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In parameter estimation, it is common place to design a linearly constrained minimum variance estimator (LCMVE) to tackle the problem of estimating an unknown parameter vector in a linear regression model. So far, the LCMVE has been mainly studied in the context of stationary constraints in stationary or non-stationary environments, giving rise to well-established recursive adaptive implementations when multiple observations are available. In this communication, provided that the additive noise sequence is temporally uncorrelated, we determine the family of non-stationary constraints leading to LCMVEs which can be computed according to a predictor/corrector recursion similar to the Kalman Filter. A particularly noteworthy feature of the recursive formulation introduced is to be fully adaptive in the context of sequential estimation as it allows at each new observation to incorporate or not new constraints. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Localization based on the reception of radio-frequency waveforms is a crucial problem in many civilian or military applications. It is also the main objective of all Global Navigation Satellite System (GNSS). Given delayed and Doppler shifted replicas of the satellites transmitted signals, the most widespread approach consists in a suboptimal two-step procedure. First, estimate the delays and Dopplers from each satellite independently, then estimate the user position and speed thanks to a Least Square (LS) minimization. More accurate and robust techniques, such as a direct Maximum Likelihood (ML) maximization, that exploit the links in between the different channels exist but suffer from an heavy computational burden that prevent their use in real time applications. Two-steps procedures with an appropriate Weighted LS (WLS) minimization are shown to be asymptotically equivalent to the ML procedure. In this paper, we develop a closed-form expression of this WLS asymptotically efficient solution. We show that this simple expression is the sum of two terms. The first one, depending on the pseudo-ranges is the widespread used WLS solution. The second one is a Doppler-aided corrective term that should be taken into account to improve the position estimation when the observation time increases. [ABSTRACT FROM AUTHOR]

The ordered values of a sample of observations are called the order statistics of the sample and are among the most important functions of a set of random variables in probability and statistics. However the study of ordered estimates seems to have been overlooked in maximum-likelihood estimation. Therefore it is the aim of this communication to give an insight into the relevance of order statistics in maximum-likelihood estimation by providing a second-order statistical prediction of ordered normally distributed estimates. Indeed, this second-order statistical prediction allows to refine the asymptotic performance analysis of the mean square error (MSE) of maximum likelihood estimators (MLEs) of a subset of the parameters. A closer look to the bivariate case highlights the possible impact of estimates ordering on MSE, impact which is not negligible in (very) high resolution scenarios. [ABSTRACT FROM AUTHOR]

In many radar applications extended targets appear together with point targets. An important problem in particular for tracking algorithms is to rapidly detect these extended targets and to determine their centroid and shape. In a previous paper we have developed and characterized estimators for the centroid and extension of an extended target based on the generalized monopulse ratio when the target is modeled as a number of independent Swerling I?II point scatterers. However, the Swerling I?II model is a very rough approximation for the behavior of a real complex target. The work presented here extends the statistical characterizations to mixtures of independent Swerling I?II point scatterers and deterministic Swerling 0 point scatterers which may, however, vary arbitrarily in time. Such a mixture of deterministic and stochastic scatter model is able to provide good approximations of real target radar cross sections. [ABSTRACT FROM AUTHOR]

In many radar applications extended targets appear together with point targets. An important problem in particular for tracking algorithms is to rapidly detect these extended targets and to determine their centroid and shape. We present a detector for extended targets and maximum-likelihood (ML) estimators for the centroid and the extension which are based only on the outputs of the generalized monopulse ratio. The generalized monopulse ratio is based on the sum and difference beams generated from the digital outputs of arbitrary subarrays of a planar array. It is therefore applicable to various kinds of modern array antennas. The extended target is characterized by an ellipse in the azimuth-elevation plane. The statistical performance of the centroid and the extension estimators is characterized by analytically their mean and variance. Numerical simulations indicate that these statistical characterizations are accurate and that small extended targets can be detected early. The statistical performance measures can therefore also be used for parametric system studies. [ABSTRACT FROM AUTHOR]

