4,911 research outputs found
Numerical Analysis of the Non-uniform Sampling Problem
We give an overview of recent developments in the problem of reconstructing a
band-limited signal from non-uniform sampling from a numerical analysis view
point. It is shown that the appropriate design of the finite-dimensional model
plays a key role in the numerical solution of the non-uniform sampling problem.
In the one approach (often proposed in the literature) the finite-dimensional
model leads to an ill-posed problem even in very simple situations. The other
approach that we consider leads to a well-posed problem that preserves
important structural properties of the original infinite-dimensional problem
and gives rise to efficient numerical algorithms. Furthermore a fast multilevel
algorithm is presented that can reconstruct signals of unknown bandwidth from
noisy non-uniformly spaced samples. We also discuss the design of efficient
regularization methods for ill-conditioned reconstruction problems. Numerical
examples from spectroscopy and exploration geophysics demonstrate the
performance of the proposed methods
Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis
Recently, compressed sensing techniques in combination with both wavelet and
directional representation systems have been very effectively applied to the
problem of image inpainting. However, a mathematical analysis of these
techniques which reveals the underlying geometrical content is completely
missing. In this paper, we provide the first comprehensive analysis in the
continuum domain utilizing the novel concept of clustered sparsity, which
besides leading to asymptotic error bounds also makes the superior behavior of
directional representation systems over wavelets precise. First, we propose an
abstract model for problems of data recovery and derive error bounds for two
different recovery schemes, namely l_1 minimization and thresholding. Second,
we set up a particular microlocal model for an image governed by edges inspired
by seismic data as well as a particular mask to model the missing data, namely
a linear singularity masked by a horizontal strip. Applying the abstract
estimate in the case of wavelets and of shearlets we prove that -- provided the
size of the missing part is asymptotically to the size of the analyzing
functions -- asymptotically precise inpainting can be obtained for this model.
Finally, we show that shearlets can fill strictly larger gaps than wavelets in
this model.Comment: 49 pages, 9 Figure
Multipath Parameter Estimation from OFDM Signals in Mobile Channels
We study multipath parameter estimation from orthogonal frequency division
multiplex signals transmitted over doubly dispersive mobile radio channels. We
are interested in cases where the transmission is long enough to suffer time
selectivity, but short enough such that the time variation can be accurately
modeled as depending only on per-tap linear phase variations due to Doppler
effects. We therefore concentrate on the estimation of the complex gain, delay
and Doppler offset of each tap of the multipath channel impulse response. We
show that the frequency domain channel coefficients for an entire packet can be
expressed as the superimposition of two-dimensional complex sinusoids. The
maximum likelihood estimate requires solution of a multidimensional non-linear
least squares problem, which is computationally infeasible in practice. We
therefore propose a low complexity suboptimal solution based on iterative
successive and parallel cancellation. First, initial delay/Doppler estimates
are obtained via successive cancellation. These estimates are then refined
using an iterative parallel cancellation procedure. We demonstrate via Monte
Carlo simulations that the root mean squared error statistics of our estimator
are very close to the Cramer-Rao lower bound of a single two-dimensional
sinusoid in Gaussian noise.Comment: Submitted to IEEE Transactions on Wireless Communications (26 pages,
9 figures and 3 tables
- …