Nonlinear regression plays a crucial role in various engineering applications. For the sake of mathematical tractability and ease of implementation, most of the existing inference procedures are derived under the assumption of independent and identically distributed (i.i.d.) Gaussian-distributed data. However, real-world situations often deviate from this assumption, with the true data generating process being a correlated, heavy-tailed and non-Gaussian one. The paper aims at providing the Misspecified Cramér–Rao Bound (MCRB) on the Mean Squared Error (MSE) of any unbiased (in a proper sense) estimator of the parameters of a nonlinear regression model derived under the i.i.d. Gaussian assumption in the place of the actual correlated, non-Gaussian data generating process. As a special case, the MCRB for an uncorrelated, i.i.d. Complex Elliptically Symmetric (CES) data generating process under Gaussian assumption is also provided. Consistency and asymptotic normality of the related Mismatched Maximum Likelihood Estimator (MMLE) will be discussed along with its connection with the Nonlinear Least Square Estimator (NLLSE) inherent to the nonlinear regression model. Finally, the derived theoretical findings will be applied in the well-known problem of time-delay and Doppler estimation for GNSS.

Location-based services, safety-critical applications, and modern intelligent transportation systems require reliable, continuous, and precise positioning, navigation, and timing (PNT) information. Global Navigation Satellite Systems (GNSS) are the main source of positioning data in open sky conditions; however, their vulnerabilities to radio interferences and signal propagation limit their use in challenging environments. Consequently, enhancing conventional GNSS-based PNT solutions to incorporate additional sensing modalities and exploit other available signals of opportunity has become necessary for continuous and reliable navigation. Articles in the special issue span detection methods, estimation algorithms, signal optimization, and the application of machine learning, providing comprehensive insights into enhancing navigation and positioning accuracy.

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

Interferences pose a significant risk to applications that rely on global navigation satellite systems (GNSSs). They have the potential to degrade GNSS performance and even result in service disruptions. The most notable type of intentional interference is characterized by a constant modulus, such as chirp and tone interferences. These interferences have a straightforward structure, leading to the creation of complex circles when attempting to identify their contribution. To address the interference and improve the situation, we calculate the maximum likelihood estimator for the relevant parameters (time delay and Doppler shift) while considering the presence of these latent variables. To achieve this, we employ the expectation–maximization algorithm, which has previously demonstrated its effectiveness in similar scenarios. Experiments conducted using synthetic signals confirm the efficiency of the proposed algorithm.

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.

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.

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.

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.

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.

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.