**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Publication# Calcul stochastique appliqué aux problèmes de détection des signaux aléatoires

1999

EPFL thesis

EPFL thesis

Abstract

Signal detection is one of the basic problems in statistical communication theory, and has many applications to contemporary technology, whether in engineering, medical science, or the environment. The most difficult problems are those involving random signals, and it is these types of signals that are found in applications to complex systems (the ocean, the atmosphere, the ecosystem). What is known of the subject at the present time is insufficient in that it suffers from mathematical restrictions which are difficult to justify in practice, is limited in the types of noises that can be accommodated, which do not cover the noises one meets in nature, and is based on algorithms whose behavior is not sufficiently understood. The broad aim of this thesis is to solve some of the problems that are open in that area of research. As detection of a non-Gaussian stochastic signal in additive and dependent Gaussian noise can be viewed as the canonical detection problem for active sonar in a reverberation-limited environment, and that this detection problem, except for a multiplicity restriction, is, mathematically, the problem nearest to a satisfactory solution, the first part of the thesis deals with the definition and the properties of a form of the Itô stochastic integral that must be tailored to remove the multiplicity restriction mentioned above. On the way some interesting connections with other forms of the stochastic integral are investigated. The second part of the thesis is devoted to the derivation of the likelihood ratio which acts as an universal detector, still within the framework of a stochastic signal in dependent Gaussian noise. The solution of the Gaussain noise problem is based on a representation of the noise and signal-plus-noise processes as superpositions of causal filters acting on noise and signal-plus-noise processes that are semimartingales, for the treatment of which stochastic calculus is the most efficient tool. The decomposition used is the Cramér-Hida decomposition which is particularly suited to the handling of Gaussian processes, though it is much more broadly valid. The last part of the thesis is a study of the possibility to extend the method that works for Gaussian noise to situations for which the noise is no longer Gaussian. A likelihood formula is obtained for noises that are non anti- cipative transformations of the sum of a Wiener process and an independent Poisson martingale.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (40)

Related MOOCs (19)

Related publications (162)

Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.

Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces.

Itô calculus

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: where H is a locally square-integrable process adapted to the filtration generated by X , which is a Brownian motion or, more generally, a semimartingale.

Digital Signal Processing [retired]

The course provides a comprehensive overview of digital signal processing theory, covering discrete time, Fourier analysis, filter design, sampling, interpolation and quantization; it also includes a

Digital Signal Processing

Digital Signal Processing is the branch of engineering that, in the space of just a few decades, has enabled unprecedented levels of interpersonal communication and of on-demand entertainment. By rewo

Digital Signal Processing I

Basic signal processing concepts, Fourier analysis and filters. This module can
be used as a starting point or a basic refresher in elementary DSP

Victor Panaretos, Neda Mohammadi Jouzdani

Functional data are typically modeled as sample paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The smoothness assumption is imposed to allow for ...

Daniel Kuhn, Bahar Taskesen, Cagil Kocyigit

Linear-Quadratic-Gaussian (LQG) control is a fundamental control paradigm that is studied in various fields such as engineering, computer science, economics, and neuroscience. It involves controlling a system with linear dynamics and imperfect observations ...

2023Sabine Süsstrunk, Radhakrishna Achanta, Mahmut Sami Arpa, Martin Nicolas Everaert, Athanasios Fitsios

There is a bias in the inference pipeline of most diffusion models. This bias arises from a signal leak whose distribution deviates from the noise distribution, creating a discrepancy between training and inference processes. We demonstrate that this signa ...

2024