Cramer–Rao lower bounds for change points in additive and multiplicative noise

The paper addresses the problem of determining the Cramer–Rao lower bounds (CRLBs) for noise and change-point parameters, for steplike signals corrupted by multiplicative and/or additive white noise. Closed-form expressions for the signal and noise CRLBs are first derived for an ideal step with a known change point. For an unknown change-point, the noise-free signal is modeled by a sigmoidal function parametrized by location and step rise parameters. The noise and step change CRLBs corresponding to this model are shown to be well approximated by the more tractable expressions derived for a known change-point. The paper also shows that the step location parameter is asymptotically decoupled from the other parameters, which allows us to derive simple CRLBs for the step location. These bounds are then compared with the corresponding mean square errors of the maximum likelihood estimators in the pure multiplicative case. The comparison illustrates convergence and efficiency of the ML estimator. An extension to colored multiplicative noise is also discussed

Detection and Estimation of Abrupt Changes contaminated by Multiplicative Gaussian Noise

The problem of abrupt change detection has received much attention in the literature. The Neyman Pearson detector can be derived and yields the well-known CUSUM algorithm, when the abrupt change is contaminated by an additive noise. However, a multiplicative noise has been observed in many signal processing applications. These applications include radar, sonar, communication and image processing. This paper addresses the problem of abrupt change detection in presence of multiplicative noise. The optimal Neyman Pearson detector is studied when the abrupt change and noise parameters are known. The parameters are unknown in most practical applications and have to be estimated. The maximum likelihood estimator is then derived for these parameters. The maximum likelihood estimator performance is determined, by comparing the estimate mean square errors with the Cramer Rao Bounds. The Neyman Pearson detector combined with the maximum likelihood estimator yields the generalized likelihood ratio detector

Time-scale analysis of abrupt changes corrupted by multiplicative noise

Multiplicative Abrupt Changes (ACs) have been considered in many applications. These applications include image processing (speckle) and random communication models (fading). Previous authors have shown that the Continuous Wavelet Transform (CWT) has good detection properties for ACs in additive noise. This work applies the CWT to AC detection in multiplicative noise. CWT translation invariance allows to define an AC signature. The problem then becomes signature detection in the time-scale domain. A second-order contrast criterion is defined as a measure of detection performance. This criterion depends upon the first- and second-order moments of the multiplicative process's CWT. An optimal wavelet (maximizing the contrast) is derived for an ideal step in white multiplicative noise. This wavelet is asymptotically optimal for smooth changes and can be approximated for small AC amplitudes by the Haar wavelet. Linear and quadratic suboptimal signature-based detectors are also studied. Closed-form threshold expressions are given as functions of the false alarm probability for three of the detectors. Detection performance is characterized using Receiver Operating Characteristic (ROC) curves computed from Monte-Carlo simulations

Towards localisation with Doppler radar

In this thesis the author introduces a novel method for Geo Localisation via Doppler Radar. The area of research is in the three dimensional space using amplitude and magnitude measurements. Geo Localisation in mobile applications is a useful technology that enables monitoring and gathering information about objects of interest

