## 2015 |

Stinner, Markus; Olmos, Pablo M Finite-Length Performance of Multi-Edge Protograph-Based Spatially Coupled LDPC Codes Inproceedings 2015 IEEE International Symposium on Information Theory (ISIT), pp. 889–893, IEEE, Hong Kong, 2015, ISBN: 978-1-4673-7704-1. Abstract | Links | BibTeX | Tags: binary erasure channel, Block codes, Couplings, Decoding, Error analysis, finite length performance, finite-length performance, graph theory, Iterative decoding, low density parity check codes, multiedge protograph, parity check codes, spatially coupled LDPC codes, spatially-coupled LDPC codes, Steady-state @inproceedings{Stinner2015, title = {Finite-Length Performance of Multi-Edge Protograph-Based Spatially Coupled LDPC Codes}, author = {Markus Stinner and Pablo M Olmos}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7282583}, doi = {10.1109/ISIT.2015.7282583}, isbn = {978-1-4673-7704-1}, year = {2015}, date = {2015-06-01}, booktitle = {2015 IEEE International Symposium on Information Theory (ISIT)}, pages = {889--893}, publisher = {IEEE}, address = {Hong Kong}, abstract = {The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies.}, keywords = {binary erasure channel, Block codes, Couplings, Decoding, Error analysis, finite length performance, finite-length performance, graph theory, Iterative decoding, low density parity check codes, multiedge protograph, parity check codes, spatially coupled LDPC codes, spatially-coupled LDPC codes, Steady-state}, pubstate = {published}, tppubtype = {inproceedings} } The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies. |

Olmos, Pablo M; Urbanke, Rudiger L A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes Journal Article IEEE Transactions on Information Theory, 61 (6), pp. 3164–3184, 2015, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, Journal, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015bb, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Pablo M Olmos and Rudiger L Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7086074}, doi = {10.1109/TIT.2015.2422816}, issn = {0018-9448}, year = {2015}, date = {2015-06-01}, journal = {IEEE Transactions on Information Theory}, volume = {61}, number = {6}, pages = {3164--3184}, abstract = {Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.}, keywords = {asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, Journal, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes}, pubstate = {published}, tppubtype = {article} } Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints. |

Olmos, Pablo M; Urbanke, Rudiger L IEEE Transactions on Information Theory, 61 (6), pp. 3164–3184, 2015, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015c, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Pablo M Olmos and Rudiger L Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7086074}, doi = {10.1109/TIT.2015.2422816}, issn = {0018-9448}, year = {2015}, date = {2015-06-01}, journal = {IEEE Transactions on Information Theory}, volume = {61}, number = {6}, pages = {3164--3184}, abstract = {Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.}, keywords = {asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes}, pubstate = {published}, tppubtype = {article} } |

Olmos, Pablo M; Mitchell, David G M; Costello, Daniel J Analyzing the Finite-Length Performance of Generalized LDPC Codes Inproceedings 2015 IEEE International Symposium on Information Theory (ISIT), pp. 2683–2687, IEEE, Hong Kong, 2015, ISBN: 978-1-4673-7704-1. Abstract | Links | BibTeX | Tags: BEC, binary codes, binary erasure channel, Block codes, Codes on graphs, Decoding, Differential equations, error probability, finite-length generalized LDPC block codes, finite-length performance analysis, generalized LDPC codes, generalized peeling decoder, GLDPC block codes, graph degree distribution, graph theory, Iterative decoding, parity check codes, protographs @inproceedings{Olmos2015b, title = {Analyzing the Finite-Length Performance of Generalized LDPC Codes}, author = {Pablo M Olmos and David G M Mitchell and Daniel J Costello}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7282943}, doi = {10.1109/ISIT.2015.7282943}, isbn = {978-1-4673-7704-1}, year = {2015}, date = {2015-06-01}, booktitle = {2015 IEEE International Symposium on Information Theory (ISIT)}, pages = {2683--2687}, publisher = {IEEE}, address = {Hong Kong}, abstract = {In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved.}, keywords = {BEC, binary codes, binary erasure channel, Block codes, Codes on graphs, Decoding, Differential equations, error probability, finite-length generalized LDPC block codes, finite-length performance analysis, generalized LDPC codes, generalized peeling decoder, GLDPC block codes, graph degree distribution, graph theory, Iterative decoding, parity check codes, protographs}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved. |

Salamanca, Luis; Murillo-Fuentes, Juan José; Olmos, Pablo M; Perez-Cruz, Fernando; Verdu, Sergio Approaching the DT Bound Using Linear Codes in the Short Blocklength Regime Journal Article IEEE Communications Letters, 19 (2), pp. 123–126, 2015, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding @article{Salamanca2014bb, title = {Approaching the DT Bound Using Linear Codes in the Short Blocklength Regime}, author = {Luis Salamanca and Juan José Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6957577}, doi = {10.1109/LCOMM.2014.2371032}, issn = {1089-7798}, year = {2015}, date = {2015-02-01}, journal = {IEEE Communications Letters}, volume = {19}, number = {2}, pages = {123--126}, abstract = {The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits.}, keywords = {binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding}, pubstate = {published}, tppubtype = {article} } The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits. |

## 2014 |

Salamanca, Luis; Murillo-Fuentes, Juan José; Olmos, Pablo M; Perez-Cruz, Fernando; Verdu, Sergio Near DT Bound Achieving Linear Codes in the Short Blocklength Regime Journal Article IEEE Communications Letters, PP (99), pp. 1–1, 2014, ISSN: 1089-7798. @article{Salamanca2014bb, title = {Near DT Bound Achieving Linear Codes in the Short Blocklength Regime}, author = {Luis Salamanca and Juan José Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6957577}, issn = {1089-7798}, year = {2014}, date = {2014-01-01}, journal = {IEEE Communications Letters}, volume = {PP}, number = {99}, pages = {1--1}, abstract = {The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density paritycheck (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 103. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits.}, keywords = {binary erasure channel, Channel Coding, Complexity theory, finite blocklength regime, LDPC codes, Maximum likelihood decoding, ML decoding, parity check codes, random coding}, pubstate = {published}, tppubtype = {article} } |

