Comments on "A fast converging series expansion for the 2-D periodic Green's function based on perfectly matched layers"

For original paper, see Rogier and De Zutter (IEEE Trans. Microwave Theory Tech., vol.52, p.1199-206, Apr. 2004).     

Authors' reply [to comments on 'A fast converging series expansion for the 2-D periodic Green's function based on perfectly matched layers']

The present authors reply to the paper by Chew (IEEE Trans. Microwave Theory Tech., vol.52, no.10, p.2450-1, Oct. 2004) which commented on the original paper by Rogier and De Zutter (IEEE Trans. Microwave Theory Tech., vol.52, p.1199-206, Apr. 2004).     

A fast multipole method for layered media based on the application of perfectly matched layers - the 2-D case

An efficient fast multipole method (FMM) formalism to model scattering from two-dimensional (2-D) microstrip structures is presented. The technique relies on a mixed potential integral equation (MPIE) formulation and a series expression for the Green functions, based on the use of perfectly matched layers (PML). In this way, a new FMM algorithm is developed to evaluate matrix-vector multiplications arising in the iterative solution of the scattering problem. Novel iteration schemes have been implemented and a computational complexity of order O(N) is achieved. The theory is validated by means of several illustrative, numerical examples. This paper aims at elucidating the PML-FMM-MPIE concept and can be seen as a first step toward a PML based multilevel fast multipole algorithm (MLFMA) for 3-D microstrip structures embedded in layered media.     

Efficient computation of the 2-D Green's function for 1-D periodic structures using the Ewald method

The Ewald method is applied to accelerate the evaluation of the Green's function of an infinite periodic phased array of line sources. The Ewald representation for a cylindrical wave is obtained from the known representation for the spherical wave, and a systematic general procedure is applied to extend previous results. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. Singularities and convergence rates are analyzed, and a recipe for selecting the Ewald splitting parameter /spl epsiv/ is given to handle both low and high frequency ranges. Indeed, it is shown analytically that the choice of the standard optimal splitting parameter /spl epsiv//sub 0/ will cause overflow errors at high frequencies. Numerical examples illustrate the results and the sensitivity of the Ewald representation to the splitting parameter /spl epsiv/.     

Efficient computation of the 2-D Green's function for 1-D periodic line sources in planar multilayered media

Periodic Green's function (PGF) of an infinite 1-D periodic array of line sources in planar multilayered media is calculated by the use of the of the 3-level discrete complex image method (DCIM) together with the surface wave pole (SWP) extraction. The Ewald's method is then applied to accelerate the evaluation of the PGF. The recipe to choose the Ewald's splitting parameter is also given. It is shown that the Ewald's transformation needs a few terms to approximate the PGF with high accuracy. The SWP contributions, which are extracted before the DCIM application, are added to this approximation. The spatial domain surface wave pole contributions are observed to be highly oscillatory and converge to zero. The numerical results are provided to demonstrate the efficiency and the accuracy of the procedure followed.     

A perfectly matched layer for the 3-D wave equation in the timedomain

In this paper, a three-dimensional (3-D) PML for the 3-D scalar wave equation is proposed for applications in practical finite difference time-domain schemes such as the time-domain wave-potential (TDWP) technique and the time-domain scalar wave equation approaches to the analysis of optical waveguides. The theoretical formulation is based on the stretched coordinates approach. It is shown that this PML is suitable for the termination of open problems as well as for port terminations in high-frequency circuit problems. New PML conductivity profile is proposed, which offers lower reflections in a wider frequency band in comparison with the commonly used profiles     

A new representation for the Green's function of multilayer media based on plane wave expansion

In this paper, a new representation for the space domain Green's function of general multilayer media is presented. This approach is based on an efficient plane wave expansion of the source incident field. Using the transmission line model for the multilayer medium, the effect of the layered medium on the incident plane waves is determined by a transmission coefficient, and the amplitude and phase of each plane wave at the field point are obtained. The total field is evaluated by summing the resultant plane waves at the field point. The main advantage of the proposed method is that the spatial domain Green's function of multilayer media can be obtained easily as a summation of simple exponential functions, without the need for Sommerfeld integration or complex image approximation. The plane wave approximation is independent of the parameters of the medium and is valid over a wide frequency range. Furthermore, the exponential form of the plane wave solution makes it possible to compute the method of moments matrix elements analytically for most important types of basis functions in multilayer problems.     

