International audience

The Modified Cramér-Rao Bound (MCRB) proves to be of significant importance in non-standard estimation scenarios, when in addition to unknown deterministic parameters to be estimated, observations also depend on random nuisance parameters. Given the interest of applications that involve estimation on Lie Groups (LGs), as well as the relevance of nonstandard estimation problems in many practical scenarios, the main concern in this communication is to derive an intrinsic MCRB on LGs (LG-MCRB). For this purpose, a modified unbiasedness constraint must be defined, yielding a modified Barankin Bound. A closed-form formula of the LG-MCRB is then provided for a LG Gaussian model on $S O(2)$, representing $2 D$ rotation matrices, while considering non-Gaussian random nuisance parameters. The validity of this expression is then assessed through numerical simulations, and compared with the intrinsic CRB on LGs for a simplified illustrative scenario, involving a concentrated Gaussian prior distribution on the random nuisance parameters.

It is well-known that the Modified Cramér-Rao Bound (MCRB) holds particular value in nonstandard deterministic estimation scenarios. Specifically, it proves invaluable when, in addition to estimating deterministic parameters, one needs to determine the probability density function (p.d.f) of the data through the marginalization of a joint p.d.f over random variables. In general, this process of marginalization is mathematically intractable, which restricts the utility of the conventional CRB. This limitation is especially pertinent in the context of discrete-time Markovian dynamic systems. However, we demonstrate that for such systems, the MCRB can be computed recursively with minimal computational burden, provided certain mild regularity conditions are met for the random nuisance parameters. Although this computational advantage may entail a degree of looseness in the bound, we present evidence showcasing the practical relevance of the proposed expressions in a scenario where the MCRB and CRB align.

In this communication, we derive a new intrinsic Cramér-Rao bound for both parameters and observations lying on Lie groups. The expression is obtained by using the intrinsic properties of Lie groups. An exact expression is obtained for the case where parameters and observations are in SE(2), the semi-direct Lie group of 2D rotation and 2D translation. To support the discussion, the proposed bound is numerically validated for a Lie group Gaussian model on SE(2).

This contribution explores the optimality properties of integrity monitoring fault detectors by exploiting the hypothesis testing invariance theory. The focus is on three fault detectors widely used in GNSS: the Generalized Likelihood Ratio Test (GLRT), the Least Squares (LS) residuals method, and the Solution Separation (SS) test statistics. The GLRT has been shown to be uniformly most powerful invariant for linear Gaussian models, and the single-state SS test statistic has been proven to be the optimal detector which minimizes the so-called worst-case integrity risk, if the LS estimator is used to estimate the unknown state vector. This work aims i) to make the connection between these two optimal detectors within the invariance framework, and ii) to establish the conditions for their equivalence in the case of a single alternative faulty hypothesis.

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.

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.

The derivation of tight estimation lower bounds is a key player to design and assess the performance of new estimators. Considering a generic band-limited signal formulation and constant transmitter to receiver propagation delay, we propose a novel compact closed-form expression of the Cramér-Rao bound for time-delay estimation. This new formulation, especially easy to use, allows to derive the best (lowest) Cramér-Rao bound for a band-limited signal of given length and energy, which provides an estimation performance loss metric. These results are illustrated with two representative band-limited signals.

One of the major challenges in Bayesian filtering is the curse of dimensionality. The quadrature Kalman filter (QKF) is the method of choice in many real-life Gaussian problems, but its computational complexity increases exponentially with the dimension of the state. As a promising solution to overcome the filter limitations in such scenarios, we further explore the multiple state-partitioning approach, which considers the partition of the original space into several subspaces, with the goal to apply a low-dimensional filter at each partition. In this contribution, the key idea is to take advantage of the estimation uncertainty provided by the QKF to improve the interaction among filters and avoid the point estimate approximation performed in the original Multiple QKF (MQKF). The new filter formulation, named Improved MQKF, considers Gauss-Hermite quadrature rules to propagate the subspaces of interest, together with cubature rules for marginalization purposes. The nested quadrature-cubature approximation provides robustness and improves the filter performance. Simulation results for a multiple target tracking scenario are provided to support the discussion.

This contribution deals with robust carrier phase tracking in Global Navigation Satellite Systems, where the ultimate goal is to obtain accurate and robust phase estimates under non-nominal conditions, such as high dynamics, strong fading and ionospheric scintillation. Within this framework, an Interacting Multiple Model approach, using a bank of parallel Kalman-based filters with different dynamic state models, is proposed to cope with signals corrupted by severe ionospheric scintillation. In the proposed formulation, the time-varying and correlated scintillation phase is introduced into the dynamic system using an AR(1) model. Simulation results are provided to show the enhanced robustness and improved accuracy of the proposed approach, with respect to state-of-the-art carrier phase tracking techniques.

This paper considers the problem of ultra-tight GNSS/INS integration. We propose a new approach, deriving the direct relation between Inertial Measurement Unit (IMU) measurements and synchronization parameters, used in the trilateration algorithm to compute the position of the receiver. We take into account the IMU's eventual biased behavior by introducing it into the state representation. We use a recently-developed, square-root derivative-free Gaussian nonlinear filter to solve the estimation problem.

This paper shows the applicability of recently-developed Gaussian nonlinear filters to sensor data fusion for positioning purposes. After providing a brief review of Bayesian nonlinear filtering, we specially address square-root, derivative-free algorithms based on the Gaussian assumption and approximation rules for numerical integration, namely the Gauss--Hermite quadrature rule and the cubature rule. Then, we propose a motion model based on the observations taken by an Inertial Measurement Unit, that takes into account its possibly biased behavior, and we show how heterogeneous sensors (using time-delay or received-signal-strength based ranging) can be combined in a recursive, online Bayesian estimation scheme. These algorithms show a dramatic performance improvement and better numerical stability when compared to typical nonlinear estimators such as the Extended Kalman Filter or the Unscented Kalman Filter, and require several orders of magnitude less computational load when compared to Sequential Monte Carlo methods, achieving a comparable degree of accuracy.

This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (CPM) over a time-selective slowly varying flat Rayleigh fading channel. We propose an expectation-maximization (EM) algorithm for path delay estimation in a Kalman smoother framework. The time-varying complex gain is modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic Bayesian system in a state-space form that allows the application of EM algorithm in the context of unobserved data for obtaining an estimate of the path delay. This is used with Kalman smoother for state estimation. We derive analytically a closed-form expression of the modified hybrid Cramer-Rao bound (MHCRB) for path delay and complex gain parameters. Finally, some numerical examples are presented to illustrate the performance of the proposed algorithm compared to the conventional generalized correlation method and to the MHCRB.

This paper deals with the on-line estimation of a dynamical carrier phase and a frequency offset in a digital receiver. We consider a Brownian phase evolution with a linear drift in a data aided scenario. The proposed study is relative to the use of an oversampled signal model after matched filtering, leading to a coloured reception noise and a non-stationary power signal. We derive a closed-form expression of the hybrid Cramer-Rao bound (HCRB) for this estimation problem. We use a binary offset carrier (BOC) function as shaping pulse. Our numerical results show the potential gain of using the oversampled signal for estimating the dynamical phase and frequency offset, obtaining better performances than using a classical synchronizer.

In this paper, we present an application of the Extended Kalman Filter for the on-line estimation of a dynamical carrier phase offset. The novel approach implies deriving the filter in an oversampled scenario in a digital receiver. We consider a Brownian phase evolution in a Data Aided scenario. Our numerical results using a BOC shaping pulse show that using the oversampled signal for estimating the phase offset we can obtain better performances than using a classical synchronizer.