Modeling And Detection Of Uterine Contractions Using Magnetomyography

In this dissertation, we develop a novel mathematical framework for modeling and analyzing uterine contractions using biomagnetic measurements. The study of myometrium contractility during pregnancy is relevant to the field of reproductive assessment. Its clinical importance is grounded in the need for a better understanding of the bioreproduction mechanisms. For example, in the last decade the number of preterm labors has increased significantly. Preterm birth can cause health problems or even be fatal for the fetus if it happens too early, and, at the same time, it imposes significant financial burdens on health care systems. Therefore, it is critical to develop models and statistical tools that help to monitor non-invasively the uterine activities during pregnancy. We derive a forward electromagnetic model of uterine contractions during pregnancy. Existing models of myometrial contractions approach the problem either at an organ level or lately at a cellular level. At the organ level, the models focus on generating contractile forces that closely resemble clinical measurements of normal intrauterine pressure during contractions in labor. At the cellular level, the models focus on predicting the changes of ionic concentrations in a uterine myocyte during a contraction, and, as a consequence, on modeling the transmembrane potential evolution as a function of time. In this work, we propose an electromagnetic modeling approach taking into account electrophysiological and anatomical knowledge jointly at the cellular, tissue, and organ levels. Our model aims to characterize myometrial contractions using magnetomyography: MMG) and electromyography: EMG) at different stages of pregnancy. In particular, we introduce a four-compartment volume conductor geometry, and we use a bidomain approach to model the propagation of the myometrium transmembrane potential on the human uterus. The bidomain approach is given by a set of reaction-diffusion equations. The diffusion part of the equations governs the spatial evolution of the transmembrane potential, and the reaction part is given by the local ionic current cell dynamics. Here we introduce a modified version of the Fitzhugh-Nagumo: FHN) equation for modeling ionic currents in each myocyte, assuming a plateau-type transmembrane potential. We incorporate the anisotropic nature of the uterus by considering conductivity tensors in our model. In particular, we propose a general approach to design the conductivity-tensor orientation and to estimate the conductivity-tensor values in the extracellular and intracellular domains for any uterine shape. We use finite element methods: FEM) to solve our model, and we illustrate our approach by presenting a numerical example to model a uterine contraction at term. Our results are in good agreement with the values reported in the experimental technical literature, and these are potentially important as a tool for helping in the characterization of contractions and for predicting labor. We propose an automatic, robust, single-channel statistical detector of uterine MMG contractions. One common restriction of previous techniques is that algorithm parameters, such as the detection threshold and the window length of analysis need to be calibrated experimentally, based on a particular data set. Therefore, the detection performance might change from patient to patient, for example, because of differences in the pregnancy stage and tissue conductivities. In contrast, the proposed algorithm does not require the use of a sliding window of analysis, and the detection threshold is determined analytically; thus, it does not need to be calibrated. Our detection algorithm consists of two stages: In the first stage, we segment the measurements using a multiple change-point estimation algorithm and assuming a piecewise constant time-varying autoregressive model of the measurements; In the second stage, we apply the non-supervised K-means cluster algorithm to classify each time segment, using the RMS and FOZC as candidate features. As a result a discrete-time binary decision signal is generated indicating the presence of a contraction. Moreover, since each single channel detector provides local information regarding the presence of a contraction, we propose a spatio-temporal estimator of the magnetic activity generated by uterine contractions. The algorithm, when evaluated with real MMG measurements, detects uterine activity much earlier than the patient begins to sense it. It also enables visualizing the relative location of the origin of uterine contraction and quantifying the amount of energy delivered during a contraction. These results are important in obstetrics, e.g., as a tool for helping to characterize contractions and to predict labor. For the aforementioned problem of multiple change-point estimation, a class of one-dimensional segmentation, we also compute fundamental mathematical results for minimal bounds on mean-square error estimation. Indeed, if an estimator is available, the evaluation of its performance depends on knowing whether it is optimal or if further improvement is still possible. In our segmentation problem the parameters are discrete therefore the conventional Cramer-Rao bound does not apply. Hence, we derive Barankin-type lower bounds, the greatest lower bound on the covariance of any unbiased estimator, which are applicable to discrete parameters. The computation of the bound is challenging, as it requires finding the supremum on a finite set of symmetric matrices with respect to the Loewner ordering, which is not a lattice order. Therefore, we discuss the existence of the supremum, propose a minimal upper-bound by using tools from convex geometry, and compute closed-form solutions for the Barankin information matrix for several distributions. The results have broad biomedical applications, such as DNA sequence segmentation, MEG and EEG segmentation, and uterine contraction MMG detection, and they also have applications for signal segmentation in general, such as speech segmentation and astronomical data analysis

Performances de détection et de localisation des terminaux « SAR » dans le contexte de transition MEOSAR

Le système Cospas-Sarsat est un système de recherche et de sauvetage à l’échelle mondiale qui fonctionne à l’aide de satellites en orbite basse et de satellites en orbite géostationnaire. La constellation de satellites actuelle est en cours de remplacement par des satellites en orbite moyenne qui couvrent de plus grandes zones de la surface de la Terre permettant des alertes quasi instantanées. L’objectif de cette thèse est d’étudier les performances de localisation de ce nouveau système, qui a été nommé système MEOSAR (Medium Earth Orbit Search and Rescue). Nous étudions d’abord la qualité de la liaison entre la balise de détresse, le satellite, et la station de réception au sol à l’aide d’un bilan de liaison. Ensuite, nous proposons un modèle de signal basé sur des fonctions sigmoïdes afin de modéliser les transitions douces du signal de détresse. Pour ce modèle, les performances de localisation (en terme de bornes de Cramér-Rao et de la variance d’estimateurs) sont étudiées pour l’estimation de position de la balise, et pour l’estimation de différents paramètres, y compris le temps d’arrivée, la fréquence d’arrivée et la durée du symbole. Ensuite, nous étudions l’impact de l’ajout d’information a priori sur la période symbole et sur le temps de montée du signal, qui proviennent des tolérances autorisées sur les spécifications des balises de détresse. Nous étudions également l’erreur introduite par l’ajout de bruit de phase caractéristique des oscillateurs des balises, et nous considérons l’amélioration de l’estimation de position en prenant en compte les multiples émissions de la balise de détresse. Finalement, les performances de localisation du système MEOSAR sont données pour les balises de détresse de deuxième génération, qui sont en cours de développement, et qui utilisent une modulation avec étalement de spectre. ABSTRACT : Cospas-Sarsat is an international search and rescue system that operates using low-orbit satellites and geostationary satellites. The current satellite constellation is being replaced by medium Earth orbit satellites which will cover larger areas of the surface of the Earth, permitting almost instantaneous alerts. The objective of this thesis is to study the localization performance of this new system, named MEOSAR (Medium Earth Orbit Search and Rescue). We first study the quality of the link between the beacon, the satellite and the ground receiving station through a link budget. Then, we propose a signal model based on sigmoidal functions to model the smooth transitions of the distress signal. For this model, the localization performance (in terms of Cramér-Rao bounds and estimator variances) is studied for the estimation of the beacon position and for different parameters including the time of arrival, the frequency of arrival and the symbol width. Then, we study the impact of adding prior information on the symbol width and the signal rise time, which are constructed from the allowed tolerances on the beacon specifications. We also investigate the error introduced by the addition of oscillator phase noise, and we show how the position estimation can be improved by taking into account multiple emissions of the beacon. Finally, the localization performance of the MEOSAR system is studied for second generation beacons, which are being developed using spread spectrum modulation