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Bit error rate (BER) is an important figure of merit to evaluate the performance of a communication system. Analyzing the BER of a linear-time-invariant system has been extensively studied. However, analyzing the BER of nonlinear circuits and systems is challenging because it cannot rely on linear time invariant (LTI) principles, while an exhaustive nonlinear simulation is computationally prohibitive. To find BER, an exhaustive approach requires nonlinear simulations of $2^{m}$ bit patterns for a channel of $m$ -bit memory, where $m$ can be larger than 100 in today’s high-speed and high-performance design. In this work, we develop a fast and accurate method to analyze the BER of large-scale nonlinear circuits. Only $O(k)$ nonlinear simulations are required, with $k$ much less than $2^{m}$ and independent of $2^{m}$. While performing $O(k)$ nonlinear simulations only, we find a method to determine the probability density function of the nonlinear responses, from which accurate BER results can be obtained. An error assessment method is also developed to evaluate the true error of the nonlinear signaling analysis without the need for knowing the entire nonlinear responses of the channel. Simulations of large-scale real-world nonlinear circuits have demonstrated the accuracy, efficiency, and capacity of the proposed method. A BER as low as 10-56 is accurately predicted. [ABSTRACT FROM AUTHOR]

Signaling analysis of nonlinear high-speed circuits is challenging because it cannot rely on linear time-invariant principles, whereas an exhaustive nonlinear simulation is computationally prohibitive for low bit error rates. To find the worst-case eye, an exhaustive approach requires nonlinear simulations of $2^m$ bit patterns for a channel of $m$ -bit memory. In this article, we represent the nonlinear responses to the $2^m$ inputs by a rank- $k$ matrix, where $k$ denotes the number of distinct waveforms in the responses for a prescribed accuracy, which is much smaller than $2^m$. We further develop a fast full cross approximation algorithm to find the rank- $k$ model with a low complexity independent of $2^m$. Simulations of large-scale real-world nonlinear circuits with an over 100-bit channel memory demonstrate the accuracy, efficiency, and capacity of the proposed work. [ABSTRACT FROM AUTHOR]

Conventional methods for signaling analysis are based on linear time invariant (LTI) assumptions. In this paper, we develop a method for efficient and accurate signaling analysis of nonlinear circuits. This method retains the nonlinear simulation as it is instead of linearizing the original nonlinear problem or adopting a simplified nonlinear model, thereby ensuring accuracy. Meanwhile, it does not suffer from the drawbacks of the brute-force nonlinear simulation in computational efficiency. Being a generic method for signaling analysis, the method is applicable to all kinds of nonlinearities as well as the LTI systems. Simulations of realistic industry nonlinear circuits have demonstrated the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

With a method developed in this work, we find the solution of the original electric field integral equation (EFIE) at an arbitrary frequency where the EFIE breaks down due to low frequencies and/or dense discretizations. This solution is equally rigorous at frequencies where the EFIE does not break down and is independent of the basis functions used. We also demonstrate, both theoretically and numerically, the fact that although the problem is commonly termed low-frequency breakdown, the solution at the EFIE breakdown can be dominated by fullwave effects instead of just static or quasi-static physics. The accuracy and efficiency of the proposed method is demonstrated by numerical experiments involving inductance, capacitance, RCS extraction, and a multiscale example with a seven-orders-of-magnitude ratio in geometrical scales, at all breakdown frequencies of an EFIE. In addition to the EFIE, the proposed method is also applicable to other integral equations and numerical methods for solving Maxwell's equations. [ABSTRACT FROM PUBLISHER]

Full-wave solutions of Maxwell's equations break down at low frequencies. Existing methods for solving this problem either are inaccurate or incur additional computational cost. In this paper, a fast full-wave finite-element-based solution is developed to eliminate the low-frequency breakdown problem in a reduced system of order one. It is applicable to general 3-D problems involving ideal conductors as well as nonideal conductors immersed in inhomogeneous, lossless, lossy, and dispersive materials. The proposed method retains the rigor of a theoretically rigorous full-wave solution recently developed for solving the low-frequency breakdown problem, while eliminating the need for an eigenvalue solution. Instead of introducing additional computational cost to fix the low-frequency breakdown problem, the proposed method significantly speeds up the low-frequency computation. [ABSTRACT FROM PUBLISHER]

It has been observed that finite element based solutions of full-wave Maxwell's equations break down at low frequencies. In this paper, we present a theoretically rigorous method to fundamentally eliminate the low-frequency breakdown problem. The key idea of this method is that the original frequency-dependent deterministic problem can be rigorously solved from a generalized eigenvalue problem that is frequency independent. In addition, we found that the zero eigenvalues of the generalized eigenvalue problem cannot be obtained as zeros because of finite machine precision. We hence correct the inexact zero eigenvalues to be exact zeros. The validity and accuracy of the proposed method have been demonstrated by the analysis of both lossless and lossy problems having on-chip circuit dimensions from dc to high frequencies. The proposed method is applicable to any frequency. Hence it constitutes a universal solution of Maxwell's equations in a full electromagnetic spectrum. The proposed method can be used to not only fundamentally eliminate the low-frequency breakdown problem, but also benchmark the accuracy of existing electromagnetic solvers at low frequencies including static solvers. Such a benchmark does not exist yet because full-wave solvers break down while static solvers involve theoretical approximations. [ABSTRACT FROM AUTHOR]

An effective method is developed in this work to extend the validity of a full-wave finite-element-based solution down to dc for general 3-D problems. In this method, we accurately decompose the Maxwell's system at low frequencies into two subsystems in the framework of a full-wave-based solution. One has an analytical frequency dependence, whereas the other can be solved at frequencies as low as dc. Thus, we bypass the numerical difficulty of solving a highly ill-conditioned and even singular system at low frequencies. In addition, we provide a theoretical analysis on the conditioning of the matrices of the original coupled Maxwell's system and the decomposed system. We show that the decomposed system is well conditioned, and also positive definite at dc. The validity and accuracy of the proposed method have been demonstrated by extraction of state-of-the-art on-chip integrated circuits at frequencies as low as dc. [ABSTRACT FROM PUBLISHER]