Study of an integral equation arising in detection theory

This paper considers the solution of a Fredholm equation occurring in detection theory problems. A solution procedure, based on solving differential equations with nonmixed boundary conditions, is described for the case when the kernel of the integral equation is known to be the output covariance of a linear finite-dimensional system excited by white noise. Solutions with discontinuities are considered.     

On filtered binary processes

The problem of calculating the probability density function of the output of anRCfilter driven by a binary random process with intervals generated by an equilibrium renewal process is studied. New integral equations, closely related to McFadden's original integral equations, are derived and solved by a matrix approximation method and by iteration. Transformations of the integral equations into differential equations are investigated and a new closed-form solution is obtained in one special case. Some numerical results that compare the matrix and iteration solutions with both exact solutions and approximate solutions based upon the Fokker-Planck equation are presented.     

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.     

Levinson-like and Schur-like fast algorithms for solvingblock-slanted Toeplitz systems of equations arising in wavelet-basedsolution of integral equations

The Wiener-Hopf integral equation of linear least-squares estimation of a wide-sense stationary random process and the Krein integral equation of one-dimensional (1-D) inverse scattering are Fredholm equations with symmetric Toeplitz kernels. They are transformed using a wavelet-based Galerkin method into a symmetric “block-slanted Toeplitz (BST)” system of equations. Levinson-like and Schur-like fast algorithms are developed for solving the symmetric BST system of equations. The significance of these algorithms is as follows. If the kernel of the integral equation is not a Calderon-Zygmund operator, the wavelet transform may not sparsify it. The kernel of the Krein and Wiener-Hopf integral equations does not, in general, satisfy the Calderon-Zygmund conditions. As a result, application of the wavelet transform to the integral equation does not yield a sparse system matrix. There is, therefore, a need for fast algorithms that directly exploit the (symmetric block-slanted Toeplitz) structure of the system matrix and do not rely on sparsity. The first such O(n²) algorithms, viz., a Levinson-like algorithm and a Schur (1917) like algorithm, are presented. These algorithms are also applied to the factorization of the BST system matrix. The Levinson-like algorithm also yields a test for positive definiteness of the BST system matrix. The results obtained are directly applicable to the problem of constrained deconvolution of a nonstationary signal, where the locations of the smooth regions of the signal being deconvolved are known a priori     

Considerations on Matrix Methods and Estimation of Their Errors

Two-dimensional field equations are reduced to Fredholm integral equations of the second kind. The integral equations are solved by matrix methods. The convergence of the matrix solutions is discussed. The matrix methods are applied to calculating the cutoff wavenumbers of waveguides. A method of estimating the errors is proposed. A method of correcting the matrix solutions is described and applied to a field problem in which the boundary is large compared with the wavelength. It is pointed out that for the commonest method of solving integral equations numerically (the method of subsections), the accuracy depends strongly on the position in each subsection of the point to which the field is referred. The dependence of the error on position is examined quantitatively.     

The well—posedness of the formulation of diffraction problems reduced to Fredholm integral equations of the first kind with a smooth kernel

The well-posedness of diffraction problems that are reduced to Fredholm integral equations of the first kind with a smooth kernel is analyzed. The auxiliary source method and the method of extended boundary conditions, both of which involve solution of Fredholm integral equations of the first kind with a smooth kernel, are applied to show for specific examples that algorithms of calculation of all physically significant quantities—the scattering pattern, the field at an arbitrary spatial point except current-carrier points, etc.—are quite stable and allow for computation of the aforementioned quantities with a preassigned accuracy.     

A Chebyshev Approximation Method for Microstrip Problems

The quasi-static TEM mode of a microstrip line may be obtained approximately from the solution of Laplace's equation subject to certain boundary conditions. The Green's function approach leads to the solution of a Fredholm integral equation with a logarithmic singularity in the kernel. It is shown that if the charge distribution on the strip is expanded in terms of Chebyshev polynomials then the integrals arising from the logarithmic term may be evaluated in closed from, and the integral equation may be approximated closely by a set of algebraic equations. The method is applied to numerous open and shielded configurations of strips and couple-strips in the presence of dielectrics. Numerical results are compared with exact results whenever possible and with results from previous authors. Design curves are presented for particular shielded couple-strip configurations.     

