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Journal Article•DOI•

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

Alper Demir1•
Koç University1
01 Dec 2002-IEEE Transactions on Circuits and Systems I-regular Papers (IEEE)-Vol. 49, Iss: 12, pp 1782-1791
TL;DR: In this paper, a stochastic characterization of phase noise in oscillators due to colored noise sources is presented, and the resulting spectrum of the oscillator output with phase noise as characterized.
Abstract: 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.
Citations
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Journal Article•DOI•
An On-Chip CMOS Relaxation Oscillator With Voltage Averaging Feedback

[...]

Yusuke Tokunaga1, Shiro Sakiyama1, Akinori Matsumoto1, Shiro Dosho1•
Panasonic1
07 Jun 2010-IEEE Journal of Solid-state Circuits
TL;DR: An on-chip CMOS relaxation oscillator with voltage averaging feedback using a reference proportional to supply voltage is presented and achieves 7x reduction in accumulated jitter (at 1500th cycle) as compared to a oscillator without VAF.
Abstract: An on-chip CMOS relaxation oscillator with voltage averaging feedback using a reference proportional to supply voltage is presented. A voltage-averaging feedback (VAF) concept is proposed to overcome conventional relaxation oscillator problems such as sensitivity to comparator delay, aging, and flicker noise of current sources. A test-chip with typical frequency of 14.0 MHz was fabricated in a 0.18 μm standard CMOS process and measured frequency variations of ±0.16 % for supply changes from 1.7 to 1.9 V and ±0.19% for temperature changes from -40 to 125°C. The prototype draws 25 μA from a 1.8 V supply, occupies 0.04 mm2, and achieves 7x reduction in accumulated jitter (at 1500th cycle) as compared to a oscillator without VAF.

179 citations

Cites background from "Phase noise and timing jitter in os..."

  • ...A Lorentzian-like finite flat level of phase noise spectrum near the carrier frequency is well known [10], [11], [12], which originates from the fact that the total power of oscillation is invariant....

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Journal Article•DOI•
Computing Timing Jitter From Phase Noise Spectra for Oscillators and Phase-Locked Loops With White and $1/f$ Noise

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Alper Demir1•
Koç University1
18 Sep 2006-IEEE Transactions on Circuits and Systems I-regular Papers
TL;DR: A unified analysis of the relationships between time-domain jitter and various spectral characterizations of phase noise is presented and 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.
Abstract: 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

177 citations

Cites background or methods from "Phase noise and timing jitter in os..."

  • ...[1], [2] proceeds without postulating that the phase noise in the oscillator can be modeled as the output of an ideal integrator....

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  • ...For the spectrum of the oscillator away from the carrier, the white noise sources contribute a term that has a frequency dependence, and the colored noise sources contribute terms that have a frequency dependence as multiplied with the spectral density of the colored noise source [2], [4]....

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  • ...We were able to make this observation based on the results obtained in [1], [2] summarized before....

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  • ...The derivation of the above results and the numerical techniques for computing these coefficients have been covered extensively in [1] and [2]....

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  • ...With timing noise characterized as above, the oscillator output with phase noise is a stationary process, and its spectral density is given by [2]...

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Journal Article•DOI•
Phase Noise in LC Oscillators: A Phasor-Based Analysis of a General Result and of Loaded $Q$

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David Murphy1, Jacob Rael2, Asad A. Abidi1•
University of California, Los Angeles1, Broadcom2
01 Jun 2010-IEEE Transactions on Circuits and Systems I-regular Papers
TL;DR: This work analyzes the negative-gm LC model and presents a simple equation that quantifies output noise resulting from phase fluctuations, and derives an expression for output Noise resulting from amplitude fluctuations.
Abstract: Recent work by Bank, and Mazzanti and Andreani has offered a general result concerning phase noise in nearly-sinusoidal inductance-capacitance (LC) oscillators; namely that the noise factor of such oscillators (under certain achievable conditions) is largely independent of the specific operation of individual transistors in the active circuitry. Both use the impulse sensitivity function (ISF). In this work, we show how the same result can be obtained by generalizing the phasor-based analysis. Indeed, as applied to nearly-sinusoidal LC oscillators, we show how the two approaches are equivalent. We analyze the negative-gm LC model and present a simple equation that quantifies output noise resulting from phase fluctuations. We also derive an expression for output noise resulting from amplitude fluctuations. Further, we extend the analysis to consider the voltage-biased LC oscillator and fully differential CMOS LC oscillator, for which the Bank's general result does not apply. Thus we quantify the concept of loaded Q.

132 citations

Cites background from "Phase noise and timing jitter in os..."

  • ...While lacking the rigor of mathematically involved analyses [2]–[ 4 ], the linear-time variant (LTV) approach to analyzing noise in oscillators has gained the most traction in the circuit design community....

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Journal Article•DOI•
Calculation of the Performance of Communication Systems From Measured Oscillator Phase Noise

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M. Reza Khanzadi1, Dan Kuylenstierna1, Ashkan Panahi1, Thomas Eriksson1, Herbert Zirath1 •
Chalmers University of Technology1
06 Jan 2014-IEEE Transactions on Circuits and Systems
TL;DR: A direct connection between oscillator measurements and optimal communication system performance, in terms of measured single-side band PN spectrum, and the resulting error vector magnitude (EVM) due to PN, is mathematically derived and analyzed.
Abstract: Oscillator phase noise (PN) is one of the major problems that affect the performance of communication systems. In this paper, a direct connection between oscillator measurements, in terms of measured single-side band PN spectrum, and the optimal communication system performance, in terms of the resulting error vector magnitude (EVM) due to PN, is mathematically derived and analyzed. First, a statistical model of the PN, considering the effect of white and colored noise sources, is derived. Then, we utilize this model to derive the modified Bayesian Cramer-Rao bound on PN estimation, and use it to find an EVM bound for the system performance. Based on our analysis, it is found that the influence from different noise regions strongly depends on the communication bandwidth, i.e., the symbol rate. For high symbol rate communication systems, cumulative PN that appears near carrier is of relatively low importance compared to the white PN far from carrier. Our results also show that 1/f 3 noise is more predictable compared to 1/f 2 noise and in a fair comparison it affects the performance less.

122 citations

Cites background from "Phase noise and timing jitter in os..."

  • ...A true Wiener process does not take into account colored (correlated) noise sources [36] and cannot describe frequency and time-domain properties of PN properly [35], [37], [38]....

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  • ...PN modeling has been investigated extensively in the circuits and systems community over the past decades [34], [36], [39]–[49]....

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Journal Article•DOI•
A Spectral Model for RF Oscillators With Power-Law Phase Noise

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Arsenia Chorti1, Mike Brookes2•
University of Southampton1, Imperial College London2
18 Sep 2006-IEEE Transactions on Circuits and Systems I-regular Papers
TL;DR: It is shown that despite its lack of stationarity it is possible to derive a closed form expression for its effect on an oscillator PSD and that the oscillator output can be considered to be wide-sense stationary.
Abstract: In this paper, we apply correlation theory methods to obtain a model for the near-carrier oscillator power-spectral density (PSD). Based on the measurement-driven representation of phase noise as a sum of power-law processes, we evaluate closed form expressions for the relevant oscillator autocorrelation functions. These expressions form the basis of an enhanced oscillator spectral model that has a Gaussian PSD at near-carrier frequencies followed by a sequence of power-law regions. New results for the effect of white phase noise, flicker phase noise and random walk frequency modulated phase noise on the near-carrier oscillator PSD are derived. In particular, in the case of 1/f phase noise, we show that despite its lack of stationarity it is possible to derive a closed form expression for its effect on an oscillator PSD and show that the oscillator output can be considered to be wide-sense stationary

117 citations

Cites methods from "Phase noise and timing jitter in os..."

