Time-domain techniques in the singularity expansion method

New methods of determining the natural frequencies and natural modes of a structure as characterized in the singularity expansion method (SEM) are presented. The method is based on the time-domain scattering equation which can be east in the form of a matrix difference equation. The homogeneous solution of the difference equation is a series of exponentials as found in the SEM representation. The natural frequency and mode solutions may be obtained either from the determinant of a matrix sum, which is similar to the current frequency-domain search method, or by an eigenvalue approach as in systems theory. The latter has shown promise for efficient SEM computations. An example of each approach is presented.     

Transient coupling analysis using the singularity expansion method

The problem of electromagnetic coupling between objects has been extensively studied within the context of continuous-wave (CW) signals. The problem of coupling between objects for transient excitation has not received as much attention. This paper addresses the problem of transient coupling using the singularity expansion method (SEM). Specifically, the proper physical interpretation of the SEM modes of a system of objects is discussed within the context of observed transient surface currents and scattered fields. It is seen that the SEM modes of the system of objects are global quantities and hence have no clear physical interpretation prior to times when global modes can be established. It is also apparent that the early-time response of the system can be expressed using the SEM modes of the individual objects for some time periods. These observations are important for the effective use of target identification schemes based on natural resonance phenomena in multiple target situations as well as the proper use of SEM in EMC coupling analyses     

Transient scattering of an L-shaped wire using the singularity expansion method

A bent thin wire in the shape of an L is analyzed in free space to determine the transient scattering characteristics by the application of the singularity expansion method (SEM) to a coupled set of Hallén-like equations. The principal results of a parametric study to determine the effect of the bend position on the complex resonant frequencies are reported here. The frequency and the time domain responses are given for several cases of plane wave illumination using the SEM and results are compared to the conventional Fourier transform method.     

The singularity expansion method and its application to targetidentification

The singularity expansion method (SEM) for quantifying the transient electromagnetic (EM) scattering from targets illuminated by pulsed EM radiation is reviewed. SEM representations for both induced currents and scattered fields are presented. Natural-resonance-based target identification schemes, based upon the SEM, are described. Various techniques for the extraction of natural-resonance modes from measured transient response waveforms are reviewed. Particular attention is given to the aspect-independent (extinction) E-pulse and (single-mode) S-pulse discriminant waveforms which, when convolved with the late-time pulse response of a matched target, produce null or mono-mode responses, respectively, through natural-mode annihilation. Extensive experiment results for practical target models are included to validate the E-pulse target discrimination technique. Finally, anticipated future extensions and areas requiring additional research are identified     

The singularity expansion method as applied to perpendicular crossed wires

The singularity expansion method (SEM) has been applied to determine the current and charge induced on a system of perpendicular crossed cylinders. The SEM characteristics of this structure have been studied as the various parameters are varied. The time domain response of one particular geometry has been obtained by SEM and compared to that determined by the more conventional frequency domain analysis and Fourier inversion.     

Impulse response of a conducting sphere based on singularity expansion method

An approximate method based on the singularity expansion method is developed to determine the impulse response of a conducting sphere.     

On the relationship between the singularity expansion method and the mathematical theory of scattering

The so-called natural frequencies in the singularity expansion method (SEM) consist of two nonintersecting sets of wavenumber parameters: namely, a set of interior resonant frequencies which occur on the negative imaginary axis of thesplane and a set of those which reside in the left half of thesplane off the imaginary axis. It is shown that only the latter set can be interpreted as intrinsic to the scatterer. It corresponds to the set of complex poles of the scattering matrix of the problem and also to the set of complex eigenvalues of the related exterior homogeneous boundary value problem. Finally, some doubts about the eigenmode expansion method (EEM) formalism are raised, and a possible justification, based on nonself-adjoint theory, is suggested.     

Analysis and synthesis of an impedance-loaded loop antenna using the singularity expansion method

The singularity expansion method (SEM) is used to represent the current on a loaded loop antenna. The shift in the poles of the loop due to impedance loading can be analyzed using contour plots in the complex frequency plane of the Fourier modal impedance transfer functions. The same plot may also be used to determine a loading function which will yield a specified pole pattern leading to frequency or time domain synthesis. A simple example of time-domain synthesis is presented.     

Equivalent circuit synthesis for a loop antenna based on the singularity expansion method

The singularity expansion method (SEM) is used to derive active equivalent circuits for a plane-wave-excited loop antenna immersed in a lossless and homogeneous medium. The circuits consist of simple modules associated with the dominant pole pairs of the loop, and circuit element values are expressed in terms of the SEM data. Particular attention is paid to the issue of physical realizability. The synthesized circuits have been analyzed under various excitation and loading conditions by a general-purpose circuit analysis code, and the results agree favorably with the responses of the loop computed by other means     

A note on the representation of scattered fields as a singularity expansion

The representation of the electromagnetic field scattered by a perfectly conducting finite-extent scatterer immersed in a lossless medium as a singularity expansion is considered. While the analytic properties of the temporal Laplace transform of the surface currents residing on such an object have received a great deal of attention, the properties of the scattered fields have not. It is shown that the representation of the transform of the scattered field must include an exponential entire function except for observation points in the forward-scattered direction. Explicit time domain representations that are counterpart to the Laplace domain representation are constructed and are shown to embody, in the early time, temporal variation besides that of the damped sinusoidal factors intrinsic to the singularity expansion. An important practical consequence of this more complicated time variation arises in connection with the application of the singularity expansion for target classification purposes and is commented upon herein.     

