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We demonstrate time and frequency transfer using a White Rabbit (WR) time transfer system over millimeter-wave (mm-wave) 71–76 GHz carriers. To validate the performance of our system, we present overlapping Allan deviation (ADEV), time deviation (TDEV), and phase statistics. Over mm-wave carriers, we report an ADEV of ${7.1}\times {10}^{-{12}}$ at 1 s and a TDEV of

We investigate the application of Ramsey spectroscopy for the development of a microcell atomic clock based on coherent population trapping (CPT). The dependence of the central Ramsey-CPT fringe properties on key experimental parameters is first studied for optimization of the clock’s short-term frequency stability. The sensitivity of the clock frequency to light-shift effects is then studied. In comparison with the continuous-wave (CW) regime case, the sensitivity of the clock frequency to laser power variations is reduced by a factor up to 14 and 40.3 for dark times of 150 and $450~\mu \text{s}$ , respectively, at the expense of intensity 3.75 times higher for short-term stability optimization. The dependence of the clock frequency on the microwave power is also reduced in the Ramsey case. We demonstrate that the Ramsey-CPT interrogation improves the clock Allan deviation for averaging times higher than 100 s. With a dark time of $450~\mu \text{s}$ , a clock fractional frequency stability of $3.8 \times 10^{-12}$ at 104 s is obtained, in comparison with the level of $8\times 10^{-11}$ obtained in the standard CW case, in similar environmental conditions. These results demonstrate that Ramsey-based interrogation protocols might be an attractive approach for the development of chip-scale atomic clocks (CSACs) with enhanced mid- and long-term stability.

This paper investigates the ultimate limits of White Rabbit (WR), an high-accuracy time distribution system based on field-programmable gate array (FPGA). The knowledge of such limits is essential for new emerging applications that are evaluating WR. In this paper, we identify and study the key elements in the WR synchronization: the digital dual mixer time difference phase detector and the Gigabit Ethernet transceiver. The benchmarks and experimental analysis of these key elements allow us to determine the WR switch (WRS) performance limits and evaluate their evolution with newer FPGAs. The identified performance limits are achievable by the present-day generation of WRS. The ultimate limits of short-term synchronization performance due to FPGA implementation have been derived through analysis and then demonstrated using the existing WRS enhanced with an additional daughterboard. This combination (WRS and daughterboard) achieves a tenfold improvement in terms of phase noise, jitter, and short-term stability with respect to the current WR performance. Both phase detectors and Gigabit transceivers have a similar phase noise contribution equal to a short-term stability of modified Allan deviation 4E−13 at $\tau = 1$ s (dominated by flicker phase modulation noise).

Laser intensity noise is currently recognized as one of the main factors limiting the short-term stability of vapor-cell clocks. In this article, we propose a signal theory approach to estimate the contribution of the laser intensity fluctuations to the short-term stability of vapor-cell clocks working in a pulsed regime. Specifically, given the laser intensity noise spectrum, an analytical expression is derived to evaluate its impact on the clock Allan deviation (ADEV). The theory has been tested for two intensity noise spectra of interest in clock applications: white frequency noise and flicker noise. The predicted results turn out to be in good agreement with experiments performed with a prototype of pulsed optically pumped Rb-cell clock, and can be extended to other compact clocks.

In this paper, we have presented the possibility of time and frequency (T&F) distribution in two generations of dense wavelength-division-multiplexing (DWDM) networks: the older one, equipped with dispersion compensation fiber (DCF) modules, and the newest, without in-line chromatic dispersion compensation (dedicated for coherent signals). The experiments were performed in a 1500-km loop arranged in the PIONIER production network, with T&F signals regarded as so-called “alien wavelength” network service. In the newest DWDM version, we observed very good stability of delivered signals: modified Allan deviation approach 10−16 for averaging longer than 104 s (for 10-MHz frequency signal), and time deviation below 15 ps for averaging up to 105 s for 1 PPS time signal. These results show that the DWDM alien wavelength service can be used for high-demanding applications like cesium fountains comparisons. Results achieved for the former version of DWDM were about one magnitude worse for a long-term comparison, but it can still be useful for less demanding applications. We found that the main reason for relatively poor results observed in the older generation of DWDM is the impact of the DCFs used in this DWDM approach.

This paper introduces the $\Omega $ 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 $\Omega $ 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 $\Omega $ counter to the traditional $\Pi $ and $\Lambda $ counters. The LR exhibits the optimum rejection of white phase noise, superior to that of the $\Pi $ and $\Lambda $ 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 $\tau $ , the variance is proportional to $1/\tau ^{2}$ for the $\Pi $ counter, and to $1/\tau ^{3}$ for both the $\Lambda $ and $\Omega $ counters. However, the $\Omega $ counter has the smallest possible variance, 1.25 dB smaller than that of the $\Lambda $ counter. The $\Omega $ 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].