Olmos, Pablo M; Mitchell, David G M; Truhachev, Dmitry; Costello, Daniel J Improving the Finite-Length Performance of Long SC-LDPC Code Chains by Connecting Consecutive Chains Inproceedings 8th IEEE International Symposium on Turbo Codes &amp; Iterative Information Processing, pp. 72–76, IEEE, Bremen, 2014. Abstract | Links | BibTeX | Tags: Decoding, Error analysis, error probability, Information processing, parity check codes, Turbo codes @inproceedings{Olmos2014, title = {Improving the Finite-Length Performance of Long SC-LDPC Code Chains by Connecting Consecutive Chains}, author = {Pablo M Olmos and David G M Mitchell and Dmitry Truhachev and Daniel J Costello}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6955088}, year = {2014}, date = {2014-01-01}, booktitle = {8th IEEE International Symposium on Turbo Codes &amp; Iterative Information Processing}, pages = {72--76}, publisher = {IEEE}, address = {Bremen}, abstract = {We propose a novel encoding/transmission scheme called continuous chain (CC) transmission that is able to improve the finite-length performance of a system using long spatially coupled low-density parity-check (SC-LDPC) code chains. First, we show that the decoding of SC-LDPC code chains is more reliable for shorter chain lengths, i.e., the scaling between block error rate and gap to threshold is more favorable for shorter chains. This motivates the use of CC transmission in which, instead of transmitting a sequence of independent codewords from a long SC-LDPC chain, we connect multiple chains in a layered format, where encoding, transmission, and decoding are now performed in a continuous fashion. Finally, we show that CC transmission can be implemented with only a small increase in decoding complexity or delay with respect to a system employing a single SC-LDPC code chain for transmission}, keywords = {Decoding, Error analysis, error probability, Information processing, parity check codes, Turbo codes}, pubstate = {published}, tppubtype = {inproceedings} } We propose a novel encoding/transmission scheme called continuous chain (CC) transmission that is able to improve the finite-length performance of a system using long spatially coupled low-density parity-check (SC-LDPC) code chains. First, we show that the decoding of SC-LDPC code chains is more reliable for shorter chain lengths, i.e., the scaling between block error rate and gap to threshold is more favorable for shorter chains. This motivates the use of CC transmission in which, instead of transmitting a sequence of independent codewords from a long SC-LDPC chain, we connect multiple chains in a layered format, where encoding, transmission, and decoding are now performed in a continuous fashion. Finally, we show that CC transmission can be implemented with only a small increase in decoding complexity or delay with respect to a system employing a single SC-LDPC code chain for transmission |

Stinner, Markus; Olmos, Pablo M Analyzing Finite-length Protograph-Based Spatially Coupled LDPC Codes Inproceedings 2014 IEEE International Symposium on Information Theory, pp. 891–895, IEEE, Honolulu, 2014, ISBN: 978-1-4799-5186-4. Abstract | Links | BibTeX | Tags: binary erasure channel, covariance analysis, covariance evolution, Decoding, degree-one check nodes, Error analysis, finite-length protograph, mean evolution, Monte Carlo methods, parity check codes, peeling decoding, protograph-based SC-LDPC codes, spatially coupled low-density parity-check codes, stable decoding phase, Steady-state, Vectors @inproceedings{Stinner2014, title = {Analyzing Finite-length Protograph-Based Spatially Coupled LDPC Codes}, author = {Markus Stinner and Pablo M Olmos}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6874961}, isbn = {978-1-4799-5186-4}, year = {2014}, date = {2014-01-01}, booktitle = {2014 IEEE International Symposium on Information Theory}, pages = {891--895}, publisher = {IEEE}, address = {Honolulu}, abstract = {The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite-length scaling laws for these constructions are given and verified by numerical simulations.}, keywords = {binary erasure channel, covariance analysis, covariance evolution, Decoding, degree-one check nodes, Error analysis, finite-length protograph, mean evolution, Monte Carlo methods, parity check codes, peeling decoding, protograph-based SC-LDPC codes, spatially coupled low-density parity-check codes, stable decoding phase, Steady-state, Vectors}, pubstate = {published}, tppubtype = {inproceedings} } The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite-length scaling laws for these constructions are given and verified by numerical simulations. |

Cespedes, Javier; Olmos, Pablo M; Sanchez-Fernandez, Matilde; Perez-Cruz, Fernando Improved Performance of LDPC-Coded MIMO Systems with EP-based Soft-Decisions Inproceedings 2014 IEEE International Symposium on Information Theory, pp. 1997–2001, IEEE, Honolulu, 2014, ISBN: 978-1-4799-5186-4. Abstract | Links | BibTeX | Tags: Approximation algorithms, Approximation methods, approximation theory, Channel Coding, channel decoder, communication complexity, complexity, Complexity theory, Detectors, encoding scheme, EP soft bit probability, EP-based soft decision, error statistics, expectation propagation, expectation-maximisation algorithm, expectation-propagation algorithm, Gaussian approximation, Gaussian channels, LDPC, LDPC coded MIMO system, Low Complexity receiver, MIMO, MIMO communication, MIMO communication systems, MIMO receiver, modern communication system, multiple input multiple output, parity check codes, per-antenna soft bit probability, posterior marginalization problem, posterior probability computation, QAM constellation, Quadrature amplitude modulation, radio receivers, signaling, spectral analysis, spectral efficiency maximization, symbol detection, telecommunication signalling, Vectors @inproceedings{Cespedes2014b, title = {Improved Performance of LDPC-Coded MIMO Systems with EP-based Soft-Decisions}, author = {Javier Cespedes and Pablo M Olmos and Matilde Sanchez-Fernandez and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6875183}, isbn = {978-1-4799-5186-4}, year = {2014}, date = {2014-01-01}, booktitle = {2014 IEEE International Symposium on Information Theory}, pages = {1997--2001}, publisher = {IEEE}, address = {Honolulu}, abstract = {Modern communications systems use efficient encoding schemes, multiple-input multiple-output (MIMO) and high-order QAM constellations for maximizing spectral efficiency. However, as the dimensions of the system grow, the design of efficient and low-complexity MIMO receivers possesses technical challenges. Symbol detection can no longer rely on conventional approaches for posterior probability computation due to complexity. Marginalization of this posterior to obtain per-antenna soft-bit probabilities to be fed to a channel decoder is computationally challenging when realistic signaling is used. In this work, we propose to use Expectation Propagation (EP) algorithm to provide an accurate low-complexity Gaussian approximation to the posterior, easily solving the posterior marginalization problem. EP soft-bit probabilities are used in an LDPC-coded MIMO system, achieving outstanding performance improvement compared to similar approaches in the literature for low-complexity LDPC MIMO decoding.}, keywords = {Approximation algorithms, Approximation methods, approximation theory, Channel Coding, channel decoder, communication complexity, complexity, Complexity theory, Detectors, encoding scheme, EP soft bit probability, EP-based soft decision, error statistics, expectation propagation, expectation-maximisation algorithm, expectation-propagation algorithm, Gaussian approximation, Gaussian channels, LDPC, LDPC coded MIMO system, Low Complexity receiver, MIMO, MIMO communication, MIMO communication systems, MIMO receiver, modern communication system, multiple input multiple output, parity check codes, per-antenna soft bit probability, posterior marginalization problem, posterior probability computation, QAM constellation, Quadrature amplitude modulation, radio receivers, signaling, spectral analysis, spectral efficiency maximization, symbol detection, telecommunication signalling, Vectors}, pubstate = {published}, tppubtype = {inproceedings} } Modern communications systems use efficient encoding schemes, multiple-input multiple-output (MIMO) and high-order QAM constellations for maximizing spectral efficiency. However, as the dimensions of the system grow, the design of efficient and low-complexity MIMO receivers possesses technical challenges. Symbol detection can no longer rely on conventional approaches for posterior probability computation due to complexity. Marginalization of this posterior to obtain per-antenna soft-bit probabilities to be fed to a channel decoder is computationally challenging when realistic signaling is used. In this work, we propose to use Expectation Propagation (EP) algorithm to provide an accurate low-complexity Gaussian approximation to the posterior, easily solving the posterior marginalization problem. EP soft-bit probabilities are used in an LDPC-coded MIMO system, achieving outstanding performance improvement compared to similar approaches in the literature for low-complexity LDPC MIMO decoding. |

## 2013 |

Salamanca, Luis; Murillo-Fuentes, Juan Jose; Olmos, Pablo M; Perez-Cruz, Fernando Improving the BP Estimate over the AWGN Channel Using Tree-Structured Expectation Propagation Inproceedings 2013 IEEE International Symposium on Information Theory, pp. 2990–2994, IEEE, Istanbul, 2013, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: Approximation algorithms, Approximation methods, AWGN channels, BEC, belief propagation decoding, BI-AWGN channel, binary additive white Gaussian noise channel, binary erasure channel, BP estimation, Channel Coding, Complexity theory, error rate reduction, error statistics, Expectation, finite-length codes, Iterative decoding, LDPC codes, LDPC decoding, low-density parity-check decoding, Maximum likelihood decoding, parity check codes, posterior distribution, Propagation, TEP algorithm, tree-structured expectation propagation algorithm, trees (mathematics) @inproceedings{Salamanca2013, title = {Improving the BP Estimate over the AWGN Channel Using Tree-Structured Expectation Propagation}, author = {Luis Salamanca and Juan Jose Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6620774}, issn = {2157-8095}, year = {2013}, date = {2013-01-01}, booktitle = {2013 IEEE International Symposium on Information Theory}, pages = {2990--2994}, publisher = {IEEE}, address = {Istanbul}, abstract = {In this paper, we propose the tree-structured expectation propagation (TEP) algorithm for low-density parity-check (LDPC) decoding over the binary additive white Gaussian noise (BI-AWGN) channel. By approximating the posterior distribution by a tree-structure factorization, the TEP has been proven to improve belief propagation (BP) decoding over the binary erasure channel (BEC). We show for the AWGN channel how the TEP decoder is also able to capture additional information disregarded by the BP solution, which leads to a noticeable reduction of the error rate for finite-length codes. We show that for the range of codes of interest, the TEP gain is obtained with a slight increase in complexity over that of the BP algorithm. An efficient way of constructing the tree-like structure is also described.}, keywords = {Approximation algorithms, Approximation methods, AWGN channels, BEC, belief propagation decoding, BI-AWGN channel, binary additive white Gaussian noise channel, binary erasure channel, BP estimation, Channel Coding, Complexity theory, error rate reduction, error statistics, Expectation, finite-length codes, Iterative decoding, LDPC codes, LDPC decoding, low-density parity-check decoding, Maximum likelihood decoding, parity check codes, posterior distribution, Propagation, TEP algorithm, tree-structured expectation propagation algorithm, trees (mathematics)}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we propose the tree-structured expectation propagation (TEP) algorithm for low-density parity-check (LDPC) decoding over the binary additive white Gaussian noise (BI-AWGN) channel. By approximating the posterior distribution by a tree-structure factorization, the TEP has been proven to improve belief propagation (BP) decoding over the binary erasure channel (BEC). We show for the AWGN channel how the TEP decoder is also able to capture additional information disregarded by the BP solution, which leads to a noticeable reduction of the error rate for finite-length codes. We show that for the range of codes of interest, the TEP gain is obtained with a slight increase in complexity over that of the BP algorithm. An efficient way of constructing the tree-like structure is also described. |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Tree-Structure Expectation Propagation for LDPC Decoding Over the BEC Journal Article IEEE Transactions on Information Theory, 59 (6), pp. 3354–3377, 2013, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: Algorithm design and analysis, Approximation algorithms, Approximation methods, BEC, belief propagation, Belief-propagation (BP), binary erasure channel, Complexity theory, decode low-density parity-check codes, Decoding, discrete memoryless channels, expectation propagation, finite-length analysis, LDPC codes, LDPC decoding, parity check codes, peeling-type algorithm, Probability density function, random graph evolution, Tanner graph, tree-structure expectation propagation @article{Olmos2013b, title = {Tree-Structure Expectation Propagation for LDPC Decoding Over the BEC}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6451276}, issn = {0018-9448}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {59}, number = {6}, pages = {3354--3377}, abstract = {We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). Expectation propagation generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regimes over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization.}, keywords = {Algorithm design and analysis, Approximation algorithms, Approximation methods, BEC, belief propagation, Belief-propagation (BP), binary erasure channel, Complexity theory, decode low-density parity-check codes, Decoding, discrete memoryless channels, expectation propagation, finite-length analysis, LDPC codes, LDPC decoding, parity check codes, peeling-type algorithm, Probability density function, random graph evolution, Tanner graph, tree-structure expectation propagation}, pubstate = {published}, tppubtype = {article} } We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). Expectation propagation generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regimes over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization. |

