3,913 research outputs found
Estimation of Sparse MIMO Channels with Common Support
We consider the problem of estimating sparse communication channels in the
MIMO context. In small to medium bandwidth communications, as in the current
standards for OFDM and CDMA communication systems (with bandwidth up to 20
MHz), such channels are individually sparse and at the same time share a common
support set. Since the underlying physical channels are inherently
continuous-time, we propose a parametric sparse estimation technique based on
finite rate of innovation (FRI) principles. Parametric estimation is especially
relevant to MIMO communications as it allows for a robust estimation and
concise description of the channels. The core of the algorithm is a
generalization of conventional spectral estimation methods to multiple input
signals with common support. We show the application of our technique for
channel estimation in OFDM (uniformly/contiguous DFT pilots) and CDMA downlink
(Walsh-Hadamard coded schemes). In the presence of additive white Gaussian
noise, theoretical lower bounds on the estimation of SCS channel parameters in
Rayleigh fading conditions are derived. Finally, an analytical spatial channel
model is derived, and simulations on this model in the OFDM setting show the
symbol error rate (SER) is reduced by a factor 2 (0 dB of SNR) to 5 (high SNR)
compared to standard non-parametric methods - e.g. lowpass interpolation.Comment: 12 pages / 7 figures. Submitted to IEEE Transactions on Communicatio
Atomic norm denoising with applications to line spectral estimation
Motivated by recent work on atomic norms in inverse problems, we propose a
new approach to line spectral estimation that provides theoretical guarantees
for the mean-squared-error (MSE) performance in the presence of noise and
without knowledge of the model order. We propose an abstract theory of
denoising with atomic norms and specialize this theory to provide a convex
optimization problem for estimating the frequencies and phases of a mixture of
complex exponentials. We show that the associated convex optimization problem
can be solved in polynomial time via semidefinite programming (SDP). We also
show that the SDP can be approximated by an l1-regularized least-squares
problem that achieves nearly the same error rate as the SDP but can scale to
much larger problems. We compare both SDP and l1-based approaches with
classical line spectral analysis methods and demonstrate that the SDP
outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix
Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.Comment: 27 pages, 10 figures. A preliminary version of this work appeared in
the Proceedings of the 49th Annual Allerton Conference in September 2011.
Numerous numerical experiments added to this version in accordance with
suggestions by anonymous reviewer
On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
Among the many ways to model signals, a recent approach that draws
considerable attention is sparse representation modeling. In this model, the
signal is assumed to be generated as a random linear combination of a few atoms
from a pre-specified dictionary. In this work we analyze two Bayesian denoising
algorithms -- the Maximum-Aposteriori Probability (MAP) and the
Minimum-Mean-Squared-Error (MMSE) estimators, under the assumption that the
dictionary is unitary. It is well known that both these estimators lead to a
scalar shrinkage on the transformed coefficients, albeit with a different
response curve. In this work we start by deriving closed-form expressions for
these shrinkage curves and then analyze their performance. Upper bounds on the
MAP and the MMSE estimation errors are derived. We tie these to the error
obtained by a so-called oracle estimator, where the support is given,
establishing a worst-case gain-factor between the MAP/MMSE estimation errors
and the oracle's performance. These denoising algorithms are demonstrated on
synthetic signals and on true data (images).Comment: 29 pages, 10 figure
Revisiting maximum-a-posteriori estimation in log-concave models
Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation
methodology in imaging sciences, where high dimensionality is often addressed
by using Bayesian models that are log-concave and whose posterior mode can be
computed efficiently by convex optimisation. Despite its success and wide
adoption, MAP estimation is not theoretically well understood yet. The
prevalent view in the community is that MAP estimation is not proper Bayesian
estimation in a decision-theoretic sense because it does not minimise a
meaningful expected loss function (unlike the minimum mean squared error (MMSE)
estimator that minimises the mean squared loss). This paper addresses this
theoretical gap by presenting a decision-theoretic derivation of MAP estimation
in Bayesian models that are log-concave. A main novelty is that our analysis is
based on differential geometry, and proceeds as follows. First, we use the
underlying convex geometry of the Bayesian model to induce a Riemannian
geometry on the parameter space. We then use differential geometry to identify
the so-called natural or canonical loss function to perform Bayesian point
estimation in that Riemannian manifold. For log-concave models, this canonical
loss is the Bregman divergence associated with the negative log posterior
density. We then show that the MAP estimator is the only Bayesian estimator
that minimises the expected canonical loss, and that the posterior mean or MMSE
estimator minimises the dual canonical loss. We also study the question of MAP
and MSSE estimation performance in large scales and establish a universal bound
on the expected canonical error as a function of dimension, offering new
insights into the good performance observed in convex problems. These results
provide a new understanding of MAP and MMSE estimation in log-concave settings,
and of the multiple roles that convex geometry plays in imaging problems.Comment: Accepted for publication in SIAM Imaging Science
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