Representation of electromagnetic fields over arbitrary surfaces bya finite and nonredundant number of samples

It is shown that the electromagnetic (EM) field, radiated or scattered by bounded sources, can be accurately represented over a substantially arbitrary surface by a finite number of samples even when the observation domain is unbounded. The number of required samples is nonredundant and essentially coincident with the number of degrees of freedom of the field. This result relies on the extraction of a proper phase factor from the field expression and on the use of appropriate coordinates to parameterize the domain. It is demonstrated that the number of degrees of freedom is independent of the observation domain and depends only on the source geometry. The case of spheroidal sources and observation domains with rotational symmetry is analyzed in detail and the particular cases of spherical and planar sources are explicitly considered. For these geometries, precise and fast sampling algorithms of central type are presented, which allow an efficient recovery of EM fields from a nonredundant finite number of samples. Such algorithms are stable with respect to random errors affecting the data     

A physical optics correction for backscattering from curved surfaces

The conventional physical optics (PO) approximation used to calculate the scattered fields from a conducting body leads to incorrect scattered fields even in the specular reflection region. This is true even when all other subdominant terms are assumed to be absent. The inaccuracy stems from the fact that the surface currents used in the PO approximation suddenly truncate at the shadow boundary of curved surfaces resulting in an erroneous contribution. Expressions for these shadow boundary (end point) contributions are presented in this paper. It is shown that if these end point contributions are subtracted from the conventional PO results, one obtains a better representation of the true scattered field. For illustration, backscattered fields from various conducting bodies are computed using the corrected PO solution and are compared with the exact scattered fields.     

The truncation error in the application of sampling series toelectromagnetic problems

Truncation error interpolation methods used in the evaluation and representation of scattered fields are discussed. Three main points are considered: the choice of the sampling point lattice: the choice of the type of sampling expansion: and the determination of the number of terms of the sampling series that must be retained to insure a negligible truncation error. The aim is to provide a clear reference frame for the applications which allow the optimal choice for the reconstruction algorithm. The cases of bounded and square integrable band-limited functions are separately dealt with in order to exploit the properties of each method. A general analysis of the multidimensional truncation error is carried out. No particular assumption on the sampling expansion is made, except for that concerning the use of a rectangular sampling lattice. Simple and effective expressions relating the multidimensional error to the corresponding one-dimensional bounds are derived. The one-dimensional error is then analyzed in detail with reference to the most relevant case of central interpolation. By very simple methods, error bounds for all the sampling expansions explicitly considered in the literature are derived in a systematic way. Expressions providing the values of the sampling rate and number of retained samples which minimize the overall computer time and memory requirement are derived. Some numerical examples are presented and briefly discussed     

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.     

New Insight into the Classical Macdonald Physical Optics Approximation

The physical optics approximation is widely used in analysis of antennas and scattering problems for electromagnetic and acoustic waves. The present paper investigates the nature of this classical approximation. It is shown that the scattered field in this approximation can be separated into two parts: the reflected field, containing all reflected rays and beams, and the shadow radiation, responsible for the Fresnel diffraction and the forward scattering. This observation elucidates the physics behind the fundamental diffraction law related to the total power scattered by large reflecting objects. It also clarifies the diffraction limit for reduction of scattering by absorbing materials.     

Optimal interpolation of radiated fields over a sphere

An optimal sampling interpolation algorithm is developed that allows the accurate recovery of scattered or radiated fields over a sphere from a minimum number of samples. Using the concept of the field equivalent (spatial) bandwidth, a central interpolation scheme is developed to compute the field in &thetas;, φ coordinates, starting from its samples. The maximum allowable sample spacing and error upper bounds are also rigorously derived. Several simulated examples of pattern reconstruction are presented, for both the cases of field and power pattern interpolation. The interpolation error, as a function of the retained sample number, has been also evaluated and compared with the theoretical upper bounds. The algorithm stability versus randomly distributed errors added to the exact samples is demonstrated     

Non-redundant implementation of method of auxiliary sources for smooth 2D geometries

An effective technique for the evaluation of the field scattered by large, smooth PEC profiles, based on the method of auxiliary sources is proposed. A criterion for positioning source and test point, a typical issue of this method, is proposed, showing that the scattering problem can be solved by using a number of unknowns slightly greater than two per wavelength, essentially coincident with the number of degrees of freedom of the scattered field. 2D numerical examples are presented and discussed.     

Approximation to duration-limited functions by sampling sums

According to the classical sampling theorem, any band-limited function can be exactly represented by its sampling expansion. It is shown that for duration-limited functions f satisfying certain smoothness properties this representation holds approximately. No assumptions upon the Fourier transform of f are needed. Estimates for the approximation error are given.     

