Impedance Matrix Compression Research Articles

Transient Analysis of EM Pulse Penetration Into a Conducting Layer Using Wavelet-Based Implicit TDIE and Iterative IMC Technique

The penetration of a wide-band electromagnetic pulse into a conducting layer is formulated in terms of an implicit time-domain integral equation and cast into matrix form via the method of moments. Though such a wide-band problem indeed calls for a time-domain solution, one may argue that the frequency-dependent attenuation of the field penetrating the conductor could suggest a frequency-domain approach that would greatly reduce the number of unknowns. The proposed via media is to use spatio-temporal wavelet functions instead of the standard pulse basis. Owing to their multiresolution property, both in the spatial and temporal domains, these functions can span the field propagating inwardly with substantially fewer terms. The reduction in the number of basis functions used is effected by the impedance matrix compression technique, which automatically omits the basis functions whose coefficients would be insignificant due to the attenuation. Reducing the number of basis functions renders the matrix equation much smaller and the overall solution far more efficient.

Wavelet-based analysis of transient electromagnetic wave propagation in photonic crystals.

Photonic crystals and optical bandgap structures, which facilitate high-precision control of electromagnetic-field propagation, are gaining ever-increasing attention in both scientific and commercial applications. One common photonic device is the distributed Bragg reflector (DBR), which exhibits high reflectivity at certain frequencies. Analysis of the transient interaction of an electromagnetic pulse with such a device can be formulated in terms of the time-domain volume integral equation and, in turn, solved numerically with the method of moments. Owing to the frequency-dependent reflectivity of such devices, the extent of field penetration into deep layers of the device will be different depending on the frequency content of the impinging pulse. We show how this phenomenon can be exploited to reduce the number of basis functions needed for the solution. To this end, we use spatiotemporal wavelet basis functions, which possess the multiresolution property in both spatial and temporal domains. To select the dominant functions in the solution, we use an iterative impedance matrix compression (IMC) procedure, which gradually constructs and solves a compressed version of the matrix equation until the desired degree of accuracy has been achieved. Results show that when the electromagnetic pulse is reflected, the transient IMC omits basis functions defined over the last layers of the DBR, as anticipated.

Improved impedance matrix compression (IMC) technique for efficient wavelet‐based method of moments solution of scattering problems

AbstractThe iterative impedance matrix compression (IMC) method iteratively constructs and solves a reduced version of the method of moments (MoM) impedance matrix based on analysis of the error in fulfilling the original matrix equation. Hence, it is possible that some of the selected basis functions, while dominant in the analyzed error, will have very little weight in the solution itself. Here, we propose a backward elimination step, which is aimed at discarding these superfluous basis functions. To demonstrate this idea, we use the scattering problem of a TMz plane wave by a conducting cylinder of square cross‐section. The resultant compression ratio is in good agreement with the theoretical limit of the compression ratio obtained a posteriori, based on the exact MoM solution. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 40: 275–280, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11351

Extension of impedance matrix compression method with wavelet transform for 2‐D conducting and dielectric scattering objects due to oblique plane‐wave incidence

AbstractThe impedance matrix compression (IMC) technique is applied to analyze the square method–of‐moments (MoM) matrix arising from the surface integral equation for 2‐D conducting and dielectric objects with oblique plane‐wave incidence. The induced current components are processed separately by using wavelet basis functions. The comparison between cylinders with circular and square cross section is presented to show the effectiveness of IMC, which depends on the geometry of the object and the wavelet transform matrix. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 34: 53–56, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10371

An adaptive multiresolution approach to the simulation of planar structures

An efficient approach for the full-wave analysis of printed structures is presented. It is based on the use of vector multiresolution (MR) functions in conjunction with the impedance matrix compression (IMC) technique, which leads to a reduced set of iteratively selected basis functions. The multilevel structure of the functions makes the matrix compression possible and also allows its further sparsification, with the subsequent reduction of the computational time and the matrix memory occupancy. Numerical results confirm the efficiency of the technique.

Impedance matrix compression using effective quadrature filter

An effective quadrature mirror filter (QMF) proposed by Vaidyanathan has been used to solve 2D scattering problems. QMF has been popular for some time in digital signal processing, under the names of multirate sampling, wavelets, etc. In this work, the impulse response coefficients of QMF were used to construct the wavelet transform matrix. Using the matrix to transform the impedance matrices of 2D scatterers produces highly sparse moment matrices that can be solved efficiently. Such a presentation provides better sparsity than the celebrated and widely used Daubechies wavelets. These QMF coefficients are dependent on the filter parameters such as transition bandwidth and filter length. It was found that the sharper the transition bandwidth, the greater the reduction in nonzero elements of the impedance matrix. It also can be applied in the wavelet packet algorithm to further sparsify the impedance matrix. Numerical examples are given to demonstrate the effectiveness and validity of our finding.

Analysis of scattering by surfaces using a wavelet-transformed triangular-patch model

When analyzing electromagnetic scattering by three-dimensional perfectly conducting bodies of arbitrary shape, the surface modeling is often affected by triangulation, and in turn, triangular-patch basis functions are used for expanding the unknown surface current. In this paper, we apply a wavelet transformation to transform the triangular-patch basis functions to a new set of basis functions, which can be interpreted as wavelet combinations of the original basis functions. The new basis functions can lead to a matrix representation of the operator equation that is more localized and which, by proper thresholding, can be rendered sparse. Alternatively, they can lead, via an impedance matrix compression (IMC) approach, to a matrix equation that is compressed (rank reduced). The solution of either the sparse or the compressed matrix equation can yield fairly accurate results with less computational effort. The dependence of the results on ordering of the original triangular-patch basis functions prior to the transformation is discussed. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 21: 359–365, 1999.

