Convolution Theorem Research Articles

Some Estimates for Certain q-analogs of Gamma Integral Transform Operators

The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.

Far field ring beam generation based on 3-bit encoded metasurface

Abstract The encoding metasurface establishes a bridge between the physical and digital worlds, ushering in a new era of manipulating electromagnetic waves and realizing programmable metamaterials through digital coding sequences. This "digital metasurface," relying on binary logic, significantly simplifies the design process, thereby enhancing the flexibility and efficiency of controlling electromagnetic waves. While most encoding metasurfaces control beamforming for pencil-shaped beams, we propose a 3-bit encoding metasurface with a "well structure" in the microwave band.The 3-bit encoding metasurface features a more extensive encoding sequence, offering increased degrees of freedom and flexibility in manipulating the direction of electromagnetic wave propagation. Its symmetric design features polarization-insensitive characteristics, suitable for generating annular beams by varying gradient-encoded numbers radially. This approach enables the production of linearly polarized omnidirectional radiation within the desired elevation angle range. Additionally, adjustments in quantity and orientation of the produced annular beams are achieved through Fourier convolution theorem. This type of annular beams holds promise for applications in various fields, including wireless radio broadcasting and wireless local area networks.

EENet: An effective and efficient network for single image dehazing

While numerous solutions leveraging convolutional neural networks and Transformers have been proposed for image dehazing, there remains significant potential to improve the balance between efficiency and reconstruction performance. In this paper, we introduce an efficient and effective network named EENet, designed for image dehazing through enhanced spatial–spectral learning. EENet comprises three primary modules: the frequency processing module, the spatial processing module, and the dual-domain interaction module. Specifically, the frequency processing module handles Fourier components individually based on their distinct properties for image dehazing while also modeling global dependencies according to the convolution theorem. Additionally, the spatial processing module is designed to enable multi-scale learning. Finally, the dual-domain interaction module promotes information exchange between the frequency and spatial domains. Extensive experiments demonstrate that EENet achieves state-of-the-art performance on seven synthetic and real-world datasets for image dehazing. Moreover, the network’s generalization ability is validated by extending it to image desnowing, image defocus deblurring, and low-light image enhancement.

Notes on q-Gamma Operators and Their Extension to Classes of Generalized Distributions

This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients of sequences, and establishes certain convolution theorems. The convolution theorems are utilized to accomplish q-equivalence classes of generalized distributions called q-Boehmians. Consequently, the q-gamma operators are therefore extended to the generalized spaces and performed to coincide with the classical integral operator. Further, the generalized q-gamma integral is shown to be linear, sequentially continuous and continuous with respect to some involved convergence equipped with the generalized spaces.

Analytical model development for vertical medium/deep ground heat exchangers with U-tube(s)

The U-shaped ground heat exchanger (UGHE) offers an effective solution for building heating and cooling purposes. In the practical applications of vertical UGHEs, the borehole wall temperature varies along the borehole depth due to the uneven initial ground temperature and transient heat transfer process. The assumption of a uniform temperature along the borehole wall in the existing quasi-3D model of UGHE causes a non-negligible error in the calculation of fluid temperature, especially for the cases of shallow-medium boreholes. Thus, two new analytical models are established to calculate the fluid temperatures for the single U-shaped GHE (1-UGHE) and double U-shaped GHE (2-UGHE), considering the depth-dependent borehole wall temperatures. The convolution theorem is employed to solve the mathematical problem of the nonlinear condition on the borehole wall, and the Laplace transformation is used for getting the final temperature expressions. The new analytical models are compared and verified by the on-site experiments. The results show that the two models exhibit promising prediction accuracies and similar variation trends in fluid temperatures. Then, parametrical and economical analyses for 1−/2- UGHEs have been carried out. Finally, the geographical thermal potentials of 1−/2- UGHEs across China have been comparatively investigated and the corresponding thermal map-based application potential recommendations have been given.

Frequency-Domain and Spatial-Domain MLMVN-Based Convolutional Neural Networks

This paper presents a detailed analysis of a convolutional neural network based on multi-valued neurons (CNNMVN) and a fully connected multilayer neural network based on multi-valued neurons (MLMVN), employed here as a convolutional neural network in the frequency domain. We begin by providing an overview of the fundamental concepts underlying CNNMVN, focusing on the organization of convolutional layers and the CNNMVN learning algorithm. The error backpropagation rule for this network is justified and presented in detail. Subsequently, we consider how MLMVN can be used as a convolutional neural network in the frequency domain. It is shown that each neuron in the first hidden layer of MLMVN may work as a frequency-domain convolutional kernel, utilizing the Convolution Theorem. Essentially, these neurons create Fourier transforms of the feature maps that would have resulted from the convolutions in the spatial domain performed in regular convolutional neural networks. Furthermore, we discuss optimization techniques for both networks and compare the resulting convolutions to explore which features they extract from images. Finally, we present experimental results showing that both approaches can achieve high accuracy in image recognition.

Coupled 2<i>D</i>-Fractional Wavelet Transform

We introduce a new fractional wavelet transform using the convolution for the coupled fractional Fourier transform. We derive that the fractional wavelet transform satisfies all the expected properties such as Parseval identity, inversion formula, convolution theorem. We also characterize the range of the fractional wavelet transform on . Finally, we establish the uncertainty principle for the fractional wavelet transform.

Design of composite stealth cloak based on anomalous reflections and vortex singularities

In this paper, an ultra-thin (0.072λ) composite stealth cloak is proposed to cover a cloak with a center height of 98.88 mm and an inclination angle of 25° to achieve the dual stealth effect of scattering and gain. Firstly, based on the Generalized Snell theorem, the designed anomalous reflective metasurface has been shown to have a good invisibility in the 14–30 GHz region in simulations as well as in experiments. Secondly, the 3-bit metasurface unit is utilized to generate vortex beams with different number of modes, whose electric field strength at the vortex singularity is 0, which achieves energetically stealth. Finally, the phases of the above two metasurfaces are superimposed using the convolution theorem, and the designed composite stealth cloak significantly reduces the scattering amplitude and reconstructs the wavefront over a wide frequency range when a circularly polarized wave is incident vertically. Based on the above study, the stealth cloak provides a feasible solution for application to radar and microwave antenna stealth systems.

Offset Linear Canonical Stockwell Transform for Boehmians

In this article, we construct a Boehmian space using the convolution theorem that contains the offset linear canonical Stockwell transforms (OLCST) of all square-integrable Boehmians. It is also proven that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Furthermore, it is one-to-one, linear, and continuous with respect to Δ-convergence as well as Δ-convergence. Subsequently, we introduce a discrete variant of the OLCST. Ultimately, we broaden the application of the presented work by examining the OLCST within the domain of almost periodic functions.

Convolution Theorem for (p,q)-Gamma Integral Transforms and Their Application to Some Special Functions

This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning (p,q)-analogs of the gamma integrals to unit step functions as well as first- and second-order (p,q)-differential operators. In addition, it presents a pair of (p,q)-convolution products for the specified (p,q)-analogs and establishes two (p,q)-convolution theorems.

Discrete octonion linear canonical transform associated with finite-length function

In this paper, the discrete octonion linear canonical transform (DOCLCT) is presented. According to the definition of DOCLCT, its shift transform and modulation transform are explored. In addition, based on a new convolution operator, we obtain the convolution theorem of DOCLCT. The DOCLCT can be derived by the three-dimensional (3D) linear canonical transform. So, the fast algorithm of 3D DOCLCT is discussed. Finally, Heisenberg’s uncertainty principle associated with DOCLCT is established.

Nonuniform Sampling in Lp-Subspaces Associated with the Multi-Dimensional Special Affine Fourier Transform

In this paper, the sampling and reconstruction problems in function subspaces of Lp(Rn) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval’s relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the Lp-stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a Lp-subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations.

Dual-domain faster Fourier convolution based network for MR image reconstruction

Deep learning methods for fast MRI have shown promise in reconstructing high-quality images from undersampled multi-coil k-space data, leading to reduced scan duration. However, existing methods encounter challenges related to limited receptive fields in dual-domain (k-space and image domains) reconstruction networks, rigid data consistency operations, and suboptimal refinement structures, which collectively restrict overall reconstruction performance. This study introduces a comprehensive framework that addresses these challenges and enhances MR image reconstruction quality. Firstly, we propose Faster Inverse Fourier Convolution (FasterIFC), a frequency domain convolutional operator that significantly expands the receptive field of k-space domain reconstruction networks. Expanding the information extraction range to the entire frequency spectrum according to the spectral convolution theorem in Fourier theory enables the network to easily utilize richer redundant long-range information from adjacent, symmetrical, and diagonal locations of multi-coil k-space data. Secondly, we introduce a novel softer Data Consistency (softerDC) layer, which achieves an enhanced balance between data consistency and smoothness. This layer facilitates the implementation of diverse data consistency strategies across distinct frequency positions, addressing the inflexibility observed in current methods. Finally, we present the Dual-Domain Faster Fourier Convolution Based Network (D2F2), which features a centrosymmetric dual-domain parallel structure based on FasterIFC. This architecture optimally leverages dual-domain data characteristics while substantially expanding the receptive field in both domains. Coupled with the softerDC layer, D2F2 demonstrates superior performance on the NYU fastMRI dataset at multiple acceleration factors, surpassing state-of-the-art methods in both quantitative and qualitative evaluations.

