26,190 research outputs found
Efficient implementation of linear programming decoding
While linear programming (LP) decoding provides more flexibility for
finite-length performance analysis than iterative message-passing (IMP)
decoding, it is computationally more complex to implement in its original form,
due to both the large size of the relaxed LP problem, and the inefficiency of
using general-purpose LP solvers. This paper explores ideas for fast LP
decoding of low-density parity-check (LDPC) codes. We first prove, by modifying
the previously reported Adaptive LP decoding scheme to allow removal of
unnecessary constraints, that LP decoding can be performed by solving a number
of LP problems that contain at most one linear constraint derived from each of
the parity-check constraints. By exploiting this property, we study a sparse
interior-point implementation for solving this sequence of linear programs.
Since the most complex part of each iteration of the interior-point algorithm
is the solution of a (usually ill-conditioned) system of linear equations for
finding the step direction, we propose a preconditioning algorithm to
facilitate iterative solution of such systems. The proposed preconditioning
algorithm is similar to the encoding procedure of LDPC codes, and we
demonstrate its effectiveness via both analytical methods and computer
simulation results.Comment: 44 pages, submitted to IEEE Transactions on Information Theory, Dec.
200
Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes
Linear programming (LP) decoding approximates maximum-likelihood (ML)
decoding of a linear block code by relaxing the equivalent ML integer
programming (IP) problem into a more easily solved LP problem. The LP problem
is defined by a set of box constraints together with a set of linear
inequalities called "parity inequalities" that are derived from the constraints
represented by the rows of a parity-check matrix of the code and can be added
iteratively and adaptively. In this paper, we first derive a new necessary
condition and a new sufficient condition for a violated parity inequality
constraint, or "cut," at a point in the unit hypercube. Then, we propose a new
and effective algorithm to generate parity inequalities derived from certain
additional redundant parity check (RPC) constraints that can eliminate
pseudocodewords produced by the LP decoder, often significantly improving the
decoder error-rate performance. The cut-generating algorithm is based upon a
specific transformation of an initial parity-check matrix of the linear block
code. We also design two variations of the proposed decoder to make it more
efficient when it is combined with the new cut-generating algorithm. Simulation
results for several low-density parity-check (LDPC) codes demonstrate that the
proposed decoding algorithms significantly narrow the performance gap between
LP decoding and ML decoding
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
On optimal and near-optimal turbo decoding using generalized max operator
Motivated by a recently published robust geometric programming approximation, a generalized approach for approximating efficiently the max* operator is presented. Using this approach, the max* operator is approximated by means of a generic and yet very simple max operator, instead of using additional correction term as previous approximation methods require. Following that, several turbo decoding algorithms are obtained with optimal and near-optimal bit error rate (BER) performance depending on a single parameter, namely the number of piecewise linear (PWL) approximation terms. It turns out that the known max-log-MAP algorithm can be viewed as special case of this new generalized approach. Furthermore, the decoding complexity of the most popular previously published methods is estimated, for the first time, in a unified way by hardware synthesis results, showing the practical implementation advantages of the proposed approximations against these method
Hardware Based Projection onto The Parity Polytope and Probability Simplex
This paper is concerned with the adaptation to hardware of methods for
Euclidean norm projections onto the parity polytope and probability simplex. We
first refine recent efforts to develop efficient methods of projection onto the
parity polytope. Our resulting algorithm can be configured to have either
average computational complexity or worst case
complexity on a serial processor where
is the dimension of projection space. We show how to adapt our projection
routine to hardware. Our projection method uses a sub-routine that involves
another Euclidean projection; onto the probability simplex. We therefore
explain how to adapt to hardware a well know simplex projection algorithm. The
hardware implementations of both projection algorithms achieve area scalings of
at a delay of
. Finally, we present numerical results in
which we evaluate the fixed-point accuracy and resource scaling of these
algorithms when targeting a modern FPGA
Interior Point Decoding for Linear Vector Channels
In this paper, a novel decoding algorithm for low-density parity-check (LDPC)
codes based on convex optimization is presented. The decoding algorithm, called
interior point decoding, is designed for linear vector channels. The linear
vector channels include many practically important channels such as inter
symbol interference channels and partial response channels. It is shown that
the maximum likelihood decoding (MLD) rule for a linear vector channel can be
relaxed to a convex optimization problem, which is called a relaxed MLD
problem. The proposed decoding algorithm is based on a numerical optimization
technique so called interior point method with barrier function. Approximate
variations of the gradient descent and the Newton methods are used to solve the
convex optimization problem. In a decoding process of the proposed algorithm, a
search point always lies in the fundamental polytope defined based on a
low-density parity-check matrix. Compared with a convectional joint message
passing decoder, the proposed decoding algorithm achieves better BER
performance with less complexity in the case of partial response channels in
many cases.Comment: 18 pages, 17 figures, The paper has been submitted to IEEE
Transaction on Information Theor
Low-Complexity LP Decoding of Nonbinary Linear Codes
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has
attracted much attention in the research community in the past few years. LP
decoding has been derived for binary and nonbinary linear codes. However, the
most important problem with LP decoding for both binary and nonbinary linear
codes is that the complexity of standard LP solvers such as the simplex
algorithm remains prohibitively large for codes of moderate to large block
length. To address this problem, two low-complexity LP (LCLP) decoding
algorithms for binary linear codes have been proposed by Vontobel and Koetter,
henceforth called the basic LCLP decoding algorithm and the subgradient LCLP
decoding algorithm.
In this paper, we generalize these LCLP decoding algorithms to nonbinary
linear codes. The computational complexity per iteration of the proposed
nonbinary LCLP decoding algorithms scales linearly with the block length of the
code. A modified BCJR algorithm for efficient check-node calculations in the
nonbinary basic LCLP decoding algorithm is also proposed, which has complexity
linear in the check node degree.
Several simulation results are presented for nonbinary LDPC codes defined
over Z_4, GF(4), and GF(8) using quaternary phase-shift keying and
8-phase-shift keying, respectively, over the AWGN channel. It is shown that for
some group-structured LDPC codes, the error-correcting performance of the
nonbinary LCLP decoding algorithms is similar to or better than that of the
min-sum decoding algorithm.Comment: To appear in IEEE Transactions on Communications, 201
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