The Cramér–Rao bounds (CRB) is a lower bound of great interest for system analysis and design in the asymptotic region [high signal-to-noise ratio (SNR) and/or large number of snapshots], as it is simple to calculate and it is usually possible to obtain closed form expressions. The first part of the paper is a generalization to complex parameters of the Barankin rationale for deriving MSE lower bounds, that is the minimization of a norm under a set of linear constraints. With the norm minimization approach the study of Fisher information matrix (FIM) singularity, constrained CRB and regularity conditions become straightforward corollaries of the derivation. The second part provides new results useful for system analysis and design: a general reparameterization inequality, the equivalence between reparameterization and equality constraints, and an explicit relationship between parameters unidentifiability and FIM singularity. [ABSTRACT FROM PUBLISHER]

Estimation of the direction of arrival (DOA) of a signal source by means of a monopulse estimation scheme is one of the oldest and most widely used high-precision techniques in operational tracking systems. Although the statistical performance of this estimation technique has been extensively investigated, it has never been analyzed from the viewpoint of the joint detection-estimation problem. As a consequence the historical form of the (detector-angle estimator) solution has become the usual form implemented in operational tracking systems and may restrict their accessible performance. Indeed, the application of the optimal detection theory reveals alternative (detector-angle estimator) solutions providing other relevant trade-offs in detection-estimation performances that are worth considering to optimize the performance of the tracking system according to the scenario. Both stochastic signal model (Rayleigh-type signal source) and deterministic signal model (signal source of unknown amplitude) have been investigated, leading to a novel but equally simple (detector-angle estimator) solution through which analytical statistical prediction has been derived. Comparison of detection-estimation performances of new and usual solutions is presented. [ABSTRACT FROM AUTHOR]

It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain signal-to-noise ratio (SNR) or number of snapshots. This effect is caused by outliers and is not captured by standard tools such as the Cramér-Rao bound (CRB). The search of the SNR threshold value (where the CRB becomes unreliable for prediction of maximum likelihood estimator variance) can be achieved with the help of the Barankin bound (BB), as proposed by many authors. The major drawback of the BB, in comparison with the CRB, is the absence of a general analytical formula, which compels one to resort to a discrete form, usually the Mcaulay-Seidman bound (MSB), requesting the search of an optimum over a set of test points. In this paper, we propose a new practical BB discrete form that provides, for a given set of test points, an improved SNR threshold prediction in comparison with existing approximations (MSB, Abel bound, Mcaulay-Hofstetter bound) at the expense of the computational complexity increased by a factor ⩽ (P + 1)³, where P is the number of unknown parameters. We have derived its expression for the general Gaussian observation model to be used in place of existing approximations. [ABSTRACT FROM AUTHOR]

This paper proposes a novel statistical prediction of monopulse errors [1] for a radar Swerling III-IV target embedded in noise or noise jamming where multiple observations are available. First, the study of the maximum likelihood estimator (MLE) of the complex monopulse ratio for a Swerling III-IV target embedded in spatially white noise allows us to extend the use of the MLE practical approximate form introduced by Mosca [2] for Swerling 0-I-II cases. Afterward, we derive analytical formulas for both the mean and variance of the MLE in approximate form conditioned by the usual detection step performed on the sum channel of a monopulse antenna. Last, we provide a comparison of target direction of arrival (DOA) estimation performance based on monopulse ratio estimation as a function of the Swerling model in the context of a multifunction radar. [ABSTRACT FROM AUTHOR]

A wide variety of actual processing requires a detection step, whose main effect is to restrict the set of observations available for parameter estimation. Therefore, as a contribution to the theoretical formulation of the joint detection and estimation problem, we address the derivation of lower bounds for deterministic parameters conditioned by a binary hypothesis testing problem. The main result is the introduction of a general scheme—detailed in the particular case of CRB—enabling the derivation of conditional deterministic MSE lower bounds. To prove that it is meaningful, we also show, with the help of a fundamental application, that the problem of lower bound tightness at low SNR may arise from an incorrect lower bound formulation that does not take into account the true nature of the problem under investigation: a joint detection-estimation problem. [ABSTRACT FROM AUTHOR]

