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The Allan variance was originally meant to be a characteristic of time and frequency standards, but now it is widely used in many fields, including the development of navigation devices. The paper considers different applications of the Allan variance to various tasks, among which is estimation of characteristics of navigation devices. However, sometimes the Allan variance does not seem to be very effective; besides, its physical interpretation may be vague. This is due to its empirical definition as an alternative estimate of variance. The paper considers the definition of the Allan variance based on a model of random processes with stationary increments that include both stationary and Wiener processes. In this model, the main characteristic is the structure function, whereas the Allan variance turns out to be its empirical estimate. This representation makes the statistical meaning of the Allan variance clear and explains its high effectiveness in the analysis of nonstationary signals and noise. It also allows the Allan variance to be used as a general characteristic describing stability of measuring and navigation devices, the one that is different from variance.

This research proposes a novel, autonomous, regression-based methodology for Allan variance analysis of inertial measurement unit (IMU) sensors. Current methods for Allan variance analysis have been rooted in the human-based interpretation of linear trends, referred to as the slope method. The slope method is so prolific; it is referenced among electrical and electronics engineering standards for IMU error analysis. However, the graphical nature and visual-inspection–based use of the method limit its ability to be programmed as a generalized algorithm,which hinders the autonomy desired in modern-day navigation computations. Using nonlinear regression with a ridge-regression initial guess, the proposed method is shown to produce comparable results to the gold standard slope method when using standard-length data collections and outperforms the slope method when the amount of available data is limited. This development directly enables accurate navigation solutions for all vehicles in land, air, sea, and space operations.

Microelectromechanical systems (MEMS) are core components in unmanned aerial vehicles (UAV). The precision of MEMS sensors is very important when the GPS signal is invalid. However, the precision and performance of MEMS sensors will be degraded by the changing of environment. Therefore, estimation and identification of the various noise terms existing in MEMS sensors are deemed to be necessary. The Allan variance is a common and standard method to analyze MEMS sensors, but it cannot be used to analyze the dynamic characteristics. The dynamic Allan variance (DAVAR) is a sliding version of the Allan variance. It is a practical tool that could represent the nonstationary behavior of the MEMS signal. As the DAVAR needs to estimate the Allan variance at each time epoch, the computation time grows significantly with the length of the signal series. In this paper, in the case of MEMS gyroscope on UAV, an improved fast DAVAR algorithm based on the choice of relevant time is proposed to shorten the computation time. As an experimental validation, numerical experiments are conducted under the proposed method. The results demonstrate that the improved method could greatly increase the computation speed and does not affect the accuracy of estimation.

A problem of estimating the movement and orientation of a mobile robot is examined in this paper. The strapdown inertial navigation systems are often engaged to solve this common obstacle. The most important and critically sensitive component of such positioning approximation system is a gyroscope. Thus, we analyze here the random error components of the gyroscope, such as bias instability and random rate walk, as well as those that cause the presence of white and exponentially correlated (Markov) noise and perform an optimization of these parameters. The MEMS gyroscopes of InvenSense MPU-6050 type for each axis of the gyroscope with a sampling frequency of 70 Hz are investigated, as a result, Allan variance graphs and the values of bias instability coefficient and angle random walk for each axis are determined. It was found that in the output signals of the gyroscopes there is no Markov noise and random rate walk, and the X and Z axes are noisier than the Y axis. In the process of inertial measurement unit (IMU) calibration, the correction coefficients are calculated, which allow partial compensating the influence of destabilizing factors and determining the perpendicularity inaccuracy for sensitivity axes, and the conversion coefficients for each axis, which transform the sensor source codes into the measure unit and bias for each axis. The output signals of the calibrated gyroscope are noisy and offset from zero to all axes, so processing accelerometer and gyroscope data by the alpha-beta filter or Kalman filter is required to reduce noise influence.

