Adaptive transient solution of nonuniform multiconductor transmission lines using wavelets

This paper presents a highly adaptive algorithm for the transient simulation of nonuniform interconnects loaded with arbitrary nonlinear and dynamic terminations. The discretization of the governing equations is obtained through a weak formulation using biorthogonal wavelet bases as trial and test functions. It is shown how the multiresolution properties of wavelets lead to very sparse approximations of the voltages and currents in typical transient analyzes. A simple yet effective time-space adaptive algorithm capable of selecting the minimal number of unknowns at each time iteration is described. Numerical results show the high degree of adaptivity of the proposed scheme.     

Hybridization of FDTD and device behavioral-modeling techniques [interconnected digital I/O ports]

We present a systematic methodology for the electromagnetic modeling of interconnected digital I/O ports. Digital drivers and receivers are represented through behavioral models based on radial basis functions expansions. Such a technique allows a very accurate representation of nonlinear/dynamic effects as well as switching behavior of real-world components by means of carefully identified discrete-time models. The inclusion of these models into a finite-difference time-domain solver for full-wave analysis of interconnected systems is presented. A rigorous stability analysis shows that use of nonlinear/dynamic discrete-time models can be easily integrated with standard full-wave solvers, even in the case of unmatched sampling time. A set of numerical examples illustrates the feasibility of this method.     

Wavelet analysis of a microbarograph network

This paper presents a wavelet-based algorithm for the detection, identification, and extraction of gravity waves from atmospheric pressure traces. The main data processing tool is a nonlinear adaptive filter based on the selective reconstruction of a waveform from its wavelet coefficients. The time-frequency localization of the wavelet transform provides an ideal framework for the decomposition of long-period gravity waves (30 min-6 h), which are characterized by a generally broad spectrum and few oscillation cycles. The procedure is iterative and allows the exhaustive processing of all the events present in a fixed time period. The waveform of each disturbance is reconstructed with high accuracy. This minimizes the influence of the data-processing technique on the estimate of horizontal speed and direction of propagation, obtained by maximization of the cross-correlation functions between the reconstructed waveforms at the different stations. The introduction of coherency criteria through the network of seven stations allows the authors to separate the events into two classes. The first includes the events that propagate with very small distortion through the network, while the second includes less coherent but still highly energetic events. The size of the network and the algorithm developed for the analysis is well suited for the identification and the extraction of those mesoscale disturbances that have a particularly strong influence on the weather as well as on the forecast     

The Time-Domain Vector Fitting Algorithm for Linear Macromodeling

The Time-Domain Vector Fitting (TD-VF) algorithm for macromodeling of linear multiport systems is presented. A rational approximation for the system matrix transfer function is easily derived from transient input/output port responses. Several validations illustrate the high accuracy of the method.     

Transient analysis of lossy transmission lines: an efficient approach based on the method of Characteristics

This paper is devoted to transient analysis of lossy transmission lines characterized by frequency-dependent parameters. A public dataset of parameters for three line examples (a module, a board, and a cable) is used, and a new example of on-chip interconnect is introduced. This dataset provides a well established and realistic benchmark for accuracy and timing analysis of interconnect analysis tools. Particular attention is devoted to the intrinsic consistency and causality of these parameters. Several implementations based on generalizations of the well-known method-of-characteristics are presented. The key feature of such techniques is the extraction of the line modal delays. Therefore, the method is highly optimized for long interconnects characterized by significant propagation delay. Nonetheless, the method is also successfully applied here to a short high/loss on-chip line, for which other approaches based on lumped matrix rational approximations can also be used with high efficiency. This paper shows that the efficiency of delay extraction techniques is strongly dependent on the particular circuit implementation and several practical issues including generation of rational approximations and time step control are discussed in detail.     

A Passivity Enforcement Scheme for Delay-Based Transmission Line Macromodels

This letter presents an algorithm for the enforcement of passivity in transmission-line macromodels based on the generalized method of characteristics. The algorithm is based on first-order perturbations of the solutions to a frequency-dependent eigenvalue problem. The theoretical formulation provides an extension of techniques that were available only for lumped models to the more general class of delay-based models.     

Passivity Enforcement With Relative Error Control

This paper introduces a new error control strategy in passivity enforcement schemes for linear lumped macromodels. We consider the general class of a posteriori passivity enforcement algorithms based on Hamiltonian matrix perturbation. Standard available formulations preserve the accuracy during passivity enforcement using special matrix norms associated to the controllability Gramian of the macromodel. This procedure leads to absolute error control. On the other hand, it is well known that relative error control in the macromodel is sometimes preferable, especially for structures that are characterized by small coupling coefficients or high dynamic range in their responses. Here, we present a frequency-weighting scheme leading to the definition of a modified Gramian that, when employed during passivity enforcement, effectively leads to relative error control. Several examples illustrate the reliability of the proposed technique.     

On the generation of large passive macromodels for complex interconnect structures

This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed techniques.     

Improving the convergence of vector fitting for equivalent circuit extraction from noisy frequency responses

The vector fitting (VF) algorithm has become a common tool in electromagnetic compatibility and signal integrity studies. This algorithm allows the derivation of a rational approximation to the transfer matrix of a given linear structure starting from measured or simulated frequency responses. This paper addresses the convergence properties of a VF when the frequency samples are affected by noise. We show that small amounts of noise can seriously impair or destroy convergence. This is due to the presence of spurious poles that appear during the iterations. To overcome this problem we suggest a simple modification of the basic VF algorithm, based on the identification and removal of the spurious poles. Also, an incremental pole addition and relocation process is proposed in order to provide automatic order estimation even in the presence of significant noise. We denote the resulting algorithm as vector fitting with adding and skimming (VF-AS). A thorough validation of the VF-AS algorithm is presented using a Monte Carlo analysis on synthetic noisy frequency responses. The results show excellent convergence and significant improvements with respect to the basic VF iteration scheme. Finally, we apply the new VF-AS algorithm to measured scattering responses of interconnect structures and networks typical of high-speed digital systems.