State estimation is a fundamental task in a plethora of applications. Two important classes of linear minimum mean square error (MMSE) estimators are Wiener filters (WFs) and Kalman filters (KFs), the latter being the recursive form of the former for linear discrete state-space (LDSS) dynamic models. The minimum set of uncorrelation conditions regarding states and measurements which allows to formulate a general KF form has been recently introduced in the literature, but for both WFs and KFs a standard key assumption is that the measurement noise covariance is invertible. In this contribution we provide i) an alternative derivation of the WF in the case of a non-invertible measurement covariance matrix, and ii) the proof that under the minimum set of uncorrelation conditions lately released, the KF can always be formulated irrespective of the measurement covariance matrix rank. Notice that these results also apply for Kalman predictors and smoothers, linearly constrained WFs and minimum variance distortionless response (MVDR) filters, therefore being a remarkable result of broad interest. [ABSTRACT FROM AUTHOR]

The popular Discrete Fourier Transform (DFT) is known to be a sub‐optimal frequency estimation technique for a finite transform length. In order to approach the Cramer‐Rao Lower Bound (CRLB), many refinement techniques have been considered, but little considering both zero padding or tapering, also known as windowing or apodisation. In this paper, a frequency estimator with closed‐form combination of three DFT samples is generalized to zero padding and tapered data within the class of cosine windowing. Root Mean Squared Error (RMSE) is shown to approach the CRLB in the case of a single tone signal with additive white Gaussian noise. Compared to state‐of‐the‐art techniques, the proposed algorithm improves the frequency RMSE up to 1 dB when using significant zero‐padding lengths (K ≥ 2 N) and for small to moderate SNR, which is the most challenging case for practical radar applications. [ABSTRACT FROM AUTHOR]

Global navigation satellite systems (GNSSs) play a key role in a plethora of applications, ranging from navigation and timing to Earth observation and space weather characterization. For navigation purposes, interference scenarios are among the most challenging operation conditions, with a clear impact on the maximum likelihood estimates (MLEs) of signal synchronization parameters. While several interference mitigation techniques exist, an approach for theoretically analyzing GNSS MLE performance degradation under interference, which is fundamental for system/receiver design, is lacking. 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 interference. The proposed bias and MCRB expressions are validated for a linear frequency-modulation chirp signal interference. [ABSTRACT FROM AUTHOR]

We consider deterministic parameter estimation and the situation 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. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known standard deterministic lower bounds (LBs) on the mean squared error (MSE). Actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard LB can be transformed into a modified one fitted to nonstandard deterministic estimation, at the expense of tightness however. Furthermore, these modified LBs (MLBs) appears to include the submatrix of hybrid LBs which is an LB for the deterministic parameters. Moreover, since in the nonstandard estimation, maximum likelihood estimators (MLEs) can be no longer derived, suboptimal nonstandard MLEs (NSMLEs) are proposed as being a substitute. We show that any standard LB on the MSE of MLEs has a nonstandard version lower bounding the MSE of NSMLEs. We provide an analysis of the relative performance of the NSMLEs, as well as a comparison with the MLBs for a large class of estimation problems. Last, the general approach introduced is exemplified, among other things, with a new look at the well-known Gaussian complex observation models. [ABSTRACT FROM PUBLISHER]

Monopulse is an established array processing technique for fast and accurate angle estimation. This technique has been generalized to space-time array processing of any dimension. The statistical performance of this generalized monopulse parameter estimation has been characterized for several target fluctuation models but not for all cases. This gap is filled by this paper. We derive the mean and variance of the complex and real and imaginary part of the averaged monopulse ratio for all Swerling target models, deterministic targets (Swerling 0 case), \chi^2 distributed targets with 4 degrees of freedom (Swerling 3 or 4 case) including a given detection threshold, for an arbitrary number of difference beams, and for an arbitrary number of noncoherent averaging of time snapshots. For completeness we give also the already known results for Rayleigh targets (Swerling 1 or 2 case) in the same notation. From these means and variances, the performance of all kinds of parameter estimates with the generalized monopulse formula can be calculated. Applications of these statistical descriptions are presented for planar arrays and adaptive beamforming and for space-time adaptive processing (STAP) for broadband interference suppression. From these examples some interesting conclusions can be drawn. [ABSTRACT FROM AUTHOR]