Modeling stochastic noise in inertial sensorsparticularly those used in guidance, navigation, and control applicationsinvolves characterizing the underlying noise process by inferring parameters such as random walks and drift rates from the Allan deviation plots. Fully overlapped Allan variance (FOAV) and total variance (TV) are two methods that accurately derive these parameters by observing all possible time averages, but existing implementations are computationally expensive: They require $\Theta (N^3)$ time for processing $N$ data points (see Section III ). Thus, several methods have been developed to trade accuracy in parameter estimates for reduced computational effort, including not fully overlapping AV (NFOAV), which runs in $\Theta (N^2)$ time. Our key contribution is a fast, parallelizable algorithm (Algorithm 1) to calculate FOAV and TV for generating smooth Allan deviation plots whose serial running time is $\Theta (N^2)$ , and we demonstrate improved execution times with parallel implementations. Our fast algorithm thus enables FOAV and TV to be the norm for estimating AV parameters efficiently and with high confidence.

The paper discusses the relationship between Allan variance and error variance of sensor bias estimate obtained by averaging over a certain period. Allan variance is shown to coincide with this variance in some cases. Bias estimation accuracy can be improved using nonlinear filtering methods in the conditions of uncertain error model.

The paper considers the model of measurement data with typical measurement noise included. For some noise combinations, errors in noise parameter determination are estimated using the Allan variance method. Coefficients of approximating polynomial of Allan variance curve are obtained by multiparametric optimization of the proposed nonlinear target function.

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.

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 dynamic Allan variance (DAVAR) [1] is a useful tool to track non-stationaries in the behavior of gyroscopes. It allows us to diagnose whether the stability of gyroscopes is changing with time. Unfortunately, with the length of the analyzed time series increasing, the computational time of the DAVAR grows very quickly. Consequently, it costs huge time to deal with the data. In this article, a Fast algorithm of the DAVAR for gyroscopes is developed. In order to verify its validity, three sets of simulation data are characterized by the Fast algorithm and the classical one. The Fast DAVAR could reduce the computational time dramatically when the long time series is analyzed. Furthermore, a vibration experiment with fiber optic gyroscopes has been implemented to validate the good performance of the Fast algorithm. Compared with the classical one, the Fast algorithm of the DAVAR shortens the computation time 6 times. In conclusion, the Fast algorithm of the DAVAR guarantees a dramatic reduction of the computational time and outperforms the classical DAVAR.

The Allan variance method is an effective way of analyzing gyro's stochastic noises. In the traditional implementation, the ordinary least square algorithm is utilized to estimate the coefficients of gyro noises. However, the different accuracy of Allan variance values violates the prerequisite of the ordinary least square algorithm. In this study, a weighted least square algorithm is proposed to address this issue. The new algorithm normalizes the accuracy of the Allan variance values by weighting them according to their relative quantitative relationship. As a result, the problem associated with the traditional implementation can be solved. In order to demonstrate the effectiveness of the proposed algorithm, gyro simulations are carried out based on the various stochastic characteristics of SRS2000, VG951 and CRG20, which are three different-grade gyros. Different least square algorithms (traditional and this proposed method) are applied to estimate the coefficients of gyro noises. The estimation results demonstrate that the proposed algorithm outperforms the traditional algorithm, in terms of the accuracy and stability.

The received signal strength (RSS) of wireless signals conveys important information that has been widely used in wireless communications, localization, and tracking. Traditional RSS-based research and applications model the RSS signal using a deterministic component plus a white noise term. This paper investigates the assumption of white noise to have a further understanding of the RSS signal and proposes a methodology based on the Allan variance (AVAR) to characterize it. Using AVAR, we model the RSS unknown perturbations as correlated random terms. These terms can account for both colored noise or other effects such as shadowing or small-scale fading. Our results confirm that AVAR can be used to obtain a flexible model of the RSS perturbations, as expressed by colored noise components . The study is complemented by introducing two straightforward applications of the proposed methodology: the modeling and simulation of RSS noise using Wiener processes, and RSS localization using the extended Kalman filter.

The environment influences Fiber Optic Gyroscopes (FOG) directly. Thus, FOGs in Measurement While Drilling (MWD) system exhibit a non-stationary behavior over time due to strong vibration, high-speed rotation and temperature increasing. Therefore, it is important to introduce a representation that can take the time-varying nature of FOGs’ stability into account. The Allan variance is a common and standard method to analyze FOGs, but it cannot be used to analyze the dynamic characteristics. In this effort, a method based on the dynamic Allan variance (DAVAR) was developed. Firstly, the validity of DAVAR was verified by simulated data. Secondly, the FOG working in the harsh environment was characterized by the DAVAR. According to the results, an improved experiment was implemented to verify the practical application of the DAVAR. In addition, a 2-D diagram is used for the first time to illustrate the noise characteristics of FOGs. A 2-D analysis diagram can not only separate and distinguish the stochastic noise term, but also track and reveal the changes in the behavior of FOGs in a clear and intuitive manner. In conclusion, with the new method developed by this paper, the performance of FOGs can be evaluated more comprehensively and effectively.