Optical fiber links are known as the most performing tools to transfer ultrastable frequency reference signals. However, these signals are affected by phase noise up to bandwidths of several kilohertz and a careful data processing strategy is required to properly estimate the uncertainty. This aspect is often overlooked and a number of approaches have been proposed to implicitly deal with it. Here, we face this issue in terms of aliasing and show how typical tools of signal analysis can be adapted to the evaluation of optical fiber links performance. In this way, it is possible to use the Allan variance (AVAR) as estimator of stability and there is no need to introduce other estimators. The general rules we derive can be extended to all optical links. As an example, we apply this method to the experimental data we obtained on a 1284-km coherent optical link for frequency dissemination, which we realized in Italy.

A method is presented for synchronizing the time of a clock to a remote time standard when the channel connecting the two has significant delay variation that can be described only statistically. The method compares the Allan deviation of the channel fluctuations to the free-running stability of the local clock, and computes the optimum interval between requests based on one of the three selectable requirements: 1) choosing the highest possible accuracy; 2) choosing the best tradeoff of cost versus accuracy; or 3) minimizing the number of requests to realize a specific accuracy. Once the interval between requests is chosen, the final step is to steer the local clock based on the received data. A typical adjustment algorithm, which supports both the statistical considerations based on the Allan deviation comparison and the timely detection of errors, is included as an example.

The Allan variance (AVAR) was introduced 50 years ago as a statistical tool for assessing the frequency standards deviations. For the past decades, AVAR has increasingly been used in geodesy and astrometry to assess the noise characteristics in geodetic and astrometric time series. A specific feature of astrometric and geodetic measurements, as compared with clock measurements, is that they are generally associated with uncertainties; thus, an appropriate weighting should be applied during data analysis. In addition, some physically connected scalar time series naturally form series of multidimensional vectors. For example, three station coordinates time series $X$ , $Y$ , and $Z$ can be combined to analyze 3-D station position variations. The classical AVAR is not intended for processing unevenly weighted and/or multidimensional data. Therefore, AVAR modifications, namely weighted AVAR (WAVAR), multidimensional AVAR (MAVAR), and weighted multidimensional AVAR (WMAVAR), were introduced to overcome these deficiencies. In this paper, a brief review is given of the experience of using AVAR and its modifications in processing astrogeodetic time series.

The three-cornered-hat (TCH) technique is used frequently in the timing community to estimate the stability of single frequency standards (clocks) when the only measurement data available are paired differences. In previous work, we presented an analytic method to estimate the reduced degrees of freedom produced when using the TCH estimation technique. In this paper, we extend that method to estimate the reduced degrees of freedom related to the more general case of the $M$ -cornered-hat.

The Dick effect is one of the main limitations of short-term frequency stability for most atomic clocks operating in the pulsed mode. Rather than employing better local oscillators with ultralow phase noise, using the interleaving lock technique to eliminate the dead time is a promising approach to suppress the Dick effect. In this paper, we demonstrate the suppression of the Dick effect by the interleaving lock technique in a pulsed Ramsey-coherent population trapping clock with two identical Cs atomic ensembles. The Dick effect is expected to be reduced by a factor of 24 theoretically. The measured short-term Allan deviation is improved from $1.3\times 10^{-11}$ at 1 s to $3.0\times 10^{-12}$ at 1 s. What is even more remarkable is that the Allan deviation curve exhibits ${1/{\tau }}$ slope from 0.01 to 2 s.

Routine use of one-way radiometric tracking for deep space navigation and radio science is not possible today because spacecraft frequency and time references that use state-of-the-art ultrastable oscillators introduce errors from their intrinsic drift and instability on timescales past 100 s. The Deep Space Atomic Clock (DSAC), currently under development as a NASA Technology Demonstration Mission, is an advanced prototype of a space-flight suitable, mercury-ion atomic clock that can provide an unprecedented frequency and time stability in a space-qualified clock. Indeed, the ground-based results of the DSAC space demonstration unit have already achieved an Allan deviation of $2 \times 10^{-15}$ at one day; space performance on this order will enable the use of one-way radiometric signals for deep space navigation and radio science.

