Power-law noise Identification: Application in Timing

1998

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. In this paper, 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. 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.

IEEE Transactions on Instrumentation and Measurement, 1993

Oscillator noise is generally modeled by a power law spectral density. Thus it is possible to characterize different noise sources, each of them corresponding to a particular power law. The measurement of the contribution of these sources is necessary to know their origin and to remedy these causes in order to improve oscillator performance. Usually, an estimation of the different types of noise present in a signal is obtained by using a variance. However, the sensitivity of these variances differs for each type of noise and then limits this method. On the other hand, the use of several variances, each of them more sensitive to one type of noise, permits one to notably improve the measurement accuracy. The method suggested here uses as many different variances as there are types of noise to measure. The improvement of measurement accuracy of the noise coefficient is discussed in this paper.

In this work, time series analysis techniques are used to analyze sequential, equispaced mass measurements of a Si density artifact, collected from an electromechanical transducer. Specifically, techniques such as Power Spectral Density, Bretthorst periodogram, Allan variance and Modified Allan variance can provide much insight regarding the stochastic correlations that are induced on the outcome of an experiment by the measurement system and establish criteria for the limited use of the classical variance in metrology. These techniques are used in conjunction with power law models of stochastic noise in order to characterize time or frequency regimes by pointing out the different types of frequency modulated (FM) or phase modulated (PM) noise. In the case of phase noise, only Modified Allan variance can tell between white PM and flicker PM noise. Oscillations in the system can be detected accurately with the Bretthorst periodogram. Through the detection of colored noise, which is expected to appear in almost all electronic devices, a lower threshold of measurement uncertainty is obtained and the white noise model of statistical independence can no longer provide accurate results for the examined data set.

IEEE Transactions on Circuits and Systems I-regular Papers, 2000

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 brute-force Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly.

IEEE Signal Processing Letters, 2002

2013

A method for simulating power law noise in clocks and oscillators is presented based on modification of the spectrum of white phase noise, then Fourier transforming to the time domain. Symmetric real matrices are introduced whose traces-the sums of their eigenvalues-are equal to the Allan variances, in overlapping or non-overlapping forms, as well as for the corresponding forms of the modified Allan variance. Diagonalization of these matrices leads to expressions for the probability distributions for observing a variance at an arbitrary value of the sampling or averaging interval τ , and hence for estimating confidence in the measurements. A number of applications are presented for the common power-law noises.

Mixed Design of Integrated Circuits & …, 2009

In this paper a new approach of thermal noise analysis of electronic oscillators is presented. Although nonlinear electronic oscillators are one of the most essential subcircuits in electronic systems typical design concepts for these oscillators based on ideas of linear ...

1997

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. we present can be extended [1] for the MNA (Modified Nodal Analysis) formulation (i.e., DAE formulation) given by d=dt qx + f x = 0. 2 After any small disturbance that does not persist, the system asymptotically settles back to the original limit cycle. See [2] for a precise definition of this stability notion.

2019

We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of previous phase reduced models.

Arxiv preprint arXiv:1103.5062, 2011

A method for simulating power law noise in clocks and oscillators is presented based on modification of the spectrum of white phase noise, then Fourier transforming to the time domain. Symmetric real matrices are introduced whose traces-the sums of their eigenvalues-are equal to the Allan variances, in overlapping or non-overlapping forms, as well as for the corresponding forms of the modified Allan variance. Diagonalization of these matrices leads to expressions for the probability distributions for observing a variance at an arbitrary value of the sampling or averaging interval τ , and hence for estimating confidence in the measurements. A number of applications are presented for the common power-law noises.