The interferometric fiber optic gyro (IFOG) that is inside the gyrocompass has been gradually equipped on the marine vessel and the vessel's swing is sensed by IFOG. This will affect the detection of Allan variance coefficients of IFOG while the vessel is in moorage especially in temperature variation conditions. However, the standard IFOG Allan variance noise analysis method is computationally expensive and requires massive data storage which is inappropriate for Allan variance coefficients real-time detection. In addition, for long-voyage and long-endurance requirements of the vessel in the marine environments, the ambient temperature variation causes errors that will degenerate IFOG accuracy. The vessel needs more independence including the fault diagnosis system and reconfiguration of the IFOG gyrocompass. To meet the marine vessel autonomy need, we propose a method for Allan variance coefficients detection including rate ramp R, rate stochastic walk K, bias instability B, angular stochastic walk N and quantization noise Q. In the study, we first eliminate the angular increments of vessel attitude by gravity in inertial frame in the moorage condition, and then realize the Allan variance coefficients real-time detection by adopting nonlinear adaptive Kalman filter with proposed state-space model. Simulations show that the proposed method potentially could implement the Allan variance coefficients’ real-time detection, and estimate the coefficients’ temperature variation effectively and correctly.

This article deal with the identification, by Allan variance, of the associated noises in signals from microelectromechanical (MEMS) inertial sensors accelerometer and gyroscope, whose data are used in attitude calculus of rigid bodies. An Arduino Uno microprocessor was used to collect sensor data while the software MATLAB was applied for post processing. The results were used for building error models to improve simulators and also to refine attitude estimation with Kalman filter.

Inertial sensor errors include deterministic errors and stochastic errors. Deterministic errors can be calibrated in laboratory by simple computation technique. Stochastic errors can be determined during calibration by adopting special methods because of their random character. The simplest method to determine the stochastic errors for inertial sensors is the Allan variance. This kind of method needs large data to fully characterize the stochastic errors. The normal nonoverlapped Allan variance has quite poor estimation accuracy in long cluster time. The fully overlapping Allan variance and traditional total variance have better estimation accuracy in long cluster time but are quite time consuming for large data set. The not fully overlapping Allan variance and nonoverlapped total variance are suitable for large data set to improve the estimation accuracy in long cluster time with much less time, but their accuracy is still relatively poor in comparison with not fully overlapping total variance. Whereas the not fully overlapping total variance is relatively time consuming and, compared with Allan variance, there is a bias which is not easy to be corrected. This paper proposes a sliding average Allan variance that has comparable estimation accuracy with total variance. The data are not required to extend as the total variance; thus the calculation burden could be reduced greatly. Therefore, it is more suitable for large data set. In addition this method has no bias in comparison with Allan variance, which means no bias correction is required. This method is applied to 12-h static data of three gyroscopes from a position and orientation system with good performance.

The Allan Variance (AV) is a widely used quantity in areas focusing on error measurement as well as in the general analysis of variance for autocorrelated processes in domains such as engineering and, more specifically, metrology. The form of this quantity is widely used to detect noise patterns and indications of stability within signals. However, the properties of this quantity are not known for commonly occurring processes whose covariance structure is non-stationary and, in these cases, an erroneous interpretation of the AV could lead to misleading conclusions. This paper generalizes the theoretical form of the AV to some non-stationary processes while at the same time being valid also for weakly stationary processes. Some simulation examples show how this new form can help to understand the processes for which the AV is able to distinguish these from the stationary cases and hence allow for a better interpretation of this quantity in applied cases.

