F. Vernotte

Oscillator noise analysis: multivariance measurement

Since the noise altering the output signal of oscillators may be modeled as power laws in the spectral density of frequency deviation, oscillator noise analysis is the measurement of the level of each power law noise. The principle of this new multivariance method consists of obtaining the noise-type contributions with different variances and different integration time values. All the data obtained from the different variances with the different integration times are then operated simultaneously. Thus, the most probable measurement, in the sense of least squares, is obtained for each type of noise. This method lends itself to an estimation of the uncertainty of the noise-type contribution measurement, taking into account the dispersion of the variance results. >

Enhancements to GPS operations and clock evaluations using a "total" Hadamard deviation

We describe a method based on the total deviation approach whereby we improve the confidence of the estimation of the Hadamard deviation that is used primarily in Global Positioning System (GPS) operations. The Hadamard-total deviation described in this paper provides a significant improvement in confidence indicated by an increase of 1.3 to 3.4 times the one degree of freedom of the plain Hadamard deviation at the longest averaging time. The new Hadamard-total deviation is slightly negatively biased with respect to the usual Hadamard deviation, and /spl tau/ values are restricted to less than or equal to T/3, to be consistent with the usual Hadamards definition. We give a method of automatically removing bias by a power-law detection scheme. We review the relationship between Kalman filter parameters and the Hadamard and Allan variances, illustrate the operational problems associated with estimating these parameters, and discuss how the Hadamard-total variance can improve management of present and future GPS satellite clocks.

On the measurement of frequency and of its sample variance with high-resolution counters

A frequency counter measures the input frequency nu averaged over a suitable time tau, versus the reference clock. Beside clock interpolation, modern counters improve the resolution by averaging multiple measurements highly overlapped. In the presence of white noise, the overlapping technique improves the square uncertainty from sigma nu 2 prop1/tau2 to sigmanu 2 prop1/tau. This is important because the input trigger integrates white noise over the full instrument bandwidth, which is usually of at least 100 MHz. Due to insufficient technical information, the general user is inclined to make the implicit assumption that the counter takes the bare mean. After explaining the overlapped-average mechanism, we prove that feeding a file of contiguous data into the formula of the two-sample (Allan) variance sigmay 2(tau) = E{frac12(ymacrk+1-ymacrk)2} gives the modified Allan variance mod sigmay 2 (tau). This conclusion is based on the mathematical reverse-engineering of the formulae found in technical specifications. More details are available on the web site arxiv.org, document arXiv:physics/0411227 (Rubiola, 2004), Our purpose is to warn the experimentalists against possible mistakes, and to encourage the manufacturers to explain what the instruments really do

Is the frequency noise of an oscillator of deterministic origin

We report on a new multifractal type approach of low-frequency noise in oscillators. The time sequence is interrogated by means of a multiscale local stability exponent which acts in a sliding filtering window. Its discontinuities are reflected into a binary coding in which the correlations are analyzed. The mapping of binary data into a devils staircase is used to emphasize the correlated and possibly deterministic origin of frequency fluctuations in quartz oscillators.

A new multi-variance method for the oscillator noise analysis

Since the noise altering the output signal of oscillators may be modeled as power laws in the spectral density of frequency deviation, the oscillator noise analysis is the measurement of the level of each power law noise. The principle of this new multivariance method consists in obtaining the noise type contributions with different variances and different integration time values. The data obtained from the different variances with the different integration times are then operated simultaneously. The most probable measurement, in the sense of the least squares is obtained for each type of noise. This method lends itself to estimation of uncertainty of the noise type contribution measurement, taking into account the dispersion of the variance results.<<ETX>>

Cutoff frequencies and noise power law model of spectral density: adaptation of the multivariance method for irregularly spaced timing data using the lowest-mode estimator approach

The concept of structure functions, which is an extension of the variance approach, is useful to determine the variance (the structure function) which is optimized for a type of noise and for an order of drift. The multivariance method was developed to use different variances over the same signal. It is then possible to select a set of variances in which each variance is optimized to the determination of one parameter (of one noise level, drift, or cutoff frequency). Recently, we adapted this method to irregularly spaced timing data. In this connection, we replaced the structure functions by another method of spectral density estimation: the lowest-mode estimator, introduced by J.E. Deeter and P.E. Boynton (1982, 1984) for the analysis of pulsar timing data. Different lowest-mode estimators can be constructed according to two priorities: the order of drifts that must be removed and the type of noise for which the sensitivity must be maximum. Thus, a multivariance system is developed using a set of different estimators. The details of this method are described, and the results for different signals are discussed in this paper.

