411 research outputs found
Algebraic number theory and code design for Rayleigh fading channels
Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.
Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available.
The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible
to a large audience
Performance of Linear Field Reconstruction Techniques with Noise and Uncertain Sensor Locations
We consider a wireless sensor network, sampling a bandlimited field,
described by a limited number of harmonics. Sensor nodes are irregularly
deployed over the area of interest or subject to random motion; in addition
sensors measurements are affected by noise. Our goal is to obtain a high
quality reconstruction of the field, with the mean square error (MSE) of the
estimate as performance metric. In particular, we analytically derive the
performance of several reconstruction/estimation techniques based on linear
filtering. For each technique, we obtain the MSE, as well as its asymptotic
expression in the case where the field number of harmonics and the number of
sensors grow to infinity, while their ratio is kept constant. Through numerical
simulations, we show the validity of the asymptotic analysis, even for a small
number of sensors. We provide some novel guidelines for the design of sensor
networks when many parameters, such as field bandwidth, number of sensors,
reconstruction quality, sensor motion characteristics, and noise level of the
measures, have to be traded off
Deformations of symplectic cohomology and exact Lagrangians in ALE spaces
We prove that the only exact Lagrangian submanifolds in an ALE space are
spheres. ALE spaces are the simply connected hyperkahler manifolds which at
infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be
realized as the plumbing of copies of the cotangent bundle of a 2-sphere
according to ADE Dynkin diagrams. The proof relies on symplectic cohomology.Comment: 35 pages, 3 figures, minor changes and corrected typo
Reconstruction of Multidimensional Signals from Irregular Noisy Samples
We focus on a multidimensional field with uncorrelated spectrum, and study
the quality of the reconstructed signal when the field samples are irregularly
spaced and affected by independent and identically distributed noise. More
specifically, we apply linear reconstruction techniques and take the mean
square error (MSE) of the field estimate as a metric to evaluate the signal
reconstruction quality. We find that the MSE analysis could be carried out by
using the closed-form expression of the eigenvalue distribution of the matrix
representing the sampling system. Unfortunately, such distribution is still
unknown. Thus, we first derive a closed-form expression of the distribution
moments, and we find that the eigenvalue distribution tends to the
Marcenko-Pastur distribution as the field dimension goes to infinity. Finally,
by using our approach, we derive a tight approximation to the MSE of the
reconstructed field.Comment: To appear on IEEE Transactions on Signal Processing, 200
Exact Lagrangian submanifolds in simply-connected cotangent bundles
We consider exact Lagrangian submanifolds in cotangent bundles. Under certain
additional restrictions (triviality of the fundamental group of the cotangent
bundle, and of the Maslov class and second Stiefel-Whitney class of the
Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically
indistinguishable from the zero-section. This implies strong restrictions on
their topology. An essentially equivalent result was recently proved
independently by Nadler, using a different approach.Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the
spectral sequence considerably expanded. Other minor change
Novikov-symplectic cohomology and exact Lagrangian embeddings
Let L be an exact Lagrangian submanifold inside the cotangent bundle of a
closed manifold N. We prove that if N satisfies a mild homotopy assumption then
the image of \pi_2(L) in \pi_2(N) has finite index.
We make no assumption on the Maslov class of L, and we make no orientability
assumptions. The homotopy assumption is either that N is simply connected, or
more generally that \pi_m(N) is finitely generated for each m \geq 2.
The result is proved by constructing the Novikov homology theory for
symplectic cohomology and generalizing Viterbo's construction of a transfer map
between the homologies of the free loopspaces of N and L.Comment: 27 pages; added two new sections (non-simply connected cotangent
bundles, unorientable setup). The final version is published in Geometry &
Topology 13, 200
Displacement energy of unit disk cotangent bundles
We give an upper bound of a Hamiltonian displacement energy of a unit disk
cotangent bundle in a cotangent bundle , when the base manifold
is an open Riemannian manifold. Our main result is that the displacement
energy is not greater than , where is the inner radius of ,
and is a dimensional constant. As an immediate application, we study
symplectic embedding problems of unit disk cotangent bundles. Moreover,
combined with results in symplectic geometry, our main result shows the
existence of short periodic billiard trajectories and short geodesic loops.Comment: Title slightly changed. Close to the version published online in Math
Zei
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