Stochastic errors characterize the performance of inertial sensors and indicate the potential improvements of the device. Accurate modeling of these errors can enhance the performance of inertial navigation system. The simplest and generally adopted method to model stochastic errors of inertial sensor is the normal nonoverlapped Allan variance, but its estimation accuracy decreases in long cluster time. The fully overlapping Allan variance improves the estimation accuracy in long cluster time very much, while the traditional total variance based on maximal-overlapping cluster samples further increases the estimation accuracy in long cluster time greatly. However, with these two methods, better estimation is achieved at the expense of much longer computation time. Besides, the computation burden for large dataset and many variance data points is extremely large, while the inertial sensors need large dataset with many variance data points to fully characterize their stochastic errors. This is because the correlation time for some stochastic error is very long (e.g., the correlation time is 3 h for rate random walk), and the Allan variance curve of some stochastic error is very complicated (e.g., sinusoidal noise). Consequently, the fully overlapping Allan variance and the traditional total variance are not suitable for inertial sensor stochastic error modeling. This paper proposes a not fully overlapping Allan variance which has similar estimation accuracy to fully overlapping Allan variance, and whose calculation time is greatly reduced relative to fully overlapping Allan variance. Then, the not fully overlapping concept is integrated to that of total variance to further improve the estimation accuracy in long cluster time with respect to Allan variance, and the calculation time of total variance is greatly reduced as well. This method enables high accuracy and high computational efficiency of Allan variance analysis at the same time, especially for large dataset and many variance data points analysis. Finally, the proposed methods are applied to 12-hour static data of gyroscopes and accelerometers from a position and orientation system, and their advantages are demonstrated.

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).

The dynamic Allan variance (DAVAR) measures the stability variations of precise clocks and oscillators. When an anomaly occurs, the DAVAR changes with time, its shape depending on the anomaly. In this work, we first discuss our current knowledge about the DAVAR, by focusing on the meaning of dynamic stability. Then, we extend our knowledge by obtaining additional properties of the DAVAR. Furthermore, we visually investigate the DAVAR for the main anomalies of precise clocks and oscillators. Finally, we review a variety of applications based on the DAVAR.

The dynamic Allan variance (DAVAR) is a surface which describes the stability of a high-precision clock with respect to time. When the DAVAR is evaluated from experimental data, its surface shows random fluctuations caused by the estimation process. It is fundamental to assign a statistical significance to these fluctuations, so that they can be differentiated from the variations of the surface caused by clock anomalies. First, we develop confidence surfaces to assign a statistical significance to the random fluctuations of the DAVAR estimator. Then, we introduce detection surfaces to reveal the variations of the surface caused by clock anomalies. We validate the obtained results through numerical simulations.

Stochastic serial correlations are often ignored in the statistical uncertainty (Type A) analysis of measurement values. For repeated measurements, the standard deviation of the mean is often used to specify Type-A uncertainty, but it is frequently underestimated by assuming that it is given by the standard deviation divided by the square root of the number of measurements. Methods of time-series analysis such as the Allan variance (Avar) formalism give realistic estimates of Type-A uncertainty, but they require equal time intervals between successive measurements. This requirement is difficult to satisfy in comparisons of Josephson voltage standards (JVSs) because of the instability of the voltage steps and the small number of repeated measurements. A JVS comparison was made using a completely automatic compact JVS. The time intervals were uneven; thus, we studied the effect of their irregularity on the Avar by simulating data having the same noise model and uneven time intervals as the measured data. We found that, for this JVS comparison, the Avar is only slightly affected by uneven intervals. The noise was found to be a mixture of white and 1/f noise. The latter limits the Allan deviation to 0.64 nV after 14.3 h of measurement. The method of simulating data this way should be applicable to other complex measurement situations.

It is well known that inertial navigation systems can provide high-accuracy position, velocity, and attitude information over short time periods. However, their accuracy rapidly degrades with time. The requirements for an accurate estimation of navigation information necessitate the modeling of the sensors' error components. Several variance techniques have been devised for stochastic modeling of the error of inertial sensors. They are basically very similar and primarily differ in that various signal processings, by way of weighting functions, window functions, etc., are incorporated into the analysis algorithms in order to achieve a particular desired result for improving the model characterizations. The simplest is the Allan variance. The Allan variance is a method of representing the root means square (RMS) random-drift error as a function of averaging time. It is simple to compute and relatively simple to interpret and understand. The Allan variance method can be used to determine the characteristics of the underlying random processes that give rise to the data noise. This technique can be used to characterize various types of error terms in the inertial-sensor data by performing certain operations on the entire length of data. In this paper, the Allan variance technique will be used in analyzing and modeling the error of the inertial sensors used in different grades of the inertial measurement units. By performing a simple operation on the entire length of data, a characteristic curve is obtained whose inspection provides a systematic characterization of various random errors contained in the inertial-sensor output data. Being a directly measurable quantity, the Allan variance can provide information on the types and magnitude of the various error terms. This paper covers both the theoretical basis for the Allan variance for modeling the inertial sensors' error terms and its implementation in modeling different grades of inertial sensors.

