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When observed from the ground, the frequency of the atomic clocks flying on the satellites of a Global Navigation Satellite System is referred to as apparent frequency, because it is observed through the on-board signal generation chain, the propagation path, the relativistic effects, the measurement system, and the clock estimation algorithm. As a consequence, the apparent clock frequency is affected by periodic variations of different origins such as, for example, the periodic component of the J2 relativistic effect, due to the oblateness of the earth, and the clock estimation errors induced by the orbital estimation errors. We present a detailed characterization of the periodic variations affecting the apparent frequency of the Galileo clocks, obtained by applying time–frequency analysis and other signal processing techniques on space clock data provided by the European Space Agency. In particular, we analyze one year of data from three Galileo Passive Hydrogen Masers, flying on two different orbital planes. Time–frequency analysis reveals how the spectral components of the apparent frequency change with time. For example, it confirms that the amplitude of the periodic signal due to the orbital estimation errors depends on the angle between the sun and the orbital plane. Moreover, it allows to find a more precise estimate of the amplitude of the J2 effect, in agreement with the prediction of the general theory of relativity, and it shows that such amplitude suddenly decreases when the corresponding relativistic correction is applied to the data, thus validating the analytical formula used for the correction.

The robustness and reliability of a reference time scale is becoming more and more crucial in many applications, particularly in global navigation satellite systems (GNSS) and in critical infrastructures with time synchronization needs. This paper proposes two steering algorithms aimed at generating a real-time time scale, the first based on a primary frequency standard (PFS), and the second on an ensemble of clocks. Extensive tests on real data from the Italian National Metrology Institute (INRIM) cesium fountain ITCsF2 and atomic clocks have been carried out. The results show that the PFS algorithm has higher performances, but it requires a cesium fountain, a prototype standard currently available in a few timing laboratories only, whereas the ensemble clock algorithm uses commercially available clocks. A special emphasis is put on the system robustness: data pre-processing and possible combinations of the proposed algorithms have been tested in order to cope with outage periods, while maintaining good performances in terms of stability and accuracy of the resulting time scale. Such algorithms can be used in several applications where a stable time reference is needed, such as for the generation of a local real-time realization of UTC, as well as for any GNSS reference time.

The frequency of atomic clocks flying on board the satellites of global navigation satellite systems, as can be observed on the ground, shows periodic variations of different origins. One of them is related to the J2 relativistic effect due to the Earth oblateness, whose amplitude is now measurable thanks to stable clocks like Galileo’s passive hydrogen masers. Orbital estimation errors are also translated into periodic frequency variations, whose amplitude changes in time and which are superimposed to the J2 signal. In this work we analyze the data of a Galileo satellite clock by using two different techniques, with a twofold objective: to characterize the most evident periodic variations affecting the apparent clock frequency; to estimate the amplitude of the J2 relativistic effect for a comparison with its theoretical value. This also represents a validation of the J2 correction that should be applied to the space clocks’ data, in order to improve the timekeeping/positioning performance of global navigation satellite systems.

Atomic clocks are key elements in several applications, such as global navigation satellite systems. Monitoring their behavior is therefore fundamental. Since during an anomaly the frequency spectrum of the signal made by the clock frequency measurements changes with time, we transform this signal into the time-frequency domain by computing the spectrogram. In the time-frequency domain, the spectrogram localizes and separates the anomalies, and it can be hence used to reveal the nonstationary structure of the clock signal, such as, for instance, the presence of time-varying sinusuoidal terms.

The Allan variance is the key measure for stability analysis, a fundamental tool to establish the performances of atomic clocks, which are often critical elements in many applications, such as global navigation satellite systems (GNSS). In some applications, only frequency measurements with numerous missing data are available. Because of missing data, the resulting Allan variance can be dramatically different from the expected stability behavior. We have developed an Allan variance estimator which corrects this divergence. We show the effectiveness of our estimator by applying it to frequency measurements with up to 94.4% of missing data.

Due to its small size, low weight, and low power consumption, the Rb atomic frequency standard (RAFS) is routinely the first choice for atomic timekeeping in space. Consequently, though the device has very good frequency stability (rivaling passive hydrogen masers), there is interest in uncovering the fundamental processes limiting its long-term performance, with the goal of improving the device for future space systems and missions. The ac Stark shift (i.e., light shift) is one of the more likely processes limiting the RAFS' long-term timekeeping ability, yet its manifestation in the RAFS remains poorly understood. In part, this comes from the fact that light-shift induced frequency fluctuations must be quantified in terms of the RAFS' light-shift coefficient and the output variations in the RAFS' rf-discharge lamp, which is a nonlinear inductively-couple plasma (ICP). Here, we analyze the light-shift effect for a family of 10 on-orbit Block-IIR GPS RAFS, examining decade-long records of their on-orbit f...