An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures

An efficient multilevel fast multipole algorithm (MLFMA) formalism to model radiation and scattering by/from large planar microwave structures is presented. The technique relies on an electric field integral equation (EFIE) formulation and a series expansion for the Green dyadic, based on the use of perfectly matched layers (PML). In this way, a new PML-MLFMA is developed to efficiently evaluate matrix-vector multiplications arising in the iterative solution of the scattering problem. The computational complexity of the new algorithm scales down to O(N) for electrically large structures. The theory is validated by means of several illustrative, numerical examples.     

Convergence behavior and acceleration of the Berenger and leakymodes series composing the 2-D Green's function for the microstripsubstrate

The Green's functions G_A and G_V are calculated for a two-dimensional microstrip substrate by placing the substrate into a closed perfectly-matched-layer waveguide and by performing a modal expansion in Berenger and leaky modes. It is shown that each series composing G_A and G_V has a particular convergence behavior when considering small lateral distances between the excitation and the observation points. It is then demonstrated that, by applying the Shanks transform to accelerate each series separately, a more efficient calculation for the Green's functions can be obtained than by direct computation of the series. The theory is illustrated by means of a representative example     

Efficient calculation of the Green's function for multilayeredplanar periodic structures

An efficient calculation scheme is presented for the periodic Green's function in planar multilayered structures. The proposed method is based on the static complex image conversion, and the Ewald sum technique and it converts the slowly convergent Green's function into the sum of two rapidly convergent series     

基于特征基函数法的一维理想导体粗糙海面电磁散射快速算法研究

针对传统矩量法在处理具有较多未知量的理想导体粗糙海面电磁散射问题时对计算机内存的需求过大, 耗时过长的缺陷, 文中引入了特征基函数法, 并根据Foldy-Lax多径散射方程构造特征基函数, 首先只考虑离散子域本身的自相互作用构造主要特征基函数, 然后考虑各离散子域间的互耦效应构造次要特征基函数, 最后由主要特征基函数和次要特征基函数的加权叠加构造特征基函数.通过与传统矩量法仿真结果的对比, 讨论了不同次要特征基函数的阶数或不同离散子域的个数对计算精度和计算效率的影响.仿真结果表明了本文所采用的算法能够在保证计算精度的前提下, 减少计算时间, 并能够通过离散子域尺寸的选取控制实际操作矩阵的维数.     

A note on the Cramer-Rao bound for 2-D direction finding based on2-D array

The authors present explicit expressions for the Cramer-Rao bound (CRB) for estimating the two-dimensional (2-D) direction of a single source based on 2-D arrays of identical omnidirectional sensors. Two commonly used models, random wave and unknown wave, are compared. It is shown that the CRBs for the two models have the same dependency on the array structure. A specialization of the CRB to two orthogonal uniform linear arrays (ULAs) is discussed. It is found that the joint CRBs of the direction angles based on the two orthogonal ULAs can be as low as one quarter (for a random waveform model with a large number of snapshots and low SNR) or one half (for both models with high SNR) of the CRBs based on each ULA     

基于差分扩张和平移的2D矢量地图的可逆水印

利用差分平移和差分扩张,提出一种新的应用于数字矢量地图的可逆水印算法。该算法通过调整地图中的顶点坐标进行水印嵌入。与差分扩张方法不同,该算法依次计算顶点坐标序列中任意2个相邻坐标之间差分,将差分大于K-1的向右平移,差分小于-K的向左平移,差分介于-K、K-1间的进行差分扩张并嵌入水印(K是由用户所确定的正整数),同时通过适当调整顶点的坐标使得嵌入水印后差分的改动均匀分布在2个坐标上,保证了较低的数据扰动。与Wang的算法相比,该算法在保持较低变形的基础上能够嵌入更多的水印信息。     

基于Split-Field FDTD法的近埋地无限长散射体二维算法

提出了一种基于周期结构split-field FDTD法的近埋地无限长散射体二维算法.该方法根据散射体轴向均匀性将三维split-field FDTD法转化为二维算法,减少了内存和计算量,可分析斜入射脉冲波照射下近地、埋地无限长散射体散射问题.为了进一步减小计算量,连接边界上的入射波(地上为原始入射波和反射波的叠加,地下为透射波)采用一维FDTD法引入.吸收边界采用了UPML匹配层,导出了适用于split-field FDTD算法与有耗介质匹配的UPML方程.通过数值算例,验证了此二维算法应用于近埋地无限长散射体问题的有效性和实用性.     

