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In this article, we discuss the important role of time-frequency (TF) signal representations in enhancing global navigation satellite system (GNSS) receiver performance. Both linear transforms and quadratic TF distributions (QTFDs) are considered. We review recent advances of antijam techniques that exploit the distinction in the TF signatures between the desired and undesired signals, enabling effective jammer excision with minimum distortion to the navigation signals. The characterization of jammers by their instantaneous frequencies (IFs) lends itself to sparse structures in the TF domain and, as such, invites compressive sensing (CS) and sparse reconstruction to aid in joint-variable domain jamming localization and suppression. Furthermore, the integration of the spatial domain with the TFDs, through the use of multiantenna receivers, permits the applications of space-time processing for effective jamming mitigation. The article also describes the fundamental role of TFDs in monitoring the performance of new GNSSs, including satellite clocks and ionospheric scintillation data. Real GNSS data collected in the presence of jamming are used to demonstrate the effectiveness of TF-based antijam approaches

The impulse response is a key function for the analysis and design of systems, therefore understanding its properties is a fundamental problem. To investigate the nonstationary structure of the impulse response, we derive its time–frequency representation both in the Wigner and smoothed Wigner distribution domains. First, we consider the class of single-input single-output (SISO) systems, which models a wide variety of physical phenomena. The obtained Wigner distribution for the class of SISO systems is made by frequency-translated versions of the Wigner distribution of the impulse response for a first-order system, a T-shaped function of time and frequency, plus the interference terms, which are effectively filtered out in the smoothed Wigner domain. Then, we consider the case of multiple-input multiple-output (MIMO) systems, whose impulse response is the collection of the individual responses obtained by applying an impulse to every input separately. Also in this case, the Wigner distribution of the impulse response depends on the T function. We apply our results to a two-dimensional MIMO harmonic oscillator, and to a SISO system with three resonances. Finally, we propose the key idea of a system identification method that operates directly in the time–frequency domain by fitting the smoothed T function. We show that the method works for a nonlinear oscillator with a large cubic term, as well as for a time-varying harmonic oscillator, and that it can tolerate the presence of both stationary and nonstationary noise.

Many deterministic and random physical signals can be modeled as the output of a multi-input-multi-output (MIMO) dynamical system. Since physical signals are typically nonstationary, their frequency content changes with time. To understand this time variation, we transform the MIMO system to the time-frequency domain. The result is a time-frequency MIMO dynamical system, whose input and output are the time-frequency spectra of the original input and output signals in the time domain. The time-frequency system reveals the spectral mechanisms involved in the generation of nonstationary signals. We apply our method to the case of a MIMO system with two vibrational modes and a nonstationary noise at the input. We obtain the time-frequency spectrum of the output, which shows how the spectrum of the modes changes with time. This result cannot be achieved with classical spectral techniques, because they require the input random process to be wide sense stationary.

In the rubidium atomic frequency standard (RAFS), an rf-discharge lamp produces the device's atomic signal. As a consequence of the light-shift effect, variations in the lamplight's intensity result in variations in the RAFS' output frequency. While the basic physics of the light-shift is reasonably well understood, its operational implications for global navigation satellite system (GNSS) performance is only beginning to be fully appreciated. Here, we describe first results examining decade-long histories of on-orbit GPS RAFS lamplight variations and GPS RAFS frequency variations. Our preliminary analyses have focused on one space vehicle's RAFS, and our conclusions are tempered by that present limitation. Nevertheless, our analyses suggest that a RAFS' long-term frequency stability (i.e., τ 106 sec) is likely lower-bounded by the lamp's intensity fluctuations. Moreover, considering the light-shift coefficient for this one particular RAFS over 12 years, we find that the data do not support Camparo's hypothesis regarding RAFS frequency aging and a time-varying light-shift coefficient.

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.

We characterize the nature of the instantaneous spectrum of a random system. We define the instantaneous spectrum as the Wigner spectrum of the state of the system, and we apply a method which transforms the system in the domain of the Wigner spectrum. By using this approach we prove that the instantaneous spectrum is made by three components. Two components decay with time, while a third stationary component is reached for large times.

