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Optimality and Duality of the Turbo Decoder
, 2007
"... The nearoptimal performance of the turbo decoder has been a source of intrigue among communications engineers and information theorists, given its ad hoc origins that were seemingly disconnected from optimization theory. Naturally one would inquire whether the favorable performance might be explain ..."
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The nearoptimal performance of the turbo decoder has been a source of intrigue among communications engineers and information theorists, given its ad hoc origins that were seemingly disconnected from optimization theory. Naturally one would inquire whether the favorable performance might be explained by characterizing the turbo decoder via some optimization criterion or performance index. Recently, two such characterizations have surfaced. One draws from statistical mechanics and aims to minimize the Bethe approximation to a free energy measure. The other characterization involves constrained likelihood estimation, a setting perhaps more familiar to communications engineers. The intent of this paper is to assemble a tutorial overview of these recent developments, and more importantly to identify the formal mathematical duality between the two viewpoints. The paper includes tutorial background material on the information geometry tools used in analyzing the turbo decoder, and the analysis accommodates both the parallel concatenation and serial concatenation schemes in a common framework.
Belief Propagation as a Dynamical System: The Linear Case and Open Problems
"... Systems and control theory have found wide application in the analysis and design of numerical algorithms. We present a discretetime dynamical system interpretation of an algorithm commonly used in information theory called Belief Propagation. Belief Propagation (BP) is one instance of the socalle ..."
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Systems and control theory have found wide application in the analysis and design of numerical algorithms. We present a discretetime dynamical system interpretation of an algorithm commonly used in information theory called Belief Propagation. Belief Propagation (BP) is one instance of the socalled SumProduct Algorithm and arises, e.g., in the context of iterative decoding of LowDensity ParityCheck codes. We review a few known results from information theory in the language of dynamical systems and show that the typically very high dimensional, nonlinear dynamical system corresponding to BP has interesting structural properties. For the linear case we completely characterize the behavior of this dynamical system in terms of its asymptotic inputoutput map. Finally, we state some of the open problems concerning BP in terms of the dynamical system presented.
Finite alphabet iterative decoders approaching maximum likelihood performance on the binary symmetric channel
 in Proc. Inf. Theory and Applications Workshop
, 2012
"... Abstract—We introduce a generic approach for improving the guaranteed error correction capability of regular lowdensity parity check codes. The method relies on operating (in serial or in parallel) a set of finite alphabet iterative decoders. The message passing update rules are judiciously chosen ..."
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Abstract—We introduce a generic approach for improving the guaranteed error correction capability of regular lowdensity parity check codes. The method relies on operating (in serial or in parallel) a set of finite alphabet iterative decoders. The message passing update rules are judiciously chosen to ensure that decoders have different dynamics on a specific finitelength code. The idea is that for the Binary Symmetric Channel, if some error pattern cannot be corrected by one particular decoder, there exists in the set of decoders, another decoder which can correct this pattern. We show how to select a plurality of message update rules so that the set of decoders can collectively correct error patterns on the dominant trapping sets. We also show that a set of decoders with dynamic reinitializations can approach the performance of maximum likelihood decoding for finitelength regular columnweight three codes. I.
On Robust Stability of the Belief Propagation Algorithm for LDPC Decoding
"... Abstract — The exact nonlinear loop gain of the belief propagation algorithm (BPA) in its loglikelihood ratio (LLR) formulation is computed. The nonlinear gains for regular lowdensity paritycheck (LDPC) error correcting codes can be computed exactly using a simple formula. It is shown that in some ..."
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Abstract — The exact nonlinear loop gain of the belief propagation algorithm (BPA) in its loglikelihood ratio (LLR) formulation is computed. The nonlinear gains for regular lowdensity paritycheck (LDPC) error correcting codes can be computed exactly using a simple formula. It is shown that in some neighborhood of the origin this gain is actually much smaller than the identity. Using a smallgain argument, this implies that the BPA is in fact locally inputtostate stable and produces bounded outputs for smallinnorm input LLR vectors. In a larger domain the algorithm produces at least bounded trajectories. Further it is shown that, as the block length increases, these regions exponentially shrink. Index Terms — iterative decoding; LDPC codes; dynamical system; convergence; belief propagation; smallgain theorem I.
Belief Propagation, Dykstra’s Algorithm, and Iterated Information Projections
, 2010
"... Belief propagation is shown to be an instance of a hybrid between two projection algorithms in the convex programming literature: Dykstra’s algorithm with cyclic Bregman projections, and an alternating Bregman projections algorithm. Via this connection, new results concerning the convergence and per ..."
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Belief propagation is shown to be an instance of a hybrid between two projection algorithms in the convex programming literature: Dykstra’s algorithm with cyclic Bregman projections, and an alternating Bregman projections algorithm. Via this connection, new results concerning the convergence and performance of belief propagation can be proven by exploiting the corresponding literature about the two projections algorithms it hybridizes. In this regard, it is identified that the lack of guaranteed convergence for belief propagation results from the asymmetry of its Bregman divergence by proving that when the associated hybrid projection algorithm generalization is used with a symmetric Bregman divergence instead it always converges. Additionally, by characterizing factorizations that are close to acyclic in a manner independent of their girth, a new collection of distributions for which belief propagation is guaranteed to perform well is identified using the new projection algorithm framework.
DOI: 10.1109/ICASSP.2009.4960128 NEW CRITERIA FOR ITERATIVE DECODING
, 2010
"... Iterative decoding was not originally introduced as the solution to an optimization problem rendering the analysis of its convergence very difficult. In this paper, we investigate the link between iterative decoding and classical optimization techniques. We first show that iterative decoding can be ..."
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Iterative decoding was not originally introduced as the solution to an optimization problem rendering the analysis of its convergence very difficult. In this paper, we investigate the link between iterative decoding and classical optimization techniques. We first show that iterative decoding can be rephrased as two embedded minimization processes involving the FermiDirac distance. Based on this new formulation, an hybrid proximal point algorithm is first derived with the additional advantage of decreasing a desired criterion. In a second part, an hybrid minimum entropy algorithm is proposed with improved performance compared to the classical iterative decoding. Even if this paper focus on iterative decoding for BICM, the results can be applied to the large class of turbolike decoders. Index Terms — Optimization methods, Iterative methods, Decoding. 1.