Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let $\Gamma_j$, $j \in J$, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group $G$ and study systems of the form $\cup_{j \in J}\{g_{j,p}(\cdot - \gamma)\}_{\gamma \in \Gamma_j, p \in P_j}$ with generators $g_{j,p}$ in $L^2(G)$ and with each $P_j$ being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical $\alpha$ local integrability condition ($\alpha$-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for $L^2(G)$. This generalizes results on generalized shift invariant systems, where each $P_j$ is assumed to be countable and each $\Gamma_j$ is a uniform lattice in $G$, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices $\Gamma_j$, our characterizations improve known results since the class of GTI systems satisfying the $\alpha$-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for $L^2(G)$. In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in $L^2(\mathbb{R}^n)$ are special cases of the same general characterizing equations.Comment: Minor changes (v2). To appear in Trans. Amer. Math. So

Co-compact Gabor systems on locally compact abelian groups

In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.Comment: Paper (v2) shortened. To appear in J. Fourier Anal. App

Density and duality theorems for regular Gabor frames

We investigate Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the "size" of the subgroup. From these density results we are able to extend the classical Wexler-Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time-frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new

Seguimiento de la señal de Galileo E1 OS para UAVs

Este proyecto está relacionado con el seguimiento de la señal E1 OS de Galileo. El objetivo es llevar a cabo el procesado en banda base de señales de Galileo usando Matlab, por lo que el alcance del proyecto está dentro del procesado digital de señales de radiofrecuencia. La implementación se basa en un toolbox existente desarrollado en el departamenteo de DTU Space para GPS, que ha sido adaptado para aceptar la señal de Galileo. Además de esto, se han recogido datos usando un receptor software y el toolbox ha sido probado. Asimismo, se ha llevado a cabo un estudio de multipath usando una estrategia multicorrelador. La tesis está dividida en cuatro bloques principales. El primero introduce la señal de Galileo, así como algo de teoría sobre receptores software como background para la implementación, que se describe justo después. Posteriormente, se muestra el setup para la recogida de datos junto con algunos resultados y la discusión de los mismos. Finalmente, se analiza el multipath en un capítulo separado, que consiste en una pequeña sección de teoría, las modificaciones en la implementación y la seccion de resultados