Estimation of Optimal Parameter of Regularization of Signal Recovery

In this paper there are researched regularizing properties of discretization in a space of output signals for some linear operator equation with noisy data. The essence of proposed method is selection of discretization level which is a parameter of the regularization in this context by the principle of equality of random and deterministic components of the input signal recovering error. It is shown the method, i.e. the solution which is discrete by input signal is stable to small inaccuracies in input signal. At that in case of definite level of output signal measurements inaccuracy the recovering error of input signal is unambiguously defined by input signal sampling increment that allows to select reasonably the regularization parameter for specific criterion, for example, for definite measurements inaccuracy. Specific calculations and examples are represented in explicit form for single-dimension case but this does not restricts generality of proposed method.     

Levinson-type extensions for non-Toeplitz systems

It is shown Levinson's basic principle for the solution of normal equations which are of Toeplitz form may be extended to the case where these equations do not possess this specific symmetry. The method is illustrated by application to various examples which are chosen so that the coefficient matrix possesses various symmetries. Specifically, the solution of the normal equations when the associated matrix is the doubly symmetric non-Toeplitz covariance matrix is considered. Next, the solution of extended Yule-Walker equations where the coefficient matrix is Toeplitz, but nonsymmetric is obtained. Finally, the approach is illustrated by considering the determination of the prediction error operator when the normal equations are of symmetric Toeplitz form     

Biharmonic Operators for the Waveguide Problem (Correspondence)

The inverse iteration method of analyzing an isotropic waveguide is reexamined with a view to finding the higher order modes. It is shown that the problem may be formulated in terms of a generalized biharmonic operator, which is always at least semidefinite. Several finite-difference representations are given for this operator. Of these, two possess property A, a point of fundamental importance for iterative solution by successive overrelaxation.     

The application of the conjugate gradient method for the solution of electromagnetic scattering from arbitrarily oriented wire antennas

The method of conjugate gradients is applied to the analysis of radiation from thin-wire antennas. With this iterative technique, it is possible to solve electrically large arbitrarily oriented wire structures without storing any matrices as is conventionally done in the method of moments. The basic difference between the proposed method and Galerkin's method, for the same expansion functions, is that for the iterative technique we are solving a least squares problem. Hence, as the order of the approximation is increased, the proposed technique guarantees a monotonic decrease of the least squared error (parallel AI - Yparallel^{2}), whereas Galerkin's method does not. Even though the method converges for any initial guess, a good one may significantly reduce the time of computation. Also, explicit error formulas are given for the rate of convergence of this method. Hence, any problem can be solved to a prespecified degree of accuracy. It is shown that the method has the advantage of a direct solution as the final solution is obtained in a finite number of steps. The method is also suitable for solving singular operator equations in which case the method monotonically converges to the least squares solution with minimum norm. Numerical results are presented for the thin-wire antennas and are compared with the solution obtained by the method of moments.     

用正交函数展开法解介质双球静电问题

本文讨论了一般介质边界条件经逆矢径变换后的表达式，证实对介质双球静电问题若采用逆矢径变换法求解，最终将得到本征展开系数的一个无穷维线性耦合方程组，实际求解困难，本文提出用正交函数展开法解介质双球静电问题，获得精确结果。数值结果进一步验证了Ｌａｐｌａｃｅ方程本征解的加法定理。     

A combined-source solution for radiation and scattering from a perfectly conducting body

A combined-source solution is developed for electromagnetic radiation and scattering from a perfectly conducting body. In this solution a combination of electric and magnetic currents, called the combined source, is placed on the surfaceSof the conducting body. The combined-source operator equation is obtained from theE-field boundary-value equation. It is shown that the solution to this operator equation is unique at all frequencies. The combined-field operator equation also has a unique solution, but it is not directly applicable to the aperture radiation problem. TheH-field andE-field operator equations fail to give unique solutions at frequencies corresponding to the resonant frequencies of a cavity formed by a hollow conductor of the same shape. The combined-source operator equation is solved by the method of moments. The solution, valid for a three-dimensional closed surfaceS, is then applied to a surface of revolution. Examples of numerical computations are given for a sphere, a cone-sphere, and a finite circular cylinder.     

Least square collocation as applied to the analysis of strip transmission lines

The crude method of collocation applied for the solution of the boundary-value problems has been improved upon in a least square sense for solving complex problems in applied mechanics. In this letter, the least square collocation method is applied for the solution of the electrostatic field problem of the strip transmission line enclosed in a shielded box, which is a case of mixed boundary-value problem. Compared to the other previous numerical methods, this method is much less laborious, simpler, and faster, and the results agree well with those available in the literature.     

