PPV-Based Modeling and Event-Driven Simulation of Injection-Locked Oscillators in SystemVerilog

Supply and ground noise sensitivity of a wide variety of ring and LC oscillators has been analyzed based on the perturbation projection vector (PPV) technique. The resulting PPV provides an understanding of how specific frequency content of supply/ground noise is converted to oscillator phase noise. Based on this analysis oscillators that are tolerant of supply/ground noise can be identified and used for low noise oscillator design.

Coupled oscillator networks (CONs) occur in various domains such as biology and electronics. The nonlinear phenomenon of “injection locking” leads to frequency synchronization among the oscillators in a CON. In this chapter, we first obtain simplified tools to analyze the phenomenon of injection locking and present the Generalized Adler’s equation. The Generalized Adler’s equation is applicable for analyzing the injection locking in any type of oscillator. We demonstrate the use of Generalized Adler’s equation for injection locking analysis on a ring oscillator and a Hodgkin-Huxley neuronal oscillator. Next, we describe efficient system-level simulation techniques (synchronized steady state and transient) for coupled oscillator networks using the Perturbation Projection Vector (PPV) phase macromodel of the oscillator. The coupled oscillator simulation techniques are generic in nature and can be applied for the simulation of any coupled oscillator network, like the ones found in biological or electronic systems. We apply our system-level techniques on a −G m LC (harmonic) and a Hodgkin-Huxley (non-harmonic) CON.

Injection-locked oscillators (ILO) have widespread use in wireline and wireless systems as fast-locking clock generators, phase noise filters, and frequency multipliers/ dividers, but their modeling and simulation typically resort to SPICE due to their nonlinear, time-varying characteristics. This paper presents a way to simulate ILOs in an event-driven logic simulator such as SystemVerilog, based on a perturbation projection vector (PPV) model. By expressing the continuous-time signals in analytical forms of which coefficients are updated as events, the nonlinear phase response of the ILO can be simulated in an algebraic fashion without numerical integration. The simulation methodology is demonstrated with the modeling examples of a phase-locked loop and a burst-mode clock recovery loop, each employing a injected-locked frequency divider and oscillator, respectively.

Coupled oscillator networks occur in various domains such as biology, astrophysics and electronics. In this paper, we present a comprehensive procedure for rapid and accurate simulation of large coupled oscillator networks using widely accepted, fully-nonlinear perturbation projection vector (PPV) phase macromodels. We validate our method against full simulation of 20times20 coupled network of Brusselator biochemical oscillator and obtain computational speedups of 170x over full simulation. Furthermore, we apply the method to study self-organization phenomenon of Brusselator under asymmetric coupling and time period variations.

Cyclic-coupled ring oscillators (CCRO), which consist of M ring oscillators each with N inverting stages, can be used in time-domain data converters to achieve sub-gate-delay resolution and improved phase noise performance compared to a single ring oscillator (RO). However, CCROs can oscillate in several different oscillation modes, where some modes contain overlapping phases. Such in-phase oscillations severely degrade the performance of a time-domain data converter by undermining the sub-gate-delay of the CCRO. This paper presents a design method to avoid the undesired in-phase oscillation modes, and thus achieve guaranteed maximal phase resolution regardless of the oscillation mode, by properly selecting the CCRO dimensions N and M. We show, both theoretically and with transistor-level simulations, that mode-agnostic maximum phase resolution can be ensured by selecting a prime M together with an N which is co-prime with M.

Solving parametric piecewise polynomial systems

On-chip clock generation is an attractive alternative to external quartz oscillators for low-power sensing systems. LC and ring oscillators typically use less power, and can start and stop much faster than traditional quartz oscillators, allowing systems to spend more time in low-power sleep states. A hybrid on-chip clocking scheme is evaluated in which a ring oscillator is used as the reference clock for digital processing, and an LC oscillator is used when a more stable, accurate reference is needed for digital communication. Simulations show that the fast wakeup capabilities of the hybrid clock system lead to as much as 80% reduced power in interrupt-driven applications compared to a system using a crystal reference clock source. The on-chip clock generators of the WIMS series of microsystems are described to demonstrate the concept of the hybrid clock scheme.