The frequency stability of an atomic fountain clock was significantly improved by employing an ultra-stable local oscillator and increasing the number of atoms detected after the Ramsey interrogation, resulting in a measured Allan deviation of 8.3 × 10(-14)τ(-1/2)). A cryogenic sapphire oscillator using an ultra-low-vibration pulse-tube cryocooler and cryostat, without the need for refilling with liquid helium, was applied as a local oscillator and a frequency reference. High atom number was achieved by the high power of the cooling laser beams and optical pumping to the Zeeman sublevel m(F) = 0 employed for a frequency measurement, although vapor-loaded optical molasses with the simple (001) configuration was used for the atomic fountain clock. The resulting stability is not limited by the Dick effect as it is when a BVA quartz oscillator is used as the local oscillator. The stability reached the quantum projection noise limit to within 11%. Using a combination of a cryocooled sapphire oscillator and techniques to enhance the atom number, the frequency stability of any atomic fountain clock, already established as primary frequency standard, may be improved without opening its vacuum chamber.

The Allan variance is a method to characterize stochastic random processes. The technique was originally developed to characterize the stability of atomic clocks and has also been successfully applied to the characterization of inertial sensors. Inertial navigation systems (INS) can provide accurate results in a short time, which tend to rapidly degrade in longer time intervals. During the last decade, the performance of inertial sensors has significantly improved, particularly in terms of signal stability, mechanical robustness, and power consumption. The mass and volume of inertial sensors have also been significantly reduced, offering system-level design and accommodation advantages. This paper presents a complete methodology for the characterization and modeling of inertial sensors using the Allan variance, with direct application to navigation systems. Although the concept of sensor fusion is relatively straightforward, accurate characterization and sensor-information filtering is not a trivial task, yet they are essential for good performance. A complete and reproducible methodology utilizing the Allan variance, including all the intermediate steps, is described. An end-to-end (E2E) process for sensor-error characterization and modeling up to the final integration in the sensor-fusion scheme is explained in detail. The strength of this approach is demonstrated with representative tests on novel, high-grade inertial sensors. Experimental navigation results are presented from two distinct robotic applications: a planetary exploration rover prototype and an autonomous underwater vehicle (AUV).

Power-law noise in clocks and oscillators can be simulated by Fourier transforming a modified spectrum of white phase noise. This approach has been applied successfully to simulation of the Allan variance and the modified Allan variance in both overlapping and nonoverlapping forms. When significant frequency drift is present in an oscillator, at large sampling times the Allan variance overestimates the intrinsic noise, while the Hadamard variance is insensitive to frequency drift. The simulation method is extended in this paper to predict the Hadamard variance for the common types of power-law noise. Symmetric real matrices are introduced whose traces—the sums of their eigenvalues—are equal to the Hadamard variances, in overlapping or nonoverlapping forms, as well as for the corresponding forms of the modified Hadamard variance. We show that the standard relations between spectral densities and Hadamard variance are obtained with this method. The matrix eigenvalues determine probability distributions for observing a variance at an arbitrary value of the sampling interval $\tau$ , and hence for estimating confidence in the measurements.

We discuss the minimization of the Dick effect in an optical lattice clock. We show that optimizing the time sequence of operation of the clock can lead to a significant reduction of the clock stability degradation by the frequency noise of the interrogation laser. By using a non-destructive detection of the atoms, we are able to recycle most of the atoms between cycles and consequently to strongly reduce the time spent capturing the atoms in each cycle. With optimized parameters, we expect a fractional Allan deviation better than 2E-16$\tau^{-1/2}$ for the lattice clock., Comment: 6 pages, 10 figures. Submitted to IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control

The most common time-domain measure of frequency stability, the Allan variance, is typically estimated using a frequency counter. Close examination of the operation of modern high-resolution frequency counters shows that they do not make measurements in the way commonly assumed. The consequence is that the results typically reported by many laboratories using these counters are not, in fact, the Allan variance, but a distorted representation. We elucidate the action of these counters by consideration of their operation in the Fourier domain, and demonstrate that the difference between the actual Allan variance and that delivered by these counters can be very significant for some types of oscillators. We also discuss ways to avoid, or account for, a distorted estimation of Allan variance.

The Allan variance (AVAR) is widely used to measure the stability of experimental time series. Specifically, AVAR is commonly used in space applications such as monitoring the clocks of the global navigation satellite systems (GNSSs). In these applications, the experimental data present some peculiar aspects which are not generally encountered when the measurements are carried out in a laboratory. Space clocks’ data can in fact present outliers, jumps, and missing values, which corrupt the clock characterization. Therefore, an efficient preprocessing is fundamental to ensure a proper data analysis and improve the stability estimation performed with the AVAR or other similar variances. In this work, we propose a preprocessing algorithm and its implementation in a robust software code (in MATLAB language) able to deal with time series of experimental data affected by nonstationarities and missing data; our method is properly detecting and removing anomalous behaviors, hence making the subsequent stability analysis more reliable.