Variational Solution of Integral Equations

A variational solution of the Fredholm integral equation of the first kind resulting from Laplace's equation with Dirichlet boundary conditions is discussed. Positive-definiteness of the integral operator is used to guarantee convergence. The square parallel plate capacitor is given as an example with several different types of trial functions. Special singular functions to handle known field behavior are shown to result in improved accuracy with reduced computing cost. The air-dielectric interface condition is related to a general Neumann-mixed boundary condition for which a variational method with a positive-definite integral operator is presented. Multiple boundary conditions are handled by mutually constraining separate variational expressions for each boundary condition. A T-shaped conductor on a dielectric slab, representative of quasi-static solutions of microstrip discontinuities, is presented as a three-dimensional example with multiple boundary conditions. Generally, it is shown how the finite-element method for the solution of partial differential equations may be extended to handle integral equation formulations.     

Analytic continuation considerations when using generalizedformulations for scattering problems

A generalized operator equation has been developed for a simple scattering problem for which a closed-form solution exists. The operator equation has been solved numerically via the method of moments using spatially impulsive fictitious sources as expansion functions together with a simple point-matching testing procedure. The study focused on the convergence and accuracy of the solution and examined how they are dependent on the location and distance of the fictitious sources relative to the area containing the singularities of the actual field simulated by these sources. As expected, it was found that when the actual field simulated by the fictitious impulsive sources has singularities lying between the physical boundary and the closed surface over which the sources are placed, the impulsive expansion does not yield a uniformly convergent solution. In this case, instabilities are encountered as the number of sources increases, in the sense that a small improvement of the boundary error requires a considerable change in the currents. The moment matrix is then difficult to invert and easily susceptible to large round-off errors. Conversely, if the actual field has no singularities lying between the physical boundary and the closed surface over which the sources are placed, the impulsive expansion does yield a uniformly convergent solution to any degree of precision     

A note on the choice weighting functions in the method of moments

The objective is to show from a mathematical standpoint that there are certain rules that must be followed in the choice of weighting functions used in the method of moments (MM). It is shown that for a particular problem it is the operator that dictates the method (Galerkin's method or another method such as the method of least squares) to be applied, and it is not computational considerations only. For example, it is shown that in solving Hallen's and Pocklington's equation by the method of moments, it is unnatural to choose the weighting functions which are zero at the ends of the domain of the solution. The deficiency of certain weighting functions is presented based on mathematical reasoning, and a numerical example is given to illustrate the effect of the choice of the weighting functions on the rate of the convergence of the solution.     

Generalization of the ECCS method to the multihour case

The classical ECCS method for dimensioning hierarchical telephone networks is extended to the multihour case. The problem is reformulated as a nonlinear optimization method, and the Kuhn-Tucker equations are derived. It is shown why the ECCS method cannot solve this multihour case. The ECCS method is presented as a solution technique for the Lagrangian relaxation of the multihour problem in which subgradient techniques are used for the solution of the dual. A numerical algorithm implementing these ideas is described that can handle networks with arbitrary hierarchies and allows time-of-day routing changes. Preliminary numerical results of the method are presented, showing that the multihour algorithm can reduce a network's cost when compared to currently used methods     

Note on the Inversion of the Schwarz-Christoffel Conformal Transformation

An inversion procedure, based on the methods used in proving Burmann's theorem, is used to provide an integral expression which exhibits the form of the electrostatic field explicitly in terms of the field coordinates. The method is illustrated with an example of a stepped-guide junction. The form of the field and the expression for the mode expansion coefficients are examined. The results are related to the companion problem of solving for the transverse field from a singular integral equation formulation. The two methods agree in the particular case of a two-to-one step for which special simplifications are possible. In the general case, progress in the solution of a class of double-kernel integral equations may be expected through the indirect use of the inversion of the solution obtained from the conformal transformation methods.     

Distributed equivalent sources for the analysis of multiconductortransmission lines excited by an electromagnetic field

The interaction of multiconductor lines with electromagnetic radiation is commonly studied in terms of field-induced voltage and current sources distributed along the line. The author presents the relationships between these sources and the incident fields for the general case of a transmission line with its conductors embedded in different dielectric volumes of arbitrary shape. It is shown that the sources can be expressed directly in terms of the incident fields and some vector parameters which are determined from the solution of a series of electrostatic problems with appropriate boundary conditions independent of the incident electric fields. It is noted that the multiconductor lines are suitable for direct applications in microwave transmission lines with rather arbitrary configurations     

Generalized formulations for electromagnetic scattering fromperfectly conducting and homogeneous material bodies-theory andnumerical solution

