Infinite Series Of Bessel Functions Research Articles

Real-time tracking of moving objects from scattering matrix in real-world microwave imaging

<abstract><p>The problem of the real-time microwave imaging of small, moving objects from a scattering matrix without diagonal elements, whose elements are measured scattering parameters, is considered herein. An imaging algorithm based on a Kirchhoff migration operated at single frequency is designed, and its mathematical structure is investigated by establishing a relationship with an infinite series of Bessel functions of integer order and antenna configuration. This is based on the application of the Born approximation to the scattering parameters of small objects. The structure explains the reason for the detection of moving objects via a designed imaging function and supplies some of its properties. To demonstrate the strengths and weaknesses of the proposed algorithm, various simulations with real-data are conducted.</p></abstract>

Задача о колебаниях в цепи связанных линейных осцилляторов с демпфирующей и антидемпфирующей границами

The problem of oscillations in a finite homogeneous chain of coupled harmonic oscillators is considered under special boundary conditions that ensure a stable flow of energy from one end of the chain to the other. The problem covers, as special cases, the classical problem of oscillations in a chain with free and fixed ends, as well as the problem of oscillations in a chain with absorbing and anti-absorbing boundaries. Absorbing boundary conditions are used in numerical simulation of wave propagation to minimize the influence of non-physical boundaries. An exact analytical solution to the considered problem is obtained. The dynamical system of the problem is studied. In particular, a description of the invariant subspaces of the system is given. The oscillatory properties of the system are investigated. The phenomenon of a significant increase in the amplitude of low-frequency oscillations has been discovered and studied. The problem is solved in Schrodinger variables. The solution is presented both in terms of infinite series of Bessel functions and in terms of natural oscillations (eigenmodes) of the system.

Character analogues of Cohen-type identities and related Voronoï summation formulas

In [4], B. C. Berndt and A. Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper [20], S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified K-Bessel function in light of recent results obtained by the first author and B. Maji [1]. Moreover, we provide a new expression for L(1,χ) from which we establish the positivity of L(1,χ) for any real primitive character χ. In addition, we deduce Cohen-type identities and then exhibit the Voronoï-type summation formulas for them.

On the application of MUSIC algorithm for identifying short sound-hard arcs in limited-view inverse acoustic problem

For proper application of the MUltiple SIgnal Classification (MUSIC) algorithm for imaging short, linear sound-hard arcs, one needs a priori information of the unit outward normal vector on each arc. Due to this reason, a set of directions was applied to estimate, which caused the imaging procedure’s slowness. Applying a unit vector instead of the estimation is natural to avoid this difficulty. However, no relevant mathematical theory has been developed to explain the various phenomena in simulation results. Here, we consider applying MUSIC without a priori information of the arcs. To this end, we introduce the imaging functions of MUSIC with and without information on the unit outward vector on the arcs. We analyze their mathematical structure by establishing an infinite series of Bessel functions and the range of incident/observation directions and discuss their properties. This is based on the physical factorization of the multi-static response (MSR) matrix that separates the incoming plane-wave information from the unknown density information. Numerical simulation results with noisy data are conducted for small and extended arcs to support the theoretical result.

Fast Localization of Small Inhomogeneities from Far-Field Pattern Data in the Limited-Aperture Inverse Scattering Problem

In this study, we consider a sampling-type algorithm for the fast localization of small electromagnetic inhomogeneities from measured far-field pattern data in the limited-aperture inverse scattering problem. For this purpose, we designed an indicator function based on the structure of left- and right-singular vectors of a multistatic response matrix, the elements of which were measured far-field pattern data. We then rigorously investigated the mathematical structure of the indicator function in terms of purely dielectric permittivity and magnetic permeability contrast cases by establishing a relationship with an infinite series of Bessel functions of an integer order of the first kind and a range of incident and observation directions before exploring various intrinsic properties of the algorithm, including its feasibility and limitations. Simulation results with synthetic data corrupted by random noise are presented to support the theoretical results.