The Kalman filter (KF) is known to loose its optimality properties when the model used does not perfectly match the true system. Extending the use of linear constraints to this filter recently proved to be efficient to mitigate a large class of parametric model mismatch, through the linearly constrained Kalman filter and minimum variance filter (LCKF and LCMVF). However, the asymptotic performances of these new filters are still an open question. In this work, we bring a first answer to the latter problem in the case of measurement model mismatch. We show that both LCKF and LCMVF are equivalent to unconstrained filters in which the directions of the constraints are cancelled by projection, allowing a better understanding of their asymptotic properties. In particular, the steady-state mean square error, when it exists, is derived. The consistency of the filters with respect to nonlinear mismatches is also improved via new constraints. An array processing example is provided to assess the derived formulas, the consistency and performance of the filters. [ABSTRACT FROM AUTHOR]

This work addressed the problem of miscalibration or decalibration of mobile stereo/bi-monocular camera setups. We especially focused on the context of autonomous vehicles. In real-world conditions, any optical system is subject to various mechanical stresses, caused by vibration, rough handling, collisions, or even thermal expansion. Such mechanical stresses change the stereo pair geometry, and as a consequence, the pre-calculated epipolar geometry or any geometric-based approach is no longer valid. The standard method, which consists of estimating the calibration online, fails in such harsh conditions. The proposed method was based on a robust linearly constrained state estimation technique able to mitigate the model mismatch without estimating the model parameters. Therefore, our solution was able to mitigate the errors with negligible use of additional computing resources. We propose to use a linearly constrained extended Kalman filter for a stereo-based visual odometry or simultaneous localization and mapping approach. Simulations confirmed that the method kept the system (and objects of the map) localized in real-time even with huge miscalibration errors and parameter variations. The results confirmed that the method was robust to a miscalibration of all the extrinsic calibration parameters even when the standard online calibration procedure failed. [ABSTRACT FROM AUTHOR]

Provided that one keeps in mind the Cram?er-Rao bound (CRB) limitations, that is, to become an overly optimistic lower bound when the observation conditions degrades, the CRB is a lower bound of great interest for analysis and design of a system of measurement in the asymptotic region. As a contribution, we introduce an original framework taking into account most (and possibly all) of the factors impacting the asymptotic estimation performance of the parameters of interest via equality constraints, leading to direct algebraic computations of constrained CRB. For complex systems, derivation of analytical expression of CRB is either impossible or inefficient. For application to active systems of measurement such as radar, we provide the general form of the Fisher information matrix (FIM) for multiple conditional models, which generally precludes the derivation of an analytical expression of the CRB for scenarios including interference and sensors modelling errors. We show that the proposed framework can also be used efficiently to generate new closed-form expressions of CRB, although this is not its main aim. [ABSTRACT FROM AUTHOR]

Abstract: The performance of the majority of high resolution algorithms designed for either spectral analysis or Direction-of-Arrival (DoA) estimation drastically degrades when the amplitude sources are highly correlated or when the number of available snapshots is very small and possibly less than the number of sources. Under such circumstances, only Maximum Likelihood (ML) or ML-based techniques can still be effective. The main drawback of such optimal solutions lies in their high computational load. In this paper we propose a computationally efficient approximate ML estimator, in the case of two closely spaced signals, that can be used even in the single snapshot case. Our approach relies on Taylor series expansion of the projection onto the signal subspace and can be implemented through 1D Fourier transforms. Its effectiveness is illustrated in complicated scenarios with very low sample support and possibly correlated sources, where it is shown to outperform conventional estimators. [Copyright &y& Elsevier]

In the field of asymptotic performance characterization of the conditional maximum-likelihood (CML) estimator, asymptotic generally refers to either the number of samples or the signal-to-noise ratio (SNR) value. The first case has been already fully characterized, although the second case has been only partially investigated. Therefore, this correspondence aims to provide a sound proof of a result, i.e., asymptotic (in SNR) Gaussianity and efficiency of the CML estimator in the multiple parameters case, generally regarded as trivial but not so far demonstrated. [ABSTRACT FROM AUTHOR]

Performance lower bounds are known to be a fundamental design tool in parametric estimation theory. A plethora of deterministic bounds exist in the literature, ranging from the general Barankin bound to the well-known Cramér-Rao bound (CRB), the latter providing the optimal mean square error performance of locally unbiased estimators. In this contribution, we are interested in the estimation of mixed real- and integer-valued parameter vectors. We propose a closed-form lower bound expression leveraging on the general CRB formulation, being the limiting form of the McAulay-Seidman bound. Such formulation is the key point to take into account integer-valued parameters. As a particular case of the general form, we provide closed-form expressions for the Gaussian observation model. One noteworthy point is the assessment of the asymptotic efficiency of the maximum likelihood estimator for a linear regression model with mixed parameter vectors and known noise covariance matrix, thus complementing the rather rich literature on that topic. A representative carrier-phase based precise positioning example is provided to support the discussion and show the usefulness of the proposed lower bound. [ABSTRACT FROM AUTHOR]

