My Publications

Authors: Hamish McPhee, Jean-Yves Tourneret, David Valat, Jérome Delporte, Yoan Grégoire, Philippe Paimblanc

Published in: Proceedings of the 32nd Annual European Signal Processing Conference (EUSIPCO) August 26-30, 2024, Lyon, France

Abstract: 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 Cramer-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.

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Authors: Hamish McPhee, Jean-Yves Tourneret, David Valat, Jérome Delporte, Yoan Grégoire, Philippe Paimblanc

Published in: Proceedings of the 37th Annual European Frequency and Time Forum June 25-27, 2024, Neuchâtel, Switzerland

Abstract: 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.

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Authors: Hamish McPhee, Jean-Yves Tourneret, David Valat, Philippe Paimblanc, Jérome Delporte, Yoan Grégoire

Published in: Proceedings of the 54th Annual Precise Time and Time Interval Systems and Applications Meeting January 23 - 26, 2023, Long Beach, California

Abstract: 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.

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Authors: Hamish McPhee, Lorenzo Ortega, Jordi Vilà-Valls, Eric Chaumette

Published in: Signal Processing, Volume 205, April 2023, 108872

Abstract: This work derives compact closed-form expressions of the misspecified Cramér–Rao bound and pseudo-true parameters of time-delay and Doppler for a high dynamics signal model. Those expressions are validated by analyzing the mean square error (MSE) of the misspecified maximum likelihood estimator. A noteworthy outcome of these MSE results is that, for some magnitudes of acceleration and signal-to-noise ratios, neglecting the acceleration is beneficial in the MSE sense. The variance performance improvement is obtained at the cost of a systematic error in the true parameter estimation. This can be seen as a specific case of the trade-off between bias and variance. Neglecting the acceleration can improve the Doppler estimation when the error induced on the misspecified model is less than the variance increase due to including an extra parameter to estimate. Then, for some non-zero acceleration magnitudes and short integration times, the Doppler estimation using a misspecified model outperforms a correctly specified model in the MSE sense.

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Authors: Hamish McPhee, Lorenzo Ortega, Jordi Vilà-Valls, Eric Chaumette

Published in: IEEE Transactions on Aerospace and Electronic Systems, Volume 59, Issue 1, February 2023

Abstract: The derivation of estimation lower bounds is paramount to designing and assessing the performance of new estimators. A lot of effort has been devoted to the range-velocity estimation problem, a fundamental stage on several applications, but very few works deal with acceleration, being a key aspect in high dynamics applications. Considering a generic band-limited signal formulation, we derive a new general compact form Cramér–Rao lower bound (CRB) expression for joint time-delay, Doppler stretch, and acceleration estimation. This generalizes and expands upon known delay/Doppler estimation CRB results for both wideband and narrowband signals. This new formulation, especially easy to use, is created based on baseband signal samples, making it valid for a variety of remote sensors. The new CRB expressions are illustrated and validated with representative GPS L1 C/A and linear frequency modulated chirp band-limited signals. The mean-square error of a misspecified estimator (conventional delay/Doppler) is compared with the derived bound. The comparison indicates that for some acceleration ranges the misspecified estimator outperforms a well-specified estimator that accounts for acceleration.

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Authors: Hamish McPhee, Lorenzo Ortega, Jordi Vilà-Valls, Eric Chaumette

Published in: ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

Abstract: The derivation of estimation lower bounds is paramount to design and assess the performance of new estimators. A lot of effort has been devoted to the joint distance-velocity estimation problem, but very few works deal with acceleration, being a key aspect in several high-dynamics applications. Considering a generic band-limited signal formulation, in this contribution we derive a new closed-form Cramér-Rao bound (CRB) expression for joint time-delay/Doppler/acceleration estimation. This new formulation, especially easy to use, depends only on the baseband signal samples, and can be exploited for several purposes including estimator assessment (i.e., for signal design or to derive performance loss metrics with respect to the best (lowest) CRB). These results are illustrated and validated with two representative band-limited signals, namely, a GPS L1 C/A signal and a linear frequency modulated chirp signal.

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