Salamanca, Luis; Olmos, Pablo M; Perez-Cruz, Fernando; Murillo-Fuentes, Juan Jose Tree-Structured Expectation Propagation for LDPC Decoding over BMS Channels Journal Article IEEE Transactions on Communications, 61 (10), pp. 4086–4095, 2013, ISSN: 0090-6778. Abstract | Links | BibTeX | Tags: Approximation algorithms, Approximation methods, BEC, belief propagation, binary erasure channel, binary memoryless symmetric channels, BMS channels, Channel Coding, Complexity theory, convolutional codes, convolutional low-density parity-check codes, Decoding, decoding block, expectation propagation, finite-length codes, LDPC decoding, message-passing algorithm, parity check codes, Probability density function, sparse linear codes, TEP algorithm, tree-structured expectation propagation, trees (mathematics), Vegetation @article{Salamanca2013a, title = {Tree-Structured Expectation Propagation for LDPC Decoding over BMS Channels}, author = {Luis Salamanca and Pablo M Olmos and Fernando Perez-Cruz and Juan Jose Murillo-Fuentes}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6587624}, issn = {0090-6778}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Communications}, volume = {61}, number = {10}, pages = {4086--4095}, abstract = {In this paper, we put forward the tree-structured expectation propagation (TEP) algorithm for decoding block and convolutional low-density parity-check codes over any binary channel. We have already shown that TEP improves belief propagation (BP) over the binary erasure channel (BEC) by imposing marginal constraints over a set of pairs of variables that form a tree or a forest. The TEP decoder is a message-passing algorithm that sequentially builds a tree/forest of erased variables to capture additional information disregarded by the standard BP decoder, which leads to a noticeable reduction of the error rate for finite-length codes. In this paper, we show how the TEP can be extended to any channel, specifically to binary memoryless symmetric (BMS) channels. We particularly focus on how the TEP algorithm can be adapted for any channel model and, more importantly, how to choose the tree/forest to keep the gains observed for block and convolutional LDPC codes over the BEC.}, keywords = {Approximation algorithms, Approximation methods, BEC, belief propagation, binary erasure channel, binary memoryless symmetric channels, BMS channels, Channel Coding, Complexity theory, convolutional codes, convolutional low-density parity-check codes, Decoding, decoding block, expectation propagation, finite-length codes, LDPC decoding, message-passing algorithm, parity check codes, Probability density function, sparse linear codes, TEP algorithm, tree-structured expectation propagation, trees (mathematics), Vegetation}, pubstate = {published}, tppubtype = {article} } In this paper, we put forward the tree-structured expectation propagation (TEP) algorithm for decoding block and convolutional low-density parity-check codes over any binary channel. We have already shown that TEP improves belief propagation (BP) over the binary erasure channel (BEC) by imposing marginal constraints over a set of pairs of variables that form a tree or a forest. The TEP decoder is a message-passing algorithm that sequentially builds a tree/forest of erased variables to capture additional information disregarded by the standard BP decoder, which leads to a noticeable reduction of the error rate for finite-length codes. In this paper, we show how the TEP can be extended to any channel, specifically to binary memoryless symmetric (BMS) channels. We particularly focus on how the TEP algorithm can be adapted for any channel model and, more importantly, how to choose the tree/forest to keep the gains observed for block and convolutional LDPC codes over the BEC. |

Bravo-Santos, Ángel M Polar Codes for Gaussian Degraded Relay Channels Journal Article IEEE Communications Letters, 17 (2), pp. 365–368, 2013, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: channel capacity, Channel Coding, Decoding, Encoding, Gaussian channels, Gaussian degraded relay channel, Gaussian noise, Gaussian-degraded relay channels, log-likelihood expression, Markov coding, Noise, parity check codes, polar code detector, polar codes, relay-destination link, Relays, Vectors @article{Bravo-Santos2013, title = {Polar Codes for Gaussian Degraded Relay Channels}, author = {Ángel M Bravo-Santos}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6412681}, issn = {1089-7798}, year = {2013}, date = {2013-01-01}, journal = {IEEE Communications Letters}, volume = {17}, number = {2}, pages = {365--368}, publisher = {IEEE}, abstract = {In this paper we apply polar codes for the Gaussian degraded relay channel. We study the conditions to be satisfied by the codes and provide an efficient method for constructing them. The relay-destination link is special because the noise is the sum of two components: the Gaussian noise and the signals from the source. We study this link and provide the log-likelihood expression to be used by the polar code detector. We perform simulations of the channel and the results show that polar codes of high rate and large codeword length are closer to the theoretical limit than other good codes.}, keywords = {channel capacity, Channel Coding, Decoding, Encoding, Gaussian channels, Gaussian degraded relay channel, Gaussian noise, Gaussian-degraded relay channels, log-likelihood expression, Markov coding, Noise, parity check codes, polar code detector, polar codes, relay-destination link, Relays, Vectors}, pubstate = {published}, tppubtype = {article} } In this paper we apply polar codes for the Gaussian degraded relay channel. We study the conditions to be satisfied by the codes and provide an efficient method for constructing them. The relay-destination link is special because the noise is the sum of two components: the Gaussian noise and the signals from the source. We study this link and provide the log-likelihood expression to be used by the polar code detector. We perform simulations of the channel and the results show that polar codes of high rate and large codeword length are closer to the theoretical limit than other good codes. |

