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27
Phase Noise in Oscillators: a Unifying Theory and Numerical Methods for Characterization
 IEEE Transactions on Circuits and Systems
, 2000
"... Abstract—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 f ..."
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Cited by 122 (11 self)
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Abstract—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 which we develop 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 bruteforce Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly. Index Terms—Circuit simulation, FokkerPlanck equations, nonlinear oscillators, oscillator noise, phase noise, stochastic
Efficient SteadyState Analysis based on MatrixFree KrylovSubspace Methods
, 1995
"... Gaussianelimination based shootingNewton methods, a commonly used approach for computing steadystate solutions, grow in computational complexity like N³ where N is the number of circuit equations. Just using iterative methods to solve the shootingNewton equations results in an algorithm whi ..."
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Cited by 41 (10 self)
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Gaussianelimination based shootingNewton methods, a commonly used approach for computing steadystate solutions, grow in computational complexity like N³ where N is the number of circuit equations. Just using iterative methods to solve the shootingNewton equations results in an algorithm which is still order N because of the cost of calculating the dense sensitivity matrix. Below, a matrixfree Krylovsubspace approach is presented, and the method is shown to reduce shootingNewton computational complexity to that of ordinary transient analysis. Results from several examples are given to demonstrate that the matrixfree approach is more than ten times faster than using iterative methods alone for circuits with as few as 400 equations.
Introduction to RF simulation and its application
 IEEE Journal of SolidState Circuits
, 1999
"... Abstract — RF circuits exhibit several distinguishing characteristics that make them difficult to simulate ..."
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Cited by 38 (9 self)
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Abstract — RF circuits exhibit several distinguishing characteristics that make them difficult to simulate
Efficient Methods for Simulating Highly Nonlinear MultiRate Circuits
, 1997
"... Widelyseparated time scales appear in many electronic circuits, making traditional analysis di#cult or impossible if the circuits are highly nonlinear. In this paper, an analytical formulation and numerical methods are presented for treating strongly nonlinear multirate circuits e#ectively. Multiv ..."
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Cited by 21 (5 self)
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Widelyseparated time scales appear in many electronic circuits, making traditional analysis di#cult or impossible if the circuits are highly nonlinear. In this paper, an analytical formulation and numerical methods are presented for treating strongly nonlinear multirate circuits e#ectively. Multivariate functions in the time domain are used to capture widely separated rates e#ciently, and a special partial differential equation #the MPDE# is shown to relate the multivariate forms of a circuit's signals. Timedomain and mixed frequencytime 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. 1 Introduction Consider a fast #e.g., 1Ghz# pulse train mixed with a slow #1Khz# sinusoid, or the same pulse train but with a slowlyvarying duty cycle. These are instances of multirat...
Fast Simulation Algorithms for RF Circuits
, 1996
"... RF integrated circuit designers make extensive use of simulation tools which perform nonlinear periodic steadystate analysis and its extensions. However, the computational costs of these simulation tools have restricted users from examining the detailed behavior of complete RF subsystems. Recent ..."
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Cited by 20 (3 self)
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RF integrated circuit designers make extensive use of simulation tools which perform nonlinear periodic steadystate analysis and its extensions. However, the computational costs of these simulation tools have restricted users from examining the detailed behavior of complete RF subsystems. Recent algorithmic de velopments, based on matriximplicit iterative methods, is rapidly changing this situation and providing new faster tools which can easily analyze circuits with hundreds of devices. In this paper we present these new methods by describing how they can be used to accelerate finitedifference, shootingNewton, and harmonic balance based algorithms for periodic steadystate analysis.
Simulation Methods for RF Integrated Circuits
 In Proceedings of ICCAD'97
, 1997
"... Abstract — The principles employed in the development of modern RF simulators are introduced and the various techniques currently in use, or expected to be in use in the next few years, are surveyed. Frequencyand timedomain techniques are presented and contrasted, as are steadystate and envelope t ..."
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Cited by 19 (1 self)
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Abstract — The principles employed in the development of modern RF simulators are introduced and the various techniques currently in use, or expected to be in use in the next few years, are surveyed. Frequencyand timedomain techniques are presented and contrasted, as are steadystate and envelope techniques and large and smallsignal techniques. I. RF CIRCUITS The increasing demand for lowcost mobile communication systems has greatly expanded the need for simulation algorithms that are both efficient and accurate when applied to RF communication circuits. RF circuits have several unique characteristics that are barriers to the application of traditional circuit simulation techniques. Over the last decade, researchers have developed many special purpose algorithms that overcome these barriers to provide practical simulation for RF circuits, often by exploiting the very characteristic that represented the barrier to traditional methods. Despite dramatic progress, the average design cycle of an RFIC is still twice the length of that for other ICs found in a communication system, such as a cellular phone. This represents a significant practical barrier to integration of the RF and baseband sections of a transceiver onto a single chip. Clearly, more progress is necessary. This paper is a overview of RF simulation methods that seeks to provide an understanding of how the various methods address the RF simulation problem, and how they relate to each other. It begins by describing the unique characteristics of RF circuits. The basic solution methods of transient analysis, harmonic balance, and shooting methods are presented and contrasted. Smallsignal analysis versions of both harmonic balance and shooting methods are covered. Composite methods are next. These methods apply the base
*PHDD: An Efficient Graph Representation for Floating Point Circuit Verification
 In Int'l Conf. on CAD
, 1997
"... Data structures such as *BMDs, HDDs, and K*BMDs provide compact representations for functions which map Boolean vectors into integer values, but not floating point values. In this paper, we propose a new data structure, called Multiplicative Power Hybrid Decision Diagrams (*PHDDs), to provide a comp ..."
