Jaijeet S. Roychowdhury

Phase noise in oscillators: a unifying theory and numerical methods for characterization

Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields, such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its characterization. In this paper, we develop a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism. We establish novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random. We obtain an exact nonlinear equation for phase error, which we solve without approximations for random perturbations. This leads us to a precise characterization of timing jitter and spectral dispersion, for computing of which we have developed efficient numerical methods. We demonstrate our techniques on a variety of practical electrical oscillators and obtain good matches with measurements, even at frequencies close to the carrier, where previous techniques break down. Our methods are more than three orders of magnitude faster than the brute-force Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly.

Phase noise in oscillators: a unifying theory and numerical methods for characterisation

Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its characterisation. In this paper, we develop a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism. We establish novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random. We obtain an exact, nonlinear equation for phase error, which we solve without approximations for random perturbations. This leads us to a precise characterisation of timing jitter and spectral dispersion, for computing which we develop efficient numerical methods. We demonstrate our techniques on practical electrical oscillators, and obtain good matches with measurements even at frequencies close to the carrier, where previous techniques break down.

Analyzing circuits with widely separated time scales using numerical PDE methods

Widely separated time scales arise in many kinds of circuits, e,g., switched-capacitor filters, mixers, switching power converters, etc. Numerical solution of such circuits is often difficult, especially when strong nonlinearities are present. In this paper, the author presents a mathematical formulation and numerical methods for analyzing a broad class of such circuits or systems. The key idea is to use multiple time variables, which enable signals with widely separated rates of variation to be represented efficiently. This results in the transformation of differential equation descriptions of a system to partial differential ones, in effect decoupling different rates of variation from each other. Numerical methods can then be used to solve the partial differential equations (PDEs). In particular, time-domain methods can be used to handle the hitherto difficult case of strong nonlinearities together with widely separated rates of signal variation. The author examines methods for obtaining quasiperiodic and envelope solutions, and describes how the PDE formulation unifies existing techniques for separated-time-constant problems. Several applications are described. Significant computation and memory savings result from using the new numerical techniques, which also scale gracefully with problem size.

Reduced-order modeling of time-varying systems

We present algorithms for reducing large circuits, described at SPICE-level detail, to much smaller ones with similar input-output behavior. A key feature of our method, called time-varying Pade (TVP), is that it is capable of reducing time-varying linear systems. This enables it to capture frequency-translation and sampling behavior, important in communication subsystems such as mixers and switched-capacitor filters, Krylov-subspace methods are employed in the model reduction process. The macromodels can be generated in SPICE-like or AHDL format, and can be used in both time- and frequency-domain verification tools. We present applications to wireless subsystems, obtaining size reductions and evaluation speedups of orders of magnitude with insignificant loss of accuracy. Extensions of TVP to nonlinear terms and cyclostationary noise are also outlined.

Cyclostationary noise analysis of large RF circuits with multitone excitations

This paper introduces a new, efficient technique for analyzing noise in large RF circuits subjected to true multitone excitations. Noise statistics in such circuits are time-varying, hence cyclostationary stochastic processes, characterized by harmonic power spectral densities (HPSDs), are used to describe noise. HPSDs are used to devise a harmonic-balance-based noise algorithm with the property that required computational resources grow almost linearly with circuit size and nonlinearity. Device noises with arbitrary spectra (including thermal, shot, and flicker noises) are handled, and input and output correlations, as well as individual device contributions, can be calculated. HPSD-based analysis is also used to establish the nonintuitive result that bandpass filtering of cyclostationary noise can result in stationary noise. Results from the new method are validated against Monte Carlo simulations. A large RF integrated circuit (>300 nodes) driven by a local oscillator (LO) tone and a strong RF signal is analyzed in less than two hours. The analysis predicts correctly that the presence of the RF tone leads to noise folding, affecting the circuits noise performance significantly.

Capturing oscillator injection locking via nonlinear phase-domain macromodels

Injection locking is a nonlinear dynamical phenomenon that is often exploited in electronic and optical oscillator design. Behavioral modeling techniques for oscillators that predict this phenomenon accurately are of significant scientific and practical importance. In this paper, we propose a nonlinear approach for generating small phase-domain oscillator/voltage-controlled oscillator (VCO) macromodels that capture injection locking well. Our nonlinear phase-domain macromodels are closely related to recent oscillator phase noise and jitter theories, and can be extracted efficiently by algorithm from SPICE-level descriptions of any oscillator or VCO. Using LC and ring oscillators as test cases, we confirm the ability of nonlinear phase macromodels to capture injection locking, and also obtain significant computational speedups over full SPICE-level circuit simulation. Furthermore, we show that our approach is equally effective for capturing the dynamics of transition to locking, including unlocked tones and phase jump phenomena.

Piecewise polynomial nonlinear model reduction

We present a novel, general approach towards model-order reduction (MOR) on nonlinear systems that combines good global and local approximation properties. The nonlinear system is first approximated as piecewise polynomials over a number of regions, following which each region is reduced via polynomial model-reduction methods. Our approach, dubbed PWP, generalizes recent piecewise linear approaches and ties them with polynomial-based MOR, thereby combining their advantages. In particular, reduced models obtained by our approach reproduce small-signal distortion and intermodulation properties well, while at the same time retaining fidelity in large-swing and large-signal analyses, e.g., transient simulations. Thus our reduced models can be used as drop-in replacements for time-domain as well as frequency-domain simulations, with small or large excitations. By exploiting sparsity in system polynomial coefficients, we are able to make the polynomial reduction procedure linear in the size of the original system. We provide implementation details and illustrate PWP with an example.

Efficient multi-tone distortion analysis of analog integrated circuits

This paper introduces a novel approach to analyze distortion behavior in analog integrated circuits using a nonlinear frequency-domain method. This approach circumvents the difficulties and inaccuracies associated with device modeling required for the traditional Volterra series method and can handle circuits operating in more strongly nonlinear regimes. The efficiency of the method renders the analysis of large analog blocks practical. We present examples of multi-tone distortion analyses of industrial amplifiers and continuous time filters. The trend towards higher levels of integration, particularly for wireless applications, renders this method especially useful.

Efficient methods for simulating highly nonlinear multi-rate circuits

Widely-separated time scales appear in many electronic circuits, making traditional analysis difficult or impossible if the circuits are highly nonlinear. In this paper, an analyticalformulation and numerical methods are presented for treating strongly nonlinear multi-rate circuits effectively. Multivariate functions in the time domain are used to capturewidely separated rates efficiently, and a special partial differential equation (the MPDE) is shown to relate the multivariate forms of a circuits signals. Time-domain and mixedfrequency-time simulation algorithms are presented for solving the MPDE. The new methods can analyze circuits that are both large and strongly nonlinear. Compared to traditional techniques, speedups of more than two orders of magnitude, as well as improved accuracy, are obtained.

Efficient transient simulation of lossy interconnect

The problem of transient simulation of lossy transmission lines is investigated in this paper. Two refinements are made to the existing convolution approach for the case of a single lossy line analytical formulae are derived for the line’s impulse-responses, and an accurate numerical convolution technique that utilises these formulae are devised. It is shown that a special case of lossy multiconductor lines can be decomposed into uncoupled lossy lines and linear memoryless networks, leading to a simple simulation algorithm. Simulation results on industrial circuits with single and mtdticonductor lossy lines are presented and compared with results obtained using lumped and pseudo-lumped approximations of lossy lines. The comparison indicates that the convolution technique with the above enhancements can be an order-ofmagnitude faster than lumped and pseudo-lumped segmenting techniques for equivalent or better accuracy.