The Statistics of the Cross-Spectrum and the Spectrum Average: Generalization to Multiple Instruments.

This article addresses the measurement of the power spectrum of red noise processes at the lowest frequencies, where the minimum acquisition time is so long that it is impossible to average on a sequence of data record. Therefore, averaging is possible only on simultaneous observation of multiple instruments. This is the case of radio astronomy, which we take as the paradigm, but examples may be found in other fields such as climatology and geodesy. We compare the Bayesian confidence interval of the red noise parameter using two estimators, the spectrum average and the cross-spectrum. While the spectrum average is widely used, the cross-spectrum using multiple instruments is rather uncommon. With two instruments, the cross-spectrum estimator leads to the Variance-Gamma distribution. A generalization to q devices based on the Fourier transform of characteristic functions is provided, with the example of the observation of millisecond pulsars with five radio telescopes (RTs). The simulations show that the spectrum average is by a small amount more efficient than the cross-spectrum, chiefly when the background exceeds the signal. However, some notable differences between their upper limit indicate that it should be wiser to compute both estimators.

Cross-Spectrum Measurement Statistics: Uncertainties and Detection Limit.

The cross-spectrum method consists in measuring a signal c(t) simultaneously with two independent instruments. Each of these instruments contributes to the global noise by its intrinsic (white) noise, whereas the signal c(t) that we want to characterize could be a (red) noise. We first define the real part of the cross spectrum as a relevant estimator. Then, we characterize the probability density function (pdf) of this estimator knowing the noise level (direct problem) as a Variance-gamma (VG) distribution. Next, we solve the "inverse problem" due to Bayes' theorem to obtain an upper limit of the noise level knowing the estimate. Checked by massive Monte Carlo simulations, VG proves to be perfectly reliable for any number of degrees of freedom (DOFs). Finally, we compare this method with another method using the Karhunen-Loève transform (KLT). We find an upper limit of the signal level slightly different as the one of VG since KLT better considers the available information.

KLTS: A Rigorous Method to Compute the Confidence Intervals for the Three-Cornered Hat and for Groslambert Covariance.

The three-cornered hat/Groslambert Covariance (GCov) methods are widely used to estimate the stability of each individual clock in a set of three, but no method gives reliable confidence intervals for large integration times. We propose a new KLTS (Karhunen-Loève Tansform using Sufficient statistics) method which uses these estimators to consider the statistics of all the measurements between the pairs of clocks in a Bayesian way. The resulting cumulative density function (CDF) yields confidence intervals for each clock Allan variance (AVAR). This CDF provides also a stability estimator that is always positive. Checked by massive Monte Carlo simulations, KLTS proves to be perfectly reliable even for one degree of freedom. An example of experimental measurement is given.

Three-Cornered Hat and Groslambert Covariance: A First Attempt to Assess the Uncertainty Domains.

The three-cornered hat method and the Groslambert covariance are very often used to estimate the frequency stability of each individual oscillator in a set of three oscillators by comparing them in pairs. However, no rigorous method to assess the uncertainties over their estimates has yet been formulated. In order to overcome this lack, this paper will first study the direct problem, i.e., the calculation of the statistics of the clock stability estimates by assuming known values of the true clock stabilities and then will propose a first attempt to solve the inverse problem, i.e., the assessment of a confidence interval over the true clock stabilities by assuming known values of the clock stability estimates. We show that this method is reliable from 5 equivalent degrees of freedom (EDF) and beyond.

Frequency Stability Measurement of Cryogenic Sapphire Oscillators With a Multichannel Tracking DDS and the Two-Sample Covariance.

This paper shows the first measurement of three 100-MHz signals exhibiting fluctuations from 2×10-16 to parts in 10-15 for an integration time τ between 1 s and 1 day. Such stable signals are provided by three cryogenic sapphire oscillators (CSOs) operating at about 10 GHz, also delivering the 100-MHz output via a dedicated synthesizer. The measurement is made possible by a six-channel tracking direct digital synthesizer (TDDS) and the two-sample covariance tool, used to estimate the Allan variance. The use of two TDDS channels per CSO enables high rejection of the instrument background noise. The covariance outperforms the three-cornered hat (TCH) method in that the background converges to zero "out of the box," with no need of the hypothesis that the instrument channels are equally noisy, nor of more sophisticated techniques to estimate the background noise of each channel. Thanks to correlation and averaging, the instrument background (AVAR) rolls off with a slope 1/√m , the number of measurements, down to 10-18 at τ = 104 s. For consistency check, we compare the results to the traditional TCH method beating the 10-GHz outputs down to the megahertz region. Given the flexibility of the TDDS, our methods find an immediate application to the measurement of the 250-MHz output of the femtosecond combs.

