Time-domain near-field analysis of short-pulse antennas .I.Spherical wave (multipole) expansion

The radiation from a time-dependent source distribution in free-space is analyzed using time-domain (TD) spherical wave (multipole) expansion. The multipole moment functions are calculated from the time-dependent source distribution. The series convergence rate in the near and far zone and the bounds on the near-zone reactive field are determined as functions of the source support and of the pulse length. The formulation involves a spherical transmission line representation that can be extended to more general spherical configurations. This formulation also describes the field and energy transmission mechanisms in a physically transparent fashion that will be used in a companion paper to define and explore fundamental concepts such as TD reactive energy and Q and to derive bounds on the antenna properties. Finally, the concepts discussed above are demonstrated numerically for pulsed radiation by a circular current disk     

The radiation operator

The concept of the radiation operator is introduced to assist in the analysis of various problems involving sources and their radiation fields. It gives the field outside the source region as operating on the field of a point source. Because there is a simple connection between the radiation vector describing the far-field and the radiation operator, it can be used to define fields anywhere outside the source region from their values in the far-field zone. Another important properly of the radiation operator is its ability to express sources of fields given their radiation pattern and polarization in the far zone. The source of such a field can be written in the form of radiation operator operating on a current element, the delta function source. To interpret this in terms of computable functions, existing tables of operational rules for different classes of operators can be applied. Examples of radiation operators corresponding to different sources are given together with examples of sources corresponding to given radiation field patterns. Finally it is shown that the radiation operator allows a considerable simplification to the derivation of the multipole expansion theory when compared to the classical recursion-formula derivation through spherical harmonic eigenfunctions     

电荷多极子和电流多极子

本文重点研究源区小而远场时的多极子展开问题。文中采用统一的三维广义Taylor展开,结合矢量算子理论,给出了电荷多极子与电流多极子的一系列结论。文中还讨论了多极子的具体矩阵表达和相对论四维电流矢的多极展开,概念清晰,推演规范,结论简洁,是多级展开理论的一次尝试。     

Spherical harmonics and Earth-rotation synthesis in radio astronomy

That spherical harmonic functionsY_{nm}(theta,phi)can be used to advantage in the Earth-rotation synthesis of radio astronomy maps is shown in this paper. As Earth rotates the baseline of a radio interferometer generates a cone whose angletheta, measured from theN-Spolar axis, can be varied by changing the baseline's azimuthal direction on the surface of Earth. A series of Earth-rotation measurements, at different cone angles but with baselines of equal lengthb, can be regarded as being made on a baseline sphere of radiusb, the analog of theuvplane in Fourier-type synthesis. The measured output distribution can be expanded as a spherical harmonic series on the baseline sphere. The coefficients of the series are related to the coefficients of the spherical harmonic series expansion of the source distribution on the celestial sphere by a matrix transformation. The matrix [B] is a function only of the baseline configuration (it does not vary with source declination). Inversion of the matrix leads to the solution for the source coefficients, from which a spherical harmonic map is formed of the source distribution.     

General reflection rule for electromagnetic multipole fields on aplane interface

The general rule for reflection of a vector 2'-pole field on a plane interface between two media of different dielectric properties is established starting from the expansion of the spherical multipole field as a linear combination of inhomogeneous vector plane waves. In fact, by considering vector multipole fields that satisfy the radiation condition at infinity we are able to define a matrix that effects their reflection on the plane interface. Such a matrix can also be used to reflect a superposition of many 2'-pole fields and so can be useful to describe the effect of a plane surface near to a specified source or to a scattering particle     

Spherical harmonics microwave algorithm for shape and location reconstruction of breast cancer tumor

A reconstruction algorithm to simultaneously estimate the shape and location of three-dimensional breast cancer tumor is presented and its utility is analyzed. The approach is based on a spherical harmonic decomposition to capture the shape of the tumor. We combine a gradient descent optimization method with a direct electromagnetic solver to determine the coefficients in the harmonic expansion as well as the coordinates of the center of the tumor. The results demonstrate the potential advantage of collecting data using a multiple-view/tomographic-type strategy. We show how the order of the harmonic expansion must be increased to capture increasingly "irregularly" shaped tumors and explore the resulting increase in the central processing unit (CPU) time required by the algorithm. Our approach shows accurate reconstruction of the tumor image regardless of the source polarization. This work demonstrates the promise of the algorithm when used on data corrupted with Gaussian noise and when perfect knowledge of the tumor electrical properties is not available.     

