Perturbation Projection Vector Research Articles

Broadband reduction of phase noise in a spatially extended tunnel‐diode oscillator through multiple self‐injection locking

SummaryA self‐injection locking scheme is proposed to reduce phase noise in a spatially extended oscillator that uses a tunnel‐diode (TD) transmission line. When a linear transmission line is connected to each cell of the TD line, a part of the oscillating edge flows through the linear line. This wave on the linear line is reflected at its far end and flows back to the point of connection of the TD and linear transmission lines. By matching the time taken by the reflected wave on the linear transmission line to reach the point of connection with the time taken by the forward‐moving oscillating edge to return to the point, the edge can be synchronized with the wave that flows back through each linear line. This multiple self‐injection locking significantly reduces the phase noise of the oscillating edge. In this work, the technique of phase noise reduction is explained through numerical phase noise analysis using the perturbation projection vector of the system. Several experimental observations were made to validate the reduction of the phase noise in the oscillating edge traveling on the TD line.

Generalized Analytical Equations for Injected Ring Oscillator With $RC$ -Load

In this paper, a new efficient circuit level analysis is introduced considering the effect of all harmonics and multi-injection for locking and pulling phenomena in $N$ -stage differential and single-ended ring oscillators with the $RC$ -load. Analytical equations are derived for the locking range in both single- and multi-injection cases. At the beginning, using a nonlinear approach, the voltage amplitude and frequency of $RC$ -load free running ring oscillators are derived as simple closed-form equations. Then, to analyze the mentioned phenomena, differential equations describing the circuit behavior are derived by circuit level equations. Furthermore, analyses are extended for general multi-injections with arbitrary amplitude and excess phase for different output nodes. The simplest and most accurate analytical equation is achieved for the locking range. In contrary to few previous works, the proposed analysis is derived directly from equations governing ring oscillators so that we can expect results with more accuracy without any tedious and time-consuming pre-processing like perturbation projection vector. To evaluate the analysis, many simulations have been performed. Simulation results validate presented analytical equations, which provide useful design insights. Absolute percent error of all proposed equations in comparison with simulation results is better than 20%.

PPV Modeling of Memristor-Based Oscillators and Application to ONN Pattern Recognition

An efficient perturbation projection vector (PPV) modeling for a memristor-based oscillator is proposed and then applied to the simulation of the oscillatory neural network (ONN) for pattern recognition. Compared with the existing model, which is very time consuming, this novel modeling method features a fast and accurate simulation of a large-scale network with many oscillators. The proposed modeling is based on a highly efficient abstraction of the phase sensitivity characteristic of the memristor-based oscillator, i.e., its PPV, which enables reducing the complexity of the ONN description, speeding up the simulation considerably while retaining comparable precision.

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

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to $2\sim 30\times$ that of transistor-level Spectre simulations with the same accuracy.

Hierarchical abstraction of weakly coupled synchronized oscillator networks

SummaryWe provide a constructive and numerically implementable proof that synchronized groups of coupled, self‐sustaining oscillators can be represented as a single effective Perturbation Projection Vector (PPV) (or Phase Response Curve) phase macromodel – in other words, that a group of synchronized oscillators behaves as a single effective oscillator with respect to external influences. This result constitutes a foundation for understanding and predicting synchronization/timing hierarchically in large, complex systems that arise in nature and engineering. We apply this result hierarchically to networks of synchronized oscillators, thereby enabling their efficient and scalable analysis. We illustrate our theory and numerical methods with examples from electronics (networks of three‐stage ring oscillators), biology (Fitzhugh–Nagumo neurons) and mechanics (pendulum clocks). Our experiments demonstrate that effective PPVs extracted hierarchically can capture emergent phenomena, such as pattern formation, in coupled oscillator networks. Copyright © 2015 John Wiley & Sons, Ltd.

Phase-Noise Analysis and Simulation of LC Oscillator-Based Injection-Locked Frequency Dividers

This letter proposes a phase-noise analysis of injection-locked frequency dividers that are based on LC oscillators. The proposed analysis method relies on the concept of the perturbation projection vector and allows one to investigate how the output phase noise is affected by the amplitude and the frequency of the input signal. Closed-form expressions for the output phase-noise spectrum under different injection conditions are provided and validated against the periodic noise analysis of a commercial circuit simulator. Indeed, the proposed semianalytical method can provide insights and guidelines to improve the circuit design.

Computing the Perturbation Projection Vector of Oscillators via Frequency Domain Analysis

This paper describes an original computational procedure to extract the perturbation projection vector of oscillators by means of a frequency domain technique. A key feature of the method is that it relies on the periodic transfer function analysis, which is available in most circuit simulators, and thus it can easily be exploited by oscillator designers. The accuracy of the proposed extraction procedure is verified for two relevant oscillator topologies.

Model Comparison for $1/f$ Noise in Oscillators With and Without AM to PM Noise Conversion

A new generalized 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> Kurokawa noise analysis applicable to both low- and high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> oscillators is proposed for 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> phase modulation (PM) and amplitude modulation (AM) noise. A theoretical correspondence between the new generalized Kurokawa theory and the impulse sensitivity function and the perturbation projection vector (PPV) is also derived in the absence and presence of AM to PM noise conversion. The importance of AM to PM noise conversion effect is investigated next using a modified Van der Pol oscillator with fourth-order nonlinear capacitances and a harmonic short. The same analytic solution for the phase noise is derived for both the new generalized 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> Kurokawa and the PPV analysis and is found to match the results from the conversion matrix method in two commercial harmonic-balance circuit simulators.