Scattering by a thin wire parallel to a ground plane using the singularity expansion method

A thin wire parallel to a ground plane is analyzed using moment methods to find the complex natural resonant frequencies and modal current distributions. The resonant frequencies are found as functions of the wire radius and distance above the ground. The complex frequencies appear to be divisible into two classes: 1) those associated with the resonant length of the dipole and 2) those associated with interactions of the dipole and its image. For large dipole radii, the two classes interact and some anomalies, which appear, are investigated. The time domain response to a step-function plane wave is computed using the singularity expansion method and the results compare favorably with the Fourier transform solution.     

Combined phase singularity and wavefront analysis for optical maps of ventricular fibrillation

Much of the research into the mechanisms of ventricular fibrillation (VF) employs high-resolution mapping of electrical activation and recovery patterns. We previously developed a method for analyzing electrically mapped VF patterns that was based on identifying individual VF wavefronts. We now introduce a related method designed to take into account the information on repolarization that is present in optically mapped VF data. The new method first converts raw fluorescence data to an angular variable that tracks the phase of the mapped tissue through the depolarization-repolarization cycle. We define wavefronts in this context as isolines of phase that terminate either at boundaries or at singular points within the phase field. These singularities are the pivots of functional reentry and are important determinants of VF patterns. We parameterize VF by constructing data structures that describe wavefronts and singularities and also maintain wavefront-wavefront, wavefront-singularity, and singularity-singularity relationships. We describe one important application of this parameterization, which is to identify, localize, and characterize the importance of occurrences of propagation block during VF.     

The far-field response of a step-excited linear antenna using SEM [singularity expansion method]

Previous investigation of the use of the singularity expansion method (SEM) for analyzing antennas or scatterers has concentrated on the behavior of currents and charges on a conducting body. In this paper the previous work is extended by defining the far-field natural modes for the linear antenna. The transient radiation pattern is then computed for a step-excited center-fed linear antenna.     

The singularity expansion method applied to perpendicular crossed wires over a perfectly conducting ground plane

The singularity expansion method (SEM) has been applied to determine natural resonances of a set of perpendicular crossed wires over a perfectly conducting ground plane. The variation of the natural resonances and the mode and coupling vectors have been studied as parameters of the system varied.     

Application of the matrix pencil method for estimating the SEM(singularity expansion method) poles of source-free transient responsesfrom multiple look directions

The matrix pencil method has been utilized for estimating the natural resonances from different transient responses recorded along multiple look directions as a function of time after the incident field has passed the structure. The novelty of this article is that a single estimate for all the poles are done utilizing multiple transient waveforms emanating from the structure along multiple look directions. The SEM poles are independent of the angle at which the transient response is recorded. The only difference between the various waveforms are that the residues at the various poles are of different magnitudes. Some of the residues may even be zero for some of the poles indicating that the contribution from certain SEM poles may not be significant along that look direction. Here all the waveforms are utilized providing a single estimate for the poles without performing an arithmetic mean of the various waveforms     

Theoretical and practical aspects of singularity and eigenmode expansion methods

In recent years a vast amount of literature has been written on singularity and eigenmode expansion methods (SEM and EEM). The main practical purpose is to produce a theory which allows us to identify a flying (or stationary) target from the observed transient field scattered by the target. (The other purposes include representation and computation of the transient field and construction of equivalent circuits according to the object's electromagnetic response.) This problem is discussed. Described are 1) the basic starting points of the engineering approach to the problem and the extent to which they are consistent from the mathematical point of view, 2) the material which has been rigorously established in the field, 3) the important points in practice, and 4) the unsolved mathematical problems in the field. This work may save efforts by other scientists and engineers, since the results demonstrate the kind of research that can be used in practice and the kind which is of mostly theoretical interest.     

On the analysis of scattering and antenna problems using the singularity expansion technique

In this paper, the possibility of analyzing scattering and antenna problems from a singularity expansion point of view is discussed. As an example of the method, a thin-wire scatterer is considered by first determining the locations of the exterior natural resonant frequencies and then constructing the time response of the current on the body, much in the same manner as in classical circuit theory. The numerical techniques used will be presented, and some advantages of the singularity expansion method over the other conventional ways of treating this problem will be mentioned.     

A novel interpretation of Prony's method

The two key steps in Prony's method are interpreted as two key approximation problems. In the first, a nonhomogeneous, constant coefficient, linear difference equation is fitted to the samples. This difference equation is fitted so that the nonhomogeneous part of the difference equation is minimized in the least-squares sense. For the second problem, a homogeneous solution to the difference equation is obtained that minimizes the particular solution. This homogeneous solution is used to approximate the general solution     

An additional physical interpretation in the Luneburg-Kline expansion

The Luneburg-Kline (LK) expansion provides an asymptotic representation of the reflected field from a smooth surface in inverse powers of k. The physical interpretation of the first term in the expansion has long been recognized. The physical significance of the second term is suggested here. Through analysis of the expansion for a parabolic, circular, and elliptic cylinders, further geometric information is seen other than the radius of curvature at the specular point. The other geometric information is the distance between the specular point and the incident shadow boundary.