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.

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.

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.

As a noise analysis method for inertial sensors, the traditional Allan variance method requires the storage of a large amount of data and manual analysis for an Allan variance graph. Although the existing online estimation methods avoid the storage of data and the painful procedure of drawing slope lines for estimation, they require complex transformations and even cause errors during the modeling of dynamic Allan variance. To solve these problems, first, a new state-space model that directly models the stochastic errors to obtain a nonlinear state-space model was established for inertial sensors. Then, a neural-extended Kalman filter algorithm was used to estimate the Allan variance coefficients. The real noises of an ADIS16405 IMU and fiber optic gyro-sensors were analyzed by the proposed method and traditional methods. The experimental results show that the proposed method is more suitable to estimate the Allan variance coefficients than the traditional methods. Moreover, the proposed method effectively avoids the storage of data and can be easily implemented using an online processor.

The Allan variance is used to characterize stochastic serial correlations in DC electrical measurements when using instruments such as nanovoltmeters and Zener voltage standards. For such applications, it is important to be able to specify the confidence intervals for the estimated Allan variances. By repeating sets of 4096 or 8192 observations 600 times or more in succession, the experimental sampling distribution of the Allan variance is obtained and found to agree well with the chi-square distribution calculated from the number of degrees of freedom estimated from the data. An expression is obtained for approximating the number of degrees of freedom as a function of sampling time; this can then be used to estimate confidence intervals.

The stability of a fiber optic gyroscope (FOG) in measurement while drilling (MWD) could vary with time because of changing temperature, high vibration, and sudden power failure. The dynamic Allan variance (DAVAR) is a sliding version of the Allan variance. It is a practical tool that could represent the non-stationary behavior of the gyroscope signal. Since the normal DAVAR takes too long to deal with long time series, a fast DAVAR algorithm has been developed to accelerate the computation speed. However, both the normal DAVAR algorithm and the fast algorithm become invalid for discontinuous time series. What is worse, the FOG-based MWD underground often keeps working for several days; the gyro data collected aboveground is not only very time-consuming, but also sometimes discontinuous in the timeline. In this article, on the basis of the fast algorithm for DAVAR, we make a further advance in the fast algorithm (improved fast DAVAR) to extend the fast DAVAR to discontinuous time series. The improved fast DAVAR and the normal DAVAR are used to responsively characterize two sets of simulation data. The simulation results show that when the length of the time series is short, the improved fast DAVAR saves 78.93% of calculation time. When the length of the time series is long ( 6 × 10 5 samples), the improved fast DAVAR reduces calculation time by 97.09%. Another set of simulation data with missing data is characterized by the improved fast DAVAR. Its simulation results prove that the improved fast DAVAR could successfully deal with discontinuous data. In the end, a vibration experiment with FOGs-based MWD has been implemented to validate the good performance of the improved fast DAVAR. The results of the experience testify that the improved fast DAVAR not only shortens computation time, but could also analyze discontinuous time series.

An exact solution to the well-known integral transform that relates the spectral density of the instantaneous fractional frequency deviation, S/sub y/(f), to the Allan variance, /spl sigma//sub y//sup 2/ (/spl tau/), is presented for the case of a power-law representation of S/sub y/(f). The approximate solution to this integral transform, which is found throughout the literature, is also derived. A graphical convergence analysis is presented, showing the range of applicability of the approximate solution. The results reinforce the use of the approximate solution, which converges quickly to the exact solution under virtually all reasonable measurement conditions.

When analyzing nanovoltmeter measurements, stochastic serial correlations are often ignored and the experimental standard deviation of the mean is assumed to be the experimental standard deviation of a single observation divided by the square root of the number of observations. This is justified only for white noise. This paper demonstrates the use of the power spectrum and the Allan variance to analyze data, identify the regimes of white noise, and characterize the performance of digital and analog DC nanovoltmeters. Limits imposed by temperature variations, 1/f noise and source resistance are investigated.

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.

