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###### Article de journal

## A Robust Time Scale for Space Applications Using the Student’s t-distribution

**Metrologia Journal (accepted manuscript online 2 September 2024).**

In this article, the principles of robust estimation are applied to the standard basic time scale equation to obtain a new method of assigning weights to clocks. Specifically, the Student’s t-distribution is introduced as a new statistical model for an ensemble of clocks that are experiencing phase jumps, frequency jumps or anomalies in their measurement links. The proposed robust time scale is designed to mitigate the effects of these anomalies without necessarily identifying them, but through applying a method of robust estimation for the parameters of a Student’s t-distribution. The proposed time scale algorithm using the Student’s t-distribution (ATST) is shown to achieve comparable robustness to phase jumps, frequency jumps, and anomalies in the measurements with respect to the AT1 oracle time scale. The AT1 oracle is a special realization of the AT1 time scale which corrects all anomalies by having prior knowledge of their occurrences. The similar performance of ATST and AT1 oracle suggests that the ATST algorithm is efficient for obtaining robustness with no prior knowledge or detection of the occurrences of anomalies.

Traitement du signal et des images / Autre

###### Article de conférence

## Misspecified Cramer-Rao Bounds for Anomalous Clock Data in Satellite Constellations

**In Proc. 32nd EUropean SIgnal Processing COnference (EUSIPCO), Lyon, France, August 26-30, 2024.**

Robust estimation methods are useful in mitigating the impact of anomalies in clock data. Such anomalous clock data is assumed to be well modeled by a Student’s t-distribution. This paper derives a lower bound on the performance of the misspecified Gaussian model using the theory of the Misspecified Cram´er-Rao bound (MCRB). The results of these derivations are verified by analyzing the Mean Square Error (MSE) of the misspecified Gaussian Maximum Likelihood Estimator (MLE) when using data generated by the Student’s t-distribution. The derived MCRB indicates a constraint on the MSE when assuming a Gaussian distribution. The MLE for the mean of the Student’s t-distribution is obtained with an Expectation maximization algorithm and is shown to obtain a lower MSE than the MCRB and hence, the misspecified estimator. This indicates an improvement in performance if anomalous clock data is appropriately accounted for in the statistical model.

Traitement du signal et des images / Localisation et navigation

## Exploiting Redundant Measurements for Time Scale Generation in a Swarm of Nanosatellites

**In Proc. European Frequency and Time Forum (EFTF), Neufchâtel, Switzerland, June 25-27, 2024.**

Traitement du signal et des images / Localisation et navigation

## Exploiting Redundant Measurements for Time Scale Generation in a Swarm of Nanosatellites

**In Proc. European Frequency and Time Forum (EFTF), Neufchâtel, Switzerland, June 25-27, 2024.**

The computation of a common reference time for a swarm of nanosatellites is restricted by the quality and availability of the timing measurements made with inter-satellite links. The presence of anomalies or absence of communication links is demonstrated to harm the stability of the time scale. The Least Squares (LS) estimator is introduced as a method of preprocessing measurement noise by using all available clock comparisons in the swarm. This estimator also provides filtered measurements when inter-satellite links are missing as long as each satellite maintains at least one link with another. Anomaly detection and removing corrupted satellite links are shown to be compatible with the LS estimator to mitigate the impact of anomalous measurements. When a satellite becomes completely isolated for some period of time, a correction at the beginning and the end of the isolation period are both detailed. The correction is simple and just requires resetting the weights of missing clocks and clocks being reintroduced. Continuity is shown to be maintained when a large portion of clocks are removed and later reintroduced at the same time.

Traitement du signal et des images / Localisation et navigation

## A Robust Time Scale Based on Maximum Likelihood Estimation

**In Proc. Institute of Navigation Precise Time and Time Interval Systems and Applications (PTTI), Long Beach, California-USA, January 23-26, 2023.**

This paper introduces a new statistical model for clock phases assuming a multivariate Gaussian distribution for the clock phase deviations from a common time scale. This model allows us to derive a maximum likelihood estimator for the clock phases, which is consistent with the current methods of computing a common time scale for a collection of clocks. Detailing a statistical model of the clock phases, which assumes a Gaussian distribution allows us to find the MLE for each clock’s phase deviation from a common time scale. For verification, the MLE for the clock phases is shown to be consistent with the result of the existing basic time scale equation. The statistical distribution of the frequency states resulting from this statistical model is Gaussian over a window of past time instants. This property can be used to design a new time scale based on the maximum likelihood estimator of frequency and frequency variances that are alternatives to the exponential filters designed for AT1. With the appropriate number of past frequency samples, this MLE has identical performance to the optimal AT1 algorithm in a nominal context. The statistical distribution of the frequency when the clock suffers a phase jump anomaly is then identified as a Student’s t-distribution. The Student’s t-distribution models the statistics of datasets contaminated by outliers, leading to the derivation of a different MLE that is robust to those outliers. The time scale using the robust MLE provides estimates of each clock’s frequency and frequency variance that are unaffected by phase jump anomalies and improves the long-term frequency stability when each clock in the ensemble experiences phase jump anomalies within some window of time.