Summary: Standard state estimation techniques, ranging from the linear Kalman filter (KF) to nonlinear extended KF (EKF), sigma‐point or particle filters, assume a perfectly known system model, that is, process and measurement functions and system noise statistics (both the distribution and its parameters). This is a strong assumption which may not hold in practice, reason why several approaches have been proposed for robust filtering, mainly because the filter performance is particularly sensitive to different model mismatches. In the context of linear filtering, a solution to cope with possible system matrices mismatch is to use linear constraints. In this contribution we further explore the extension and use of recent results on linearly constrained KF for robust nonlinear filtering under both process and measurement model mismatch. We first investigate how linear equality constraints can be incorporated within the EKF and derive a new linearly constrained extended KF (LCEKF). Then we detail its use to mitigate parametric modeling errors in the nonlinear process and measurement functions. Numerical results are provided to show the performance improvement of the new LCEKF for robust vehicle navigation. [ABSTRACT FROM AUTHOR]

In many signal processing applications, conducting a coherent integration over the whole observation duration is not possible due to uncontrolled phase evolutions. In such a case one may refer to Post Detection Integration that consists in wisely combine shorter time integration outputs. In this paper, we propose a new detector that improves the state of art ones. Compared with the GLRT from which it is derived, it consists in a closed-form expression. Moreover, it is based on a simple linear phase assumption and can be used, by the way, in a wide area of applications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• The paper derives a new class of recursive linearly constrained minimum variance estimators (LCMVE). • These LCMVEs provide additional robustness to modeling errors through the incorporation of non-stationary linear constraints. • These LCMVEs are fully adaptive as it allows the addition of optional constraints that can be triggered by a preprocessing of each new observation. • This article considers the general class of partially coherent signal (PCS) sources. • Both errors on system matrices and noise statistics uncertainty are considered. This article introduces a new class of recursive linearly constrained minimum variance estimators (LCMVEs) that provides additional robustness to modeling errors. To achieve that robustness, a set of non-stationary linear constraints are added to the standard LCMVE that allow for a closed form solution that becomes appealing in sequential implementations of the estimator. Indeed, a key point of such recursive LCMVE is to be fully adaptive in the context of sequential estimation as it allows optional constraints addition that can be triggered by a preprocessing of each new observation or external information on the environment. This methodology has significance in the popular problem of linear regression among others. Particularly, this article considers the general class of partially coherent signal (PCS) sources, which encompasses the case of fully coherent signal (FCS) sources. The article derivates the recursive LCMVE for this type of problems and investigates, analytically and through simulations, its robustness against mismatches on linear discrete state-space models. Both errors on system matrices and noise statistics uncertainty are considered. An illustrative multi-channel array processing example is treated to support the discussion, where results in different model mismatched scenarios are provided with respect to the standard case with only FCS sources. [ABSTRACT FROM AUTHOR]

In this article, we first derive a general intrinsic Barankin bound (IBB) for unknown parameters lying on Lie groups (LGs), and its intrinsic McAulay–Seidman bound (IMSB) approximation. Second, the IMSB expression is used to revisit the intrinsic Cramér–Rao bound (ICRB) on LGs. Indeed, an analytic expression of the ICRB, which is a special IMSB case, is obtained from the latter. Finally, closed-form expressions for both IMSB and ICRB are obtained for Euclidean and LG observation models depending on parameters lying in S O (3) and S E (3). The validity of the these IMSB and ICRB expressions, with respect to the intrinsic mean square error, is shown via numerical simulations to support the discussion. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• Derivation of a closed-form misspecified Cramér Rao bound. • Derivation of pseudo-true parameters for specific operating scenario. • Presentation of bound so users can also compute it for any signal samples. • Comparison of misspecified signal model with model that includes acceleration. • Examples of high dynamics scenarios that do not require estimation of acceleration. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the context of multiple change-points estimation, performance analysis of estimators such as the maximum likelihood is often difficult to assess since the regularity assumptions are not met. Focusing on the estimators variance, one can, however, use lower bounds on the mean square error. In this paper, we derive the so-called Weiss–Weinstein bound (WWB) that is known to be an efficient tool in signal processing to obtain a fair overview of the estimation behavior. Contrary to several works about performance analysis in the change-point literature, our study is adapted to multiple changes. First, useful formulas are given for a general estimation problem whatever the considered distribution of the data. Second, closed-form expressions are given in the cases of Gaussian observations with changes in the mean and/or the variance, and changes in the mean rate of a Poisson distribution. Furthermore, a semidefinite programming formulation of the minimization procedure is given in order to compute the tightest WWB. Specifically, it consists of finding the unique minimum volume covering the set constituted by hyperellipsoid elements that are generated using the derived candidate WWB matrices w.r.t. the so-called Loewner partial ordering. Finally, simulation results are provided to show the good behavior of the proposed bound. [ABSTRACT FROM PUBLISHER]