The stability of an atomic clock can change with time due to several factors, such as temperature, humidity, radiations, aging, and sudden breakdowns. The dynamic Allan variance, or DAVAR, is a representation of the time-varying stability of an atomic clock, and it can be used to monitor the clock behavior. Unfortunately, the computational time of the DAVAR grows very quickly with the length of the analyzed time series. In this article, we present a fast algorithm for the computation of the DAVAR, and we also extend it to the case of missing data. Numerical simulations show that the fast algorithm dramatically reduces the computational time. The fast algorithm is useful when the analyzed time series is long, or when many clocks must be monitored, or when the computational power is low, as happens onboard satellites and space probes.

When an anomaly occurs in an atomic clock, its stability and frequency spectrum change with time. The variation with time of the stability can be evaluated with the dynamic Allan variance. The variation with time of the frequency spectrum can be described with the spectrogram, a time–frequency distribution. We develop a method that uses the dynamic Allan variance and the spectrogram to detect and to identify the typical anomalies of an atomic clock. We apply the method to simulated data.

Atomic clocks are ultra-precise time references, and because of this fact in the past 20 years they have found a fundamental application in navigation problems. The error in the localization of the user is highly dependent on the clock stability: variations of few nanoseconds in the clock phase result in an increase of the localization error by a factor of meters. Unfortunately atomic clocks are nonstationary, and their stability changes with time. We have recently proposed the DAVAR, or dynamic Allan variance, a representation of the instantaneous stability of an atomic clock. In this paper we will discuss the definition of the DAVAR and we will apply it to simulated nonstationary data, to prove its validity in clock noise characterization.

The number of applications where precise clocks play a key role is steadily increasing, satellite navigation being the main example. Precise clock anomalies are hence critical events, and their characterization is a fundamental problem. When an anomaly occurs, the clock stability changes with time, and this variation can be characterized with the dynamic Allan variance (DAVAR). We obtain the DAVAR for a series of common clock anomalies, namely, a sinusoidal term, a phase jump, a frequency jump, and a sudden change in the clock noise variance. These anomalies are particularly common in space clocks. Our analytic results clarify how the clock stability changes during these anomalies.

We study the case of an atomic clock whose parameters are time dependent. This results in a time variation of the frequency spectrum of the clock error that we are able to track using time–frequency techniques. The clock is modelled by a standard stochastic differential equation, and several cases of time-varying coefficients are studied, including, amongst others, sudden and periodic variations. The time–frequency technique used to estimate the instantaneous power spectrum is the sliding Welch's estimator. We present numerical simulations that prove the effectiveness of the approach.

In Global Navigation Satellite Systems (GNSS) the role of atomic clocks is essential for the determination of the user position. Nowadays, several satellite navigation systems are operating and different clock technologies have been employed on board satellites, taking benefit of the improvements reached during the last tens of years. The analysis of GNSS clock performance is thus crucial for ensuring the GNSS positioning and timing capabilities. We performed an analysis on the clock estimates generated by the Information-Analytical Centre [1] (IAC) of the Russian Federal Space Agency, which provides a service similar to the one of the International GNSS Service (IGS). Therefore, we collected and processed the RINEX for clock files containing satellite clock estimates for GPS and GLONASS constellations, from 2008 to 2013: with our statistical tools [2] we performed an analysis of the performances of the different space clock technologies employed on board different GNSSs. In particular, we focused the attention on the frequency behaviour and the frequency stability, evaluating also the stationarity of these characteristics, with the aim of classifying and analysing the clock anomalies. In this paper, the attention is mostly focused on the change in clock noise variance.

