Time Scale Approach for Chirp Detection

International audienceTwo different approaches for joint detection and estimation of signals embedded in stationary random noise are considered and compared, for the subclass of amplitude and frequency modulated signals. Matched filter approaches are compared to time-frequency and time scale based approaches. Particular attention is paid to the case of the so-called " power-law chirps " , characterized by monomial and polynomial amplitude and frequency functions. As target application, the problem of gravitational waves at interferometric detectors is considered

Perturbative methods in theory of phase gratings

Perturbative methods are generally invoked for problems in which there is a small parameter. In the theory of phase gratings, the small parameter is the modulation amplitude of the refractive index, and the classical perturbative method is then the Born approximation. But it is well-known that the Born approximation fails at the Bragg resonance, however small the modulation amplitude is. In this paper a perturbative method is presented, which is working at Bragg resonance as well. A sequence of numbers (called the eigenvalues of the problem) are introduced; they depend on the geometrical configuration (incidence angle, grating parameters). It is shown that the Bragg resonance occurs if (and only if) two eigenvalues become equal. These eigenvalues – and the corresponding solutions of the equations – can be expanded in powers of the modulation amplitude. The expansions are different according to whether the corresponding eigenvalue is simple or double. Explicit formulae or algorithms are given. Computing programs have been written from them. These programs are efficient

Application d'une théorie alternative de la diffraction pour l'étude des profils de modulation des réseaux holographiques de volume

In this contribution we present a numerical study of the influence of the refractive index modulation profile onto several parameters of interest such that diffraction efficiency, spectral selectivity, or angular selectivity. The most widely used materials (silver halides and dichromated gelatin) have a strongly non-linear response at high exposures. Many papers have been published about this subject. Solymar and his group [22 - 24] have studied in detail the formation of a modulation profile for the refractive index in the region of saturation. For linear exposures the profile will be sinusoidal, but for exposures with saturation, it will be non-sinusoidal. We have selected a series of possible profiles, which correspond to the properties of the materials at saturation, as they have been studied in [22-24]. For these models of profiles we have carried out a complete set of numerical computations, using programs of our own. These programs are based on a complete and exact solution of the Maxwell equations. The programs have been used for the computation of the efficiencies and of the angular selectivity at Bragg incidence for the different models of non-sinusoidal profiles. We were interested mainly by the optimal cases; indeed the efficiencies at a given order depend strongly on the thickness and on the modulation amplitude: for each given thickness, there is a precise value of the modulation amplitude at which the efficiency is maximum; conversely, for each given amplitude of modulation, there is a precise value of the thickness at which the efficiency is maximum. We have then studied the dependence of the efficiency and of the angular selectivity with respect to the profile, in the neighbourhood of these optimal cases. This study should explain theoretically the way by which the efficiency depends on the exposure: it is due not only to the fact that the modulation amplitude increases with the exposure, but also to the fact that after having reached saturation, the profiles gets a non-sinusoidal form for which the theoretical efficiency is better than for the sinusoidal from. We present only a theoretical analysis; for the analysis of the correlation between the exposure and the form of the resulting profile, we follow the work of Solymar and his group.Nous présentons une étude numérique de l'influence du profil de modulation de l'indice de réfraction, sur quelques paramètres tels que: l'efficacité de diffraction, la sélectivité spectrale et angulaire. Les matériaux photosensibles les plus utilisés en holographie à savoir les halogénures d'argent et la gélatine bichromatée, sont fortement non linéaires pour des expositions élevées. Nous avons choisi une série de profils correspondant à des expositions différentes. Pour ces profils nous avons fait une étude numérique complète en utilisant une théorie alternative de la diffraction dans les milieux modulés [1]

Détermination du profil de modulation des réseaux holographiques de phase

The complete knowledge of the geometrical and physic-chemical parameters of a periodically modulated volume material, permits the determination of the diffraction picture. Such a purely mathematical and numerical determination is of great scientific and technological interest. The modulation profile (given by its Fourier coefficients) is one of these parameters. It can be determinated a posteriori only by the measurement of the different diffracted intensities at different orders. Starting from this idea, we can achieve a new method (theoritically exact) which permits the study of the diffraction of an electromagnetic plane wave by a dielectric grating. This method leads to the numerical treatment of ordinary differential equation with variable – by periodic – coefficients. The method is presented here for the classical case of a wave with electric polarization parallel to the grating. For the analysis of the modulation profile, we have considered realistic models of profiles, contrary to the current models, which have only a numerical existence. In order to achieve our experimental work, we have developed two experimental set-up: the first for the recording and the second for the analysis of diffractive elements. The whole set-up can be directed with the aid of a software from a personal computer. The validity of results are discussed.La connaissance parfaite des paramètres géométriques et physico-chimiques d'un matériau de volume modulé périodiquement, donne une bonne connaissance de la figure de diffraction. Celle-ci a une grande importance scientifique et technologique. Le profil de modulation (donné par ses coefficients de Fourier) est l'un de ces paramètres qui ne peut être déterminé a posteriori qu'à partir de la répartition d'intensité entre les différents ordres. A partir de cette idée, nous avons établi une méthode exacte permettant l'étude de la diffraction d'une onde plane électromagnétique par un réseau diélectrique, qui conduit au traitement numérique d'une équation différentielle à coefficients variables. La méthode est donnée pour le cas classique où le champ électrique est parallèle au plan du réseau. Pour analyser l'influence du profil de modulation, on considère des modèles mathématiques réalistes et non (comme il est courant dans la littérature) des modèles physiques irréalisables. Pour notre travail expérimental nous avons mis au point deux montages : le premier pour l'enregistrement et le second pour l'analyse. Ce dernier est totalement automatisé et piloté par un ordinateur. Les résultats obtenus sont largement discutés

Analyse de formes par moiré

We present a mathematical analysis of moiré phenomena for shape recognition. The basic theoretical concept - and tool - will be the $contour function$. We show that the mathematical analysis is greatly simplified by the systematic recourse to this tool. The analysis presented permits a simultaneous treatment of two different modes of implementing the moiré technique : the direct mode (widely used and well-known), and the converse mode (scarcely used). The converse mode consists in computing and designing a grating especially for one model of object, in such a manner that if (and only if) the object is in conformity with the prescribed model, the resulting moiré fringes are parallel straight lines. We give explicit formulas and algorithms for such computations.
Nous présentons une analyse mathématique du moiré permettant une reconnaissance des formes. Le concept théorique de base est celui de “ fonction de contour ”. Nous montrons que l'analyse mathématique est simplifiée en faisant appel à ces fonctions. De plus, la méthode proposée permet de traiter d'une manière unifiée les deux différents modes d'utilisation des techniques de moiré : le mode direct (le plus utilisé et le mieux connu), et le moiré inverse, qui consiste, pour un modèle d'objet donné, à calculer et réaliser un réseau spécifique, tel que si (et seulement si) un objet est conforme au modèle, les franges de moiré obtenues seront des lignes droites parallèles. Nous proposons des formules explicites et des algorithmes pour ces traitements.