Characterizing changes in the noise statistics of GNSS space clocks with the dynamic Allan variance

The dynamic Allan variance (DAVAR) is a tool for the characterization of precise clocks. Monitoring anomalies of precise clocks is essential, especially when they are employed onboard the satellites of a global navigation satellite system (GNSS). When an anomaly occurs, the DAVAR changes with time, its shape depending on the type of anomaly occurred. We obtain the analytic DAVAR for a change of variance in the clock noise, an anomaly with critical effects on the clock performances. This result is helpful when the clock health is monitored by observing the DAVAR

Nonlinear Transformation of Differential Equations into Phase Space

Time-frequency representations transform a one-dimensional function into a two-dimensional function in the phase-space of time and frequency. The transformation to accomplish is a nonlinear transformation and there are an infinite number of such transformations. We obtain the governing differential equation for any two-dimensional bilinear phase-space function for the case when the governing equation for the time function is an ordinary differential equation with constant coefficients. This connects the dynamical features of the problem directly to the phase-space function and it has a number of advantages

Approximation of the Wigner Distribution for Dynamical Systems Governed by Differential Equations

A conceptually new approximation method to study the time-frequency properties of dynamical systems characterized by linear ordinary differential equations is presented. We bypass solving the differential equation governing the motion by writing the exact Wigner distribution corresponding to the solution of the differential equation. The resulting equation is a partial differential equation in time and frequency. We then show how it lends itself to effective approximation methods because in the time frequency plane there is a high degree of localization of the signal. Numerical examples are given and compared to exact solutions

Generating a real-time time scale making full use of the available frequency standards

We propose a time-scale algorithm for the automated generation of a real-time time scale, making full use of the frequency standards available in a typical time laboratory. The time-scale algorithm is made by a pre-processing stage, a steering algorithm, and a post-processing stage. In particular, in this work we propose a set of three different steering algorithms, running in parallel and eventually producing a unique steering correction to be applied to a master clock. Each algorithm is based on a different steering reference, namely, a primary frequency standard, an ensemble clock, and the Coordinated Universal Time (UTC), or its rapid version, UTCr. Pre- and post-processing stages help to provide robustness and to cope with data gaps. The proposed algorithms have been extensively and successfully tested off-line, on real data from the time laboratory of the Italian National Institute of Metrological Research (INRiM), where an on-line test has also been performed in the period May-October 2019. Then, since the mid of January 2020, the time-scale algorithm has been applied for the generation of the Italian legal time scale, UTC(IT). We show here the results of the off-line tests and of the 5-month on-line test. The proposed strategy can be used wherever a stable, accurate, and robust time reference is needed, e.g. for a local realization of UTC in a laboratory k, UTC(k), or for generating the reference system time of a global navigation satellite system (GNSS)

Instantaneous spectrum estimation of event-based densities

We present a method for obtaining a time-varying spectrum that is particularly suited when the data are in event-based form. This form arises in many areas of science and engineering, and especially in astronomy, where one has photon counting detectors. The method presented consists of three procedures. First, estimating the density using the kernel method; second, highpass filtering the manifestly positive density; finally, obtaining the time-frequency distribution with a modified Welch′s method. For the sake of validation event-based data are generated from a given distribution and the proposed method is used to construct the time-frequency spectrum and is compared to the original density. The results demonstrate the effectiveness of the method

A time-frequency relationship between the Langevin equation and the harmonic oscillator

We derive a simple relationship between the Wigner distribution of the Green's function of the Langevin equation and of the harmonic oscillator. This relationship shows that the Wigner distribution of the Green's function of the harmonic oscillator consists of the sum of two terms obtained by translating the Wigner distribution of the Green's function of the Langevin equation at the resonant frequencies of the harmonic oscillator, plus an interference term. This result paves the way for a simplification of the time-frequency representation of differential equations, as well as for a better understanding and filtering of interference terms