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###### Article de journal

## On the asymptotic performance of time-delay and Doppler estimation with a carrier modulated by a band-limited signal

**EURASIP Journal on Advances in Signal Processing, vol. 2024 (1), pp. 47, April, 2024.**

Time-delay and Doppler estimation is crucial in various engineering fields, as estimating these parameters constitutes one of the key initial steps in the receiver’s operational sequence. Due to its importance, several expressions of the Cramér–Rao Bound (CRB) and Maximum Likelihood Estimation (MLE) have been derived over the years. Previous contributions started from the assumption that the transmission process introduces an unknown phase, which hindered the explicit consideration of the time-delay parameter in the carrier-phase component in theoretical derivations. However, this contribution takes into account this additional term under the assumption that such an unknown phase is inferred and compensated for. This new condition leads to the derivation of a novel MLE. Subsequently, a closed-form expression of the achievable Mean Squared Error (MSE) for the time-delay and Doppler parameters is provided for the asymptotic region, assuming the signal is band-limited. Both expressions are validated via Monte Carlo simulations. This analysis reveals five distinct regions of operation of the MLE, refining existing knowledge and providing valuable insights into time-delay estimation

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

## Approximate Maximum Likelihood Time-Delay Estimation for Two Closely Spaced Sources

**Signal processing, vol. 210, 109056, September, 2023.**

The study of ground reflections of Global Navigation Satellite System (GNSS) signals, as in GNSS Reflectometry (GNSS-R) can lead to the receiver height estimation. The latter is estimated by comparing the time of arrival difference between the direct and reflected signals, also called path separation. In ground-based scenarios, this path separation can be very small, inducing important interference between paths, which makes it difficult to correctly obtain altimetry products. The path separation estimation can be obtained by a brute force dual source maximum likelihood estimator (2S-MLE), but this solution has a large computational cost. On the other hand, the path separation is so small that a number of approximations can be done. In this study, a third order Taylor approximation of the dual source likelihood criterion is proposed to reduce its complexity. The proposed algorithm performance is compared to the non approximated 2S-MLE for the estimation of the path separation, and to a standard single source processing for the estimation of the direct signal time-delay. These results, along with the corresponding lower bounds, prove that the proposed approach may be of interest for two applications: ground-based GNSS-R altimetry (or radar with low elevation targets) and GNSS multipath mitigation.

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

## GNSS Channel Coding Structures for Fast Acquisition Signals in Harsh Environment Conditions.

**Navigation-Journal of Navigation, doi: 10.33012/navi.585, September, 2023.**

In this article, we investigate on a new method to jointly design the navigation message with an error correcting scheme. This joint design exploits the "carousel" nature of the broadcasted navigation message and allows both: i) to reduce the Time To First Fix (TTFF) and ii) to enhance the error correcting performances under favorable and challenging channel conditions. We show that the joint design requires error correcting schemes characterized by Maximum Distance Separable (MDS) and the full diversity properties. Those error correcting codes are referred to as Root LowDensity Parity Check (Root-LDPC) codes and they can efficiently operate on block varying channels, enabling the efficient and rapid recovery of information over possibly non ergodic channels. Finally, in order to ensure the data demodulation performance over harsh condition, we propose Root-LDPC codes endowed with the nested property, which allows to inherently adapt the channel coding rate depending on the number of received blocks. The proposed error correcting joint design is then simulated and compared with the well-known GPS L1C subframe 2 structure under several transmission scenarios. Simulations showthatwe can have some enhancement of the error correction performance and a reduction of the TTFF for some scenarios.

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

## On GNSS Synchronization Performance Degradation under Interference Scenarios: Bias and Misspecified Cramér-Rao Bounds

**Navigation, vol. 70 (4), pp. navi.606, July, 2023.**

Global navigation satellite systems (GNSS) are a key player in a plethora of applications, ranging from navigation and timing, to Earth observation or space weather characterization. For navigation purposes, interference scenarios are among the most challenging operation conditions, which clearly impact the maximum likelihood estimates (MLE) of the signal synchronization parameters. While several interference mitigation techniques exist, a theoretical analysis on the GNSS MLE performance degradation under interference, being fundamental for system/receiver design, is a missing tool. The main goal of this contribution is to provide such analysis, by deriving closed-form expressions of the misspecified Cramér-Rao (MCRB) bound and estimation bias, for a generic GNSS signal corrupted by an interference. The proposed bias and MCRB expressions are validated for a linear frequency modulation chirp signal interference.