The phase-noise spectrum determines the Allan variance by a well-known integral formula. It is shown here that unique inversion of this formula is not possible in general because the mapping from spectrum to Allan variance is not one-to-one. A necessary and sufficient condition for two distinct phase spectra to have the same Allan variance is given.

A method to investigate the characteristics of noise of the adsorption–desorption phenomena is presented. The method is based on the Allan variance which allows the evaluation, in counting experiments, of noise spectral densities also in case of 1/f or Lorentzian noises. A first application to quartz microbalances coated with metalloporphyrins interacting with volatile compounds is here reported. The results indicate the presence of a 1/f noise component in adsorption phenomena.

Publication in the conference proceedings of EUSIPCO, Lisbon, Portugal, 2014

Wavelets have recently been a subject of great interest in geophysics, mathematics and signal processing. The discrete wavelet transform can be used to decompose a time series with respect to a set of basis functions, each one of which is associated with a particular scale. The properties of a time series at different scales can then be summarized by the wavelet variance, which decomposes the variance of a time series on a scale by scale basis. The wavelet variance corresponding to some of the recently discovered wavelets can provide a more accurate conversion between the time and frequency domains than can be accomplished using the Allan variance. This increase in accuracy is due to the fact that these wavelet variances give better protection against leakage than does the Allan variance. >

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.

Fast frequency fluctuations of a 780 nm grating-stabilized diode laser are directly observed by the measurement of the Allan variance. Time intervals as short as 10 ns and up to 1 s are realized by means of two independent time interval counters. Frequency fluctuations with the laser locked to a Doppler-free resonance transition of atomic rubidium are examined in detail.

A real-time frequency stability measurement system for semiconductor lasers was developed. Since the frequency of the input signal is measured successively without clearing the counter, measurements of the Allan variance made with this system are more accurate than those made with conventional instruments. The Allan variance can be measured for integration times tau from 1 mu s to 10000 s, and the number N of measured frequencies averaged over the integration time tau can be arbitrarily selected up to N=707 for each integration time. The highest measurable frequency was 90 MHz. It was demonstrated experimentally that this system can be used for measurements of the frequency stability of semiconductor lasers. >

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.

Some of the first acoustic well logs were done in the 1950s by Vogel [l]. The tool (commonly called a sonde) used for these logs consisted of one transmitter and one or two receivers. At that time only a rudimentary knowledge of the acoustics of the borehole was known; however, it was known that the compressional wave in the formation would be the first to effect the received signal. Using this fact, the logs were developed to measure the compressional wave travel time through the formation. The present-day sonde employs two transmitters and two receivers, in order to obtain a more accurate measure of compressional travel times. It is apparent that compressional travel time is only a part of the information that is available from the acoustic log. Observing the time trace produced at a receiver, several different occurrences can be seen. These occurrences are the results of various wave types, including the compressional wave, the shear wave, and others. The characteristics of the various waves and how they relate to the surrounding formation is the subject of a great deal of research. A relationship between acoustic travel times and the porosity of the rock formation has been found. Also the ratio of the shear and compressional wave velocities can be related to the lithology of the surrounding formation. This paper explores techniques for the detection and measurement of the compressional and shear waves. First the Generalized Allan Variance is used to detect the compressional wave. The detection of the shear wave is complicated by the trailing end of the compressional wave, which covers the shear wave arrival. The arrival of the shear wave can be detected by applying nonstationary spectral analysis to the wavetrain and identifying the spectral character associated with the wave arrival. A second technique for measuring the shear wave velocity involving the processing of wavetrains from multiple receivers is described.

A new online method is presented for estimation of the angular randomwalk and rate randomwalk coefficients of inertial measurement unit gyros and accelerometers. In the online method, a state-space model is proposed, and recursive parameter estimators are proposed for quantities previously measured from offline data techniques such as the Allan variance method. The Allan variance method has large offline computational effort and data storage requirements. The technique proposed here requires no data storage and computational effort of approximately 100 calculations per data sample.

The performance of an Allan variance measuring system was drastically improved by employing time interval analysis incorporating a beat frequency method. It was used to evaluate the performance of a heterodyne optical phase-locked loop with a very low optical frequency tracking error of 0.4 mHz at the integration time of 70 s. Advantages of the system are precise measurement for highly stable frequency sources with good reproducibility and simple structure