We present DAGSENS, a new approach to parametric transient sensitivity analysis of Differential Algebraic Equation systems (DAEs), such as SPICE-level circuits. The key ideas behind DAGSENS are, (1) to represent the entire sequence of computations from DAE parameters to the objective function (whose sensitivity is needed) as a Directed Acyclic Graph (DAG) called the “sensitivity DAG”, and (2) to compute the required sensitivites efficiently by using dynamic programming techniques to traverse the DAG. DAGSENS is simple, elegant, and easy-to-understand compared to previous approaches; for example, in DAGSENS, one can switch between direct and adjoint sensitivities simply by reversing the direction of DAG traversal. Also, DAGSENS is more powerful than previous approaches because it works for a more general class of objective functions, including those based on “events” that occur during a transient simulation (e.g., a node voltage crossing a threshold, a phase-locked loop (PLL) achieving lock, a circuit signal reaching its maximum/minimum value, etc.). In this paper, we demonstrate DAGSENS on several electronic and biological applications, including high-speed communication, statistical cell library characterization, and gene expression.

Many subsystems encountered in communication systems can be modeled as linear periodic time-varying (LPTV) systems. In this paper, we present a novel structure preserving reduced-order modeling algorithm for LPTV systems. A key advance of our approach is that it preserves the periodic time-varying structure during the reduction process, thus resulting in reduced LPTV systems. Unlike prior LPTV model order reduction (MOR) techniques which recast the LPTV systems to artificial linear time-invariant (LTI) systems and apply LTI MOR techniques for reduction, our structure preserving algorithm uses a time-varying projection directly on the original LPTV systems. Our approach always produces a smaller system than the original system, which was not valid for previous LPTV MOR techniques. We validate the proposed technique with several circuit examples, demonstrating significant size reductions and excellent accuracy.

Envelope-following methods face special challenges when applied to oscillators because of their fundamental property of dynamically changing frequencies. In this paper, we present a novel and robust approach for oscillator envelope following. Our method combines, unifies, and extends ideas from two prior oscillator envelope-following approaches, namely, Petzold's method and the warped multitime partial differential equation. Our technique uses two extra system unknowns, as well as two extra ldquophase conditionrdquo equations, to track quantities related to dynamical frequency/time-period changes. These advances confer significant robustness, without appreciable computational overhead. We validate our method on LC, ring, and crystal oscillators, accurately predicting frequency and amplitude modulations, as well as transient startup envelopes. Speedups of one to two orders of magnitude are obtained over traditional alternatives.

Standard small-signal analysis methods for circuits break down for oscillators because small-input perturbations result in arbitrarily large-output changes, thus invalidating fundamental assumptions for small-signal analysis. In this paper, we propose a novel oscillator ac approach remedying this situation, thus restoring validity and rigour to small-signal analysis of oscillators. Our approach centers around a novel general equation formulation for circuits that we term the Generalized Multitime Partial Differential Equations (GeMPDE). While this formulation is broadly applicable to any kind of circuit or dynamical system, we show that it has unique advantages for oscillators in that small-input perturbations now lead to small output ones, thus making small-signal analysis valid. A key feature of our approach is to solve for bivariate-frequency variables with the help of novel augmenting-phase-condition equations. Unlike prior oscillator-analysis methods, which require special handling of the phase mode, our GeMPDE-based small-signal analysis provides both amplitude and frequency characteristics in a unified manner and is applicable to any kind of oscillator described by differential equations. We obtain speedups of 1-2 orders of magnitude over the transient-simulation approach commonly used today by designers for oscillator-perturbation analysis

In this paper, we propose a novel envelope-following method which is uniformly applicable to both non-autonomous and oscillatory circuits. A key feature of our technique is the use of an efficient minimum least squares solution technique to solve an underdetermined envelope system directly. This leads to a general purpose approach which is much easier to solve than previous phase condition based envelope-following method, improving numerically robustness dramatically. We validate our method on a variety of autonomous and non-autonomous circuits, including a PLL in transition to lock. The new method provides speedups of 1-2 orders of magnitude over transient simulation, while obtaining results that are equally or more accurate.

We present a generalization of standard AC analysis to oscillators by exploiting least-squares solution techniques. This provides an attractive alternative to the current practice of employing transient simulation for small signal analysis of oscillators. Unlike phase condition based oscillator analysis techniques, which suffer from numerical artifacts, the least-squares approach of this paper results in a robust and efficient oscillator AC technique. We validate our method on LC and ring oscillators, obtaining speedups of 1-3 orders of magnitude over transient simulation, and 4-6×over phase-condition-based techniques.