Traitement du signal et des images / Localisation et navigation

## Cycle Slip Detection in Galileo Widelane Signals Tracking

**In Proc. 31st International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+), Miami, Florida, USA, September 24 - 28, 2018.**

Precise positioning (based on Precise Point Positioning, PPP, or Real Time Kinematics, RTK) is steadily gaining momentum. The main difficulty when using carrier phase measurements remains to correctly estimate their ambiguities: not only is it a computationally intensive process, but it can be affected by cycle slips (CS), which are brutal variations in ambiguity values, due to receiver’s dynamics or unfortunate reception events. As GNSS constellations are now able to provide users with signals on three different frequencies, the concept of Triple Carrier Ambiguity Resolution has become widespread. It typically relies on the use of widelane signals, which are combinations of raw signals and are defined as to have an apparent wavelength much higher than original signals, thus making accelerating the ambiguity fixing process and reducing the frequency of cycle slips. However, CS may remain a problem for the availability of precise positioning services. The present paper therefore focuses on a cycle slip detection method, based on a hypothesis test. The main idea consists in using both code and widelane phase measurements to compute a geometry- and ionospheric-free test vector, theoretically containing only noise and possible cycle slips. The latter can be detected by looking for brutal changes on the average of the test vector. Performance is assessed on simulated and Rinex data.

Traitement du signal et des images / Localisation et navigation

## Robust Estimation of Time Information in GALILEO Navigation Messages

**In Proc. European Workshop on GNSS Signals and Signal Processing, Neubiberg, Germany, December 5-6, 2013**

This paper proposes methods designed to optimize and robustify the demodulation of the Time Of Week information contained in the GALILEO navigation message. The TOW is crucial to the determination of user’s Position-Velocity-Time and is broadcasted several times in each message frame. The redundancy and predictability of the successive TOW values can be used to reduce the probability of a demodulation error. Three methods are proposed to take advantage of this, two are empirical methods, the third consists in considering the sequence of TOWs and determining the values which maximize reception probability.

Traitement du signal et des images et Communications numériques / Localisation et navigation

## Investigation of CSK as a Candidate for Future GNSS Signals

**In Proc. 6th European Workshop on GNSS Signals and Signal Processing (EWGNSS 2013), Munich, Germany, December 2013.**

This paper presents an analysis on the implementation on a GNSS signal of a Code Shift Keying (CSK) modulation: an orthogonal M-ary modulation specifically designed to increase the bandwidth efficiency of direct-sequence spread spectrum (DS-SS) signals. Two decoding methods are presented as suitable candidates to be implemented by a CSK modulation with a LDPC channel code: classical sequential decoding and Bit-interleaved coded Modulation – Iterative Decoding (BICM-ID). Afterwards, this paper presents the methodology used to construct CSK signals which increase the useful bit rate with respect to a BPSK signal but maintaining the same symbol rate. This methodology includes the calculation and comparison of signal demodulation performances in AWGN and mobile channels, the generation of CSK symbols allowing the desired bit rate and the determination of the codeword durations. Proposals for real signals have been made. Finally, this paper analyses the impact of processing a CSK modulated signal on a GNSS receiver with respect to a BPSK signal. This analysis includes the increase of complexity of the demodulator block and the possible performance degradation of the acquisition and, the carrier and code delay tracking.

Traitement du signal et des images

## Ship Localization Using AIS Signals Received by Satellites

**In Proc. European Signal and Image Processing Conference (EUSIPCO), Marrakech, Morocco, September 9-13, 2013.**

This paper addresses the problem of ship localization by using the messages received by satellites and transmitted by the automatic identification system (AIS). In particular, one considers the localization of ships that do not transmit their actual position in AIS signals. The proposed localization method is based on the least squares algorithm and uses the differences of times of arrival and the carrier frequencies of the messages received by satellite. A modification of this algorithm is proposed to take into account the displacement model of the ships as additional measurements. This modification shows a significant localization improvement.

Traitement du signal et des images / Localisation et navigation et Systèmes spatiaux de communication

## Modelling New Indoor Propagation Models for WLAN Based on Empirical Results

**In 11th IEEE International Conference on Computer Modelling and Simulation (UKSIM'09), pp. 585-588, Cambridge, United Kingdom, March 25-27 2009.**

This paper presents the modelling of new WLAN models for different indoor environments. This work was carried out in the frame of the FIL project which is funded by the French research agency ANR, in collaboration with Thales Alenia Space France. Based on the standard Opnet models for WLAN nodes, the propagation loss estimation for these types of environment has been improved. We derive an empirical model for spatial registration patterns of mobile users as they move within a TeSA Labs wireless local area network (WLAN) environment and register signal power from different access points. Such a model can be very useful in a variety of simulation studies of the performance of mobile wireless systems, to address issues such as resource management and mobility management protocols. We base the model on extensive experimental data from a TeSA Labs 2.4 GHz WiFi LAN installation. We divide the empirical data available to us into training and test data sets, develop the model based on the training set, and evaluate it against the test set. The new scenarios used to simulate these new propagation models are shown. Finally, results, conclusions and further work are given.

Traitement du signal et des images et Communications numériques / Localisation et navigation

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