Statistics of atomic frequency standards
01 Feb 1966-Vol. 54, Iss: 2, pp 221-230
TL;DR: In this paper, a theoretical analysis of the relationship between the expectation value of the standard deviation of the frequency fluctuations for any finite number of data samples and the infinite time average value of a standard deviation is presented.
Abstract: A theoretical development is presented which results in a relationship between the expectation value of the standard deviation of the frequency fluctuations for any finite number of data samples and the infinite time average value of the standard deviation, which provides an invariant measure of an important quality factor of a frequency standard. A practical and straightforward method of determining the power spectral density of the frequency fluctuations from the variance of the frequency fluctuations, the sampling time, the number of samples taken, and the dependence on system bandwidth is also developed. Additional insight is also given into some of the problems that arise from the presence of "flicker noise" (spectrum proportional to |ω|-1) modulation of the frequency of an oscillator. The theory is applied in classifying the types of noise on the signals of frequency standards made available at NBS, Boulder Laboratories, such as: masers (both H and N15H 3 ), the cesium beam frequency standard employed as the U. S. Frequency Standard, and rubidium gas cells. "Flicker noise" frequency modulation was not observed on the signals of masers for sampling times ranging from 0.1 second to 4 hours. In a comparison between the NBS hydrogen maser and the NBS III cesium beam, uncorrelated random noise was observed on the frequency fluctuations for sampling times extending to 4 hours; the fractional standard deviations of the frequency fluctuations were as low as 5 parts in 1014.
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01 Feb 2006TL;DR: Wavelet analysis of finite energy signals and random variables and stochastic processes, analysis and synthesis of long memory processes, and the wavelet variance.
Abstract: 1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.
2,734 citations
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Abstract: "Quantum sensing" describes the use of a quantum system, quantum properties or quantum phenomena to perform a measurement of a physical quantity Historical examples of quantum sensors include magnetometers based on superconducting quantum interference devices and atomic vapors, or atomic clocks More recently, quantum sensing has become a distinct and rapidly growing branch of research within the area of quantum science and technology, with the most common platforms being spin qubits, trapped ions and flux qubits The field is expected to provide new opportunities - especially with regard to high sensitivity and precision - in applied physics and other areas of science In this review, we provide an introduction to the basic principles, methods and concepts of quantum sensing from the viewpoint of the interested experimentalist
1,878 citations
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TL;DR: A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced and is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions.
Abstract: A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.
1,034 citations
Cites background from "Statistics of atomic frequency stan..."
...There is, however, a time-domain estimator with better properties known as theAllan variance[7], which consists in measuring the expectation of squared difference of averages of the data within windows of length...
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TL;DR: In this paper, the authors review the frequency-domain description of a mode-locked laser and the connection between the pulse phase and the frequency spectrum in order to provide a basis for understanding how the absolute frequencies can be determined and controlled.
Abstract: Recently there has been a remarkable synergy between the technologies of precision laser stabilization and mode-locked ultrafast lasers. This has resulted in control of the frequency spectrum produced by mode-locked lasers, which consists of a regular comb of sharp lines. Thus such a controlled mode-locked laser is a ``femtosecond optical frequency comb generator.'' For a sufficiently broad comb, it is possible to determine the absolute frequencies of all of the comb lines. This ability has revolutionized optical frequency metrology and synthesis. It has also served as the basis for the recent demonstrations of atomic clocks that utilize an optical frequency transition. In addition, it is having an impact on time-domain applications, including synthesis of a single pulse from two independent lasers. In this Colloquium, we first review the frequency-domain description of a mode-locked laser and the connection between the pulse phase and the frequency spectrum in order to provide a basis for understanding how the absolute frequencies can be determined and controlled. Using this understanding, applications in optical frequency metrology and synthesis and optical atomic clocks are discussed. This is followed by a brief overview of how the comb technology is affecting and will affect time-domain experiments.
965 citations
Cites methods from "Statistics of atomic frequency stan..."