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Cited by 13 (1 self)
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Data structures such as *BMDs, HDDs, and K*BMDs provide compact representations for functions which map Boolean vectors into integer values, but not floating point values. In this paper, we propose a new data structure, called Multiplicative Power Hybrid Decision Diagrams (*PHDDs), to provide a compact representation for functions that map Boolean vectors into integer or floating point values. The size of the graph to represent the IEEE floating point encoding is linear with the word size. The complexity of floating point multiplication grows linearly with the word size. The complexity of floating point addition grows exponentially with the size of the exponent part, but linearly with the size of the mantissa part. We applied *PHDDs to verify integer multipliers and floating point multipliers before the rounding stage, based on a hierarchical verification approach. For integer multipliers, our results are at least 6 times faster than *BMDs. Previous attempts at verifying floating point multipliers required manual intervention. We verified floating point multipliers before the rounding stage automatically.
Analysing circuits with widely separated time scales using numerical PDE methods
 IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications
, 2001
"... Abstract—Widely separated time scales arise in many kinds of circuits, e.g., switchedcapacitor filters, mixers, switching power converters, etc. Numerical solution of such circuits is often difficult, especially when strong nonlinearities are present. In this paper, we present a mathematical formul ..."
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Cited by 12 (3 self)
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Abstract—Widely separated time scales arise in many kinds of circuits, e.g., switchedcapacitor filters, mixers, switching power converters, etc. Numerical solution of such circuits is often difficult, especially when strong nonlinearities are present. In this paper, we present 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, timedomain methods can be used to handle the hitherto difficult case of strong nonlinearities together with widely separated rates of signal variation. We examine methods for obtaining quasiperiodic and envelope solutions, and describe how the PDE formulation unifies existing techniques for separatedtimeconstant problems. Several applications are described. Significant computation and memory savings result from using the new numerical techniques, which also scale gracefully with problem size. Index Terms—Multitime partial differential equations, widely separated time scales. I.
ComputerAided Circuit Analysis Tools for RFIC Simulation: Algorithms, Features, and Limitations
 IEEE Trans. on Circuits and Systems II: analog and digital signal processing
, 2000
"... The design of the radio frequency (RF) section in a communication integrated circuit (IC) is a challenging problem. Although several computeraided analysis tools are available for RFIC design, they are not effectively used, because there is a lack of understanding about their features and limitatio ..."
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Cited by 11 (3 self)
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The design of the radio frequency (RF) section in a communication integrated circuit (IC) is a challenging problem. Although several computeraided analysis tools are available for RFIC design, they are not effectively used, because there is a lack of understanding about their features and limitations. These tools provide fast simulation of RFIC's. However, no single tool delivers a complete solution for RFIC design. This paper describes the shortcomings of conventional SPICElike simulators and the analyses required for RF applications with an emphasis on accurate and efficient simulation of distortion and noise. Various analysis methods, such as harmonic balance, shooting method, mixed frequencytime methods, and envelope methods, that are currently available for RFIC simulation are presented. Commercial simulators are compared in terms of their functionalities and limitations. The key algorithmic features and the simulatorspecific terminology are described. Index TermsCircuit simulation, cyclostationary noise, distortion, envelope method, frequencydomain methods, harmonic distortion, intermodulation, linear timevarying analysis, mixed frequencytime methods, mixer noise, noise, periodic steadystate, phase noise, quasiperiodic steadystate, RFIC simulation, SPICE harmonic balance, shooting method, timedomain methods. I.
Independent and Interdependent Latch Setup/Hold Time Characterization via NewtonRaphson Solution and Euler Curve Tracking of StateTransition Equations
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 2008
"... Interdependent characterization of latch setup/hold times is a core component of techniques for pessimism reduction via Setup/Hold Interdependence Aware Static Timing Analysis (SHIASTA) [1], [2]. We present an efficient and novel method for such characterization, by formulating the interdependent s ..."
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Cited by 9 (0 self)
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Interdependent characterization of latch setup/hold times is a core component of techniques for pessimism reduction via Setup/Hold Interdependence Aware Static Timing Analysis (SHIASTA) [1], [2]. We present an efficient and novel method for such characterization, by formulating the interdependent setuphold time problem as an underdetermined nonlinear equation h(τs,τh) =0, which we derive from the latch’s statetransition function. We solve this equation numerically using a MoorePenrose Newton method. Further, we use nullspace information from the Newton’s Jacobian matrix to efficiently find constantclocktoQ contours (in the setup/hold time plane), via an EulerNewton curve tracing procedure. We validate the method on TSPC and C 2 MOS registers, obtaining speedups of more than 20 × over prior approaches while achieving superior accuracy. This speedup increases linearly with the precision with which curve tracing is desired. In view of the importance and large computational expense of latch characterization in industry today, the new technique represents a significant enabling technology for dramatically speeding up industrial timing closure flows.