Method of Hybrid Approximations for Modelling of Multidimensional Nonlinear Systems

In this paper we propose a new approach to the constructive mathematical representation of nonlinear systems transforming stochastic signals. The approach is based on a combination of a new best approximation technique and a new iterative procedure. For each iteration, the approximation is constructed as a polynomial operator of degree r which minimizes the mean–squared error between a desired output signal and the output signal of the approximating system. We show that this hybrid technique produces a computationally efficient and flexible method for modelling of nonlinear systems. The method has two degrees of freedom, the degree r of the approximating operator and the number of iterations, to decrease the associated error.     

Exact sampling results for some classes of parametric nonbandlimited 2-D signals

We present sampling results for certain classes of two-dimensional (2-D) signals that are not bandlimited but have a parametric representation with a finite number of degrees of freedom. While there are many such parametric signals, it is often difficult to propose practical sampling schemes; therefore, we will concentrate on those classes for which we are able to give exact sampling algorithms and reconstruction formulas. We analyze in detail a set of 2-D Diracs and extend the results to more complex objects such as lines and polygons. Unlike most multidimensional sampling schemes, the methods we propose perfectly reconstruct such signals from a finite number of samples in the noiseless case. Some of the techniques we use are already encountered in the context of harmonic retrieval and error correction coding. In particular, singular value decomposition (SVD)-based methods and the annihilating filter approach are both explored as inherent parts of the developed algorithms. Potentials and limitations of the algorithms in the noisy case are also pointed out. Applications of our results can be found in astronomical signal processing, image processing, and in some classes of identification problems.     

Sampling method using prefiltered band-limited Green's functionsfor the solution of electromagnetic integral equations

In the context of far-field scattering, the spatial spectrum of fields or currents on a scatterer may be approximated as band-limited. On this premise one can greatly simplify the computation of radiation integrals by eliminating in advance those spectral components of the Green's function not expected to be present in the fields. This approach enables one to dispense with the notion of basis functions and, instead, represent fields and currents as well as the impedance matrix representing the Green's function entirely in terms of a sparse set of samples, thereby obviating the need for integration. Moreover, by taking full advantage of the band-limited character of the solution, this sampling theoretical approach yields an efficient representation with a minimal number of degrees of freedom. The method was explored for scattering from two-dimensional perfect electric conductors including ellipsoidal cylinders and flat strips, in both polarizations, using uniform as well as nonuniform sampling, and yielded surprisingly good results     

High-frequency scattering from a wedge with impedance facesilluminated by a line source. II. Surface waves

For pt.I see ibid., vol.37, no.2, p.212-18 (1989). In Part I a rigorous integral representation for the field scattered at a finite distance from the edge of an impedance wedge when it is illuminated by a line source was derived. It was shown that the total field can be expressed as the sum of the geometrical optics (GO) field, the field diffracted by the edge, and terms related to the excitation of surface waves. The double spectral integral representation for the diffracted field was asymptotically evaluated there, in the case in which no surface wave can be supported by the two faces of the wedge. In particular, the high-frequency solution was expressed in the special format of the uniform geometrical theory of diffraction (UTD). Here, field contributions related to the surface wave excitation mechanism are examined. By a convenient asymptotic approximation of the integrals, a high-frequency solution which is uniform with respect to aspects of both incidence and observation is obtained. Moreover, this solution has useful symmetry properties so that it explicitly exhibits reciprocity. Numerical results are presented to show the relevance of the surface wave terms in the evaluation of the field     

Wave field singularity aspects in large-size scatterers and inverseproblems

It is known that any scattered wave field carrying energy into infinity must have source singularity centers within a bounded space. Otherwise, the scattered field should be identically equal to zero everywhere (Kupradze 1935). In this paper, attention is paid to localization of these singularities under the assumption that every scattered wave is determined uniquely by its own singularities. Investigations have shown that these singularities are distributed as "bright centers" and the distance between them depends on frequency. To determine the position (localization) of the scattered wave field singularities, the functions describing converging and diverging waves are used. Based on these concepts and the method of auxiliary sources, an efficient numerical method to reconstruct a field up to its singularities is suggested. The localization of singularities is used for partial representation of the scattered fields, which reduces significantly the number of unknowns in describing the scattering process and leading into optimized inverse scattering problem solutions     

A data compression technique for scattered fields from complextargets

A data compression technique to efficiently compress the scattered fields from complex targets is presented. Since at high frequencies the scattered fields from complex radar targets exhibit a highly localized behavior, the data compression technique is based on extracting and then sampling the responses associated with individual scattering mechanisms. The compression technique preserves the angle as well as the frequency variation of individual scattering mechanisms and, thus, leads to small reconstruction error. The proposed technique, however, is a lossy process in the sense that a tradeoff exists between the compression ratio and the data reconstruction error. The smaller the data reconstruction error, the smaller the compression ratio. The effectiveness of the technique is demonstrated by compressing measured scattered fields of a complex target     