Salamanca, Luis; Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Tree Expectation Propagation for ML Decoding of LDPC Codes over the BEC Journal Article IEEE Transactions on Communications, 61 (2), pp. 465–473, 2013, ISSN: 0090-6778. Abstract | Links | BibTeX | Tags: approximate inference, Approximation algorithms, Approximation methods, BEC, binary codes, binary erasure channel, code graph, Complexity theory, equivalent complexity, Gaussian elimination method, Gaussian processes, generalized tree-structured expectation propagatio, graphical message-passing procedure, graphical models, LDPC codes, Maximum likelihood decoding, maximum likelihood solution, ML decoding, parity check codes, peeling decoder, tree expectation propagation, tree graph, Tree graphs, tree-structured expectation propagation, tree-structured expectation propagation decoder, trees (mathematics) @article{Salamanca2013b, title = {Tree Expectation Propagation for ML Decoding of LDPC Codes over the BEC}, author = {Luis Salamanca and Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6384612}, issn = {0090-6778}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Communications}, volume = {61}, number = {2}, pages = {465--473}, abstract = {We propose a decoding algorithm for LDPC codes that achieves the maximum likelihood (ML) solution over the binary erasure channel (BEC). In this channel, the tree-structured expectation propagation (TEP) decoder improves the peeling decoder (PD) by processing check nodes of degree one and two. However, it does not achieve the ML solution, as the tree structure of the TEP allows only for approximate inference. In this paper, we provide the procedure to construct the structure needed for exact inference. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph. The GTEP decoder upon completion either provides the unique ML solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ML solution. We also explain the algorithm as a Gaussian elimination method, relating the GTEP to other ML solutions. Compared to previous approaches, it presents an equivalent complexity, it exhibits a simpler graphical message-passing procedure and, most interesting, the algorithm can be generalized to other channels.}, keywords = {approximate inference, Approximation algorithms, Approximation methods, BEC, binary codes, binary erasure channel, code graph, Complexity theory, equivalent complexity, Gaussian elimination method, Gaussian processes, generalized tree-structured expectation propagatio, graphical message-passing procedure, graphical models, LDPC codes, Maximum likelihood decoding, maximum likelihood solution, ML decoding, parity check codes, peeling decoder, tree expectation propagation, tree graph, Tree graphs, tree-structured expectation propagation, tree-structured expectation propagation decoder, trees (mathematics)}, pubstate = {published}, tppubtype = {article} } We propose a decoding algorithm for LDPC codes that achieves the maximum likelihood (ML) solution over the binary erasure channel (BEC). In this channel, the tree-structured expectation propagation (TEP) decoder improves the peeling decoder (PD) by processing check nodes of degree one and two. However, it does not achieve the ML solution, as the tree structure of the TEP allows only for approximate inference. In this paper, we provide the procedure to construct the structure needed for exact inference. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph. The GTEP decoder upon completion either provides the unique ML solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ML solution. We also explain the algorithm as a Gaussian elimination method, relating the GTEP to other ML solutions. Compared to previous approaches, it presents an equivalent complexity, it exhibits a simpler graphical message-passing procedure and, most interesting, the algorithm can be generalized to other channels. |

## 2012 |

Salamanca, Luis; Murillo-Fuentes, Juan Jose; Olmos, Pablo M; Perez-Cruz, Fernando Tree-Structured Expectation Propagation for LDPC Decoding over the AWGN Channel Inproceedings 2012 IEEE International Workshop on Machine Learning for Signal Processing, pp. 1–6, IEEE, Santander, 2012, ISSN: 1551-2541. Abstract | Links | BibTeX | Tags: additive white Gaussian noise channel, Approximation algorithms, Approximation methods, approximation theory, AWGN channel, AWGN channels, belief propagation solution, Bit error rate, Decoding, error floor reduction, finite-length regime, Gain, Joints, LDPC decoding, low-density parity-check decoding, pairwise marginal constraint, parity check codes, TEP decoder, tree-like approximation, tree-structured expectation propagation, trees (mathematics) @inproceedings{Salamanca2012, title = {Tree-Structured Expectation Propagation for LDPC Decoding over the AWGN Channel}, author = {Luis Salamanca and Juan Jose Murillo-Fuentes and Pablo M Olmos and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6349716}, issn = {1551-2541}, year = {2012}, date = {2012-01-01}, booktitle = {2012 IEEE International Workshop on Machine Learning for Signal Processing}, pages = {1--6}, publisher = {IEEE}, address = {Santander}, abstract = {In this paper, we propose the tree-structured expectation propagation (TEP) algorithm for low-density parity-check (LDPC) decoding over the additive white Gaussian noise (AWGN) channel. By imposing a tree-like approximation over the graphical model of the code, this algorithm introduces pairwise marginal constraints over pairs of variables, which provide joint information of the variables related. Thanks to this, the proposed TEP decoder improves the performance of the standard belief propagation (BP) solution. An efficient way of constructing the tree-like structure is also described. The simulation results illustrate the TEP decoder gain in the finite-length regime, compared to the standard BP solution. For code lengths shorter than n = 512, the gain in the waterfall region achieves up to 0.25 dB. We also notice a remarkable reduction of the error floor.}, keywords = {additive white Gaussian noise channel, Approximation algorithms, Approximation methods, approximation theory, AWGN channel, AWGN channels, belief propagation solution, Bit error rate, Decoding, error floor reduction, finite-length regime, Gain, Joints, LDPC decoding, low-density parity-check decoding, pairwise marginal constraint, parity check codes, TEP decoder, tree-like approximation, tree-structured expectation propagation, trees (mathematics)}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we propose the tree-structured expectation propagation (TEP) algorithm for low-density parity-check (LDPC) decoding over the additive white Gaussian noise (AWGN) channel. By imposing a tree-like approximation over the graphical model of the code, this algorithm introduces pairwise marginal constraints over pairs of variables, which provide joint information of the variables related. Thanks to this, the proposed TEP decoder improves the performance of the standard belief propagation (BP) solution. An efficient way of constructing the tree-like structure is also described. The simulation results illustrate the TEP decoder gain in the finite-length regime, compared to the standard BP solution. For code lengths shorter than n = 512, the gain in the waterfall region achieves up to 0.25 dB. We also notice a remarkable reduction of the error floor. |