Theoretical Identification of Coupling Effect and Performance Analysis of Single-Source Direct Sampling Method

Although the direct sampling method (DSM) has demonstrated its feasibility in identifying small anomalies from measured scattering parameter data in microwave imaging, inaccurate imaging results that cannot be explained by conventional research approaches have often emerged. It has been heuristically identified that the reason for this phenomenon is due to the coupling effect between the antenna and dipole antennas, but related mathematical theory has not been investigated satisfactorily yet. The main purpose of this contribution is to explain the theoretical elucidation of such a phenomenon and to design an improved DSM for successful application to microwave imaging. For this, we first survey traditional DSM and design an improved DSM, which is based on the fact that the measured scattering parameter is influenced by both the anomaly and the antennas. We then establish a new mathematical theory of both the traditional and the designed indicator functions of DSM by constructing a relationship between the antenna arrangement and an infinite series of Bessel functions of integer order of the first kind. On the basis of the theoretical results, we discover various factors that influence the imaging performance of traditional DSM and explain why the designed indicator function successfully improves the traditional one. Several numerical experiments with synthetic data support the established theoretical results and illustrate the pros and cons of traditional and designed DSMs.

Direct sampling method for identifying magnetic inhomogeneities in limited-aperture inverse scattering problem

The direct sampling method (DSM) in limited-aperture inverse scattering problems for transverse electric (TE) polarization is considered. Based on the asymptotic expansion formula in the presence of small targets, we demonstrate that the indicator function of DSM can be represented by an infinite series of Bessel functions of integer order and correspondingly examine various properties of DSM. Simulation results with noisy data are created to support theoretical results. Based on the identified structure of the indicator function, we design a new indicator function of DSM, analyze its mathematical structure, and generate simulation results to demonstrate its effectiveness and improvement.

Localization of Small Anomalies via the Orthogonality Sampling Method from Scattering Parameters

We investigate the application of the orthogonality sampling method (OSM) in microwave imaging for a fast localization of small anomalies from measured scattering parameters. For this purpose, we design an indicator function of OSM defined on a Lebesgue space to test the orthogonality relation between the Hankel function and the scattering parameters. This is based on an application of the Born approximation and the integral equation formula for scattering parameters in the presence of a small anomaly. We then prove that the indicator function consists of a combination of an infinite series of Bessel functions of integer order, an antenna configuration, and material properties. Simulation results with synthetic data are presented to show the feasibility and limitations of designed OSM.

Fast Imaging of Short Perfectly Conducting Cracks in Limited-Aperture Inverse Scattering Problem

In this paper, we consider the application and analysis of subspace migration technique for a fast imaging of a set of perfectly conducting cracks with small length in two-dimensional limited-aperture inverse scattering problem. In particular, an imaging function of subspace migration with asymmetric multistatic response matrix is designed, and its new mathematical structure is constructed in terms of an infinite series of Bessel functions and the range of incident and observation directions. This is based on the structure of left and right singular vectors linked to the nonzero singular values of MSR matrix and asymptotic expansion formula due to the existence of cracks. Investigated structure of imaging function indicates that imaging performance of subspace migration is highly related to the range of incident and observation directions. The simulation results with synthetic data polluted by random noise are exhibited to support investigated structure.

Application and analysis of direct sampling method in real-world microwave imaging

In this paper, a direct sampling method (DSM) is designed for a real-time detection of small anomalies from scattering parameters measured by a small number of dipole antennas. Applicability of the DSM is theoretically demonstrated by proving that its indicator function can be represented in terms of an infinite series of Bessel functions of integer order, Hankel function of order zero, and the antenna configurations. Experiments using real-data then demonstrate both the effectiveness and limitations of this method.

Models for optical crosstalk in lens-based free-space optical interconnects system that use Gaussian beams as information carriers

We introduce three crosstalk models in lens-based free-space optical interconnects systems. In these models, we use the Gaussian beams as the information carriers. In the first and the second models, the resulted finite integral of the optical field at the detectors plane was solved by expanding the lens’ finite aperture in terms of complex Gaussian functions and in terms of an infinite series of Bessel functions, respectively. In the third proposed model, the resulted finite integral was solved by approximating the Gaussian function in the integrand by zeroth-order and first-order Bessel functions without the need to expand the aperture function. We show that the third proposed model compares favorably with the other models with superior computational efficiency.

Real-time microwave imaging of unknown anomalies via scattering matrix

We consider an inverse scattering problem to identify the locations or shapes of unknown anomalies from scattering parameter data collected by a small number of dipole antennas. Most of researches does not considered the influence of dipole antennas but in the experimental simulation, they are significantly affect to the identification of anomalies. Moreover, opposite to the theoretical results, it is impossible to handle scattering parameter data when the locations of the transducer and receiver are the same in real-world application. Motivated by this, we design an imaging function with and without diagonal elements of the so-called scattering matrix. This concept is based on the Born approximation and the physical interpretation of the measurement data when the locations of the transducer and receiver are the same and different. We carefully explore the mathematical structures of traditional and proposed imaging functions by finding relationships with the infinite series of Bessel functions of integer order. The explored structures reveal certain properties of imaging functions and show why the proposed method is better than the traditional approach. We present the experimental results for small and extended anomalies using synthetic and real data at several angular frequencies to demonstrate the effectiveness of our technique.