用于计算任意截面二维电磁散射的多心柱谐函数展开法

本文提出了一种计算任意截面二维电磁散射的方法多心柱谐函数展开法。它用多点的柱谐波函数展开散射场,代替柱谐函数展开中一点展开,并用最小均方边界误差准则处理边界条件。多极子法和柱谐函数展开法是该方法的两个特例。     

Rigorous analysis of radiating structures using finite elements and spherical modes expansion

A new rigorous method for analysing radiating structures, using the finite elements method inside a sphere and an expansion of fields in spherical harmonics in the exterior domain, is presented here. The studied structure, represented as a multipole, is entirely described by its scattering matrix at the interfaces. The radiation patterns and input impedances of several types of antennas are presented and compared to the ones obtained from other numerical methods or measurements.     

Generalized Sommerfeld integrals and field expansions in two-medium half-spaces

Time harmonic modal electromagnetic fields in two-medium half-spaces are investigated. For practical and numerical considerations, the primary sources of the modal fields are chosen to be the spherical multipoles, and the potential vectors are z-directed. It is shown that modal fields of such combination are not able to represent a conventional spherical modal field. The horizontal rotating potentials are added to ensure proper representation and fast convergence. The recurrence relations which transform the spherical Hankel-Legendre functions into the Fourier-Bessel integrals are derived. The secondary fields of the Sommerfeld type are obtained for all spherical multipole sources and the added horizontally rotating potentials. The combination of the modal fields are capable of representing arbitrary electromagnetic fields resulting from radiation and scattering problems.     

A diagonalized multilevel fast multipole method with spherical harmonics expansion of the k-space Integrals

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.     

快速球谐函数展开的并行算法设计及实现

球谐函数展开是一种广泛应用的数学方法.在研究Rokhlin-Tygert球谐函数展开快速算法（RT算法）的基础上,扩充并完善了RT算法,建立了所有阶连带Legendre多项式展开系数的计算过程,进而研究了该算法的MPI并行策略并给出了RT并行算法.数值实验表明,RT算法的计算量随三角截断波数的增大而急剧增长;MPI并行技术能够有效提高其运算速度,较好地提升算法的整体性能,但并行效率会受多核处理器cache大小和访存带宽的限制.     

Numerical evaluation of dyadic diffraction coefficients andbistatic radar cross sections for a perfectly conducting semi-infiniteelliptic cone

In this paper, the scattering of electromagnetic waves by a perfectly conducting semi-infinite elliptic cone is treated. The exact solution of this boundary value problem in problem-adapted spheroconal coordinates in the form of a spherical multipole expansion is of poor convergence if both the source point and the field point are far away from the cone's tip. Therefore, an appropriate sequence transformation of these series expansions (we apply the Shanks transformation) is necessary to numerically determine the dyadic diffraction coefficients and bistatic radar cross sections (RCS) for an arbitrary elliptic cone. Our far-field data for an elliptic cone, a circular cone, and a plane angular sector are compared with some other results obtained with the aid of quite different methods     

