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.

Accurately determining signal time-delay is crucial across various domains, such as localization and communication systems. Understanding the achievable optimal estimation performance of such technologies, especially during design phases, is essential for benchmarking purposes. One common approach is to derive bounds like the Cramér-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 implemented 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.

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.

International audience

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.

Maximum Likelihood (ML) frequency estimation of a single tone in noise is known to be a computationally intensive task that does not cope with many real-time and embedded hardware architectures. Thereby, many sub-optimal techniques, based on approximations, have been proposed in the literature. In this paper, we show that the ML criterion can be solved directly, using an appropriate two-step procedure. The closed-form solution is shown to be asymptotically equivalent to the ML. Moreover, its formulation is very close to the popular Fitz's expression, with a slight correction. Numerical simulations show that the proposed scheme is very close to the ML.

Accurate and real-time position estimates are cru-cial for mobile robots. This work focuses on ranging-based positioning systems, which rely on distance measurements between known points, called anchors, and a tag to localize. The topology of the network formed by the anchors strongly influences the tag's localizability, i.e., its ability to be accurately localized. Here, the tag and some anchors are supposed to be carried by robots, which allows enhancing the positioning accuracy by planning the anchors' motions. We leverage Bayesian Cramer-Rao Lower Bounds (CRLBs) on the estimates' covariance in order to quantify the tag's localizability. This class of CRLBs can capture prior information on the tag's position and take it into account when deploying the anchors. We propose a method to decrease a potential function based on the Bayesian CRLB in order to maintain the localizability of the tag while having some prior knowledge about its position distribution. Then, we present a new experiment highlighting the link between the localizability potential and the precision expected in practice. Finally, two real-time anchor motion planners are demonstrated with ranging measurements in the presence or absence of prior information about the tag's position.

Position estimation in Multi-Robot Systems (MRS) relies on relative angle or distance measurements between the robots, which generally deteriorate as distances increase. Moreover, the localization accuracy is strongly influenced both by the quality of the raw measurements but also by the overall geometry of the network. In this paper, we design a cost function that accounts for these two issues and can be used to develop motion planning algorithms that optimize the localizability in MRS, i.e., the ability of individual robots to localize themselves accurately. This cost function is based on computing new Cramér Rao Lower Bounds characterizing the achievable positioning performance with range and angle measurements that deteriorate with increasing distances. We describe a gradient-based motion-planning algorithm for MRS deployment that can be implemented in a distributed manner, as well as a non-myopic strategy to escape local minima. Finally, we test the proposed methodology experimentally for range measurements obtained using ultra-wide band transceivers and illustrate the improvements resulting from leveraging the more accurate measurement model in the robot placement algorithms.

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.

Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurements. Indeed, most of factors impacting the asymptotic estimation performance of the parameters of interest can be taken into account via equality constraints. In this communication, we introduce a new constrained Cramér-Rao- like bound for observations where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. In this setting, it is now possible to consider random equality constraints, i.e., equality constraints on the unknown deterministic parameters depending on the random parameters, which can not be addressed with the usual constrained Cramér-Rao bound. The usefulness of the proposed bound is illustrated by way of a coupled canonical polyadic model with linear constraints applied to the hyperspectral super-resolution problem.

It has been shown lately that any "standard" Bayesian lower bound (BLB) on the mean squared error (MSE) of the Weiss-Weinstein family (WWF) admits a "tighter" form which upper bounds the "standard" form. Applied to the Bayesian Cramér-Rao bound (BCRB), this result suggests to redefine the concept of efficient estimator relatively to the tighter form of the BCRB, an update supported by a noteworthy example. This paper lays the foundation to revisit some Bayesian estimation problems where the BCRB is not tight in the asymptotic region.

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.

The identification of nonlinear Wiener models (NWMs) for deterministic inputs and Gaussian noise is studied. We show that the nonparametric kernel regression estimation of the nonlinearity of a NWM (based on the Nadaraya-Watson kernel estimator) can be formulated as a parametric estimation problem leading to a Gaussian conditional observation model. This property allows us to derive the maximum likelihood estimators of the unknown parameters of the NWM, as well as the associated Cramér-Rao (CR) bounds. We finally derive a CR-like bound on the global mean squared error (MSE) of the estimated nonlinearity of a NWM. Numerical results obtained for a pulse wave input are presented and compared to the ones based on the Nadaraya-Watson kernel estimator.