...The time record of the beat frequency can be used to determine the Allan variance (Allan, 1966) that displays the frequency noise vs its characteristic time scales....
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TL;DR: A detailed second-order analysis is carried out for wavelet coefficients of FBM, revealing a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of F BM can be estimated.
Abstract: Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out. >
934 citations
References
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TL;DR: The theory of generalised functions and their Fourier transforms is discussed in this paper. But the analysis of Fourier transform is limited to the case of generalized functions, and it is not suitable for generalised function analysis.
Abstract: 1. Introduction 2. The theory of generalised functions and their Fourier transforms 3. Definitions, properties and Fourier transforms of particular generalised functions 4. The asymptotic estimation of Fourier transforms 5. Fourier series Index.
1,327 citations
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01 Feb 1966TL;DR: In this article, the effects of finite observation time on the frequency and phase stability of a servo-controlled oscillator with respect to a given quartz oscillator and an atomic reference are analyzed.
Abstract: Precision quartz oscillators have three main sources of noise contributing to frequency fluctuations: thermal noise in the oscillator, additive noise contributed by auxiliary circuitry such as AGC, etc., and fluctuations in the quartz frequency itself as well as in the reactive elements associated with the crystal, leading to an f-1type of power spectral density in frequency fluctuations. Masers are influenced by the first two types of noise, and probably also by the third. The influence of these sources of noise on frequency fluctuation vs. averaging time measurements is discussed. The f-1-spectral density leads to results that depend on the length of time over which the measurements are made. An analysis of the effects of finite observation time is given. The characteristics of both passive and active atomic standards using a servo-controlled oscillator are discussed. The choice of servo time constant influences the frequency fluctuations observed as a function of averaging time and should be chosen for best performance with a given quartz oscillator and atomic reference. The conventional methods of handling random signals, i.e., variances, autocorrelation, and spectral densities, are applied to the special case of frequency and phase fluctuations in oscillators, in order to obtain meaningful criteria for specifying oscillator frequency stability. The interrelations between these specifications are developed in the course of the paper.
388 citations
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01 Feb 1966TL;DR: In this paper, a detailed analysis of the calibration procedure showed that the third finite difference of the phase is closely related to the clock errors and that quartz crystal oscillators exhibit a "flicker" or |ω|-1type of noise modulating the frequency of the oscillator.
Abstract: Since most systems that generate atomic time employ quartz crystal oscillators to improve reliability, it is essential to determine the effect on the precision of time measurements that these oscillators introduce. A detailed analysis of the calibration procedure shows that the third finite difference of the phase is closely related to the clock errors. It was also found, in agreement with others, that quartz crystal oscillators exhibit a "flicker" or |ω|-1type of noise modulating the frequency of the oscillator. The method of finite differences of the phase is shown to be a powerful means of classifying the statistical fluctuations of the phase and frequency for signal generators in general. By employing finite differences it is possible to avoid divergences normally associated with flicker noise spectra. Analysis of several cesium beam frequency standards have shown a complete lack of the |ω|-1type of noise modulation.
89 citations
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01 Jan 1958
83 citations
01 Jun 1972
TL;DR: The method of finite differences of the phase is shown to be a powerful means of classifying the statistical fluctuations of thephase and frequency for signal generators in general and by employing finite differences it is possible to avoid divergences normally associated with flicker noise spectra.
Abstract: Since most systems that generate atomic time employ quartz crystal oscillators to improve reliability, it is essential to determine the effect on the precision of time measurements that these oscillators introduce. A detailed analysis of the calibration procedure shows that the third finite difference of the phase is closely related to the clock errors. It was also found, in agreement with others, that quartz crystal oscillators exhibit a "flicker" or |ω|-1type of noise modulating the frequency of the oscillator. The method of finite differences of the phase is shown to be a powerful means of classifying the statistical fluctuations of the phase and frequency for signal generators in general. By employing finite differences it is possible to avoid divergences normally associated with flicker noise spectra. Analysis of several cesium beam frequency standards have shown a complete lack of the |ω|-1type of noise modulation.
80 citations