The origins of the Allan variance trace back 50 years ago to two seminal papers, one by Allan (1966) and the other by Barnes (1966). Since then, the Allan variance has played a leading role in the characterization of high-performance time and frequency standards. Wavelets first arose in the early 1980s in the geophysical literature, and the discrete wavelet transform (DWT) became prominent in the late 1980s in the signal processing literature. Flandrin (1992) briefly documented a connection between the Allan variance and a wavelet transform based upon the Haar wavelet. Percival and Guttorp (1994) noted that one popular estimator of the Allan variance—the maximal overlap estimator—can be interpreted in terms of a version of the DWT now widely referred to as the maximal overlap DWT (MODWT). In particular, when the MODWT is based on the Haar wavelet, the variance of the resulting wavelet coefficients—the wavelet variance—is identical to the Allan variance when the latter is multiplied by one-half. The theory behind the wavelet variance can thus deepen our understanding of the Allan variance. In this paper, we review basic wavelet variance theory with an emphasis on the Haar-based wavelet variance and its connection to the Allan variance. We then note that estimation theory for the wavelet variance offers a means of constructing asymptotically correct confidence intervals (CIs) for the Allan variance without reverting to the common practice of specifying a power-law noise type a priori . We also review recent work on specialized estimators of the wavelet variance that are of interest when some observations are missing (gappy data) or in the presence of contamination (rogue observations or outliers). It is a simple matter to adapt these estimators to become estimators of the Allan variance. Finally we note that wavelet variances based upon wavelets other than the Haar offer interesting generalizations of the Allan variance.

We present a long-term chip scale stabilization scheme for optoelectronic oscillators (OEOs) based on a rubidium coherent population trapping (CPT) atomic resonator. By locking a single mode of an OEO to the 85Rb 3.035-GHz CPT resonance utilizing an improved phase-locked loop (PLL) with a PID regulator, we achieved a chip scale frequency stabilization system for the OEO. The fractional frequency stability of the stabilized OEO by overlapping Allan deviation reaches ${6}.{2} \times {1}{{0}^{-11}}$ (1 s) and $\sim\!{1}.{45} \times {10}^{ -11}$ (1000 s). This scheme avoids a decrease in the extra phase noise performance induced by the electronic connection between the OEO and the microwave reference in common injection locking schemes. The total physical package of the stabilization system is ${20}\;\text {cm}^{3}$ and the total power consumption is 400 mW, which provides a chip scale and portable frequency stabilization approach with ultra-low power consumption for OEOs.

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.

The papers in this special issue focus on the Allan Variance (also referred to as two-sample variance), a measurement technique that determeines the frequency stability in amplifiers, clocks, and oscillators.

We propose and demonstrate the phase-stabilized transfer of a chirped frequency as a tool for synchronization and time transfer. Technically, this is done by evaluating remote measurements of the transferred, chirped frequency. The gates of the frequency counters, here driven by a 10-MHz oscillation derived from a hydrogen maser, play a role analogous to the 1-pulse per second (PPS) signals usually employed for time transfer. In general, for time transfer, the gates consequently must be related to the external clock. Synchronizing observations based on frequency measurements, on the other hand, only requires a stable oscillator driving the frequency counters. In a proof of principle, we demonstrate the suppression of symmetrical delays, such as the geometrical path delay. We transfer an optical frequency chirped by around 240 kHz/s over a fiber link of around 149 km. We observe an accuracy and simultaneity, as well as a precision (Allan deviation, 18,000 s averaging interval) of the transferred frequency of around 2 × 10(-19). We apply chirped frequency transfer to remote measurements of the synchronization between two counters' gate intervals. Here, we find a precision of around 200 ps at an estimated overall uncertainty of around 500 ps. The measurement results agree with those obtained from reference measurements, being well within the uncertainty. In the present setup, timing offsets up to 4 min can be measured unambiguously. We indicate how this range can be extended further.

In this paper, we present the concept of accessing the signal at some midpoint of a frequency dissemination system with stabilized propagation delay, which allows building the point-to-multipoint frequency dissemination network. In the first experiments with a 160 km-long fiber link composed of a field-deployed optical cable and fibers spooled in the lab, exposed to both diurnal and seasonal temperature variations, in the access node, we obtained the Allan deviation of a 10- MHz frequency signal of about 3 × 10(-17) and the time deviation not greater than 2 ps for 10(5) s averaging.

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.

