Accurate modeling of lossy nonuniform transmission lines by using differential quadrature methods

This paper discusses an efficient numerical approximation technique, called the differential quadrature method (DQM), which has been adapted to model lossy uniform and nonuniform transmission lines. The DQM can quickly compute the derivative of a function at any point within its bounded domain by estimating a weighted linear sum of values of the function at a small set of points belonging to the domain. Using the DQM, the frequency-domain Telegrapher's partial differential equations for transmission lines can be discretized into a set of easily solvable algebraic equations. DQM reduces interconnects into multiport models whose port voltages and currents are related by rational formulas in the frequency domain. Although the rationalization process in DQM is comparable with the Pade approximation of asymptotic waveform evaluation (AWE) applied to transmission lines, the derivation mechanisms in these two disparate methods are significantly different. Unlike AWE, which employs a complex moment-matching process to obtain rational approximation, the DQM requires no approximation of transcendental functions, thereby avoiding the process of moment generation and moment matching. Due to global sampling of points in the DQM approximation, it requires far fewer grid points in order to build accurate discrete models than other numerical methods do. The DQM-based time-domain model can be readily integrated in a circuit simulator like SPICE.     

An efficient surface-impedance boundary condition for thin wires of finite conductivity

To include dispersive loss into a sub-cell thin-wire model, a new implementation of a surface-impedance boundary condition (SIBC) is proposed. The surface-impedance function is approximated in the frequency domain by a series of first-order rational functions allowing straightforward transform into time domain. The contribution of the wire radius is included in the surface-impedance function extending the model to very thin or poorly conducting metal wires. Further, the SIBC includes the direct-current (dc) resistance. The approximation for the SIBC is chosen such that the dependence on the wire radius and conductivity can be removed prior to the computation of the approximation coefficients. Consequently, the wire radius and conductivity can be varied without re-computing the coefficients. The proposed model is compared with an existing SIBC wire model based on the high-frequency approximation and Prony's method, NEC-2 generated reference data, and analytical results. The results indicate enhanced accuracy at a reduced computational cost as compared with an existing model.     

同轴接头时域反射特性的频域测量方法研究

首先介绍了只用有频域测量功能的矢量网络分析仪来判断半刚性同轴电缆接头的时域反射特性的基本原理和基本计算公式。为了方便在实际的半刚性同轴电缆接头的质量检测工作中的应用,提出一个前述计算方法的近似模型,由这两种计算方法比较,近似模型的误差在10~(-3)量级内,可见即使在较严格的质检中,使用该近似模型也是有效的。     

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.     

Equivalent-circuit interconnect modeling based on the fifth-order differential quadrature methods

This paper introduces an efficient and passive discrete modeling technique for estimating signal propagation delays through on-chip long interconnects that are represented as distributed RLC transmission lines. The proposed delay model is based on a less frequently used numerical approximation technique, called the differential quadrature method (DQM). The DQM can compute the partial derivative of a function at any arbitrary point located within a prespecified closed domain of the function by quickly estimating the weighted linear sum of values of the function at a relatively small set of well-chosen grid points within the domain. By using the fifth-order DQM, a new approximation framework is constructed in this paper for discretizing the distributed RLC interconnect and thereafter modeling its delay. Due to high efficiency of DQM approximation, the proposed framework requires only few grid points to achieve good accuracy. The presented equivalent-circuit model appears like the ones derived by the finite difference (FD) method. However, it has higher accuracy and less internal nodes than generated by the FD-based modeling. The fifth-order DQM modeling technique is shown to preserve passivity. It has linear forms that are compatible with the passive order-reduction algorithm for linear network. Numerical experiments show that the proposed modeling approach leads to high accuracy as well as high efficiency.     

