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

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

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

We describe a method based on the total deviation approach whereby we improve the confidence of the estimation of the Hadamard deviation that is used primarily in global positioning system (GPS) operations. The Hadamard-total deviation described in this paper provides a significant improvement in confidence indicated by an increase of 1.3 to 3.4 times the one degree of freedom of the plain Hadamard deviation at the longest averaging time. The new Hadamard-total deviation is slightly negatively biased with respect to the usual Hadamard deviation, and tau values are restricted to less than or equal to T/3, to be consistent with the usual Hadamard's definition. We give a method of automatically removing bias by a power-law detection scheme. We review the relationship between Kalman filter parameters and the Hadamard and Allan variances, illustrate the operational problems associated with estimating these parameters, and discuss how the Hadamard-total variance can improve management of present and future GPS satellite clocks.     

A domain-specific language to design false data injection tests for air traffic control systems

International Journal on Software Tools for Technology Transfer - The ADS-B—automatic dependent surveillance-broadcast—technology requires aircraft to broadcast their position and...     

Statistical biases and very-long-term time stability analysis

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

A Bayesian method for oscillator stability analysis

The power spectral density of frequency fluctuations of an oscillator is generally modeled as a sum of power laws with integer exponents (from -2 to +2). However, a power law with a fractional exponent may exist. We propose a method for measuring the level of such a noise process and determining the probability density of the exponent. This yields a criterion for compatibility with an integer exponent. This method is based on a Bayesian approach called the reference analysis of Bernardo-Berger. The application to a sequence of frequency measurement from a quartz oscillator illustrates this paper.     

Approche multivariance du bruit de fréquence des oscillateurs

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