  • ...This approach was adopted in [18] for the analysis of noise sources in the phase of an oscillator....

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References
More filters
Book•
The Fokker-Planck equation

[...]

Hannes Risken
01 Jan 1984

5,597 citations

"Phase noise and timing jitter in os..." refers methods in this paper

  • ...The partial integro-differential equation (14) for the timevarying marginal PDF of is a generalization of a partial differential equation known as the Fokker‐Planck equation [4], [ 5 ] derived for the PDF of satisfying (10) when is a white-noise process, which is given as...

    [...]

Book•
Analysis and Design of Analog Integrated Circuits

[...]

Paul R. Gray, Robert G. Meyer
01 Jan 1977
TL;DR: In this article, the authors combine bipolar, CMOS and BiCMOS analog integrated circuits into a unified treatment that stresses their commonalities and highlights their differences, and provide valuable insights into the relative strengths and weaknesses of these important technologies.
Abstract: The Fifth Edition of this academically rigorous text provides a comprehensive treatment of analog integrated circuit analysis and design starting from the basics and through current industrial practices. The authors combine bipolar, CMOS and BiCMOS analog integrated-circuit design into a unified treatment that stresses their commonalities and highlights their differences. The comprehensive coverage of the material will provide the student with valuable insights into the relative strengths and weaknesses of these important technologies.

4,717 citations

"Phase noise and timing jitter in os..." refers background or methods in this paper

  • ...The waveform for base‐emitter voltage is used in calculating the contribution of the and the burst noise source connected between the base and the emitter of the transistor in its noise model [ 6 ]....

    [...]

  • ...The source of burst noise is not fully understood, although it has been shown to be related to the presence of heavy-metal ion contamination [ 6 ]....

    [...]

  • ...where is a constant for a particular device, is the current through the device, is a constant in the range 0.5 to 2, and is the 3-dB bandwidth of the Lorentzian spectrum [ 6 ]....

    [...]

Book•
Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences

[...]

Crispin W. Gardiner
01 Jan 1985
TL;DR: The Handbook of Stochastic Methods as mentioned in this paper covers the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes.
Abstract: The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes Strong emphasis is placed on systematic approximation methods for solving problems Stochastic adiabatic elimination is newly formulated The book contains the "folklore" of stochastic methods in systematic form and is suitable for use as a reference work

3,261 citations

Journal Article•
Handbook of stochastic methods for physics, chemistry and the natural sciences, second edition

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C. W. Gardiner
01 Sep 1986-Applied Optics
TL;DR: The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes.
Abstract: The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes. Strong emphasis is placed on systematic approximation methods for solving problems. Stochastic adiabatic elimination is newly formulated. The book contains the \"folklore\" of stochastic methods in systematic form and is suitable for use as a reference work.

2,383 citations

Journal Article•DOI•
Phase noise in oscillators: a unifying theory and numerical methods for characterization

[...]

Alper Demir1, Amit Mehrotra2, Jaijeet Roychowdhury1•
Alcatel-Lucent1, University of Illinois at Urbana–Champaign2
01 May 2000-IEEE Transactions on Circuits and Systems I-regular Papers
TL;DR: In this paper, the authors developed a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism, and established novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random.
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 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.

1,226 citations

"Phase noise and timing jitter in os..." refers background or methods or result in this paper

  • ...A brief review of previous work on phase noise is given in [ 1 ]....

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  • ...We carried out a rigorous analysis of this case in [ 1 ] and obtained the following results....

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  • ...In recent publications [ 1 ] and [2], we presented a theory and numerical methods for practical characterization of phase noise in oscillators with white-noise sources....

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  • ...In [ 1 ], we concretized the above observation for white-noise perturbations, which we summarize in Section III....

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  • ...We developed numerical methods [ 1 ], [2], both in time and frequency domain, for the efficient computation of the periodic Floquet vector in (3), which is sufficient to characterize both spectral spreading and timing jitter in oscillators....

    [...]

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