Atomic clocks are fundamental elements of a Global Navigation Satellite System (GNSS). Currently, several GNSSs are operational and the different clock technologies employed onboard their satellites benefit from the technological improvements achieved during the last decades. To ensure the timing capabilities needed for correct positioning, the analysis of GNSS clock performances is essential. We have recently started a performance analysis for GPS and GLONASS clocks, by using the satellite clock estimates produced by the Information-Analytical Centre (IAC) of the Russian Federal Space Agency. In this paper we discuss a few preliminary results of this analysis. We analyze the time deviation, frequency deviation, and frequency stability of the Cesium and Rubidium clocks onboard three GPS and GLONASS satellites. The obtained results highlight the presence of two common space clock anomalies, namely, deterministic oscillations and frequency jumps. Our final goal is to build a detailed statistics of the clock anomalies for all GNSSs.

Frequency jumps are common anomalies in atomic clocks aboard navigation system satellites. These anomalous behaviors must be detected quickly and accurately to minimize the impact on user positioning. We develop a detector for frequency jumps based on the Kalman filter. Numerical simulations show that the detector is fast, with high probability of detection and low probability of false alarms. It also has a low computational cost because it takes advantage of the recursive nature of the Kalman filter. Therefore, it can be used in applications in which little computational power is available, such as aboard navigation system satellites.

Precise timing has several applications, and their number is steadily increasing. Atomic clocks are the main actors of this phenomenon, a consequence of the fact that they provide the most accurate time measurements currently available. By definition, the accuracy of a clock is its deviation from the Coordinated Universal Time (UTC), the time standard based on the International Atomic Time (TAI), which is obtained by averaging more than 300 atomic clocks throughout the world [1]. The most precise atomic clock presently available guarantees a time error in the determination of the second that is equivalent to±1s in 10 million years.

We propose an algorithm for the detection of frequency jumps in atomic frequency standards. The algorithm is based on the innovation generated by the Kalman filter. When the magnitude of the innovation is larger than a given threshold, an anomaly is detected. We characterize the performance of our method for the case of a Rubidium clock by estimating, through numerical simulations, the probability of false alarm, the probability of detection, and the latency time on a large number of numerical simulations.

Due to their stability atomic clocks represent the core of navigation systems such as GPS and the future European Galileo system. To identify possible anomalies, it is fundamental to detect when the clock stability varies with time. The dynamic Allan variance (DAVAR) makes this monitoring process possible. We extend the DAVAR to the case of a time series with missing data, and we analyze the presence of periodic behaviors, two common phenomena in space clocks.

This article reviews applications of the Kalman filter to atomic timing. The objectives of the article are twofold - present diverse applications and concepts in a consistent fashion, both in regard to notation and mathematical concepts. and develop the key ideas in a tutorial form by introducing the basic concepts and then applications.we apply the Kalman filter to clock estimation, clock monitoring, and time-scale definition. Furthermore, the GPS composite clock algorithm along with numerical simulations is described. Finally, the advantages and criticalities of the application of the Kalman filter to atomic timing, highlighting issues that are worth investigating are pointed out and on which the time and frequency community is currently working.

We present and discuss the dynamic Allan variance, a measure of the time-varying stability of an atomic clock. First, the dynamic Allan variance is mathematically defined, then its behavior is extensively tested on simulated and experimental data. The results prove the validity and the effectiveness of the proposed new tool.

The dynamic Allan variance (DAVAR) is a tool that allows to understand if the stability of an atomic clock is changing with time. Since an anomaly in the clock behavior generates a change in its stability, the DAVAR can be used to monitor the performances of a clock. In this paper we present a fast algorithm for the computation of the DAVAR, which outperforms the classical computational method.

The frequency spectrum of the atomic clock noise can change with time for several reasons. We develop a method to represent the time–frequency spectrum of clock noise. The method can detect and identify clock anomalies, since they generate non-stationarities in the time–frequency spectrum. We apply the method to numerical and experimental data.

The two-state model provides an effective description of the atomic clock noise. We present the construction of the two-state model, which is useful to understand the statistics of clock noise. Furthermore, we derive the discrete-time version of the model, which can be used for numerical simulations. We illustrate our results through numerical examples.

The stability of the atomic clocks on board the satellites of a navigation system should remain constant with time. In reality there are numerous physical phenomena that make the behavior of the clocks a function of time, and for this reason we have recently introduced the dynamic Allan variance (DAVAR), a measure of the time-varying stability of an atomic clock. In this paper we discuss the dynamic Allan variance for phase and frequency jumps, two common nonstationarities of atomic clocks. The analysis of both numerical simulations and experimental data proves that the dynamic Allan variance is an effective way of characterizing nonstationary behaviors of atomic clocks.