In smooth support vector machine (SSVM), the plus function must be approximated by some smooth function, and the approximate error will affect the classification ability. This paper studies the smooth approximation to the plus function by piecewise polynomials. First, some standard piecewise polynomial smooth approximation problems are formulated. Then, the existence and uniqueness of solution for these problems are proved and the analytic solutions are achieved. The comparison between the results in this paper and the previous ones shows that the piecewise polynomial functions in this paper achieve better approximation to the plus function

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In practice, we are interested in approximating a function f(x) near x = a by a polynomial of degree M. Other issues involved in approximation are how good the approximation is and over what interval can we expect the approximation to be good. In this paper we present approximation that is realized as piecewise polynomial function where in each interval ƒ(x) is approximated by a Taylor polynomial of low order. The piecewise polynomial approximation can be effectively implemented using Farrow structure.

We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well-approximated by piecewise polynomial functions. Let ƒ be the density function of an arbitrary univariate distribution, and suppose that ƒ is OPT-close in Li- distance to an unknown piecewise polynomial function with t interval pieces and degree d. For any γ > 0, our algorithm draws n = Õγ(t(d + 1)/ ∊2) samples from ƒ, runs in time Õ(n), and with probability at least 9/10 outputs an Ογ (t)-piecewise degree-d hypothesis H that is (3 + γ) · OPT + ∊ close to f. Our approximation factor almost matches the best known information-theoretic (but computationally inefficient) upper bound of 3.Our general algorithm yields (nearly) sample- optimal and nearly-linear time estimators for a wide range of structured distribution families over both continuous and discrete domains in a unified way. For most of our applications, these are the first sample-optimal and nearly-linear time estimators in the literature. As a consequence, our work resolves the sample and computational complexities of a broad class of inference tasks via a single “meta-algorithm”. Moreover, we demonstrate that our algorithm performs very well in experiments.Our algorithm consists of three levels: (i) At the top level, we employ an iterative greedy algorithm for finding a good partition of the real line into the pieces of a piecewise polynomial. (ii) For each piece, we show that the sub-problem of finding a good polynomial fit on the current interval can be solved efficiently with a separation oracle method. (iii) We reduce the task of finding a separating hyperplane to a combinatorial problem and design a nearly-linear algorithm for this problem. Combining these three procedures gives a density estimation algorithm with the claimed guarantees.

In this paper, we present a simple analytical equation for capturing phase errors in 3-stage ring oscillators. The model, based on a simple but useful idealization of the ring oscillator, is provably exact for small noise perturbations. Despite its simplicity and purely analytical form, our model correctly captures the time- dependent sensitivity of oscillator phase to external perturbations. It is thus well suited for estimating both qualitative and quantitative features of ring oscillator phase response to internal noises, as well as to power, ground and substrate interference. The nonlinear nature of the model makes it suitable for predicting injection locking as well. Comparisons of the new model with existing phase models are provided, and its application for correct prediction of thermal jitter demonstrated. Requiring knowledge only of the amplitude and frequency of the oscillator, the model is ideally suited for early design exploration at the system and circuit levels.

The paper is concerned with some characteristics of approximation by piecewise polynomial functions with nonfixed knots in various integral metrics Lp(a,b), 0 <p ≤∞ .

A multi-level discontinuous Galerkin method for solving the stationary Navier–Stokes equations

Timing jitter is a concern in high frequency oscillators; the presence of timing jitter will degrade system performance in many high speed applications. In the first part of the paper, the authors have simulated the timing jitter due to CMOS device noise in a nine-stage CMOS differential ring oscillator, and a methodology to efficiently simulate timing jitter has been developed. Simulation results show that the variation of absolute jitter due to flicker noise has t-dependence while for white noise it has t/sup 0.5/-dependence; these are consistent with accepted theory. Two important parameters, cycle jitter and cycle-to-cycle jitter, used to describe jitter performance can be obtained from simulation. Simulation results are also compared with measurement results, and it is shown that simulation results are very close to measurement results. All these serve to verify the validity of this technique. In the second part of the paper, the authors have employed this methodology and investigated the timing jitter in silicon BJT/or SiGe HBT ECL ring oscillators, and they have shown that BJT/or SiGe HBT oscillators have lower jitter compared to their CMOS counterparts. The methodology described in the paper is also applicable to other types of clock generator and oscillators such as LC oscillators, as well as other kinds of noise source such as power supply and substrate noise.