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 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 define the generalized Wiener process as the output of a first-order differential equation when the input is an arbitrary stochastic input. This is in contrast to the standard Wiener process, where the input is white Gaussian noise. We obtain a simple explicit result for any input wide sense stationary random process, namely, that the Wigner spectrum of the output random process is the product of the power spectrum of the input process times a simple universal function of time and frequency. We also obtain the impulse response function wherein the output of the generalized process is expressed as the impulse response function integrated with the input random process. Various limiting values are derived. We apply the method to the case where the driving stochastic process has an exponentially decaying autocorrelation function

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.

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.

This paper presents a new method for the classification of nonlinear systems based on the time–frequency analysis of the free oscillations. In particular, oscillators described by a nonlinear second-order differential equation are considered. The nonlinearity is due to a nonlinear function of the state variable, which produces free oscillations with a time-variant spectrum. The classification is based on two different estimates of the fundamental frequency of the free oscillations. The comparison between the two estimates allows a very efficient system classification. Both the proposed estimators and the way of combining the two estimates are new. Validation tests have shown the high performances of the proposed technique.

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.

We obtain the time-frequency response of the output of a dynamical system when the input belongs to a class of common nonstationary signals, namely, an impulse, a linear chirp, a causal sinusoid, and a short duration sinusoid. The obtained results clarify how the system processes the time-varying frequencies of the input signal to generate the time-frequency spectrum of the output. All analytic results are exact. The solution is obtained by developing a method which can be used to evaluate the output of a dynamical system for complex combinations of nonstationary inputs. We show numerical examples which prove that the response of a system to a nonstationary input is made by a series of events occurring in the joint time-frequency domain.

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.

The frequency spectrum is a fundamental quantity for the analysis and design of a system. When a system is turned on or off, the frequency spectrum of its output changes with time. We define the transient spectrum as the time-frequency spectrum of the system output during the transient phase. We obtain the exact transient spectrum for a wide class of random systems, and we formulate it with respect to the classical power spectrum, which is reached for large times. We apply the developed method to the case of a harmonic oscillator and of a system with two resonances.

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.

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

The aim of the adjustable bandwidth concept (ABC) is to enhance nonstationary signals in noise by bringing out the main features so that they be effectively used in detection and classification algorithms. We show that the method transforms a time-frequency distribution by means of individual time-frequency transformations into a set of final time-frequency distributions. Each step of the procedure can be understood and formulated in terms of the kernel method. Explicit expressions are derived and examples are given. The formulation suggests further enhancements of the method.

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.

We present a new method to study linear systems that exhibit time-varying characteristics and are governed by ordinary differential equations. The method transforms the differential equation into the governing equation for the ambiguity function and to the time-frequency domain. The advantage of the method is that in the new domain an immense simplification occurs, both mathematically and physically. The method applies to both deterministic and stochastic differential equations. The approach is applied to the classical RC filter driven by white noise. We obtain a new characterization of the output spectra.

Cohen, L., Galleani, L., Hedges, R., Hughes, D., Loughlin, P., and Suter, B., Time–Frequency Analysis of a Variable Stiffness Model for Fault Development, Digital Signal Processing 12 (2002) 429–440 Over the past 50 years it has been found that many real systems have spectra which are changing in time. In this article we explore the possibility of describing such systems with harmonic and nonlinear oscillator equations whose coefficients are time dependent. We show that time–frequency analysis offers a powerful method to study such systems.

Recently an European Consortium of 15 partners from 7 different countries, started to work on a research project, DEMETRA (DEMonstrator of EGNSS services based on Time Reference Architecture), funded by the European Union in the frame of the Horizon 2020 program, aiming to develop and experiment time services based on the European Global Navigation Satellite System (GNSS). DEMETRA aims to be a prototype of an European time dissemination service, based on the timing signal of the European Galileo system, adding particular features like certification, calibration, or integrity, that could be of interest to some specific users like traffic control, energy distribution, finance, telecommunication, and scientific institutions.