Two-way satellite time and frequency transfer (TWSTFT) using geostationary telecommunication satellites is widely used in the timing community today and has also been chosen as the primary means to effect synchronization of elements of the ground segment of the European satellite navigation system Galileo. We investigated the link performance in a multistation network based on operational parameters such as the number of simultaneously transmitting stations, transmit and receive power, and chip rates of the pseudorandom noise modulation of the transmitted signals. Our work revealed that TWSTFT through a "quiet" transponder channel (2 stations transmitting only) leads to a measurement noise, expressed by the 1 pps jitter, reduced by a factor of 1.4 compared with a busy transponder carrying signals of 12 stations. The frequency transfer capability expressed by the Allan deviation is reduced at short averaging times by the same amount. At averaging times of >1 d, no such reduction could be observed, which points to the fact that other noise sources dominate at such averaging times. We also found that higher transmit power increases the carrier-to-noise density ratio at the receive station and thus entails lower jitter but causes interference with other station's signals. In addition, the use of lower chip rates, which could be accommodated by a reduced assigned bandwidth on the satellite transponder, is not recommended. The 1 pps jitter would go up by a factor of 2.5 when going from 2.5 MCh/s to 1 MCh/s. The 2 Galileo precise timing facilities (PTFs) can be included in the currently operated network of 12 stations in Europe and all requirements on the TWSTFT performance can be met, provided that suitable ground equipment will be installed in the Galileo ground segment.

Total variance is a statistical tool developed for improved estimates of frequency stability at averaging times up to one-half the test duration. As a descriptive statistic, total variance performs an exact decomposition of the sample variance of the frequency residuals into components associated with increasing averaging times. As an estimator of Allan variance, total variance has greater equivalent degrees of freedom and lesser mean square error than the standard unbiased estimator has.

A cryocooled compensated sapphire oscillator (CSO), developed for the Cassini Ka-band Radio Science experiment, and operating in the 7 K-10 K temperature range, was demonstrated to show ultra-high stability of /spl sigma//sub y/=2.5/spl times/10/sup -15/ for measuring times 200 seconds /spl les//spl tau//spl les/600 seconds using a hydrogen maser as a reference. CSO-1 and CSO-3 are now both operational with new low noise receivers. We have made initial phase noise and Allan deviation measurements that show more than 10 times stability improvement over the hydrogen maser for measuring times 1 second /spl les//spl tau//spl les/ 10 seconds, and indicate performance for the individual units of /spl sigma//sub y//spl ap/3/spl times/10/sup -15/ for measuring times from 10 to 1000 seconds. Phase noise is reduced by 20 to 28 dB over the design offset frequency range from 1 Hz to 40 Hz. Receiver design is also discussed.

Frequency flicker of quartz resonators can be derived from the measurement of S/sub /spl phi// (f), i.e., the power spectrum density of phase fluctuations /spl phi/. The interferometric method appears to be the best choice to measure the phase fluctuations of the quartz resonators because of its high sensitivity in the low power conditions, which is required for this type of resonator. Combining these two ideas, we built an instrument suitable to measure the frequency flicker floor of the quartz resonators, and we measured the stability of some 10-MHB high performance resonators as a function of the dissipated power. The stability limit of our instrument, described in terms of Allan deviation /spl sigma//sub y/(/spl tau/), is of some 10/sup -14/.

A spectral interpretation using the frequency sensitivity of the Allan variance (Avar) and Thêo-Hybrid (ThêoH) is used to determine falpha noise, or "power-law noise." ThêoH has narrower chi-square confidence than Avar; consequently, ThêoH provides significantly better determination of falpha noise types at long term. Furthermore, ThêoH has even narrower confidence than chi-square. Because the algorithms used to calculate these confidence intervals are computationally intensive, we have constructed an empirical formula that approximates confidence intervals as the percent error for ThêoH.

In this paper, we describe a laboratory proto-type of pulsed optically pumped clock based on a rubidium vapor cell with buffer gas. The measured frequency stability (overlapping Allan deviation) is /spl sigma//sub y/(/spl tau/) = 3 /spl times/ 10/sup -12/ /spl tau/ /sup -/1/2 and the level of 4 /spl times/ 10/sup -14/ is reached for averaging time of /spl tau/ = 3 /spl times/ 10/sup 4/ s. For the same set of data, the statistical tool Theol predicts a frequency stability of 2 /spl times/ 10/sup -14/ for /spl tau/ = 10/sup 5/ s. This result confirms the theoretical predictions regarding this kind of frequency standard and makes it very attractive for satellite navigation and space applications in which a simple and reliable implementation is required, and the short and medium term stability (till one day) is the main concern.

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 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.