The use of carrier phase data is the main driver for high-precision Global Navigation Satellite Systems (GNSS) positioning solutions, such as Real-Time Kinematic (RTK). However, carrier phase observations are ambiguous by an unknown number of cycles, and their use in RTK relies on the process of mapping real-valued ambiguities to integer ones, so-called Integer Ambiguity Resolution (IAR). The main goal of IAR is to enhance the position solution by virtue of its correlation with the estimated integer ambiguities. With the deployment of new GNSS constellations and frequencies, a large number of observations is available. While this is generally positive, positioning in medium and long baselines is challenging due to the atmospheric residuals. In this context, the process of solving the complete set of ambiguities, so-called Full Ambiguity Resolution (FAR), is limiting and may lead to a decreased availability of precise positioning. Alternatively, Partial Ambiguity Resolution (PAR) relaxes the condition of estimating the complete vector of ambiguities and, instead, finds a subset of them to maximize the availability. This article reviews the state-of-the-art PAR schemes, addresses the analytical performance of a PAR estimator following a generalization of the Cramér–Rao Bound (CRB) for the RTK problem, and introduces Precision-Driven PAR (PD-PAR). The latter constitutes a new PAR scheme which employs the formal precision of the (potentially fixed) positioning solution as selection criteria for the subset of ambiguities to fix. Numerical simulations are used to showcase the performance of conventional FAR and FAR approaches, and the proposed PD-PAR against the generalized CRB associated with PAR problems. Real-data experimental analysis for a medium baseline complements the synthetic scenario. The results demonstrate that (i) the generalization for the RTK CRB constitutes a valid lower bound to assess the asymptotic behavior of PAR estimators, and (ii) the proposed PD-PAR technique outperforms existing FAR and PAR solutions as a non-recursive estimator for medium and long baselines. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is well known that the standard state estimation technique performance is particularly sensitive to perfect system knowledge, where the underlying assumptions are: (i) Process and measurement functions and parameters are known, (ii) inputs are known, and (iii) noise statistics are known. These are rather strong assumptions in real-life applications; therefore, a robust filtering solution must be designed to cope with model misspecifications. A possible way to design robust filters is to exploit linear constraints (LCs) within the filter formulation. In this contribution we further explore the use of LCs, derive a linearly constrained extended Kalman filter (LCEKF) for systems affected by non-additive noise and system inputs, and discuss its use for model mismatch mitigation. Numerical results for a robust tracking and navigation problem are provided to show the performance improvement of the proposed LCEKF, with respect to state-of-the-art techniques, that is, a benchmark EKF without mismatch and a misspecified EKF not accounting for the mismatch. [ABSTRACT FROM AUTHOR]

Evaluating the time-delay, Doppler effect and carrier phase of a received signal is a challenging estimation problem that was addressed in a large variety of remote sensing applications. This problem becomes more difficult and less understood when the signal is reflected off one or multiple surfaces and interferes with itself at the receiver stage. This phenomenon might deteriorate the overall system performance, as for the multipath effect in Global Navigation Satellite Systems (GNSS), and mitigation strategies must be accounted for. In other applications such as GNSS reflectometry (GNSS-R) it may be interesting to estimate the parameters of the reflected signal to deduce the geometry and the surface characteristics. In either case, a better understanding of this estimation problem is directly brought by the corresponding lower performance bounds. In the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér-Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this article, we derive a new compact CRB expression for the joint time-delay and Doppler estimation in a dual source context, considering a band-limited signal and its specular reflection. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. This extends existing results in the single source context and opens the door to a plethora of usages to be discussed in the article. The proposed CRB expressions are validated in two representative navigation and radar examples. [ABSTRACT FROM AUTHOR]