Evaluation of edge effects in slot arrays using the geometricaltheory of diffraction

The geometric theory of diffraction (GTD) is applied to evaluate efficiently the coupling coefficient associated with the design of a waveguide-fed longitudinal shunt slot arrays. The coupling coefficient is proportional to the reaction integral between the field of a slot and the equivalent current distribution of another slot. Using an approximate form of the Green's function for the wedge, it is shown that the edge-diffraction field due to each slot is practically equal to the field of a suitable `mirror image' of such slot. In this way the actual coupling coefficient can be decomposed into a sum of coupling coefficients between slots on an infinite ground plane. The latter can be evaluated very efficiently so that inclusion of edge effects does not slow down the design procedure. The same approach also allows the computation of the relation between the self-admittance with and without the edge. Some test cases are provided which show that the overall error of this approximation can be neglected since it is comparable with the error due to mechanical tolerances     

Implementation of Bilateral Control System Based on Acceleration Control Using FPGA for Multi-DOF Haptic Endoscopic Surgery Robot

Bilateral control systems are strongly required to apply to endoscopic surgeries. It is necessary that the system has enough degrees of freedom for applying various operation procedures including endoscopic surgery. When the degrees of freedom of the system are increased, the amount of control calculation is also increased, and it is hard to keep sampling periods short. The bilateral control systems, however, require comparatively shorter sampling periods, particularly the control system based on an acceleration control system. Hence, it is a difficult issue to increase the degrees of freedom of the bilateral control system. In this paper, the sampling period is kept short in the multi-degree of freedom system by using field programmable gate arrays as processors. The bilateral control system based on the acceleration control system is implemented in a robot system, which has 12 DOF, and some experimental results are shown, and the errors between the response of the master and slave robots are discussed.      

Scattering by an arbitrary cylinder at a plane interface: broadsideincidence

The problem of the determination of the fields scattered by an infinite dielectric cylinder of arbitrary cross section located at the interface between two semi-finite dielectric media is reduced to the solution of integral equations for unknown functions defined on the boundaries. These boundary functions are chosen so as to minimize their number. The incident field is that of a plane monochromatic wave. The derivation of the integral equations is given for the transverse electric (TE) mode for a dielectric cylinder and for a perfectly conducting cylinder. The exact electromagnetic fields are obtained from the solutions of the integral equations by integration, and the radar cross section can be computed from the far-field approximation. Sample outputs of the computer programs that implement this solution are shown     

One Dimensional Inverse Dielectric Profiling of Embedded Slabs by a Quadratic Approximation

In this paper the inverse problem of the determination of the dielectric profile of a slab embedded into a homogeneous half space starting from the knowledge of the scattered field for different illumination angles is considered. A comparison between the features of a linear and a quadratic distorted approximation of the scattering problem is carried out, in order to determine the kind of profiles that can be reconstructed. In particular, the investigation of the direct problem reveals that, under the linear approximation, only a limited number of Fourier harmonics of the profile give contribution to the scattered field. The use of the quadratic approximation allows to change the number and order of the harmonic that can be imaged. The dependence of the results on the dielectric permittivity of the host medium is also highlighted. Numerical examples support the theoretical considerations.     

A general technique to correct probe position errors in planarnear-field measurements to arbitrary accuracy

A general theoretical procedure is presented to remove known probe-position errors when planar near-field data are transformed to the far field. The measured data are represented as a Taylor series whose terms contain the error function and the ideal spectrum of the antenna. This representation is then assumed to be an actual near-field existing on an error-free regularly spaced two-dimensional scan plane. By inverting the Taylor series, the ideal spectrum in terms of the measured data and the position errors is obtained. The solution is given by an infinite series of an error operator acting on data containing errors of measurement. This error operator is the Taylor series without the zeroth-order term. The nth order approximation to the ideal near-field of the antenna can be explicitly constructed by inspection of the structure of the error operator. Convergence of the approximation is examined. Computer simulations are used to demonstrate the excellent results obtained with this technique, as well as to demonstrate the convergence of the error-corrected fields to the true solution     

The radar cross section of dielectric disks

A solution is presented for the backscatter (monostatic) radar cross section of dielectric disks of arbitrary shape, thickness, and dielectric constant. The result is obtained by employing a Kirchhoff-type approximation to obtain the fields inside the disk. The internal fields induce polarization and conduction currents from which the scattered fields and the radar cross section can be computed. The solution for the radar cross section obtained in this manner will be shown to agree with known results in the special cases of normal incidence, thin disks, and perfect conductivity. It will also be shown that the solution can be written as a product of the reflection coefficient of an identically oriented slab times the physical optics solution for the backscatter cross section of a perfectly conducting disk of the same shape. This result follows directly from the Kirchhoff-type approximation without additional assumptions.