presumed to lower the activation energy barrier separating reactants from products, but the precise mechanism remains unresolved. Here we examine the temperature dependence of the enzyme-catalyzed dissociation of p-nitrophenyl-α-D-glucopyranoside (pNPG), a chromogenic analog for maltose, isomaltose, and sucrose disaccharide sugars, into p-nitrophenol (pNP) and glucose (monosaccharide). The enzymes of interest are the wild type and mutant forms of glucosidase MalL produced by the probiotic bacterium Bacillus subtilis. The per-enzyme production rates k(T) for the pNPG→ glucose reaction all show a characteristic temperature profile with an Arrhenius-like (approximately exponential) slow acceleration at low temperatures, rising through a point of inflexion to reach a maximum, then turning over to decline steeply towards zero production at high temperatures. This asymmetric profile is found to be well fitted by convolving an exponential growth function f(T) with a Gaussian temperature distribution g(T) to produce an exponentially modified Gaussian function h(T). To give a physical interpretation of the convolution components, we make the temperature mapping Θ≡T_{ref}-T where T_{ref} marks the temperature at which a given mutant becomes fully denatured (unfolded) and therefore inactive, then convert the convolution components to probability density functions which obey the convolution theorem of statistics. Working in Θ space, we identify f(Θ) as the density function for an Arrhenius-like transition from ground-state A to metastable-state B, and g(Θ) as the Gaussian distribution of offset-temperature fluctuations for the metastable state. By mapping the standard thermodynamic relations for temperature and energy fluctuations to the enzyme frame of reference, we are able to derive an expression for the lifetime for the metastable B state. For the 15 enzyme experiments, we obtain a mean value 〈Δt〉≳(29.0±1.3)×10^{-15}s, in remarkably good agreement with the ∼30-fs estimate for the period of glycosidic bond oscillations extracted from published infrared spectroscopy. We suggest that the metastable B state provides a low-energy target that has the effect of lowering the activation energy barrier by presenting an alternative axis for the reaction coordinate.

Analytical model of heat transfer of U-shaped borehole heat exchanger under convolution theory and its optimal heat extraction analysis

Middle deep borehole heat exchanger (DBHE) has become widely known as one of the main ways of geothermal energy development and utilization, and the use of middle DBHE system for heating buildings has received much attention. More accurate analytical models for a coaxial borehole heat exchanger (CBHE) have been built using the convolution theorem, however, a U-shaped deep borehole heat exchanger (U-DBHE) has not been reported. Therefore, in this paper, to obtain more accurate fluid temperature distribution, an analytical model of middle DBHE considering geological stratification phenomenon is developed by using convolution theorem. The relative error of the analytical model is verified within 5% by numerical simulation. At the same time, the interaction between the three factors was analyzed and studied by response surface method: mass flow, inlet temperature and geothermal gradient on heat extraction and heat loss rate. The results reveal that for thermal extraction, the significance levels of the three factors are from large to small: geothermal gradient > inlet temperature > mass flow, for heat loss rate, the two factors of flow rate and inlet temperature are significant, and the impact of geothermal gradient is not significant. In this engineering project, when the inlet water temperature is 10 °C, the flow rate is 26.65 kg/s and the geothermal gradient is 4.5 °C/hm, there is a maximum heat extraction of 2424.16 kW.

Digital coding metasurface for Multi-Beam and Multi-Mode OAM in Full-Space

A digital coding metasurface is proposed to generate multi-beam and multi-mode orbital angular momentum (OAM) wave in the full space. The 3-bit coding elements are designed for multifunctional integrated digital coding metasurface (MIDCMS) based on the propagation phase theory. The variable-sized phase-shift scheme is applied on metal structures to realize different functionalities in linearly polarized (LP) incidence. Furthermore, the phase distribution of MIDCMS is determined by the pattern convolution theorem. The transmitted and reflected wavefronts are manipulated simultaneously through MIDCMS. As a proof of concept, a metasurface device was fabricated and measured for the four-beam OAM with +1 mode in transmission space and the four-beam OAM with −1 mode in reflection space. Both experimental results and simulated values verify that the proposed MIDCMS can effectively realize the full-space multi-beam multi-mode OAM generation and improve the capacity of information channels, which has potential applications for comprehensive implementations in the future wireless communication systems.