• GNSS positioning precision derivation. • Use of Doppler information to improve positioning precision. • In harsh environments, Doppler measurements are all the more favourable. The main objective of Global Navigation Satellite Systems (GNSS) is to precisely locate a receiver based on the reception of radio-frequency waveforms broadcasted by a set of satellites. Given delayed and Doppler shifted replicas of the known transmitted signals, the most widespread approach consists in a two-step algorithm. First, the delays and Doppler shifts from each satellite are estimated independently, and subsequently the user position and velocity are computed as the solution to a Weighted Least Squares (WLS) problem. This second step conventionally uses only delay measurements to determine the user position, although Doppler is also informative. The goal of this paper is to provide simple and meaningful expressions of the positioning precision. These expressions are analysed with respect to the standard WLS algorithms, exploiting the Doppler information or not. We can then evaluate the performance improvement brought by a joint frequency and delay positioning procedure. Numerical simulations assess that using Doppler information is indeed effective when considering long observation times, and particularly useful in challenging scenarios such as urban canyons (constrained satellite visibility) or near indoor situations (weak signal conditions which need long integration times), thus providing new insights for the design of robust and high-sensitivity receivers. [ABSTRACT FROM AUTHOR]

Since time-delay, Doppler effect and phase estimation are fundamental tasks in a plethora of engineering fields, tractable lower performance bounds for this problem are key tools of broad interest for a large variety of remote sensing applications. In the large sample regime and/or the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér–Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this contribution, we introduce firstly a new compact CRB expression for the joint time-delay and Doppler stretch estimation, considering a generic delayed and dilated band-limited signal. This generalizes known results for both wideband signals and the standard narrowband signal model where the Doppler effect on the band-limited baseband signal is not considered and amounts to a frequency shift. General compact closed-form CRB expressions for the amplitude and phase are also provided. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. The new CRB expressions are validated in a positioning case study, both using synthetic and real data. These results show that the maximum likelihood estimator converges to the CRB at high signal-to-noise ratios, which confirms the exactness of the CRB. The CRB is further validated by comparing the ambiguity function and its 2nd order Taylor expansion where the perfect match also proves its exactness. [ABSTRACT FROM AUTHOR]

Global Navigation Satellite Systems (GNSS) are the main source of position, navigation, and timing (PNT) information and will be a key player in the next-generation intelligent transportation systems and safety-critical applications, but several limitations need to be overcome to meet the stringent performance requirements. One of the open issues is how to provide precise PNT solutions in harsh propagation environments. Under nominal conditions, the former is typically achieved by exploiting carrier phase information through precise positioning techniques, but these methods are very sensitive to the quality of phase observables. Another option that is gaining interest in the scientific community is the use of large bandwidth signals, which allow obtaining a better baseband resolution, and therefore more precise code-based observables. Two options may be considered: (i) high-order binary offset carrier (HO-BOC) modulations or (ii) the concept of GNSS meta-signals. In this contribution, we assess the time-delay and phase maximum likelihood (ML) estimation performance limits of such signals, together with the performance translation into the position domain, considering single point positioning (SPP) and RTK solutions, being an important missing point in the literature. A comprehensive discussion is provided on the estimators' behavior, the corresponding ML threshold regions, the impact of good and bad satellite constellation geometries, and final conclusions on the best candidates, which may lead to precise solutions under harsh conditions. It is found that if the receiver is constrained by the receiver bandwidth, the best choices are the L1-M or E6-Public Regulated Service (PRS) signals. If the receiver is able to operate at 60 MHz, it is recommended to exploit the full-bandwidth Galileo E5 signal. In terms of robustness and performance, if the receiver can operate at 135 MHz, the best choice is to use the GNSS meta-signals E5 + E6 or B2 + B3, which provide the best overall performances regardless of the positioning method used, the satellite constellation geometry, or the propagation conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]