Integral equation method in problems of electromagnetic-wave reflection from inhomogeneous interfaces between media

Problems on reflection of a plane electromagnetic wave from various irregular interfaces between media are studied by the integral equation method in the cases of two- and three-dimensional incident electromagnetic field. The reflecting surfaces are meant as periodic transparent interfaces between two media and plane boundaries with locally inhomogeneous and transparent sections. The boundary value problems for the system of Maxwell’s equations in an infinite domain with an irregular boundary are reduced to Fredholm or singular integral equations, depending on the problem considered. Numerical algorithms for solving such integral equations are developed. Results of calculation of currents induced on inhomogeneities and characteristics of the electric field in the far zone are presented.Problems on reflection of a plane electromagnetic wave from various irregular interfaces between media are studied by the integral equation method in the cases of two- and three-dimensional incident electromagnetic field. The reflecting surfaces are meant as periodic transparent interfaces between two media and plane boundaries with locally inhomogeneous and transparent sections. The boundary value problems for the system of Maxwell’s equations in an infinite domain with an irregular boundary are reduced to Fredholm or singular integral equations, depending on the problem considered. Numerical algorithms for solving such integral equations are developed. Results of calculation of currents induced on inhomogeneities and characteristics of the electric field in the far zone are presented.     

A linear discretization of the volume conductor boundary integralequation using analytically integrated elements [electrophysiologyapplication]

A method is presented to compute the potential distribution on the surface of a homogeneous isolated conductor of arbitrary shape. The method is based on an approximation of a boundary integral equation as a set linear algebraic equations. The potential is described as a piecewise linear or quadratic function. The matrix elements of the discretized equation are expressed as analytical formulas.     

Fredholm resolvents, Wiener-Hopf equations, and Riccati differential equations

We shall show that the solution of Fredholm equations with symmetric kernels of a certain type can be reduced to the solution of a related Wiener-Hopf integral equation. A least-squares filtering problem is associated with this equation. When the kernel has a separable form, this related problem suggests that the solution can be obtained via a matrix Riccati differential equation, which may be a more convenient form for digital computer evaluation. The Fredholm determinant is also expressed in terms of the solution to the Riccati equation; this formula can also be used for the numerical determination of eigenvalues. The relations to similar work by Anderson and Moore and by Schumitzky are also discussed.     

Applications of the Maxwellian Circuits to Linear Wire Antennas and Scatterers

The recently proposed theory of Maxwellian circuits is demonstrated for applications to linear wire scatterers as well as to linear antennas. It is shown that for each integral equation of thin wire type, there exist coupled linear ordinary differential equations of currents and voltages, the solutions of which are identical to the integral equation, if the same boundary conditions of the integral equation are applied. The subsequence is that the coupled differential equations can be interpreted as equivalent circuit of new type named Maxwellian circuit. The equivalent circuit can provide physical insights to design engineers and computational advantages for broadband calculations. The highlight of this paper is to show both theoretically and numerically that the Maxwellian circuit components depend only on the geometry of the problem, not on the excitation or boundary conditions at the terminals.     

On the application of numerical methods to Hallen's equation

The so-called Hallen integral equation for the current on a finite linear antenna center-driven by a delta-function generator takes two forms depending on the choice of kernel. The two kernels are usually referred to as the exact and the approximate or reduced kernel. With the approximate kernel, the integral equation has no solution. Nevertheless, the same numerical method is often applied to both forms of the integral equation. In this paper, the behavior of the numerical solutions thus obtained is investigated, and the similarities and differences between the two numerical solutions are discussed. The numerical method is Galerkin's method with pulse functions. We first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length. In this case, the method yields an infinite Toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter. The infinite system is solved exactly for nonzero pulse width; the exact solution is then developed asymptotically for the case where the pulse width is small. When the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna, the latter problem is greatly facilitated. For the approximate kernel, the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the Pocklington type     