Olmos, Pablo M; Salamanca, Luis; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando On the Design of LDPC-Convolutional Ensembles Using the TEP Decoder Journal Article IEEE Communications Letters, 16 (5), pp. 726–729, 2012, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: belief propagation decoding, binary erasure channel, channel capacity, Complexity theory, convolutional codes, convolutional LDPC codes, Decoding, design, Error analysis, finite-length analysis, Iterative decoding, LDPC-convolutional ensemble design, LDPCC code decoding, low-density parity-check convolutional code, parity check codes, tree-expectation propagation decoder, tree-structured expectation propagation, window-sliding scheme @article{Olmos2012b, title = {On the Design of LDPC-Convolutional Ensembles Using the TEP Decoder}, author = {Pablo M Olmos and Luis Salamanca and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6168872}, issn = {1089-7798}, year = {2012}, date = {2012-01-01}, journal = {IEEE Communications Letters}, volume = {16}, number = {5}, pages = {726--729}, abstract = {Low-density parity-check convolutional (LDPCC) codes asymptotically achieve channel capacity under belief propagation (BP) decoding. In this paper, we decode LDPCC codes using the Tree-Expectation Propagation (TEP) decoder, recently proposed as an alternative decoding method to the BP algorithm for the binary erasure channel (BEC). We show that, for LDPCC codes, the TEP decoder improves the BP solution with a comparable complexity or, alternatively, it allows using shorter codes to achieve similar error rates. We also propose a window-sliding scheme for the TEP decoder to reduce the decoding latency.}, keywords = {belief propagation decoding, binary erasure channel, channel capacity, Complexity theory, convolutional codes, convolutional LDPC codes, Decoding, design, Error analysis, finite-length analysis, Iterative decoding, LDPC-convolutional ensemble design, LDPCC code decoding, low-density parity-check convolutional code, parity check codes, tree-expectation propagation decoder, tree-structured expectation propagation, window-sliding scheme}, pubstate = {published}, tppubtype = {article} } Low-density parity-check convolutional (LDPCC) codes asymptotically achieve channel capacity under belief propagation (BP) decoding. In this paper, we decode LDPCC codes using the Tree-Expectation Propagation (TEP) decoder, recently proposed as an alternative decoding method to the BP algorithm for the binary erasure channel (BEC). We show that, for LDPCC codes, the TEP decoder improves the BP solution with a comparable complexity or, alternatively, it allows using shorter codes to achieve similar error rates. We also propose a window-sliding scheme for the TEP decoder to reduce the decoding latency. |

Salamanca, Luis; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Bayesian Equalization for LDPC Channel Decoding Journal Article IEEE Transactions on Signal Processing, 60 (5), pp. 2672–2676, 2012, ISSN: 1053-587X. Abstract | Links | BibTeX | Tags: Approximation methods, Bayes methods, Bayesian equalization, Bayesian estimation problem, Bayesian inference, Bayesian methods, BCJR (Bahl–Cocke–Jelinek–Raviv) algorithm, BCJR algorithm, Channel Coding, channel decoding, channel equalization, channel equalization problem, Channel estimation, channel state information, CSI, Decoding, equalisers, Equalizers, expectation propagation, expectation propagation algorithm, fading channels, graphical model representation, intersymbol interference, Kullback-Leibler divergence, LDPC, LDPC coding, low-density parity-check decoder, Modulation, parity check codes, symbol posterior estimates, Training @article{Salamanca2012b, title = {Bayesian Equalization for LDPC Channel Decoding}, author = {Luis Salamanca and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6129544}, issn = {1053-587X}, year = {2012}, date = {2012-01-01}, journal = {IEEE Transactions on Signal Processing}, volume = {60}, number = {5}, pages = {2672--2676}, abstract = {We describe the channel equalization problem, and its prior estimate of the channel state information (CSI), as a joint Bayesian estimation problem to improve each symbol posterior estimates at the input of the channel decoder. Our approach takes into consideration not only the uncertainty due to the noise in the channel, but also the uncertainty in the CSI estimate. However, this solution cannot be computed in linear time, because it depends on all the transmitted symbols. Hence, we also put forward an approximation for each symbol's posterior, using the expectation propagation algorithm, which is optimal from the Kullback-Leibler divergence viewpoint and yields an equalization with a complexity identical to the BCJR algorithm. We also use a graphical model representation of the full posterior, in which the proposed approximation can be readily understood. The proposed posterior estimates are more accurate than those computed using the ML estimate for the CSI. In order to illustrate this point, we measure the error rate at the output of a low-density parity-check decoder, which needs the exact posterior for each symbol to detect the incoming word and it is sensitive to a mismatch in those posterior estimates. For example, for QPSK modulation and a channel with three taps, we can expect gains over 0.5 dB with same computational complexity as the ML receiver.}, keywords = {Approximation methods, Bayes methods, Bayesian equalization, Bayesian estimation problem, Bayesian inference, Bayesian methods, BCJR (Bahl–Cocke–Jelinek–Raviv) algorithm, BCJR algorithm, Channel Coding, channel decoding, channel equalization, channel equalization problem, Channel estimation, channel state information, CSI, Decoding, equalisers, Equalizers, expectation propagation, expectation propagation algorithm, fading channels, graphical model representation, intersymbol interference, Kullback-Leibler divergence, LDPC, LDPC coding, low-density parity-check decoder, Modulation, parity check codes, symbol posterior estimates, Training}, pubstate = {published}, tppubtype = {article} } We describe the channel equalization problem, and its prior estimate of the channel state information (CSI), as a joint Bayesian estimation problem to improve each symbol posterior estimates at the input of the channel decoder. Our approach takes into consideration not only the uncertainty due to the noise in the channel, but also the uncertainty in the CSI estimate. However, this solution cannot be computed in linear time, because it depends on all the transmitted symbols. Hence, we also put forward an approximation for each symbol's posterior, using the expectation propagation algorithm, which is optimal from the Kullback-Leibler divergence viewpoint and yields an equalization with a complexity identical to the BCJR algorithm. We also use a graphical model representation of the full posterior, in which the proposed approximation can be readily understood. The proposed posterior estimates are more accurate than those computed using the ML estimate for the CSI. In order to illustrate this point, we measure the error rate at the output of a low-density parity-check decoder, which needs the exact posterior for each symbol to detect the incoming word and it is sensitive to a mismatch in those posterior estimates. For example, for QPSK modulation and a channel with three taps, we can expect gains over 0.5 dB with same computational complexity as the ML receiver. |