Performance analysis of single-focus phase singularity based on elliptical reflective annulus quadrangle-element coded spiral zone plates

Optical vortices generated by the conventional vortex lens are usually disturbed by the undesired higher-order foci, which may lead to additional artifacts and thus degrade the contrast sensitivity. In this work, we propose an efficient methodology to combine the merit of elliptical reflective zone plates (ERZPs) and the advantage of spiral zone plates (SZPs) in establishing a specific single optical element, termed elliptical reflective annulus quadrangle-element coded spiral zone plates (ERAQSZPs) to generate single-focus phase singularity. Differing from the abrupt reflectance of the ERZPs, a series of randomly distributed nanometer apertures are adopted to realize the sinusoidal reflectance. Typically, according to our physical design, the ERAQSZPs are fabricated on a bulk substrate; therefore, the new idea can significantly reduce the difficulty in the fabrication process. Based on the Kirchhoff diffraction theory and convolution theorem, the focusing performance of ERAQSZPs is calculated. The results reveal that apart from the capability of generating optical vortices, ERAQSZPs can also integrate the function of focusing, energy selection, higher-order foci elimination, as well as high spectral resolution together. In addition, the focusing properties can be further improved by appropriately adjusting the parameters, such as zone number and the size of the consisted primitives. These findings are expected to direct a new direction toward improving the performance of optical capture, x-ray fluorescence spectra, and forbidden transition.

Mixed Transform Iterative Method for Solving Wave Like Equations

In this article, a hybrid method that combines Aboodh transform, variational iteration method, and the homotopyperturbation method is presented for approximate the solution of important partial differential equations that describe wave like differential equations in one, two and three dimensions. The suggested method utilizes only the initial conditions to providean analytical solution, in contrast to the method of separation of variables, that also requires boundary conditions. This method makes use of the Aboodh's transform advantageous on the Lagrange multiplier computation, hence there is no need to use the convolution theorem or any integration in a recurrence relation for the process of finding it. The obtained exact solutions are found as a convergent series with components that are simple to compute. In addition, the method needsno any assumptions that change the problem's physical nature, such as those that involve discretization, linearization, or minor factors compared to certain other techniques. Some examples are given to show how efficient and useful the method is.

Semianalytical Model of Multiphase Halbach Array Axial Flux Permanent-Magnet Motor Considering Magnetic Saturation

This paper proposes a nonlinear semi-analytical model (SAM) of the multi-phase Halbach array axial flux permanent magnet motor (AFPMM) to speed up the computation of its magnetic field. Compared to the existing analytical models, the proposed nonlinear SAM can directly consider magnetic saturation to obtain more accurate results. To this end, the multi-phase Halbach Array AFPMM is equivalent to several 2-D models by the Quasi-3D method under the Cartesian coordinate system. Then, the nonlinear SAM is developed by using the convolution theorem and the fast Fourier factorization. The proposed nonlinear SAM is studied on a five-phase Halbach array AFPMM with different rotors, and the nonlinear finite element (FE) model and experiment verify its effectiveness. The proposed SAM is computationally efficient and accurate, and it is also applicable to other types of multi-phase Halbach array PM electrical motors in Cartesian coordinates.

On uniqueness and ill-posedness for the deautoconvolution problem in the multi-dimensional case

This paper analyzes the inverse problem of deautoconvolution in the multi-dimensional case with respect to solution uniqueness and ill-posedness. Deautoconvolution means here the reconstruction of a real-valued L 2-function with support in the n-dimensional unit cube from observations of its autoconvolution either in the full data case (i.e. on ) or in the limited data case (i.e. on ). Based on multi-dimensional variants of the Titchmarsh convolution theorem due to Lions and Mikusiński, we prove in the full data case a twofoldness assertion, and in the limited data case uniqueness of non-negative solutions for which the origin belongs to the support. The latter assumption is also shown to be necessary for any uniqueness statement in the limited data case. A glimpse of rate results for regularized solutions completes the paper.

Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional free parameters. We explore the properties of 2D right-sided QOLCT, including inversion and Parseval formulas, besides its relationship with other transforms. We also examine the convolution and correlation theorems of 2D right-sided QOLCT, followed by several uncertainty principles. Additionally, we present an illustrative example of the proposed transform, demonstrating its graphical representation of a given signal and its transformed signal. Finally, we demonstrate an application of QOLCT, where it can be utilized to generalize the treatment of swept-frequency filters.

Convolution Theorem Associated with the QWFRFT

Convolution Theorem Associated with the QWFRFT

The spherical linear canonical transform: Definition and properties

The spherical Fourier transform has attracted considerable attention in the fields of acoustics, optics, and heat because of its superiority in solving practical problems- within the confines of spherical symmetry. A spherical linear canonical transform in spherical polar coordinates is investigated in this study. First, definitions of the spherical linear canonical transform and spherical linear canonical Hankel transform are proposed. Second, the relationship between the spherical linear canonical transform and spherical linear canonical Hankel transform is derived based on the orthogonality of the spherical harmonics. Finally, several essential properties of the proposed spherical linear canonical transform were obtained based on this relationship, including linearity, inversion formulas, shifts, and convolution theorems. Finally, potential applications of the spherical linear canonical transform are discussed.

Some Properties and Applications of a New General Triple Integral Transform “Gamar Transform’’

The goal of this study is to suggest a new general triple integral transform known as Gamar transform. Next, we compare the current transform to a number of existing triple integral transforms such as those by Laplace, Sumudu, Elzaki, Aboodh, and Laplace–Aboodh–Sumudu. We outline its essential properties and prove some important results, including linearity property, existence theorem, triple convolution theorem, and derivatives properties. Moreover, the proposed new transform is applied to solve some partial differential equations (PDEs) such as Laplace, Mboctara, and Wave equations. The capacity of general triple integral transforms to change PDEs into simple algebraic equations is demonstrated.

Investigation of the dynamic vertical behaviour of a floating pile embedded in unsaturated poroelastic half-space

This paper proposes an analytical solution for investigating the dynamic vertical behaviour of a floating pile in unsaturated poroelastic half-space by considering the dynamic properties of the soil below the pile base. The pile shaft resistance is derived by applying Hankel integral transformation to the governing equations, and the pile base resistance is derived by introducing the potential function and operator decomposition methods. The dynamic interaction between the soil and the pile is considered by virtue of the impedance function. To account for the mixed boundary conditions of the pile-soil interface, an analytical solution for the vertical dynamic response of a floating pile is derived in the frequency domain. Additionally, by utilizing the inverse Fourier transform and convolution theorem, a quasi-analytical solution for the velocity response of the floating pile is derived in the time domain. Furthermore, the derived solution is verified through comparisons with existing solutions available in the literature. Finally, a parametric study is performed to investigate the influence of saturation and intrinsic permeability on the dynamic flexibility coefficient of porous unsaturated half-space, the influence of saturation on the dynamic impedance of the soil layer, and slenderness ratio on the vertical behaviour of floating pile and end-bearing pile.

Analytical investigation of the coupled fractional models for immersed spheres and oscillatory pendulums

In this research, coupled fractional models for immersed spheres and oscillatory pendulums have been proposed. We deploy the Laplace transform method together with the negative binomial formula to analytically investigate the dynamics of the systems. Approximate closed-form solutions are successfully revealed with the help of the convolution theorem. Additionally, we graphically illustrate the variational effects of the fractional-orders and the coupling parameters on the paired fields.

The Convolution Theorem Involving Windowed Free Metaplectic Transform

The convolution product is widely used in many fields, such as signal processing, numerical analysis and so on; however, the convolution theorem in the domain of the windowed metaplectic transformation (WFMT) has not been studied. The primary goal of this paper is to give the convolution theorem of WFMT. Firstly, we review the definitions of the FMT and WFMT and give the inversion formula of the WFMT and the relationship between the FMT and WFMT. Then, according to the form of the classical convolution theorem and the convolution operator of the FMT, the convolution theorem in the domain of the WFMT is given. Finally, we prove the existence theorems of the proposed convolution theorem.