Analysis and Design of Injection-Locked Frequency Dividers by Means of a Phase-Domain Macromodel

This paper describes an original method to estimate the locking ranges of injection-locked frequency dividers (ILFDs) and their sensitivity to variations in parameter values. The synchronization capability of a given oscillator architecture with respect to possible injection points is explored by applying small-amplitude signals and adopting a phase-domain macromodel based on the paradigm of the perturbation projection vector. It is shown that the method provides synthesis information and guidelines that help to improve the ILFD design process and the selection of a proper injection strategy.

Synchronization Analysis of Two Weakly Coupled Oscillators Through a PPV Macromodel

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper adopts a phase-domain macromodel based on perturbation projection vector to study the synchronization effects that take place between two weakly coupled oscillators. Original closed-form expressions for the locking region of the coupled oscillators and for the common locking frequency are derived. The route to synchronization is described analytically by predicting the oscillation frequency shifts, which are induced by mutual pulling as a function of the interaction strength. Under the assumption of a weak coupling, the proposed approach can be applied to a wide class of oscillator topologies. </para>

Automated Design and Optimization of Low-Noise Oscillators

This paper presents a technique for automated design and optimization of low-noise oscillators. A sensitivity analysis for oscillators guides the reduction of oscillator noise intensities and improvement of oscillator's immunity to noise. A design-oriented approach to circuit analysis efficiently handles design constraints and reduces the dimensionality of the optimization problem. The perturbation projection vector based phase noise computation makes the proposed optimization technique general and applicable to all types of oscillators, independent of circuit topology. Several examples illustrate the benefits of the new optimization approach.

Sensitivity Analysis for Oscillators

This paper presents an analysis for calculating sensitivities of an oscillator's periodic steady state and perturbation projection vector to design, process, or environmental parameters. A general continuous-time formulation and time-domain numerical methods are described. The applications of the oscillator sensitivity analysis in design optimization, macromodeling, as well as in analyzing the impact of process variations are demonstrated through examples.

The Analytic Determination of the PPV for Second-Order Oscillators Via Time-Varying Eigenvalues

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Time-varying eigenvalues may be used to formulate a set of linearly independent solutions for an arbitrary dynamical linear time-varying system. In this brief, it is shown how these quantities are used to determine analytically the perturbation projection vector (PPV) associated to a given oscillator. The PPV can be further used to estimate the spectral and timing properties of the oscillator output subject to the presence of noise sources. For this purpose, a complete set of solutions of the variational system associated to the oscillator under consideration is generated in terms of time-varying eigenvalues. To compute these quantities, a solution for a particular form of the Riccati equation must be found. It is shown how to obtain a solution for this equation from the steady-state behavior of the oscillator. A simple example demonstrating the application of the concepts mentioned above is also provided. </para>

Physics-Based Analysis and Simulation of Phase Noise in Oscillators

A technology computer-aided design framework that can predict the phase noise spectrum of an oscillator using nonlinear perturbation analysis is developed. The device-circuit mixed-mode simulation technique based upon the shooting-Newton method is exploited to evaluate the periodic steady-state solution of the oscillator. The influence of noise sources inside the devices on the phase deviation is calculated in an efficient and accurate way using the perturbation projection vector. The period jitter and the output power spectrum can be easily obtained in this framework. As an application, the output power spectrum of a CMOS LC voltage-controlled oscillator is calculated

Computation of period sensitivity functions for the simulation of phase noise in oscillators

Accurate phase noise simulation of circuits for radio frequency applications is of great importance during the design and development of wireless communication systems. In this paper, we present an approach based on the Floquet theory for the analysis and numerical computation of phase noise that solves some drawbacks implicitly present in previously proposed algorithms. In particular, we present an approach that computes the perturbation projection vector directly from the Jacobian matrix of the shooting method adopted to compute the steady-state solution. Further, we address some problems that arise when dealing with circuits whose modeling equations do not satisfy the Lipschitz condition at least from the numerical point of view. Frequency-domain aspects of phase noise analysis are also considered and, finally, simulation results for some benchmark circuits are presented.

A reliable and efficient procedure for oscillator PPV computation, with phase noise macromodeling applications

The main effort in oscillator phase noise calculation and macromodeling lies in computing a vector function called the perturbation projection vector (PPV). Current techniques for PPV calculation use time-domain numerics to generate the system's monodromy matrix, followed by full or partial eigenanalysis. We present superior methods that find the PPV using only a single linear solution of the oscillator's time- or frequency-domain steady-state Jacobian matrix. The new methods are better suited for implementation in existing tools with harmonic balance or shooting capabilities (especially those incorporating fast variants), and can also be more accurate than explicit eigenanalysis. A key advantage is that they dispense with the need to select the correct one eigenfunction from amongst a potentially large set of choices, an issue that explicit eigencalculation-based methods have to face. We illustrate the new methods in detail using LC and ring oscillators.