Deterministic asymptotic Cramer-Rao bound for the multidimensional harmonic model

International audienceThe harmonic model sampled on a P-dimensional grid contaminated by an additive white Gaussian noise has attracted considerable attention with a variety of applications. This model has a natural interpretation in a P-order tensorial framework and an important question is to evaluate the theoretical lowest variance on the model parameter (angular-frequency, real amplitude and initial phase) estimation. A standard Mathematical tool to tackle this question is the Crame´r–Rao bound (CRB) which is a lower bound on the variance of an unbiased estimator, based on Fisher information. So, the aim of this work is to derive and analyze closed-form expressions of the deterministic asymptotic CRB associated with the M-order harmonic model of dimension P with P41. In particular, we analyze this bound with respect to the variation of parameter P

Sparsity in array processing: methods and performances

International audienceIn the last few years, we witnessed to an extraordinary and still growing development of sparse signal recovery in a wide number of applicative contexts such as communications, biomedicine, radar, microwave imaging, source localization, astronomy, seismology... In many realistic array processing applications, the sparsity nature underlying various signals/arrays has to be exploit in recovery algorithms to enhance their performances. In this special session, most recent results in estimators based on sparsity-­‐promoting criteria is proposed

MULTILINEAR SINGULAR VALUE DECOMPOSITION FOR STRUCTURED TENSORS

International audienceThe Higher-Order SVD (HOSVD) is a generalization of the Singular Value Decompo- sition (SVD) to higher-order tensors (i.e. arrays with more than two indices) and plays an important role in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a third-order tensor involves the computation of the SVD of three matrices, which are referred to as "modes", or "matrix unfoldings". In this paper, we present fast algorithms for computing the full and the rank-truncated HOSVD of third-order structured (symmetric, Toeplitz and Hankel) tensors. These algorithms are derived by considering two specific ways to unfold a structured tensor, leading to structured matrix unfoldings whose SVD can be efficiently computed1

ADAPTIVE MULTILINEAR SVD FOR STRUCTURED TENSORS

International audienceThe Higher-Order SVD (HOSVD) is a generalization of the SVD to higher-order tensors (ie. arrays with more than two indexes) and plays an important role in various domains. Unfortunately, the computational cost of this decomposition is very high since the basic HOSVD algorithm involves the computation of the SVD of three highly redundant block-Hankel matrices, called modes. In this paper, we present an ultra-fast way of computing the HOSVD of a third-order structured tensor. The key result of this work lies in the fact it is possible to reduce the basic HOSVD algorithm to the computation of the SVD of three non-redundant Hankel matrices whose columns are multiplied by a given weighting function. Next, we exploit an FFT-based implementation of the orthogonal iteration algorithm in an adaptive way. Even though for a square (I ×I ×I) tensor the complexity of the basic full-HOSVD is O(I4) and O(rI3) for its r-truncated version, our approach reaches a linear complexity of O(rI log2(I))

Asymptotic Performance for Delayed Exponential Process

International audienceThe damped and delayed sinusoidal (DDS) model can be defined as the sum of sinusoids whose waveforms can be damped and delayed. This model is suitable for compactly modeling short time events. This property is closely related to its ability to reduce the time-support of each sinusoidal component. In this correspondence, we derive exact and approximate asymptotic Cramér–Rao bounds (CRBs) for the DDS model. This analysis shows that this model has better, or at least similar, theoretical performance than the well-known exponentially damped sinusoidal (EDS) model. In particular, the performance in the DDS case is significantly improved compared to that of the EDS for closely spaced sinusoids thanks to the nonzero time delays. Consequently, we can exploit the advantageous properties of the DDS model and, in the same time, we can keep high theoretical model parameter estimation accuracy

Damped and delayed sinuosidal model for transient modeling

International audienceIn this work, we present the Damped and De- layed Sinusoidal (DDS) model, a generalization of the sinu- soidal model. This model takes into account an angular fre- quency, a damping factor, a phase, an amplitude and a time- delay parameter for each component. Two algorithms are introduced for the DDS parameter estimation using a sub- band processing approach. Finally, we derive the Cramer- Rao Bound (CRB) expression for the DDS model and a simulation-based performance analysis in the context of a noisy fast time-varying synthetic signal and in the audio transient signal modeling context

Audio Modeling based on Delayed Sinusoids

International audienceIn this work, we present an evolution of the DDS (Damped & Delayed Sinusoidal) model introduced within the framework of the general signal modeling. This model is named the Partial Damped & Delayed Sinusoidal (PDDS) model and takes into account a single time delay parameter for a set (sum) of damped sinusoids. This modi- ¯cation is more consistent with the transient audio modeling problem. We show the validity of this approach by compari- son with the well-known EDS (Exponentially Damped Sinu- soids) approach. Finally, the performances of three model high-resolution parameter estimation algorithms are com- pared on synthetic fast time-varying signals and on two typ- ical audio transients

CONTRIBUTIONS AUX METHODES DE TRAITEMENT MULTIDIMENTIONELS ET A L'ANALYSE DES PERFORMANCES EN TERMES D'ESTIMATION ET DE RESOLUTION LIMITE

Ce travail propose une synthèse de mon activité de recherche en trois parties. (1) La première approche tend à répondre à la question suivante : quelles sont la précision minimale d'estimation et la résolution limite que l'on peut espérer dans le cadre des modèles multidimensionnels (plusieurs paramètres d'intérêt par signal) ? Les applications sont nombreuses telles que le traitement d'antennes, les systèmes de communications MIMO, ou encore le radar... (2) Ma seconde thématique de recherche se situe autour des décompositions multidimensionnelles (aka. tensorielles) telles que la SVD multilinéaire dans le cadre du traitement du signal. Souvent la nature multidimensionnelle des modèles sous-jacents est ignorée dans les applications de traitement du signal pour des raisons de complexité calculatoire. Dans cet axe de recherche, il est proposé des décompositions orthogonales rapides exploitant la structuration naturelle des données (Toeplitz, Hankel, symétrique). Les applications sont nombreuses telles que l'analyse harmonique, les statistiques d'ordres supérieures ou encore les modèles de Volterra. (3) Enfin le dernier axe de recherche s'articule autour des méthodes de dissimulation de l'information mieux connues sous le nom de « tatouage robuste » ou de « stéganographie ». Dans le contexte des réseaux de communication dits surveillés, il a été proposé des algorithmes de marquage robustes aux attaques licites ou illicites, à haute capacité et statistiquement invisibles

COMMON POLE ESTIMATION WITH AN ORTHOGONAL VECTOR METHOD

International audienceIn some applications as in biomedical analysis, we encounter the problem of estimating the common poles (angularfrequency and damping-factor) in a multi-channel set-up composed as the sum of Exponentially Damped Sinusoids. In this contribution, we propose a new subspace algorithm belonging to the family of the Orthogonal Vector Methods which solves the considered estimation problem. In particular, we expose a root-MUSIC algorithm which deals with damped components for an algorithmic cost comparable to the root- MUSIC for constant modulus components. Finally, we show by means of an example, that the proposed method is efficient, especially for low SNRs