A unique feature of oscillators is that small but sustained external perturbations lead to unboundedly large changes in phase, thereby making standard harmonic balance (HB) inapplicable to realistic oscillator phase macromodels. In this paper, we rectify this situation by presenting a novel extension of HB that is capable of handling oscillator phase macro-models. Key to the new method, termed PPV-HB, is a formulation that separates unboundedly increasing phase terms from the bounded, periodic components. PPV-HB can be used not only on individual oscillators, but it also enables the application of HB-like techniques for simulating system-level equation systems composed of higher-level macromodels of blocks. We validate PPV-HB on individual oscillators and a PLL system, demonstrating excellent matches with transient simulation using phase macromodels. Speedups of 1-2 orders of magnitude are obtained, over and above additional speedups of another 2-3 orders of magnitude that stem from using macromodels (as opposed to full circuit simulation)

We present a novel method for finding periodic steady states of general classes of oscillators robustly. The new method, which we term the multi-harmonic probe (MHP) technique, generalizes the well-known technique of augmenting harmonic balance (HB) for oscillators using an external probe. By using non-sinusoidal periodic probes, MHP enhances the applicability of the standard probe method (which uses purely sinusoidal probes) to broader classes of oscillators. We thus obtain a general and robust method for the periodic steady state of any kind of oscillator. Results on LC and ring oscillator circuits are presented that testify to the efficacy of our approach.

Envelope-following methods face special challenges when applied to oscillators because of their fundamental property of dynamically-changing frequencies. In this paper, we present a novel and robust approach for oscillator envelope following. Our method combines, unifies and extends ideas from two prior oscillator envelope-following approaches, Petzold's method and the WaMPDE. Our technique uses two extra system unknowns, as well as two extra "phase condition" equations, to track quantities related to dynamical frequency/time-period changes. These advances confer significant robustness without appreciable computational overhead. We validate our method on LC, ring and crystal oscillators, predicting frequency and amplitude modulations as well as transient startup envelopes accurately. Speedups of 1-2 orders of magnitude are obtained over traditional alternatives.

Standard small-signal analysis methods for circuits break down for oscillators because small input perturbations result in arbitrarily large output changes, thus invalidating fundamental assumptions for small-signal analysis. In this paper, we propose a novel oscillator-AC (OAC) approach remedying this situation, thus restoring validity and rigour to small-signal analysis of oscillators. Our approach centers around a novel, general equation formulation for circuits that we term the GeMPDE. A key feature of our approach is to solve for bivariate frequency variables with the help of novel augmenting phase condition equations. Our GeMPDE-based small-signal analysis provides both amplitude and frequency characteristics in a unified manner and is applicable to any kind of oscillator described by differential equations. We obtain speedups of 1-2 orders of magnitude over the transient simulation approach commonly used today by designers for oscillator perturbation analysis. We also demonstrate and explain how our linearization approach captures the inherently nonlinear phenomenon of injection locking in oscillators.

We explore the stability properties of time-domain numerical methods for multitime partial differential equations (MPDEs) in detail. We demonstrate that simple techniques for numerical discretization can lead easily to instability. By investigating the underlying eigenstructure of several discretization techniques along different artificial time scales, we show that not all combinations of techniques are stable. We identify choices of discretization method and step size, along fast and slow time scales, that lead to robust, stable time-domain integration methods for the MPDE. One of our results is that applying overstable methods along one time-scale can compensate for unstable discretization along others. Our novel integration schemes bring robustness to time-domain MPDE solution methods, as we demonstrate with examples.

In this paper, we explore in detail the stability properties of time-domain numerical methods for multi-time partial differential equations (MPDEs). We demonstrate that simple techniques for numerical discretization can lead easily to instability. By investigating the underlying eigenstructure of several discretization techniques along different artificial time scales, we show that not all combinations of techniques are stable. We identify choices of discretization method and of step size along slow time scales that lead to robust, stable time-domain integration methods for the MPDE. One of our results is that applying overstable methods along one time-scale can compensate for unstable discretization along others. Our novel integration schemes bring robustness to time-domain MPDE solution methods, as we demonstrate with examples.

This paper proposes a new diffuse self-organizing map (DSOM), which is a competitive self-organizing neural network that can forms a topological map of the input vector space in a circle-shaped region. Active and inactive neurons are introduced to restrict the range of competition and the size of the final map. Lateral conduction is also introduced to calculate the learning radius and learning rate. DSOM learns in a uniform manner and is capable to learn new classes either within or after its learning stage.