## On the accuracy limits of misspecified delay-Doppler estimation

**Signal Processing, article 108872, vol. 205, April, 2023.**

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.

## Untangling first and second order statistics contributions in multipath scenarios

**Signal Processing, vol. 205, Art. no 108868, April, 2023.**

In ranging-based applications, ignoring the presence of multipath often leads to a bias upon the estimated range, which actually originates from misspecified estimation problem because the assumed data signal model, here without multipath, is not equal to the true one. Such misspecification also results in an error covariance matrix around the biased estimates, so-called pseudotrue parameters, that differs from the Cramér–Rao bound applied to the true model. This error covariance matrix can be lower bounded by a misspecified Cramér–Rao bound (MCRB). In this work, a closed-form expression of the MCRB under multipath conditions is proposed, which only depends on the baseband signal samples and both delay, Doppler and complex amplitude pseudotrue parameters. These MCRB expressions are fundamental (i) to understand and characterize the impact of multipath conditions when not taken into account, (ii) for system/signal design, and (iii) to derive new robust estimators. The proposed MCRBs are validated for a representative navigation signal, comparing the resulting bounds with the mean square error obtained by the misspecified maximum likelihood estimator with respect to the pseudotrue parameters.

Traitement du signal et des images / Localisation et navigation

## Band-limited impulse response estimation performance

**Signal Processing, vol. 208, Art. no 108998, July, 2023.**

When a signal is strongly distorted by a reflecting surface, the surface can be seen as a filter whose impulse response is convoluted with the incident signal. Depending on the application, it can be useful to estimate this impulse response in order to either compensate or interpret it. When it comes to estimation, a performance lower bound should be computed in order to better understand the performance limits of the observation model at hand. Hence, a first contribution of this work is to provide an easy-to-use closed-form Cramér–Rao bound for the proposed signal model. The validation process of this lower bound raises the problem of the size, generally unknown, of the impulse response to be estimated. A second contribution of this study is then to provide adapted theoretical and practical tools to determine the size of a given impulse response along with its estimation.

Traitement du signal et des images / Localisation et navigation

## Accounting for Acceleration – Signal Parameters Estimation Performance Limits in High Dynamics Applications

**IEEE Transactions on Aerospace and Electronic Systems, Vol 59, Issue 1, pp 610-622, February 2023.**

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 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 (LFM) chirp band-limited signals. The mean square error (MSE) 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.

## Clean-to-Composite Bound Ratio: A Multipath Criterion for GNSS Signal Design and Analysis

**IEEE Transactions on Aerospace and Electronic Systems, vol. 58, issue 6, pp. 5412-5424, December, 2022.**

Multipath is one of the most challenging propagation conditions affecting Global Navigation Satellite Systems (GNSS), which must be mitigated in order to obtain reliable navigation information. In any case, the random multipath nature makes it difficult to anticipate and overcome. Therefore, for legacy GNSS signal performance assessment, modern GNSS signal design and future GNSS-based applications, robustness to multipath is a fundamental criterion. Different multipath metrics exist in the literature, such as the multipath error envelope, usually leading to analyses only valid for a dedicated receiver/signal combination and only providing information on the bias. This paper presents a general criterion to characterize the multipath robustness of a generic band-limited signal (e.g., GNSS or radar), considering the joint delay-Doppler and phase estimation. This criterion is based on the Cramr-Rao bound, which makes it universal, regardless the receiver architecture and the signal under analysis, and provides information on the actual achievable performance in terms of estimated time-delay (i.e., pseudo-range) and Doppler frequency variances.

Traitement du signal et des images et Réseaux / Localisation et navigation

## Non-Binary PRN-Chirp Modulation: A GNSS Fast Acquisition Signal Waveform

**IEEE Communications Letters, vol. 26, Issue 9, pp. 2151-2155, September, 2022.**

In this article, we propose a new non-binary modulation which allows both Global Navigation Satellite Systems (GNSS) synchronization and the demodulation of non-binary symbols, without the need of a pilot signal, with the aim to provide a fast first position, velocity and time fix. The waveform is constructed as the product of i) a pseudo-random noise sequence with good auto-correlation and cross-correlation properties, and ii) a chirp spread spectrum family, which allows to demodulate non-binary symbols even if the signal phase is unknown. In order to demodulate the data, a bank of non-coherent matched filters is proposed. Because of the particular modulation structure, the receiver is capable to demodulate the navigation message faster while allowing the basic GNSS signal processing functionalities. Illustrative results are provided to support the discussion.

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