The dynamic Allan variance (DAVAR) method has been recently proposed as an extension of the classical Allan variance, with the aim of defining the instantaneous stability of an atomic clock. It is in fact known that atomic clocks may undergo sudden failures, are influenced by environmental factors, and they eventually age and stop working. Hence the stability of an atomic clock may vary with time, and in some demanding applications, such as navigation systems, a quick identification of clock failures is of the utmost importance. Working on the first preliminary data coming from the Galileo System Test Bed (GSTB) V2, some criticalities were encountered in the DAVAR evaluation and its graphical representation, when dealing with long periods of missing data. A new estimation method has been developed, allowing to evaluate the DAYAR in case of missing data, not equally spaced data and with large periods of missing observation. The new estimation method is presented here. The experimental results obtained by applying the improved DAVAR on the Galileo experimental satellite are also shown. Also experimental data obtained by using the two-way satellite time and frequency rransfer (TWSTFT) method are analyzed and discussed.

Atomic clocks are ultra-stable time references, and they have found fundamental applications in several fields, navigation being certainly one of the most successful. Unfortunately, atomic clocks may experience nonstationary behaviors which make their stability a function of time. Even when the change in stability is thought to be small, it can still force the system that uses the clock to fail the design requirements. In navigation systems this might turn into an unwanted degradation of the user localization error. It is therefore essential to monitor the stability of the clock, and to understand and interpret the possible nonstationary behaviors. To this aim, we have recently Introduced the dynamic Allan variance, or DAVAR, a quantity that allows to represent the evolution with time of the clock stability. In this paper we show several cases of nonstationary time series, made by the combination of random processes and deterministic signals, and we discuss their dynamic Allan variance. The ultimate goal is to be able to do the inverse operation, that is to understand and classify the possible nonstationarities directly from the DAVAR representation. Examples with experimental data are presented.

In this paper, the problem of identifying variations in the nature of atomic clock noise is addressed. Two methods are proposed. One method is based on a generalized likelihood ratio test (GLRT), and the other is based on the dynamic Allan variance (DAVAR), which is a representation of the instantaneous clock stability that is able to point out possible nonstationary behaviors. Both methods efficiently track variations in the experimental clock data and, thus, appear as suitable tools for the detection of atomic clock anomalies

In a previous paper [Galleani, 2003] we have introduced the concept of dynamic Allan variance, an extension of the classical Allan variance that is commonly used to evaluate the stability of atomic clocks. The Allan variance assumes the stationarity of the (increment of the) clock error signal, a condition that is valid for ideal clocks only. For real clocks one has to pay attention in the evaluation of the clock stability, because even for short time intervals the clock can exhibit a nonstationary behavior. Possible reasons for the lack of stationarity are sudden breakdowns, or, in the long term, clock ageing. Even cyclostationary behaviors can be observed due to daily or seasonal variation of temperature, humidity and other physical quantities that have a direct influence on the clock behavior. The main purpose of the dynamic Allan variance is to describe the variation in time of the clock stability. In this paper we give a mathematical definition of this quantity. We apply our method to simulated data and to real data coming from a rubidium clock. The results are very interesting, and they show that the proposed method can track and reveal in a clear and intuitive manner the changes in the behavior of atomic clock data

We introduce the dynamic Allan variance, a quantity that characterizes the variation in time of the stability of an atomic clock. We connect the dynamic Allan variance to the Wigner spectrum, a time-frequency representation that can reveal the time-varying frequencies generally present in the clock error noise under nonstationary conditions. We also propose a practical implementation of the dynamic Allan variance for quasi-stationary clock noises, and we show numerical results that prove the validity of our approach, both on simulated and real data.

A mathematical model for the clock phase and frequency deviation based on the theory of stochastic differential equations (SDEs) is discussed. In particular, we consider a model that includes what are called the `white and random walk frequency noises' in time metrology, which give rise in a mathematical context to a Wiener and an integrated Wiener process on the clock phase. Due to the particular simple expression of the functions involved an exact solution exists, and we determine it by considering the model in a wider theoretical context, which is suitable for generalizations to more complex instances. Moreover, we determine an iterative form for the solution, useful for simulation and further processing of clock data, such as filtering and prediction. The Euler method, generally applied in literature to approximate the solution of SDEs, is examined and compared to the exact solution, and the magnitude of the approximation is evaluated. The possible extension of the model to other noise sources, such as the flicker and white phase noises, is also sketched.