理想导体目标重建的Fourier衍射定理

本文给出推广的物理光学近似下理想导体目标的Ｆｏｕｒｉｅｒ衍射定理，其形式与介质目标的相应关系式十分相似，利用这一关系式可以由测量直线上散射场分布重建二维理想导体目标Ｆｏｕｒｉｅｒ域的空间谱值。通过简单例子讨论了这一关系式的适用性。     

修正Gram-Schmidt正交化构建电源分配网络时域宏模型

有理函数逼近法方程组病态条件数为构建电源分配网络时域宏模型带来了数值问题。该文提出利用修正Gram-Schmidt正交化处理对超定方程组系数矩阵进行正交三角分解,结合Bernoulli多项式求根方法及稳定、无源性准则可以精确构建无源时域宏模型。计算实例精确构建了自DC至1GHz频带内的宽带无源时域宏模型,从而验证了该方法的有效性。     

Entwurf nichtrekursiver Digitalfilter mit beliebigen Betrags- und Phasengängen

This paper presents an efficient multiple exchange algorithm for the frequency domain design of digital FIR filters with arbitrary magnitude and phase responses. The designed filters minimize theL ₂ norm of the weighted complex frequency domain error function subject to constraints on the resulting magnitude and phase errors. This general design criterion allows for an arbitrary trade-off between complexL ₂ approximation and Chebyshev approximation of given magnitude and phase responses. The optimization can be carried out by iteratively solving small quadratic programs. These linearly constrained subproblems can be solved using robust standard software.     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

透层天线电流积分方程的数值解

在已有的三层有耗媒质中的透层天线（ＢＰＡ）电流积分方程的基础上,给出了电流积分方程中的索末菲积分的低频近似解析公式并给出了其物理解释。利用三项式全域电流基函数,求得了透层天线的输入阻抗。采用静态场近似方法,很方便的求出了透层天线在地面的电场分布。     

Two-level wave functions of electrons in double-barrier quantum-size structures in an electric field with finite amplitude

The applicability of various two-level approximations to calculation of electron transitions in a double-barrier structure with current pumping in a resonance electric field with finite amplitude was analyzed. It is shown that the solution obtained by applying a simple iterative procedure to the perturbation method of first order in the field amplitude can be appreciably extended beyond the convergence domain of the iteration method. On the other hand, it is demonstrated that the two-level approximation itself becomes inapplicable for field amplitudes much smaller than those generally assumed. This limitation is due to the influence of satellites represented by nonresonance components of the wave function. A model of double-barrier structures with electron pumping is suggested; this model is valid in the entire domain of applicability of the two-level approximation and makes it possible to take into account arbitrary forms of barriers and perturbations.     

架空多导体传输线的时域阻抗分析

利用误差函数的拉普拉斯变换，对复深度法得出的频域阻抗直接进行拉普拉斯变换，获得了考虑土壤影响的架空多导体传输线时域自阻抗和互阻抗简单的表达式，通过Timotin公式和Pade变换的结果进行比较，验证了公式的正确性。     

The Sinc and Cosinc Transform

A new transform, the sinc and cosinc transform, uses Walsh functions to obtain the Fourier transform. This technique converts a staircase approximation of a function to a set of sinc and cosinc terms in the frequency domain that is equivalent to the Fourier transform. The calculation is slower than the fast Fourier transform (FFT) but is devoid of aliasing. The interpolation and scaling in the frequency domain are built in, and any frequency point may be chosen without changing the number or spacing of the samples in the time domain. The intervening set of coefficients is computed more rapidly than those obtained using the fast Hadamard transform.     

An approximation to alpha of a junction transistor

In this paper, a new approximation for the frequency dependence of the short-circuit current gain of a theoretical junction triode is derived, which is a rational function of frequency and convenient to use. It is shown that the approximation is in excellent agreement with the frequency response of the theoretical expression for alpha in both magnitude and phase to above the alpha cutoff frequency. The approximation is also considered in the time domain, where it is in good agreement except for small values of time corresponding to frequencies well above alpha cutoff.     