Analytical solutions for joints in an infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to time-lag loads

This paper investigates the analytical method for the infinite beam with joint parts resting on a foundation under time-lag loads. The ground is simulated as a viscoelastic foundation. The existence of joint parts in beams is explained by introducing virtual forces expressed through the Dirac delta function. Integral transformation simplifies the dynamic governing equations for standard beam sections and joint parts into an algebraic equation. The convolution theorem converts the solutions into the time domain. The analytical solutions for dynamic analysis of standard beam sections and joint parts are obtained conveniently in the frequency domain. The proposed analytical method is verified by comparing it with the traditional methods and also validated by numerical examples with time-lag loads. Parametric discussions study the effects on the dynamic response of joint parts and standard beam sections, such as the amplitude and frequency of time-lag loads and coefficient of bending stiffness reduction of joint parts. The parametric analyses emphasize the dramatic effect of critical parameters on the discontinuous deformation of beams. The damage caused by critical values of some factors is emphasized. The severe effects of discontinuous deformation induced by dynamic loads in the safety of beam structures are revealed in this paper. The differences in dynamic responses between the beam and joint part must be monitored to ensure the safety of whole beam structures.

Multifunctional 2-bit coded reconfigurable metasurface based on graphene-vanadium dioxide

In this paper, a graphene-vanadium dioxide-based reconfigurable metasurface unit structure is proposed. Using the change at a graphene Fermi energy level on the surface of the unit structure to satisfy the 2-bit coding condition, four reflection units with a phase difference of 90∘ can be discovered. The modulating impact of the multi-beam reflection wave with 1-bit coding is then confirmed. Then we study the control of a single-beam reflected wave by metasurfaces combined with a convolution theorem in a 2-bit coding mode. Finally, when vanadium dioxide is in an insulating condition, the structure can also be transformed into a terahertz absorber. It is possible to switch between a reflection beam controller and a terahertz multifrequency absorber simply by changing the temperature of the vanadium dioxide layer without retooling a new metasurface. Moreover, compared with the 1-bit coded metasurface, it increases the ability of single-beam regulation, which makes the device more powerful for beam regulation.

Independently regulating linearly and circularly polarized terahertz wave metasurface

We propose a terahertz metasurface that can independently regulate linearly circularly polarized waves. It consists of the top layer “O-O” metal pattern, polyimide layer, middle layer “I” shaped metal pattern, polyimide layer, and metal substrate from top to bottom. By using the phase principle of Pancharatnam Berry (PB) for encoding and arrangement, the metasurface generates vortex beams with different topological charges under circularly polarized terahertz wave incidence, and achieves focusing shift at different positions. Combining the convolution theorem for encoding arrangement, the metasurface can achieve focused vortex function under circularly polarized terahertz wave incidence. The designed metasurface can also generate Airy beam under linearly polarized terahertz wave incidence. The simulation results indicate that under different polarization (linear/circular polarization) terahertz wave incidence, the metasurface can achieve different functions, which provides a new approach for flexible control of terahertz waves.

Discrete Octonion Linear Canonical Transform: Definition and Properties

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established.

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

Approximate Solution of Linear Fuzzy Random Ordinary Differential Equations Using Laplace Variational Iteration Method

In this article, the Laplace transformation method in connection with the variational iteration method will be used to solve approximately fuzzy random ordinary differential equations. After that, the sequence of approximated closed form iterated solutions is derived based on the general Lagrange multiplier evaluated using the well-known convolution theorem of the Laplace transformation method. In addition, two examples are given and solved to illustrate the reliability, efficiency and applicability of the proposed method, they are simulated using computer programs with two different generations of stochastic processes, namely the Wiener process or Brownian motion, which are 1000 and 10000, respectively.

Inequalities for Means Regarding the Trigamma Function

Let G(α, β), A(α, β) and H(α, β), respectively, be the geometric mean, arithmetic mean and harmonic mean of α and β. In this paper, we prove that G(ψ′ (z), ψ′ (1/z)) ≥ π2/6, A(ψ′ (z), ψ′ (1/z)) ≥ π2/6 and H(ψ′ (z), ψ′ (1/z)) ≤ π2/6. This extends the previous results of Alzer and Jameson regarding the digamma function ψ. The mathematical tools used to prove the results include convexity, concavity and monotonicity properties of certain functions as well as the convolution theorem for Laplace transforms.