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On the accuracy limits of misspecified delay-Doppler estimation

Auteurs : Mc Phee Hamish Scott, Ortega Espluga Lorenzo, Vilà-Valls Jordi et Chaumette Eric

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

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

Untangling first and second order statistics contributions in multipath scenarios

Auteurs : Lubeigt Corentin, Ortega Espluga Lorenzo, Vilà-Valls Jordi et Chaumette Eric

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

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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.

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Traitement du signal et des images / Localisation et navigation

Band-limited impulse response estimation performance

Auteurs : Lubeigt Corentin, Ortega Espluga Lorenzo, Vilà-Valls Jordi et Chaumette Eric

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

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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.

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Traitement du signal et des images / Localisation et navigation

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

Auteurs : Mc Phee Hamish Scott, Ortega Espluga Lorenzo, Vilà-Valls Jordi et Chaumette Eric

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

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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.

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

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

Auteurs : Lubeigt Corentin, Ortega Espluga Lorenzo, Vilà-Valls Jordi, Lestarquit Laurent et Chaumette Eric

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

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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.

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

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

Auteurs : Ortega Espluga Lorenzo, Vilà-Valls Jordi et Chaumette Eric

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

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

Insights on the Estimation Performance of GNSS-R Coherent and Noncoherent Processing Schemes

Auteurs : Ortega Espluga Lorenzo, Vilà-Valls Jordi et Chaumette Eric

IEEE Geoscience and Remote Sensing Letters, vol. 19, Art no. 8012205, pp. 1-5, 2022.

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Parameter estimation is a problem of interest when designing new remote sensing instruments, and the corresponding lower performance bounds are a key tool to assess the performance of new estimators. In global navigation satellite systems reflectometry (GNSS-R), a noncoherent averaging is applied to reduce speckle and thermal noise, and subsequently the parameters of interest are estimated from the resulting waveform. This approach has been long regarded as suboptimal with respect to the optimal coherent one, which is true in terms of detection capabilities, but no analysis exists on the corresponding parameter estimation performance exploiting GNSS signals. First, we show that for certain signal models, both coherent and noncoherent Cramér-Rao bounds are equivalent, and therefore, any maximum likelihood estimation coherent/noncoherent combination scheme is efficient (optimal) at high signal-to-noise ratios. This is validated for an illustrative GNSS-R estimation problem. In addition, it is shown that considering the joint delay/Doppler/phase estimation problem, the noncoherent performance for the delay is still optimal, which is of practical importance for instance in altimetry applications.

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

On Nested Property of Root-LDPC Codes

Auteurs : Ortega Espluga Lorenzo et Poulliat Charly

IEEE Wireless Communications Letters, vol. 10, issue 5, pp. 1005-1009, May 2021.

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We investigate on binary Protograph Root-LDPC codes that can embed an interesting property, called nested property. This property refers to the ability for a coding scheme to achieve full diversity and equal coding gain for any number of received coded blocks and for any configuration of the received code blocks. One can take advantage of this property for “carousel”-type transmissions broadcasting cyclically coded information. For regular Root-LDPC codes, we show that these codes inherently have both properties over the nonergodic block fading channel and when message passing decoding is used. Then, we show that irregular Root-LDPC structures could not provide equal coding gain except if explicit design rules are enforced to ensure that the nested property is fulfilled when designing irregular Root-LDPC codes. Simulation results show that designed nested Root-LDPC codes achieve good performance and full diversity for coding rates R=1/2, R=1/3 and R=1/4.

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Communications numériques / Localisation et navigation et Systèmes spatiaux de communication

On the Impact and Mitigation of Signal Crosstalk in Ground-Based and Low Altitude Airborne GNSS-R

Auteurs : Lubeigt Corentin, Ortega Espluga Lorenzo, Vilà-Valls Jordi, Lestarquit Laurent et Chaumette Eric

Remote sensing, vol. 13, issue 6, Art. no 1085, March, 2021.

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Global Navigation Satellite System Reflectometry (GNSS-R) is a powerful way to retrieve information from a reflecting surface by exploiting GNSS as signals of opportunity. In dual antenna conventional GNSS-R architectures, the reflected signal is correlated with a clean replica to obtain the specular reflection point delay and Doppler estimates, which are further processed to obtain the GNSS-R product of interest. An important problem that may appear for low elevation satellites is signal crosstalk, that is the direct line-of-sight signal leaks into the antenna dedicated to the reflected signal. Such crosstalk may degrade the overall system performance if both signals are very close in time, similar to multipath in standard GNSS receivers, the reason why mitigation strategies must be accounted for. In this article: (i) we first provide a geometrical analysis to justify that the estimation performance is only affected for low height receivers; (ii) then, we analyze the impact of crosstalk if not taken into account, by comparing the single source conditional maximum likelihood estimator (CMLE) performance in a dual source context with the corresponding Cramér–Rao bound (CRB); (iii) we discuss dual source estimators as a possible mitigation strategy; and (iv) we investigate the performance of the so-called variance estimator, which is designed to eliminate the coherent signal part, compared to both the CRB and non-coherent dual source estimators. Simulation results are provided for representative GNSS signals to support the discussion. From this analysis, it is found that: (i) for low enough reflected-to-direct signal amplitude ratios (RDR), the crosstalk has no impact on standard single source CMLEs; (ii) for high enough signal-to-noise ratios (SNR), the dual source estimators are efficient irrespective of the RDR, then being a promising solution for any reflected signal scenario; (iii) non-coherent dual source estimators are also efficient at high SNR; and (iv) the variance estimator is efficient as long as the non-coherent part of the signal is dominant.

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Traitement du signal et des images / Localisation et navigation

GNSS Data Demodulation over Fading Environments: Antipodal and M-ary CSK Modulations

Auteurs : Ortega Espluga Lorenzo, Vilà-Valls Jordi, Poulliat Charly et Closas Pau

IET Radar, Sonar & Navigation, vol. 15, issue 2, pp. 113-127, February, 2021.

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This article investigates new strategies to compute accurate low-complexity Log Likelihood Ratio (LLR) values based on the Bayesian formulation under uncorrelated fading channels for both antipodal and CSK modulations when no Channel State Information (CSI) is available at the receiver. These LLR values are then used as input to modern error correcting schemes used in the data decoding process of last generation GNSS signals. Theoretical analysis based on the maximum achievable rate is presented for the different methods in order to evaluate the performance degradation with respect to the optimal CSI channel. Finally, Frame Error Rate (FER) simulation results are shown, validating the appropriate performance of the proposed LLR approximation methods.

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Communications numériques / Localisation et navigation et Systèmes spatiaux de communication

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