Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

A novel linear algebra method for the determination of periodic steady states of nonlinear oscillators

Periodic steady-state (PSS) analysis of nonlinear oscillators has always been a challenging task in circuit simulation. We present a new way that uses numerical linear algebra to identify the PSS(s) of nonlinear circuits. The method works for both autonomous and excited systems. Using the harmonic balancing method, the solution of a nonlinear circuit can be represented by a system of multivariate polynomials. Then, a Macaulay matrix based root-finder is used to compute the Fourier series coefficients. The method avoids the difficult initial guess problem of existing numerical approaches. Numerical examples show the accuracy and feasibility over existing methods. © 2014 IEEE.postprin

Using Volterra Series for an Estimation of Fundamental Intermodulation Products

The most precise procedure for determining the intermodulation products is to find a steady-state period of the signal first, and then to calculate its spectrum by means of the fast Fourier transform. However, this method needs time-consuming numerical integration over many periods of the faster signal even for enhanced methods for finding the steady state. In the paper, an efficient method for fast estimation of the fundamental intermodulation products is presented. The method uses Volterra series in a simple multistep algorithm which is compatible with a typical structure of the frequency-domain part of circuit simulators. The method is demonstrated by an illustrative testing circuit first, which clearly shows possible incorrect interpretation of the Volterra series. Thereafter, practical usage of the algorithm is demonstrated by fast estimation of the main intermodulation products of a low-voltage low-power RF CMOS fourquadrant multiplier

Analysis of superregenerative oscillators in nonlinear mode

Superregenerative oscillators in a nonlinear mode are investigated in detail using methodologies based on envelope transient, complemented with additional algorithms. A maximum-detection technique is applied to obtain the input-power threshold for nonlinear operation under different implementations of the quench signal. A mapping procedure enables the prediction of hangover and self-oscillation effects. It is based on the detection of the sequence of local maxima in the envelope amplitude after the application of a single input pulse. Using a contour-intersection method, and depending on the analysis time interval, it is possible to quantify the hangover effects and obtain the oscillation boundary, in terms of any two significant parameters. Then, a compact time-variant behavioral model is derived, valid in the absence of hangover and self-oscillation effects. It consists of a single time-variant Volterra kernel and is applicable provided that the amplitude transitions occur outside the sensitivity interval. Various methodologies are tested in a practical FET-based oscillator at 2.7 GHz. The prototype has been manufactured and measured, obtaining good agreement with the analysis results.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under the research project TEC2017-88242-C3-1-R

Volterra Series-Based Time-Domain Macromodeling of Nonlinear Circuits

Volterra series (VS) representation is a powerful mathematical model for nonlinear circuits. However, the difficulties in determining higher order Volterra kernels limited its broader applications. In this paper, a systematic approach that enables a convenient extraction of Volterra kernels from X-parameters is presented. A concise and general representation of the output response due to arbitrary number of input tones is given. The relationship between Volterra kernels and X-parameters is explicitly formulated. An efficient frequency sweep scheme and an output frequency indexing scheme are provided. The least square linear regression method is employed to separate different orders of Volterra kernels at the same frequency, which leads to the obtained Volterra kernels complete. The proposed VS representation based on X-parameters is further validated for time-domain verification. The proposed method is systematic and general-purpose. It paves the way for time-domain simulation with X-parameters and constitutes a powerful supplement to the existing blackbox macromodeling methods for nonlinear circuits.postprin