A survey of the near-field far-field inverse scattering inverse source integral equation

The Kirchhoff direct integration of the scalar wave equation is reviewed, and some properties of the Kirchhoff surface integral are discussed, from the perspective of the inverse scattering inverse source problem. A modified Kirchhoff surface integral is introduced, leading to a Fredholm integral equation of the first kind for the unknown sources (induced by the incident field) inside a volume in terms of the (scattered) fields on the surface enclosing this volume. The properties and physical meaning of this integral equation are discussed. A generalization of this integral equation for the vector electromagnetic wave equations is presented.     

SOLUTION OF ARBITRARY CROSS-SECTION WAVEGUIDE USING METHOD OF EIGEN WEIGHTED BOUNDARY INTEGRAL EQUATION

Based on the second kind of Green's identity,a boundary integral equation forarbitrary cross-section waveguide is transformed to a system of linear homogeneous algebraicequations by means of expansion of boundary bases and by using the eigenfunctions of a fictitiousregular boundary as weighting functions,which corresponds to less algebraic equations than BEMand simpler coefficients than the modified BEM.The numerical results for some typical metallicwaveguides are given by using the method of eigen-weighted boundary integral equation,and theyare accurate enough with fast convergence.     

A differential equation related to a random telegraph wave problem--Computer calculation of series solution

A method is developed for calculating the probability density functionp(y)of the output of anRCfilter when the input is a particular kind of random telegraph wave. The method makes use of a computer to determine the numerical values of the coefficients in two series solutions, one of which contains a logarithm, of a fourth-order linear differential equation. The constants of integration are determined from the boundary conditions by a procedure that involves a summation of the series. Representative values and curves ofp(y)are presented, and the generality of the computer method is discussed briefly,     

An integral equation approach to electrical conductance tomography

A reconstruction procedure for electrical conductance tomography developed by solving a linear Fredholm integral equation of the first kind is discussed. The integral equation is obtained from a linearized Poisson's equations. Properties of the integral equation are discussed, and problems associated with numerical solution of the equation are treated. The reconstruction requires only one matrix multiplication and therefore can be computed in a short time. Test results of the algorithm using both simulated and measured data are presented.     

Plane wave diffraction by a slit in a perfectly conducting plane and a parallel complementary strip

The diffraction of plane electromagnetic waves by the configuration formed by a slit in a perfectly conducting plane and a parallel complementary strip is investigated. The related boundary-value problem is formulated into a modified matrix Wiener-Hopf equation. The factorization of the kernel matrix is accomplished through Abrahams’ method and the modified matrix Wiener-Hopf equation is first reduced to a pair of coupled Fredholm integral equations of the second kind and then solved by iterations. Several numerical results illustrating the effects of various parameters such as the spacing between the slit and the strip and their width on the diffraction phenomena are presented.     

Diffraction From A Material Loaded Tandem Slit

In this paper, we consider plane wave diffraction by a tandem slit loaded with a homogenous material. The boundary value problem is formulated into a pair of simultaneous Wiener-Hopf equations via Fourier transformation. After decoupling these equations by elementary transformation, each modified Wiener-Hopf equation is reduced to a Fredholm integral equation of the second kind. The integral equations are then solved approximately to yield the Fourier transform of the diffracted fields. The inverse transform is evaluated asymptotically to obtain the far field expressions. Measurements and numerical simulations are also performed for several different geometry and material configurations. The analytic solutions compare well with measured and simulated results. The possibility of reducing beamwidth and increasing power coupled through the loaded tandem slit is explored.      

本征权边界积分法解任意截面波导传输问题

本文选取本征函数作为权函数,由格林第二恒等式建立边界积分方程,并采用边界基形式得到线性齐次方程组。这种方法不仅降低了系数矩阵的维数,而且使其各项仍保持为简单一项,减少了计算量.本文提供了几种常见金属波导的例子,计算结果既收敛快、又足够准确。