A generalized E-field formulation for three-dimensional scattering from perfectly conducting bodies and generalized coupled operator equations for three-dimensional scattering from material bodies are introduced. A fictitious electric current flowing on a mathematical surface enclosed inside the body is used to simulate the scattered field, and, in the material case, a fictitious electric current flowing on a mathematical surface enclosing the body is used to simulate the diffracted field inside the body. Application of the respective boundary conditions lead to operator equations to be solved for the unknown fictitious currents, which facilitates calculation of the fields in the various regions, using the magnetic vector potential integral. The existence and uniqueness of the solution are discussed. These alternative operator equations are solvable using the method of moments. The numerical solution is simple to execute, rapidly converging, and general in that bodies of smooth but otherwise arbitrary surface, both lossless and lossy, can be handled effectively. Comparison of the results with available analytic solutions demonstrates the accuracy of the moment procedure     

Self-Adjoint Variational Formulation of Problems Having Non-Self-Adjoint Operators

A systematic approach is given for deriving a variational formulation, previously stated by Stakgold, of non-self-adjoint operators from the standard quadratic functional for self-adjoint operators given by Mikhlin. If the same set of basis functions is used to approximate the solution of the operator equation and its adjoint equation, the resulting system of equations is identical to that derived from the Galerkin method. By using two differing sets of basis functions, one obtains a system of equations which corresponds to that derived from the moment method in general. As a particular and important example, the integral equation for the interface problem between differing media is considered. Compared to the method used by McDonald, Friedman, and Wexler, the present formulation involves no danger of finding a false solution, results in a simpler set of equations, requires fewer integrations, and is seen--in the case of integral equations--to correspond to the Galerkin method. It is also shown that for wave propagation through a lossy medium, which involves the solution of the non-self-adjoint complex Helmholtz equation, the resulting system of linear equations takes the same form as those for the real self-adjoint case but for the addition of complex arithmetic.     

A highly simple failure detection method for electrostatic microactuators: application to automatic testing and accelerated lifetime estimation

Reliability and testability are two important factors for the development of batch fabrication of microelectromechanical systems (MEMS) or mixed MEMS/IC applications. In that frame, an easy-to-implement electrical method of detection of failures is presented in this paper, valid for every kind of electrostatic microactuators showing pull-in behavior, with great expectation value in the case of arrayed MEMS. The method relies on the detection of a pull-in current peak at very low frequency. An experimental setup is described and results are shown obtained with prototypes of electrostatic microactuators with parallel capacitances created by deep reactive ion etching on silicon-on-insulator wafers. This method is really suitable for automation and application to on-line testing during mass production and reliability studies. The functional schematic of a test circuitry is proposed for implementation in an industrial tester or for built-in self-test purposes. Then, in order to illustrate the benefits of this method for reliability studies, it is used in conjunction with high voltage testing for accelerated lifetime measurements. A time gain factor of 20 is achieved with above-mentioned test structures.     

数值吸收边界条件在OSRC方法中的应用

本文引入一种数值吸收边界条件,将其应用于解决开放空间电磁散射问题的表面辐射条件法(OSRC)之中。本文使用这种边界条件对理想导电及均匀电磁可穿透材料圆柱散射体的表面场进行了数值计算,并与使用二阶微分算子B_2型的吸收边界条件的结果作了比较。所有结果说明,这种数值吸收边界条件特大大改善OSRC方法的精度。     

等几何分析法应用于偏心柱面静电场问题

利用等几何分析思想通过加权余量法对静电场控制方程进行弱化,推导出静电场的等几何分析方程,提出了一种分析静电场问题的高精度方法。将此方法应用于求解偏心柱面静电场问题,并对计算结果与解析解和经典的有限元方法比较,结果表明:此方法具有自由度少、精度高、收敛速度快的优点。     

On the applicability of OSRC method to homogeneous scatterers

The author investigates the application of radiation boundary conditions to two-dimensional scattering by a homogeneous penetrable cylinder. The second-order radiation boundary operator proposed by G.A. Kriegsmann et al. (1987) is applied on a conformal outer boundary enclosing the object. By taking the specific case of a lossless circular cylinder for which the exact solution exists it is shown that the on-surface radiation condition (OSRC) method does not necessarily yield accurate results for all values of material constants. Highly erroneous results are produced for certain values of permittivity and permeability, while acceptable results can be obtained for others. The author suggests that the cause of these erroneous results is related directly to the error in the reactive portion of the complex scattered power. However, as the outer boundary of the radiation condition recedes away from the object contour, the error is shown to diminish. Numerical results are presented for the bistatic echo width for both transverse electric (TE) and transverse magnetic (TM) polarizations     

Error and convergence in numerical implementations of the conjugategradient method [EM problems]

The conjugate gradient method has previously been applied in electromagnetics in two ways: to moment method matrices and directly to continuous operator equations. Numerical implementations of these two methods are shown here to be equivalent. It is concluded that the advantage of the conjugate gradient method is therefore its potential computational efficiency as a solution procedure, not its ability to achieve a more exact solution than the moment method