Direct sampling method for anomaly imaging from scattering parameter

In this paper, we develop a fast imaging technique for small anomalies located in homogeneous media from S-parameter data measured at dipole antennas. Based on the representation of S-parameters when an anomaly exists, we design a direct sampling method (DSM) for imaging an anomaly and establishing a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Simulation results using synthetic data at f=1GHz of angular frequency are illustrated to support the identified structure of the indicator function.

Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.

Topological derivative-based technique for imaging thin inhomogeneities with few incident directions

Many non-iterative imaging algorithms require a large number of incident directions. Topological derivative-based imaging techniques can alleviate this problem, but lacks a theoretical background and a definite means of selecting the optimal incident directions. In this paper, we rigorously analyse the mathematical structure of a topological derivative imaging function, confirm why a small number of incident directions is sufficient, and explore the optimal configuration of these directions. To this end, we represent the topological derivative-based imaging function as an infinite series of Bessel functions of integer order of the first kind. Our analysis is supported by the results of numerical simulations.

Performance analysis of multi-frequency topological derivative for reconstructing perfectly conducting cracks

This paper concerns a fast, one-step iterative technique of imaging extended perfectly conducting cracks with Dirichlet boundary condition. In order to reconstruct the shape of cracks from scattered field data measured at the boundary, we introduce a topological derivative-based electromagnetic imaging functional operated at several nonzero frequencies. The structure of the imaging functionals is carefully analyzed by establishing relationships with infinite series of Bessel functions for the configurations of both symmetric and non-symmetric incident field directions. Identified structure explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction. Various numerical simulations with noise-corrupted data are conducted to assess the performance, effectiveness, robustness, and limitations of the proposed technique.

New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoï, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings.The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ(12), and in 1925, S. Wigert obtained a generalization giving a formula for ζ(1k) for any even integer k≥2. We extend the work of Ramanujan and Wigert in this paper.The Voronoï summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σs(n).The modified Bessel functions Ks(x) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σs(n) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.

Sums of divisors functions and Bessel function series

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. These two identities are intimately connected with the classical circle and divisor problems, respectively. There are three possible interpretations for the double series of these identities. The first identity has been proved under all three interpretations, and the second under two of them. Furthermore, several analogues of them were established, and they were extended to Riesz sum identities as well. In this paper, we provide analogous Riesz sum identities for the weighted sums of divisors functions, and in particular two of them yield a generalization of the Riesz sum identity for r6(n), where r6(n) denotes the number of representations of n as a sum of six squares.

On the effusion time of drugs from the open pore of a spherical vesicle

Solute permeation through a spherical liposomal vesicle was analyzed using Fick’s second law and a mixed Neumann–Dirichlet boundary condition. The first-principles approach was necessary to help calculate the effusion time of a medication through a pore located on the surface of the device. An infinite series of Bessel functions represented the concentration in the Laplace domain. This method yielded closed-form expressions for the characteristic time and the Laplace-transformed fraction of drug released, which was approximated by the first term of the series. The time constant was inversely proportional to the diffusion coefficient in the system and decreased as the pore size increased. It took 4 times the effusion time to unload nearly ninety-eight percent of the pharmaceutical ingredient.

Analysis of MUSIC-type imaging functional for single, thin electromagnetic inhomogeneity in limited-view inverse scattering problem

This study analyzes the well-known MUltiple SIgnal Classification (MUSIC) algorithm to identify unknown support of thin penetrable electromagnetic inhomogeneity from scattered field data collected within the so-called multi-static response matrix in limited-view inverse scattering problems. The mathematical theories of MUSIC are partially discovered, e.g., in the full-view problem, for an unknown target of dielectric contrast or a perfectly conducting crack with the Dirichlet boundary condition (Transverse Magnetic–TM polarization) and so on. Hence, we perform further research to analyze the MUSIC-type imaging functional and to certify some well-known but theoretically unexplained phenomena. For this purpose, we establish a relationship between the MUSIC imaging functional and an infinite series of Bessel functions of integer order of the first kind. This relationship is based on the rigorous asymptotic expansion formula in the existence of a thin inhomogeneity with a smooth supporting curve. Various results of numerical simulation are presented in order to support the identified structure of MUSIC. Although a priori information of the target is needed, we suggest a least condition of range of incident and observation directions to apply MUSIC in the limited-view problem.