1,008 research outputs found
A Low Density Lattice Decoder via Non-Parametric Belief Propagation
The recent work of Sommer, Feder and Shalvi presented a new family of codes
called low density lattice codes (LDLC) that can be decoded efficiently and
approach the capacity of the AWGN channel. A linear time iterative decoding
scheme which is based on a message-passing formulation on a factor graph is
given.
In the current work we report our theoretical findings regarding the relation
between the LDLC decoder and belief propagation. We show that the LDLC decoder
is an instance of non-parametric belief propagation and further connect it to
the Gaussian belief propagation algorithm. Our new results enable borrowing
knowledge from the non-parametric and Gaussian belief propagation domains into
the LDLC domain. Specifically, we give more general convergence conditions for
convergence of the LDLC decoder (under the same assumptions of the original
LDLC convergence analysis). We discuss how to extend the LDLC decoder from
Latin square to full rank, non-square matrices. We propose an efficient
construction of sparse generator matrix and its matching decoder. We report
preliminary experimental results which show our decoder has comparable symbol
to error rate compared to the original LDLC decoder.%Comment: Submitted for publicatio
Finite Dimensional Infinite Constellations
In the setting of a Gaussian channel without power constraints, proposed by
Poltyrev, the codewords are points in an n-dimensional Euclidean space (an
infinite constellation) and the tradeoff between their density and the error
probability is considered. The capacity in this setting is the highest
achievable normalized log density (NLD) with vanishing error probability. This
capacity as well as error exponent bounds for this setting are known. In this
work we consider the optimal performance achievable in the fixed blocklength
(dimension) regime. We provide two new achievability bounds, and extend the
validity of the sphere bound to finite dimensional infinite constellations. We
also provide asymptotic analysis of the bounds: When the NLD is fixed, we
provide asymptotic expansions for the bounds that are significantly tighter
than the previously known error exponent results. When the error probability is
fixed, we show that as n grows, the gap to capacity is inversely proportional
(up to the first order) to the square-root of n where the proportion constant
is given by the inverse Q-function of the allowed error probability, times the
square root of 1/2. In an analogy to similar result in channel coding, the
dispersion of infinite constellations is 1/2nat^2 per channel use. All our
achievability results use lattices and therefore hold for the maximal error
probability as well. Connections to the error exponent of the power constrained
Gaussian channel and to the volume-to-noise ratio as a figure of merit are
discussed. In addition, we demonstrate the tightness of the results numerically
and compare to state-of-the-art coding schemes.Comment: 54 pages, 13 figures. Submitted to IEEE Transactions on Information
Theor
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