Application of the moment condition to noise simulation and to stability analysis

It is well-known that low frequency noises (flicker FM and random walk FM) are not stationary; it is not possible to define either the mean value or the (true) variance. Therefore, the use of a stationary approach yields convergence problems unless a low cut-off frequency is introduced, the physical meaning of which is not clear. The moment condition explains the link between insensitivity to drifts and convergence for low frequency noises in a stationary approach. This condition may be summarized by the following consideration: the divergence effect of a low frequency noise for the lowest frequencies induces a false drift with random drift coefficients; the lower the low cut-off frequency, the higher the variance of the coefficients of this drift. These variances may be known by theoretical calculations. The order of the drift is directly linked to the power law of the noise. The moment condition is demonstrated and applied for creating new estimators (new variances) and for simulating low frequency noises with a very low cut-off frequency.

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This paper introduces the Ω counter, a frequency counter-i.e., a frequency-to-digital converter-based on the linear regression (LR) algorithm on time stamps. We discuss the noise of the electronics. We derive the statistical properties of the Ω counter on rigorous mathematical basis, including the weighted measure and the frequency response. We describe an implementation based on a system on chip, under test in our laboratory, and we compare the Ω counter to the traditional Π and Λ counters. The LR exhibits the optimum rejection of white phase noise, superior to that of the Π and Λ counters. White noise is the major practical problem of wideband digital electronics, both in the instrument internal circuits and in the fast processes, which we may want to measure. With a measurement time τ, the variance is proportional to 1/τ2 for the Π counter, and to 1/τ3 for both the Λ and Ω counters. However, the Ω counter has the smallest possible variance, 1.25 dB smaller than that of the Λ counter. The Ω counter finds a natural application in the measurement of the parabolic variance, described in the companion article in this Journal [vol. 63 no. 4 pp. 611-623, April 2016 (Special Issue on the 50th Anniversary of the Allan Variance), DOI 10.1109/TUFFC.2015.2499325].

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This paper introduces the parabolic variance (PVAR), a wavelet variance similar to the Allan variance (AVAR), based on the linear regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the Ω frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and modified AVAR (MVAR). PVAR is good for long-term analysis because the wavelet spans over 2τ, the same as the AVAR wavelet, and good for short-term analysis because the response to white and flicker PM is 1/τ3 and 1/τ2, the same as the MVAR. After setting the theoretical framework, we study the degrees of freedom and the confidence interval for the most common noise types. Then, we focus on the detection of a weak noise process at the transition - or corner - where a faster process rolls off. This new perspective raises the question of which variance detects the weak process with the shortest data record. Our simulations show that PVAR is a fortunate tradeoff. PVAR is superior to MVAR in all cases, exhibits the best ability to divide between fast noise phenomena (up to flicker FM), and is almost as good as AVAR for the detection of random walk and drift.

Counter, a Frequency Counter Based on the Linear Regression

We analyze the Allan variance estimator as the combination of discrete-time linear filters. We apply this analysis to the different variants of the Allan variance: the overlapping Allan variance, the modified Allan variance, the Hadamard variance and the overlapping Hadamard variance. Based upon this analysis, we present a new method to compute a new estimator of the Allan variance and its variants in the frequency domain. We show that the proposed frequency domain equations are equivalent to extending the data by periodization in the time domain. Like the total variance, which is based on extending the data manually in the time domain, our frequency domain variance estimators have better statistics than the estimators of the classical variances in the time domain. We demonstrate that the previous well-know equation that relates the Allan variance to the power spectrum density (PSD) of continuous-time signals is not valid for real world discrete-time measurements and we propose a new equation that relates the Allan variance to the PSD of the discrete-time signals and allows computation of the Allan variance and its different variants in the frequency domain.