A uniform asymptotic expansion for the Green's functions used in microstrip calculations

A uniform asymptotic approximation is developed in the limit of small substrate thickness for the Green's functions used in microstrip-type problems. The author develops an approximation which is uniformly valid for all distances from the source in the limit where the substrate region is thin, which is typically the case of interest for microstrip problems. The approximation is valid for a single-layer substrate. The expansions agree with previously published near and far-field results. The approximation is compared with numerical evaluations of the exact integral solution available for the problem.<>     

Two fast architectures for the direct 2-D discrete wavelettransform

We propose two architectures for the direct two-dimensional (2-D) discrete wavelet transform (DWT). The first one is based on a modified recursive pyramid algorithm (MRPA) and performs a “"nonstandard” decomposition (i.e., Mallat's (1989) tree) of an N×N image in approximately 2N²/3 clock cycles (ccs). This result consistently speeds up other known architectures that commonly need approximately N² ccs. Furthermore, the proposed architecture is simpler than others in terms of hardware complexity. Subsequently, we show how “symmetric”/“anti-symmetric” properties of linear-phase wavelet filter bases can be exploited in order to further reduce the VLSI area. This is used to design a second architecture that provides one processing unit for each level of decomposition (pipelined approach) and performs a decomposition in approximately N²/2 ccs. In many practical cases, even this architecture is simpler than general MRPA-based devices (having only one processing unit)     

A simple 2-D/3-D method for fast analysis of patch resonators withhigh accuracy

For a patch resonator, a two-dimensional (2-D) planar circuit analysis (modal or contour integral) has advantages of being simple and fast. However, this method does not account for perturbation of the fringe fields at the edge. A three-dimensional (3-D) analysis such as integral equation moment method has the advantage of accounting for the fringe fields at the edge, however, the disadvantage of having to long computation time and high truncation error. A better way is to make use of advantages of both and discard the disadvantages. The result is the combined 2-D/3-D method described in this paper. This method requires a few seconds of computer time, but gives errors of resonance frequency only around 0.5%. The errors in general are within the tolerances of presently available analysis and experiments. Field theory reasons accounting for such fast convergent and low error results, are explained and numerical examples are given     

A fast computation method for MUSIC spectrum function based on circular arrays

The large computation amount of multiple signal classification (MUSIC) spectrum function seriously affects the timeliness of direction finding system using MUSIC algorithm, especially in the two-dimensional directions of arrival (DOA) estimation of azimuth and elevation with a large antenna array. This paper proposes a fast computation method for MUSIC spectrum. It is suitable for any circular array. First, the circular array is transformed into a virtual uniform circular array, in the process of calculating MUSIC spectrum, for the cyclic characteristics of steering vector, the inner product in the calculation of spatial spectrum is realised by cyclic convolution. The computational amount of MUSIC spectrum is obviously less than that of the conventional method. It is a very practical way for MUSIC spectrum computation in circular arrays.     

A new fast method for the reconstruction of 2-D microwave images ofrotating objects

The problem of microwave imaging of rotating objects is approached from a digital signal processing point of view. Attention is focused on the reconstruction of the two-dimensional (2-D) image of an object obtained by processing complex samples of the continuous wave (CW) signal return received for different object aspect angles covering a 2pi total variation. The effects of a polar sampling of the Fourier transform (FT) of the target reflectivity function on the impulse response of the imaging system are analyzed. A criterion for the choice of the angular sampling interval in order to control the image blurring is given. Finally, a new and fast digital signal processing method for image reconstruction is proposed. This method reconstructs high resolution images of rotating targets illuminated by a CW electromagnetic signal or by a large bandwidth transmitted signal. This algorithm is suitable for processing wideband received signal of targets observed for a large variation of the viewing angle, so it is very attractive for the reconstruction of very high spatial resolution target images in SAR/ISAR systems. To show the performance of the method we finally present and discuss the results obtained by applying the proposed technique on simulated and experimental data.