Transient response of a layer of discrete random media over adielectric half space

A solution is presented in the time domain for the scattering of a short pulse incident on a layer of discrete random media. The layer consists of identical, but randomly oriented particles situated above a homogeneous (i.e. deterministic) half-space. A solution is found for the backscattered pulse when the incident pulse is obliquely incident on the layer. The solution is obtained in the case of low particle density using the distorted Born approximation. The short pulse solution is an approximation to the impulse response (Green's function) for the layer and as such can be used to compute the response to pulses of arbitrary shape     

An efficient implementation of surface impedance boundaryconditions for the finite-difference time-domain method

An efficient way to implement the surface impedance boundary conditions (SIBC) for the finite-difference time-domain (FDTD) method is presented in this paper. Surface impedance boundary conditions are first formulated for a lossy dielectric half-space in the frequency domain. The impedance function of a lossy medium is approximated with a series of first-order rational functions. Then, the resulting time-domain convolution integrals are computed using recursive formulas which are obtained by assuming that the fields are piecewise linear in time. Thus, the recursive formulas derived here are second-order accurate. Unlike a previously published method [7] which requires preprocessing to compute the exponential approximation prior to the FDTD simulation, the preprocessing time is eliminated by performing a rational approximation on the normalized frequency-domain impedance. This approximation is independent of material properties, and the results are tabulated for reference. The implementation of the SIBC for a PEC-backed lossy dielectric shell is also introduced     

Wavelets and differential-dilation equations

It is shown how differential-dilation equations can be constructed using iterations, similar to the iterations with which wavelets and dilation equations are constructed. A continuous-time wavelet is constructed starting from a differential-dilation equation. It has compact support and excellent time domain and frequency domain localization properties. The wavelet is infinitely differentiable and therefore cannot be obtained using digital filter banks. In addition, the wavelet has excellent approximation properties. New sampling and differentiation techniques are also introduced. Results on image interpolation using the solution of the differential-dilation equation are presented. Examples are given, demonstrating the suitability of the new wavelet function for signal analysis     

DCT和DWT域音频水印幅值变化规律分析

 嵌入水印对音频信号波形的影响受到的关注不多.分析得出改变采样序列离散余弦变换(Discrete Cosine Transform,DCT)域某个交流系数时,逆DCT后幅值变化服从余弦曲线,改变离散小波变换(Discrete Wavelet Transform,DWT)域某个近似系数时,重构后采样序列幅值变化呈现为所选小波的尺度函数.并对在DWT-DCT混合域嵌入水印做了讨论.通过推导和实验,得到采样点的改变量与DCT域低频系数的改变量、DWT域近似系数的改变量及混合域低频系数的改变量成线性关系.所得结论有助于发现嵌入算法带给采样点序列的不连续变化而使含水印音频产生可听噪声,也有助于提高水印的嵌入效率.     

Chebyshev approximation problem of multidimensional IIR digital filters in the frequency domain

This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximating functions. Chebyshev approximation theory and the notion of line homotopy are used to reveal the approximation properties of this set of IIR functions. A sign condition is derived to characterize a convex stable domain in this set. This sign condition can be incorporated into the optimization of the Chebyshev simultaneous approximation. The generally sufficient global Kolmogorov criterion is shown to be a necessary condition, for the characterization of best approximation, in the considered set of approximating functions. Thus, it can be incorporated, as a stopping constraint, in the design of the optimal filter. Moreover, the local Kolmogorov criterion is shown to be also necessary for the set of approximating functions. Finally, it is proved that the best approximation is a global minimum.

     

Direct design of transitional Butterworth-Chebyshev recursive digital filters

A simple method for the approximation of an all-pole transfer function directly in the digital domain for the design of recursive digital filters is described. The transfer function of these filters, referred to as transitional Butterworth-Chebyshev filters, has properties that lie between Butterworth and Chebyshev functions.<>