The derivation of tight estimation lower bounds is a key tool to design and assess the performance of new estimators. In this contribution, first, the authors derive a new compact Cramér–Rao bound (CRB) for the conditional signal model, where the deterministic parameter's vector includes a real positive amplitude and the signal phase. Then, the resulting CRB is particularised to the delay, Doppler, phase, and amplitude estimation for band-limited narrowband signals, which are found in a plethora of applications, making such CRB a key tool of broad interest. This new CRB expression is particularly easy to evaluate because it only depends on the signal samples, then being straightforward to evaluate independently of the particular baseband signal considered. They exploit this CRB to properly characterise the achievable performance of satellite-based navigation systems and the so-called real-time kinematics (RTK) solution. To the best of the authors’ knowledge, this is the first time these techniques are theoretically characterised from the baseband delay/phase estimation processing to position computation, in terms of the CRB and maximum-likelihood estimation.

This contribution analyzes the fundamental performance limits of traditional two-step Global Navigation Satellite System (GNSS) receiver architectures, which are directly linked to the achievable time-delay estimation performance. In turn, this is related to the GNSS baseband signal resolution, i.e., bandwidth, modulation, autocorrelation function, and the receiver sampling rate. To provide a comprehensive analysis of standard point positioning techniques, we consider the different GPS and Galileo signals available, as well as the signal combinations arising in the so-called GNSS meta-signal paradigm. The goal is to determine: (i) the ultimate achievable performance of GNSS code-based positioning systems; and (ii) whether we can obtain a GNSS code-only precise positioning solution and under which conditions. In this article, we provide clear answers to such fundamental questions, leveraging on the analysis of the Cramér–Rao bound (CRB) and the corresponding Maximum Likelihood Estimator (MLE). To determine such performance limits, we assume no external ionospheric, tropospheric, orbital, clock, or multipath-induced errors. The time-delay CRB and the corresponding MLE are obtained for the GPS L1 C/A, L1C, and L5 signals; the Galileo E1 OS, E6B, E5b-I, and E5 signals; and the Galileo E5b-E6 and E5a-E6 meta-signals. The results show that AltBOC-type signals (Galileo E5 and meta-signals) can be used for code-based precise positioning, being a promising real-time alternative to carrier phase-based techniques.

This article investigates on the derivation of good log likelihood ratio (LLR) approximations under uncorrelated fading channels with partial statistical channel state information (CSI) at the receiver. While previous works focused mainly on solutions exploiting full statistical CSI over the normalized Rayleigh fading channel, in this article, a Bayesian approach based on conjugate prior analysis is proposed to derive LLR values that only uses moments of order one and two associated with the random fading coefficients. The proposed approach is shown to be a more robust method compared to the best existing approximations, since it can be performed independently of the fading channel distribution and, in most cases, at a lower complexity. Results are validated for both binary and M-ary modulations over different uncorrelated fading channels.

Accurately determining signal time-delay is crucial across various domains, such as localization and communication ystems. Understanding the achievable optimal estimation peformance of such technologies, especially during design phases, is essential for benchmarking purposes. One common approach is to derive bounds like the Cramer-Rao Bound (CRB), which directly reflects the minimum achievable estimation error for unbiased estimators. Different studies vary in their approach to deal with the degree of misalignment in the global phase originating from both the transmitter and the receiver in a single input, single output (SISO) link during time-delay estimation assessment. While some treat this phase term as unknown, others assume ideal calibration and compensation. As an alternative to these two opposing approaches, this study adopts a more balanced approach by considering that such a phase can be estimated with a defined uncertainty, a measure that could be mplemented in many practical applications. The primary contribution provided lies in the derivation of a closed-form CRB expression for this alternative signal model, which, as observed, exhibits an asymptotic behavior transitioning between the results observed in previous studies, influenced by the uncertainty assumed for the mentioned phase term.

Time-delay and Doppler estimation is an operation performed in a plethora of engineering applications. A common hypothesis underlying most of the existing works is that the noise of the true and assumed signal model follows a centered complex normal distribution. However, everyday practice shows that the true signal model may differ from the nominal case and should be modeled by a non Gaussian distribution. In this paper, we analyse the asymptotic performance of the time-delay and Doppler estimation for the non-nominal scenario where the true noise model follows a centered complex elliptically symmetric (CES) distribution and the receiver assumed that the noise model follows a centered complex normal distribution. It turns out that performance bound under the misspecified model is equal to the one obtained for the well specified Gaussian scenario. In order to validate the theoretical outcomes, Monte Carlo simulations have been carried out.

Since time-delay and phase estimation is a fundamental task in a plethora of engineering fields, several CRB and MLE expressions have been derived for the past decades. In all these previous works, a common hypothesis is that the wave transmission process introduces an unknown phase which prevents from estimating both delay and transmission phase components. By revisiting this problem, including the derivation of the MLE and the associated CRB, we show that this well-admitted assertion is not true strictly: both informations can be estimated, but generally with a sub-optimal achievable MSE in the asymptotic region. Moreover, since practical problems exist where the transmission phase can be estimated apart, adding this additionnal measure to the observation model provides a setting allowing to explore the contribution of each signal component (carrier frequency, baseband signal and transmission phase measure) to the achievable MSE of time-delay and phase estimation in the asymptotic region.

Interferences are an important threat for applications relying on Global Navigation Satellite Systems (GNSS). Interferences degrade GNSS performance, and can lead to denial of service. The most notable intentional interference family is characterized by its constant envelope, e.g. chirp and tone interferences. Due to its simple structure, the space to search the interference contribution yields to complex circles, allowing the introduction of some latent variables related to those circles. In order to mitigate the interference effect, we compute the maximum likelihood estimator of the parameters of interest (time delay and Doppler shift) in presence of those latent variables. Thus, we resort to the Expectation Maximization algorithm which has already been proved to be efficient in such cases. Experiments conducted on synthetic signals highlight the efficiency of the proposed algorithm.

This paper aims to characterize the estimation precision at the output of the GNSS receiver tracking stage. We define an original statistical modelling of the GNSS tracking loop, which can then be exploited by an optimal linear Kalman Filter (KF) in order to obtain an analytical expression of the steady-state regime. The latter is designed to encompass dynamic information of the GNSS receiver. Two observation models are of interest: the first one considers the propagation delay and Doppler parameters, and the second one also including the Doppler rate, i.e., the acceleration, which is known to be relevant for high dynamics scenarios and can easily be included into the acquisition step. Within this context, the steady-state asymptotic performance of the tracking stage is obtained by solving an algebraic discrete Riccati equation. In both cases, simulation results are provided to show the validity of the proposed approach and the resulting steady-state performance.

Global Navigation Satellite Systems (GNSS) are a key player in a plethora of applications. 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 introduce a mathematical tool to evalute the effect of any type of interference on any GNSS signal. Regarding such tool, we provide closedform expressions of the misspecified Cram´er-Rao (MCRB) bound and estimation bias, for a generic GNSS signal corrupted by an interference. The proposed expressions are used to analyze the GNSS performance degradation induced by the distance measuring equipment (DME) system.

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 estimation bias, for a generic GNSS signal corrupted by an interference. The proposed bias are validated for a tone interference and a linear frequency modulation chirp interference.