Olmos, Pablo M; Perez-Cruz, Fernando; Salamanca, Luis; Murillo-Fuentes, Juan Jose Finite-Length Analysis of the TEP Decoder for LDPC Ensembles over the BEC Inproceedings 2012 IEEE International Symposium on Information Theory Proceedings, pp. 2346–2350, IEEE, Cambridge, MA, 2012, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: Approximation methods, BEC, binary codes, binary erasure channel, Decoding, Error analysis, error probability, finite-length analysis, LDPC ensembles, low-density parity check ensembles, parity check codes, TEP decoder, Trajectory, tree-expectation propagation algorithm, waterfall region @inproceedings{Olmos2012a, title = {Finite-Length Analysis of the TEP Decoder for LDPC Ensembles over the BEC}, author = {Pablo M Olmos and Fernando Perez-Cruz and Luis Salamanca and Juan Jose Murillo-Fuentes}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6283932}, issn = {2157-8095}, year = {2012}, date = {2012-01-01}, booktitle = {2012 IEEE International Symposium on Information Theory Proceedings}, pages = {2346--2350}, publisher = {IEEE}, address = {Cambridge, MA}, abstract = {In this work, we analyze the finite-length performance of low-density parity check (LDPC) ensembles decoded over the binary erasure channel (BEC) using the tree-expectation propagation (TEP) algorithm. In a previous paper, we showed that the TEP improves the BP performance for decoding regular and irregular short LDPC codes, but the perspective was mainly empirical. In this work, given the degree-distribution of an LDPC ensemble, we explain and predict the range of code lengths for which the TEP improves the BP solution. In addition, for LDPC ensembles that present a single critical point, we propose a scaling law to accurately predict the performance in the waterfall region. These results are of critical importance to design practical LDPC codes for the TEP decoder.}, keywords = {Approximation methods, BEC, binary codes, binary erasure channel, Decoding, Error analysis, error probability, finite-length analysis, LDPC ensembles, low-density parity check ensembles, parity check codes, TEP decoder, Trajectory, tree-expectation propagation algorithm, waterfall region}, pubstate = {published}, tppubtype = {inproceedings} } In this work, we analyze the finite-length performance of low-density parity check (LDPC) ensembles decoded over the binary erasure channel (BEC) using the tree-expectation propagation (TEP) algorithm. In a previous paper, we showed that the TEP improves the BP performance for decoding regular and irregular short LDPC codes, but the perspective was mainly empirical. In this work, given the degree-distribution of an LDPC ensemble, we explain and predict the range of code lengths for which the TEP improves the BP solution. In addition, for LDPC ensembles that present a single critical point, we propose a scaling law to accurately predict the performance in the waterfall region. These results are of critical importance to design practical LDPC codes for the TEP decoder. |

Olmos, Pablo M; Perez-Cruz, Fernando; Salamanca, Luis; Murillo-Fuentes, Juan Jose Finite-Length Performance of Spatially-Coupled LDPC Codes under TEP Decoding Inproceedings 2012 IEEE Information Theory Workshop, pp. 1–6, IEEE, Lausanne, 2012, ISBN: 978-1-4673-0223-4. Links | BibTeX | Tags: asymptotic limit, belief propagation decoding, Complexity theory, convolutional codes, convolutional LDPC codes, Decoding, decoding latency, decoding threshold, erasure channel, Error analysis, error rates, finite-length analysis, finite-length performance, maximum a posteriori threshold, maximum likelihood estimation, parity check codes, regular sparse codes, spatially-coupled LDPC codes, TEP decoding, tree-structured expectation propagation, underlying regular code, very large code length, window-sliding scheme @inproceedings{Olmos2012, title = {Finite-Length Performance of Spatially-Coupled LDPC Codes under TEP Decoding}, author = {Pablo M Olmos and Fernando Perez-Cruz and Luis Salamanca and Juan Jose Murillo-Fuentes}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6404722}, isbn = {978-1-4673-0223-4}, year = {2012}, date = {2012-01-01}, booktitle = {2012 IEEE Information Theory Workshop}, pages = {1--6}, publisher = {IEEE}, address = {Lausanne}, keywords = {asymptotic limit, belief propagation decoding, Complexity theory, convolutional codes, convolutional LDPC codes, Decoding, decoding latency, decoding threshold, erasure channel, Error analysis, error rates, finite-length analysis, finite-length performance, maximum a posteriori threshold, maximum likelihood estimation, parity check codes, regular sparse codes, spatially-coupled LDPC codes, TEP decoding, tree-structured expectation propagation, underlying regular code, very large code length, window-sliding scheme}, pubstate = {published}, tppubtype = {inproceedings} } |