Analysis of truncated periodic array using two-stage wavelet-packet transformations for impedance matrix compression

A novel method of moments procedure is applied to the problem of scattering by metallic truncated periodic arrays. In such problems, the induced current shows localized behavior within the unit cell and at the same time exhibits cell-to-cell periodicity. In order to select a set of expansion functions that may account for such behavior, a two-stage basis transformation, of which the first stage is an ordinary wavelet transformation performed independently on each unit-cell, has been applied to a pulse basis. The resultant basis functions at the first stage are regrouped and retransformed to reveal the periodicity of their coefficients. Expansion functions are then iteratively selected from this newly constructed basis to form a compressed impedance matrix. The compression ratios obtained in this manner are higher than the compression ratio achieved using a basis constructed via an ordinary single-stage wavelet transformation, where compression is the ratio between the number of nonzero elements in the matrix used to solve the problem and the number of elements in the original matrix. An even higher compression is attained by considering, in addition, functions that reveal array-end related features and iteratively selecting the expansion from an overcomplete dictionary.

Wavelets in electromagnetics: the impedance matrix compression (IMC) method

The use of wavelet expansions in numerical solutions of electromagnetic frequency-domain integral equation formulations is steadily growing. In this paper we review the recently suggested impedance matrix compression (IMC) method for a more effective integration of wavelet-based transforms into existing numerical solvers. The difference between the IMC method and the previous approaches to applying wavelets in computational electromagnetics is twofold. Firstly, the transformation is effected by means of a digital filtering approach. This approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Secondly, the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet-unknown) current are retained and hence a substantially smaller number of coefficients has to be determined. A few numerical results are included to demonstrate the advantages of the presented method over the currently used ones. The feasibility of ensuring a slow growth in the number of unknowns even when there is a rapid increase in the problem complexity is shown by an illustrative example. © 1998 John Wiley & Sons, Ltd.

Wavelets in electromagnetics: the impedance matrix compression (IMC) method

The use of wavelet expansions in numerical solutions of electromagnetic frequency-domain integral equation formulations is steadily growing. In this paper we review the recently suggested impedance matrix compression (IMC) method for a more effective integration of wavelet-based transforms into existing numerical solvers. The difference between the IMC method and the previous approaches to applying wavelets in computational electromagnetics is twofold. Firstly, the transformation is effected by means of a digital filtering approach. This approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Secondly, the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet-unknown) current are retained and hence a substantially smaller number of coefficients has to be determined. A few numerical results are included to demonstrate the advantages of the presented method over the currently used ones. The feasibility of ensuring a slow growth in the number of unknowns even when there is a rapid increase in the problem complexity is shown by an illustrative example. © 1998 John Wiley & Sons, Ltd.

Impedance matrix compression (IMC) using iteratively selected wavelet basis

We present a novel approach for the incorporation of wavelets into the solution of frequency-domain integral equations arising in scattering problems. In this approach, we utilize the fact that when the basis functions used are wavelet-type functions, only a few terms in a series expansion are needed to represent the unknown quantity. To determine these dominant expansion functions, an iterative procedure is devised. The new approach combined with the iterative procedure yields a new algorithm that has many advantages over the presently used methods for incorporating wavelets. Numerical results which illustrate the approach are presented for three scattering problems.

Iterative Selection of Expansion Functions From an Overcomplete Dictionary of Wavelet Packets for Impedance Matrix Compression

The paper further develops the recently introduced idea of compressing the impedance matrix by an iterative selection of expansion functions. The improved algorithm uses an overcomplete dictionary comprising bases whose inherent properties match some features of the given scattering problem. Thus, the overcomplete dictionary offers a variety of expansion functions from which a solution-oriented spanning set for the unknown current is extracted. The number of iterations in the proposed algorithm is suppressed by selecting more than one expansion function at each iteration. This latter section is facilitated by a matching pursuit process. It is shown that the compression ratio of the resultant compressed impedance matrix is superior to the one achieved by the ordinary iterative matrix compression algorithm.

Impedance matrix compression with the use of wavelet expansions

Impedance matrix compression with the use of wavelet expansions

Impedance matrix compression (IMC) using iteratively selected wavelet basis for MFIE formulations

In this article we present a novel approach to incorporating wavelet expansions in method-of-moments (MoM) solutions for scattering problems described by a magnetic field integral equation (MFIE) formulation. In this approach, we utilize the fact that when the basis functions used are wavelet-type functions, only a few terms in a series expansion would be needed to represent the unknown quantity. An iterative procedure is suggested to determine these dominant expansion functions. The new approach combined with the iterative procedure yields a new algorithm that has many advantages over the presently used methods for incorporating wavelets. Numerical results that illustrate the approach are presented. © 1996 John Wiley & Sons, Inc.

Impedance matrix compression using adaptively constructed basis functions

Wavelet expansions have been employed recently in numerical solutions of commonly used frequency-domain integral equations. In this paper, we propose a novel method for integrating wavelet-based transforms into existing numerical solvers. The newly proposed method differs from the presently used ones in two ways. First, the transformation is affected by means of a digital filtering approach. This approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Second, the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet unknown) current are retained and subsequently derived. Numerical results for a few TM scattering problems are included to demonstrate the advantages of the proposed method over the presently used ones.