In this paper, Bayesian lower bounds (BLBs) are obtained via a general form of the Pythagorean theorem where the inner product derives from the joint or the a-posteriori probability density function (pdf). When joint pdf is considered, the BLBs obtained encompass the Weiss-Weinstein family (WWF). When a-posteriori pdf is considered, by resorting to an embedding between two ad hoc subspaces, it is shown that any “standard” BLBs of the WWF admits a “tighter” form which upper bounds the “standard” form. Interestingly enough, this latter result may explain why the “standard” BLBs of the WWF are not always as tight as expected, as exemplified in the case of the Bayesian Cramer-Rae Bound. As a consequence an updated definition of efficiency is proposed, as well as the introduction of an updated class of efficient estimators.

This paper studies the performance of the generalized likelihood ratio test (GLRT) for the conditional signal model. By conditional signal model, we mean that under both hypotheses, the observations are a linear superposition of unknown deterministic signals corrupted by additive noise, with a mixing matrix depending on an unknown deterministic parameter vector. The contribution of this work is the derivation of closed form expressions for the probabilities of false alarm and of detection of the GLRT at high signal-to-noise ratio, allowing the receiver operating characteristic of the GLRT to be computed analytically. The most general case is tackled, i.e., when the number of unknown signals and the number of unknown deterministic parameters of the mixing matrix are allowed to be different under the two hypotheses.

In statistics, the concept of a concomitant, also called the induced order statistic, arises when one sorts the members of a random sample according to corresponding values of another random sample. Indeed, multivariate order statistics induced by the ordering of linear combinations of the components arises naturally in many instances. As a contribution, we provide a general second-order statistical prediction of concomitant of order statistics for multivariate normal distribution, generalizing earlier works. We exemplify its usefulness in parametric inference via two examples related to deterministic and Bayesian estimation.

When the assumed probability distribution of the observations differs from the true distribution, the model is said to be misspecified. The key results on maximum-likelihood estimation of misspecified models have been introduced in the limit of large sample support and depend on a parameters vector solution of a computationally expensive non-linear optimization problem. As a possible strategy to circumvent these limitations, we extend the approach lately proposed by Fritsche et al [1]. It is shown that the lower bound derived in [1] is a representative of a family of lower bounds deriving from a misspecified unbiasedness constraint leading to generalized Barankin-type lower bounds. For future use, we derive the standard representative of the “Small Errors” and “Large Errors” bounds, namely the generalized CRB and the generalized McAulay-Seidman bound.

In many deterministic estimation problems, the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on additional random variables. Unfortunately, this marginalization is often mathematically intractable, which prevents from using standard maximum likelihood estimators (MLEs) or any standard lower bound on their mean squared error (MSE). To circumvent this problem, the use of joint MLEs of deterministic and random parameters are proposed as being a substitute. It is shown that, regarding the deterministic parameters: 1) the joint MLEs provide generally suboptimal estimates in any asymptotic regions of operation yielding unbiased efficient estimates, 2) any representative of the two general classes of lower bounds, respectively the Small-Error bounds and the Large-Error bounds, has a “non-standard” version lower bounding the MSE of the deterministic parameters estimate.

A fairly general class of Bayesian “large-error” lower bounds of the Weiss-Weinstein family, essentially free from regularity conditions on the probability density functions support, and for which a limiting form yields a generalized Bayesian Cramér-Rao bound (BCRB), is introduced. In a large number of cases, the generalized BCRB appears to be the Bobrovsky-Mayer-Wolf-Zakai bound (BMZB). Interestingly enough, a regularized form of the Bobrovsky-Zakai bound (BZB), applicable when the support of the prior is a constrained parameter set, is obtained. Modified Weiss-Weinstein bound and BZB which limiting form is the BMZB are proposed, in expectation of an increased tightness in the threshold region. Some of the proposed results are exemplified with a reference problem in signal processing: the Gaussian observation model with parameterized mean and uniform prior.

In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurement. However in many systems both random and non-random parameters may occur simultaneously. In this communication, we propose a constrained hybrid lower bound which take into account of equality constraint on deterministic parameters. The usefulness of the proposed bound is illustrated with an application to radar Doppler estimation

This letter investigates hybrid lower bounds on the mean square error in order to predict the so-called threshold effect. A new family of tighter hybrid large error bounds based on linear transformations (discrete or integral) of a mixture of the McAulay-Seidman bound and the Weiss-Weinstein bound is provided in multivariate parameters case with multiple test points. For use in applications, we give a closed-form expression of the proposed bound for a set of Gaussian observation models with parameterized mean, including tones estimation which exemplifies the threshold prediction capability of the proposed bound.

