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.

We develop Adaptive Cosine Estimator (ACE) type detector for non-zero mean Gaussian interference specifically for the replacement and additive target models of the hyperspectral imaging problem. We consider the case where the data under test and the training samples differ from one scaling factor on the mean and one scaling factor on the covariance matrix. We derive two-step generalized likelihood ratio tests for both the additive model and the replacement model and show that the new detectors differ in the way the mean value is removed. A real data experiment shows that they outperform the standard version.

One of the main objectives of hyperspectral image processing is to detect a given target among an unknown background. The standard data to conduct such detection is a reflectance map, where the spectral signatures of each pixel's components, known as endmembers, are associated with their abundances in the pixel. Due to the low spatial resolution of most hyperspectral sensors, such a target occupies a fraction of the pixel. A widely used model in the case of subpixel targets is the replacement model. Among the vast number of possible detectors, algorithms matched to the replacement model are quite rare. One of the few examples is the finite target matched filter (MF), which is an adjustment of the well-known MF. In this article, we derive the exact generalized likelihood ratio test for this model. This new detector can be used both with a local covariance estimation window or a global one. It is shown to outperform the standard target detectors on real data, especially for small covariance estimation windows.

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.

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.

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 this paper, we propose a distributed learning strategy in wireless sensor networks. Taking advantage of recent developments on kernel-based machine learning, we consider a new sparsification criterion for online learning. As opposed to previously derived criteria, it is based on the estimated error and is therefore is well suited for tracking the evolution of systems over time. We also derive a gradient descent algorithm, and we demonstrate its relevance to estimate the dynamic evolution of temperature in a given region.

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.

In this paper, we address the problem of detecting a signal whose associated spatial signature is subject to uncertainties, in the presence of subspace interference and broadband noise, and using multiple snapshots from an array of sensors. To account for steering vector uncertainties, we assume that the spatial signature of interest lies in a given linear subspacewhile its coordinates in this subspace are unknown. The generalized likelihood ratio test (GLRT) for the problem at hand is formulated. We show that the GLRT amounts to searching for the best direction in the subspaceafter projecting out the interferences. The distribution of the GRLT under both hypotheses is derived and numerical simulations illustrate its performance.

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. Recently, worst-case approaches which consist of protecting the array's response in an ellipsoid centered around the nominal steering vector have been proposed. They amount to generalized (i.e. non necessarily diagonal) loading of the covariance matrix. In this paper, we present a theoretical analysis of the signal to interference plus noise ratio (SINR) for this class of robust beamformers, in the presence of random steering vector errors. A closed-form expression for the SINR is derived which is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR.