Alper Demir

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.

Phase noise and timing jitter in oscillators with colored-noise sources

Phase noise or timing jitter in oscillators is of major concern in wireless and optical communications, being a major contributor to the bit-error rate of communication systems, and creating synchronization problems in other clocked and sampled-data systems. This paper presents the theory and practical characterization of phase noise in oscillators due to colored, as opposed to white, noise sources. Shot and thermal noise sources in oscillators can be modeled as white-noise sources for all practical purposes. The characterization of phase noise in oscillators due to shot and thermal noise sources is covered by a recently developed theory of phase noise due to white-noise sources. The extension of this theory and the practical characterization techniques to noise sources in oscillators, which have a colored spectral density, e.g., 1/f noise, is crucial for practical applications. In this paper, we first derive a stochastic characterization of phase noise in oscillators due to colored-noise sources. This stochastic analysis is based on a novel nonlinear perturbation analysis for autonomous systems, and a nonlocal Fokker-Planck equation we derive. Then, we calculate the resulting spectrum of the oscillator output with phase noise as characterized. We also extend our results to the case when both white and colored-noise sources are present. Our treatment of phase noise due to colored-noise sources is general, i.e., it is not specific to a particular type of colored-noise source.

Computing Timing Jitter From Phase Noise Spectra for Oscillators and Phase-Locked Loops With White and

Phase noise and timing jitter in oscillators and phase-locked loops (PLLs) are of major concern in wireless and optical communications. In this paper, a unified analysis of the relationships between time-domain jitter and various spectral characterizations of phase noise is first presented. Several notions of phase noise spectra are considered, in particular, the power-spectral density (PSD) of the excess phase noise, the PSD of the signal generated by a noisy oscillator/PLL, and the so-called single-sideband (SSB) phase noise spectrum. We investigate the origins of these phase noise spectra and discuss their mathematical soundness. A simple equation relating the variance of timing jitter to the phase noise spectrum is derived and its mathematical validity is analyzed. Then, practical results on computing jitter from spectral phase noise characteristics for oscillators and PLLs with both white (thermal, shot) and 1/f noise are presented. We are able to obtain analytical timing jitter results for free-running oscillators and first-order PLLs. A numerical procedure is used for higher order PLLs. The phase noise spectrum needed for computing jitter may be obtained from analytical phase noise models, oscillator or PLL noise analysis in a circuit simulator, or from actual measurements

1/f

1. Introduction. 2. Mathematical Background. 3. Noise Models. 4. Overview of Noise Simulation for Nonlinear Electronic Circuits. 5. Time-Domain Non-Monte Carlo Noise Simulation. 6. Noise in Free Running Oscillators. 7. Behavioral Modeling and Simulation of Phase-Locked Loops. 8. Conclusions and Future Work. References. Index.

Noise

This paper presents behavioral simulation techniques for phase/delay-locked systems. Numerical simulation algorithms are compared and the issue of numerical noise is discussed. Behavioral phase noise simulation for phase/delay-locked systems is described. The role of behavioral simulation for phase/delay-locked systems in our top-down constraint-driven design methodology, and in bottom-up verification of designs, is explained with examples. Accuracy and efficiency comparisons with other methods are made. Simulation techniques are described in the framework of phase/delay-locked systems, but simulation methodology and the results attained in this work are applicable to the behavioral simulation of mixed-mode nonlinear dynamic systems.<<ETX>>

Analysis and simulation of noise in nonlinear electronic circuits and systems

Oscillators are key components of electronic systems. Undesired perturbations, i.e. noise, in practical electronic systems adversely affect the spectral and timing properties of oscillators resulting in phase noise, which is a key performance limiting factor, being a major contributor to bit-error-rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled data systems. We first present a theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of differential algebraic equations (DAEs), which extends our recent results on perturbation analysis of autonomous ordinary differential equations (ODEs). In developing the above theory, we rely on novel results we establish for linear periodically time varying (LPTV) systems: Floquet theory for DAEs. We then use this nonlinear perturbation analysis to derive the stochastic characterization, including the resulting oscillator spectrum, of phase noise in oscillators due to colored (e.g., 1/f noise), as opposed to white noise sources. The case of white noise sources has already been treated by us in a recent publication (A. Demir et al., 1998). The results of the theory developed in this work enabled us to implement a rigorous and effective analysis and design tool in a circuit simulator for low phase noise oscillator design.

Behavioral simulation techniques for phase/delay-locked systems

Oscillators are key components of electronic systems. In RF communication systems, they are used for frequency translation of information signals and for channel selection, and in digital electronic systems, they are used as a time reference, i.e. a clock signal, in order to synchronize operations. Undesired perturbations in practical electronic systems adversely affect the spectral and timing properties of oscillators, which is a key performance limiting factor, being a major contributor to bit-error-rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled-data systems. Characterizing how perturbations affect oscillators is therefore crucial for practical applications. The traditional approach to analysing perturbed nonlinear systems (i.e. linearization) is not valid for oscillators. In this paper, we present a theory and efficient numerical methods, for non-linear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations (DAEs). Our techniques can be used in characterizing phase noise and timing jitter due to intrinsic noise in IC devices, and evaluating the effect of substrate and supply noise on the timing properties of practical oscillators. In this paper, we also establish novel results for periodically time-varying systems of linear DAEs, which we rely on in developing the above theory and the numerical methods. Copyright

Phase noise in oscillators: DAEs and colored noise sources

Gordon and Mollenauer, in their famous paper published in 1990, laid out how the interplay between the nonlinear Kerr effect in optical fibers and the amplified spontaneous-emission (ASE) noise from the optical-amplifiers results in enhanced levels of noise and degrades the performance of modulation schemes that encode information in, particularly, the phase of the optical carrier. This phenomenon has been termed as nonlinear phase noise in the literature. In this paper, we first present a comparative and critical review of previous techniques that have been proposed for the analysis of nonlinear phase noise by forming a classification framework that reveals some key underlying features. We then present a unifying theory and a comprehensive methodology and computational techniques for the analysis and characterization of nonlinear phase noise and its impact on system performance by building on and extending previous work that we identify as most favorable and systematic. In our treatment, we consider a multichannel multispan optically amplified dense wavelength-division multiplexed system and develop general techniques for the analysis of the intricate interplay among Kerr nonlinearity, chromatic dispersion, and ASE noise, and for computing the bit-error-ratio performance of differential phase-shift-keying (DPSK) systems. By means of the extensive results we present, we demonstrate and argue that correlated noise behavior plays a most significant role in understanding nonlinear phase noise and its impact on DPSK system performance.

Floquet theory and non-linear perturbation analysis for oscillators with differential-algebraic equations

The main effort in oscillator phase noise calculation lies in computing a vector function called the Perturbation Projection Vector (PPV). Current techniques for PAVE calculation use time domain numerics to generate the systems monodromy matrix, followed by full or partial eigenanalysis. We present a superior method that finds the PPV using only a single linear solution of the oscillators time- or frequency-domain steady-state Jacobian matrix. The new method is better suited for existing tools with fast harmonic balance or shooting capabilities, and also more accurate than explicit eigenanalysis. A key advantage is that it dispenses 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.