In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. As a contribution to the hybrid estimation framework, we introduce a recursive hybrid Cramér–Rao lower bound for discrete-time Markovian dynamic systems depending on unknown deterministic parameters. Additionally, the regularity conditions required for its existence and its use are clarified.

We consider Direction of Arrival (DoA) estimation in the case of two closely spaced sources. In this case, most high resolution techniques fail to estimate the two DoAs if the waveforms are highly correlated. Maximum Likelihood Estimators (MLE) are known to be more robust, but their excessive computational load limits their use in practice. In this paper, we propose an asymptotic approximation of the Unconditional Maximum Likelihood (UML) procedure in the case of a Uniform Linear Array (ULA) and two closely spaced targets. This approximation is based on an asymptotically (in the number of observations) equivalent formulation of the UML criterion, and on its Taylor series approximation for small DoA separation. This simplified procedure, which requires solving a 1D-optimization problem only, is shown to be accurate for source separation lower than half the mainlobe. Furthermore, it outperforms conventional high resolution algorithms in the case of two correlated sources.

In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. In this communication, we propose a new hybrid lower bound which, for the first time, includes the Ziv-Zakaï bound well known for its tightness in the Bayesian context (random parameters only). For the general case of parameterized mean model with Gaussian noise, closed-form expressions of the proposed bound are provided.

This article investigates hybrid lower bounds in order to predict the estimators mean square error threshold effect. A tractable and computationally efficient form is derived. This form combines the Barankin and the Weiss-Weinstein bounds. This bound is applied to a frequency estimation problem for which a closed-form expression is provided. A comparison with results on the hybrid Barankin bound shows the superiority of this new bound to predict the mean square error threshold.

The CRB is a lower bound of great interest for system analysis and design in the asymptotic region (high SNR and/or large number of snapshots), as it is simple to calculate and it is usually possible to obtain closed form expressions. It is from this perspective that the paper highlights, by means of a classical radar estimation problem, two results useful for system analysis and design: a reparameterization inequality and the equivalence between reparameterization and equality constraints.

It is well known that in non-linear estimation problems the ML estimator exhibits a threshold effect, i.e. a rapid deterioration of estimation accuracy below a certain SNR or number of snapshots. This effect is caused by outliers and is not captured by standard tools such as the Cramer-Rao bound (CRB). The search of the SNR threshold value can be achieved with the help of approximations of the Barankin bound (BB) proposed by many authors. These approximations result from a linear transformation (discrete or integral) of the uniform unbiasness constraint introduced by Barankin. Nevertheless, non-linear transformations can be used as well for some class of p.d.f. including the Gaussian case. The benefit is their combination with existing linear transformation to get tighter lower bounds improving the SNR threshold prediction.

A synthetic aperture radar scale model has been developed to improve signal processing teaching. Based on low frequency ultrasound transmission, it is a low cost demonstration kit. The overall software is directly running on Matlab® and allows easy and realtime modifications. This educational tool can be used to test different waveforms and show the effects of a real scene on the final image. It can also be used in a more advanced way to test different signal processing in order to improve image focusing or to reduce computation burden.

A wide variety of processing incorporates a binary detection test that restricts the set of observations available for parameter estimation and requires to take this statistical conditioning into account to compute the Cramer-Rao bound (CRB). Therefore, we propose a derivation of the CRB for the deterministic signal model conditioned by the energy detector widely used in signal processing applications. This derivation has lead us to introduce novel identities on some conditional expectations of complex circular Gaussian random vectors that may be useful for other derivations.

The search for an easily computable but tight approximation of the Barankin bound (BB) is important for the prediction of the signal-to-noise ratio (SNR) value where the Cramer-Rao bound (CRB) becomes unreliable for prediction of maximum likelihood estimators (MLE) variance. In this paper we propose a method for the derivation of a general class of BB approximations which has the advantage of a clear interpretation. This method suggests a new practical BB approximation, whose computational complexity does not exceed that of the CRB but which seems tighter than existing approximations

Estimation of the direction of arrival of a signal source by means of a monopulse antenna is one of the oldest and most widely used high resolution techniques. Although the statistical performance of this estimation technique has been extensively investigated, it has never been analyzed from the viewpoint of a two-sensor system. This deficiency is responsible for the form of the common solution (detector/estimator) which restricts the accessible performance. Applying the optimal detection theory to this problem, when a Raleigh-type signal source is present, shows that changing the detector is necessary to optimize the overall performance. The analytical performance of the new solution has been established , thus complementing the existing characterization of the common solution.