一种用于分布式阵列的球谐波域声源定位方法

本文提出一种适用于任意阵型和阵元指向性的球谐波域声源定位方法,能够在较宽的频域范围内,尤其是低频,提供较高的空间分辨率。水下噪声源的高分辨识别具有重大意义。传统阵列信号处理方法对低频噪声源的精确定位要求阵列具有较大孔径和较多的阵元数,导致系统过于庞大且成本较高。我们基于声场的球谐波表达和变换,采用分布在一定空间区域内的多个阵列估计该区域的球谐波系数矢量,对系数矢量进行信号处理实现声源定位。理论证明了该方法具有理想的空间选择特性。在一种特定的阵元分布下,仿真研究了该算法的方位谱估计性能以及阵元不一致性和位置误差对声源定位性能的影响。仿真结果显示,该算法在低频具有较高的空间分辨率且误差对算法定位性能的影响有限。      

Complex space multipole expansion theory with application to scattering from dielectric bodies

Classical multipole theory valid for radiation from sources with small dimensions is extended to complex space. It is shown that by locating the multipole in complex space, the first terms in multipole series can be made more dominating than when restricting to real space. Expression for the complex location is derived to minimize the effect of the second-order multipole term. With an example, radiation from a phased cubic source it is seen that the radiating characteristic is an order of magnitude better for the dipole approximation when the dipole is in complex space instead of being at the center of the cube. Simultaneously, it is demonstrated that the validity of the dipole approximation is extended fromL ll lambdato aboutL leq lambda/2,Lbeing the measure of the source. Complex location for the electric dipole approximating a dielectric spherical scatterer is found through integral equation analysis corresponding to a linear approximation of the inner field of the scatterer.     

The translation formula for vector multipole fields and therecurrence relations of the translation coefficients of scalar andvector multipole fields

We present simple alternate derivations of the translation formula for spherical vector multipole fields and the recurrence relations satisfied by the translation coefficients of spherical scalar and vector multipole fields. The derivations use the well-known and widely used spherical tensor technique to conveniently couple and decouple quantities associated with spherical angular variables, and these coupling and decoupling of angular quantities substantially simplify the algebra involved. As a consequence, the derivations are quite concise, and the resulting expressions for the recurrence relations are more compact and general than the existing ones     

Research in electrocardiography and magnetocardiography

The mathematical, physical, and engineering aspects of electrocardiography and magnetocardiography are reviewed. A brief summary of relevant physiological and clinical information is also given. The aim of the article is to provide a general perspective for engineers new to the area who want to do research. A detailed discussion of difficulties encountered in determining the heart vector and the nondipolar properties of the heart's field is included. Stress is placed on the need for development of practical ECG and MCG systems for use in the clinic. A number of research problems of current interest are pointed out.     

Transformation formulas for spherical and spheroidal multipole fields

A transformation linking spherical multipole fields with generalized spheroidal multipole fields is derived. Applications including the inverse diffraction problem for spherically scanned near-field data for sources that due to conformal considerations are efficiently described using spheroidal (either oblate or prolate) volume support regions are discussed with the aid of numerical illustrations.     

Surface Integral Equation Solutions by Hierarchical Vector Basis Functions and Spherical Harmonics Based Multilevel Fast Multipole Method

The method of moments solution of electromagnetic surface integral equation formulations for perfect electric conductors or impedance boundary objects is obtained using hierarchical vector basis functions. The singular and hypersingular integrals involved in the near field coupling matrix are computed fully numerically using adaptive singularity cancellation technique. Multilevel fast multipole method with spherical harmonics expansion of the $k$-space representations of the basis vectors and the incoming waves at the finest level is utilized for its memory and computation time efficient implementation. Improved performance of the proposed techniques in terms of accuracy, computational time, and memory requirements is demonstrated by comparing various results with some of the existing implementations available in the literature.      

Efficient Recursive Generation of the Scalar Spherical Multipole Translation Matrix

New efficient recursive procedures for generating the translation matrix of the scalar spherical multipole field are described. They are based on a new set of recurrence relations that result when the angular-momentum operator is applied to the spherical multipole field. Their efficiency and accuracy are compared analytically and through a computer experiment with those of the brute-force method and an existing recursive procedure.