The Ω Counter, a Frequency Counter Based on the Linear Regression.

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/τ(2) for the Π counter, and to 1/τ(3) 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].

Allan Variance Computed in Space Domain: Definition and Application to InSAR Data to Characterize Noise and Geophysical Signal.

The Allan variance was introduced 50 years ago for analyzing the stability of frequency standards. In addition to its metrological interest, it may be also considered as an estimator of the large trends of the power spectral density (PSD) of frequency deviation. For instance, the Allan variance is able to discriminate different types of noise characterized by different power laws in the PSD. The Allan variance was also used in other fields than time and frequency metrology: for more than 20 years, it has been used in accelerometry, geophysics, geodesy, astrophysics, and even finances. However, it seems that up to now, it has been exclusively applied for time series analysis. We propose here to use the Allan variance on spatial data. Interferometric synthetic aperture radar (InSAR) is used in geophysics to image ground displacements in space [over the synthetic aperture radar (SAR) image spatial coverage] and in time thanks to the regular SAR image acquisitions by dedicated satellites. The main limitation of the technique is the atmospheric disturbances that affect the radar signal while traveling from the sensor to the ground and back. In this paper, we propose to use the Allan variance for analyzing spatial data from InSAR measurements. The Allan variance was computed in XY mode as well as in radial mode for detecting different types of behavior for different space-scales, in the same way as the different types of noise versus the integration time in the classical time and frequency application. We found that radial Allan variance is the more appropriate way to have an estimator insensitive to the spatial axis and we applied it on SAR data acquired over eastern Turkey for the period 2003-2011. Spatial Allan variance allowed us to well characterize noise features, classically found in InSAR such as phase decorrelation producing white noise or atmospheric delays, behaving like a random walk signal. We finally applied the spatial Allan variance to an InSAR time series to detect when the geophysical signal, here the ground motion, emerges from the noise.

The Parabolic Variance (PVAR): A Wavelet Variance Based on the Least-Square Fit.

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/τ(3) and 1/τ(2), 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.

Statistical biases and very-long-term time stability analysis.

The prediction of very-long-term time stability is a key issue in various fields, such as time keeping, obviously, but also navigation and spatial applications. This is usually performed by extrapolating the measurement data obtained by estimators such as the Allan variance, modified Allan variance, Hadamard variance, etc. This extrapolation may be assessed from a fit over the variance estimates. However, this fit should be performed on the log-log graph of the estimates, which corresponds to a least-squares minimization of the relative difference between the variance estimates and the fitting curve. However, a bias exists between the average of the log of the estimates and the log of the true value of the estimated variance. This paper presents the theoretical calculation of this log-log bias based on the number of equivalent degrees of freedom of the estimates, shows simulations over a large number of realizations, and provides a reliable method of unbiased logarithmic fit. Extrapolating this fit yields a more confident assessment of the very-long-term time stability.

Stability variances: a filter approach.

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.

Composite clock including a Cs clock, an H-maser clock, and a voltage-controlled oscillator.

This article presents the project of creating a composite clock, the advances in its realization and the technical choices. The goal is to generate an output signal which combines the advantages of each clock, the long-term stability for the cesium clock, the mid-term stability for the hydrogen maser clock, and the short-term stability for the voltage-controlled oscillator (VCO). The control system is designed to transfer the stability of the best clock at a given averaging time to the VCO. This system is designed to reach a relative instability of approximately 10(-14) at 1 s, 10(-15) at 10(3) s, and 10(-14) at 10(6) s at 100 MHz, depending on the stability of the master clocks. To make this composite clock, we use a digital PLL with 2 references.

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 tau values are restricted to less than or equal to T/3, to be consistent with the usual Hadamard's 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.

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. As an example, in the case of random walk FM, the mean frequency of an oscillator does not converge if the analysis duration tends toward infinity. However, linear drifts appear if a phase sequence of random walk FM is observed over a duration smaller than the inverse of its low cut-off frequency. Moreover, the estimators, which are devoted to these non-stationary processes (i.e., the Hadamard variance), are insensitive to linear frequency drifts and converge for lower frequency noises (f(-4) FM). 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 will be demonstrated and applied for creating new estimators (new variances) and for simulating low frequency noises with a very low cut-off frequency.