## 2011 |

Olmos, Pablo M; Urbanke, Rudiger Scaling Behavior of Convolutional LDPC Ensembles over the BEC Inproceedings 2011 IEEE International Symposium on Information Theory Proceedings, pp. 1816–1820, IEEE, Saint Petersburg, 2011, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: BEC, binary codes, binary erasure channel, Bit error rate, convolutional codes, convolutional LDPC ensembles, coupled sparse graph codes, Couplings, Decoding, error probability, Iterative decoding, parity check codes, scaling behavior @inproceedings{Olmos2011, title = {Scaling Behavior of Convolutional LDPC Ensembles over the BEC}, author = {Pablo M Olmos and Rudiger Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6033863}, issn = {2157-8095}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE International Symposium on Information Theory Proceedings}, pages = {1816--1820}, publisher = {IEEE}, address = {Saint Petersburg}, abstract = {We study the scaling behavior of coupled sparse graph codes over the binary erasure channel. In particular, let 2L+1 be the length of the coupled chain, let M be the number of variables in each of the 2L+1 local copies, let ℓ be the number of iterations, let Pb denote the bit error probability, and let ∈ denote the channel parameter. We are interested in how these quantities scale when we let the blocklength (2L + 1)M tend to infinity. Based on empirical evidence we show that the threshold saturation phenomenon is rather stable with respect to the scaling of the various parameters and we formulate some general rules of thumb which can serve as a guide for the design of coding systems based on coupled graphs.}, keywords = {BEC, binary codes, binary erasure channel, Bit error rate, convolutional codes, convolutional LDPC ensembles, coupled sparse graph codes, Couplings, Decoding, error probability, Iterative decoding, parity check codes, scaling behavior}, pubstate = {published}, tppubtype = {inproceedings} } We study the scaling behavior of coupled sparse graph codes over the binary erasure channel. In particular, let 2L+1 be the length of the coupled chain, let M be the number of variables in each of the 2L+1 local copies, let ℓ be the number of iterations, let Pb denote the bit error probability, and let ∈ denote the channel parameter. We are interested in how these quantities scale when we let the blocklength (2L + 1)M tend to infinity. Based on empirical evidence we show that the threshold saturation phenomenon is rather stable with respect to the scaling of the various parameters and we formulate some general rules of thumb which can serve as a guide for the design of coding systems based on coupled graphs. |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Capacity Achieving LDPC Ensembles for the TEP Decoder in Erasure Channels Inproceedings 2011 IEEE International Symposium on Information Theory Proceedings, pp. 2398–2402, IEEE, St. Petersburg, 2011, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: BP threshold, Complexity theory, Decoding, Differential equations, erasure channels, fixed-rate code, Iterative decoding, LDPC, low-density parity-check codes, MAP capacity, MAP threshold, optimisation, Optimization, optimization problem, parity check codes, TEP decoder, tree-expectation propagation decoder @inproceedings{Olmos2011b, title = {Capacity Achieving LDPC Ensembles for the TEP Decoder in Erasure Channels}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6033993}, issn = {2157-8095}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE International Symposium on Information Theory Proceedings}, pages = {2398--2402}, publisher = {IEEE}, address = {St. Petersburg}, abstract = {In this work we address the design of degree distributions (DD) of low-density parity-check (LDPC) codes for the tree-expectation propagation (TEP) decoder. The optimization problem to find distributions to maximize the TEP decoding threshold for a fixed-rate code can not be analytically solved. We derive a simplified optimization problem that can be easily solved since it is based in the analytic expressions of the peeling decoder. Two kinds of solutions are obtained from this problem: we either design LDPC ensembles for which the BP threshold equals the MAP threshold or we get LDPC ensembles for which the TEP threshold outperforms the BP threshold, even achieving the MAP capacity in some cases. Hence, we proved that there exist ensembles for which the MAP solution can be obtained with linear complexity even though the BP threshold does not achieve the MAP threshold.}, keywords = {BP threshold, Complexity theory, Decoding, Differential equations, erasure channels, fixed-rate code, Iterative decoding, LDPC, low-density parity-check codes, MAP capacity, MAP threshold, optimisation, Optimization, optimization problem, parity check codes, TEP decoder, tree-expectation propagation decoder}, pubstate = {published}, tppubtype = {inproceedings} } In this work we address the design of degree distributions (DD) of low-density parity-check (LDPC) codes for the tree-expectation propagation (TEP) decoder. The optimization problem to find distributions to maximize the TEP decoding threshold for a fixed-rate code can not be analytically solved. We derive a simplified optimization problem that can be easily solved since it is based in the analytic expressions of the peeling decoder. Two kinds of solutions are obtained from this problem: we either design LDPC ensembles for which the BP threshold equals the MAP threshold or we get LDPC ensembles for which the TEP threshold outperforms the BP threshold, even achieving the MAP capacity in some cases. Hence, we proved that there exist ensembles for which the MAP solution can be obtained with linear complexity even though the BP threshold does not achieve the MAP threshold. |

Salamanca, Luis; Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando MAP Decoding for LDPC Codes over the Binary Erasure Channel Inproceedings 2011 IEEE Information Theory Workshop, pp. 145–149, IEEE, Paraty, 2011, ISBN: 978-1-4577-0437-6. Abstract | Links | BibTeX | Tags: binary erasure channel, Channel Coding, computational complexity, Decoding, generalized peeling decoder, generalized tree-structured expectation propagatio, graphical models, Iterative decoding, LDPC codes, MAP decoding, MAP decoding algorithm, Maximum likelihood decoding, parity check codes, TEP decoder, tree graph theory, Tree graphs, tree-structured expectation propagation, trees (mathematics) @inproceedings{Salamanca2011a, title = {MAP Decoding for LDPC Codes over the Binary Erasure Channel}, author = {Luis Salamanca and Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6089364}, isbn = {978-1-4577-0437-6}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE Information Theory Workshop}, pages = {145--149}, publisher = {IEEE}, address = {Paraty}, abstract = {In this paper, we propose a decoding algorithm for LDPC codes that achieves the MAP solution over the BEC. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), extends the idea of our previous work, the TEP decoder. The GTEP modifies the graph by eliminating a check node of any degree and merging this information with the remaining graph. The GTEP decoder upon completion either provides the unique MAP solution or a tree graph in which the number of parent nodes indicates the multiplicity of the MAP solution. This algorithm can be easily described for the BEC, and it can be cast as a generalized peeling decoder. The GTEP naturally optimizes the complexity of the decoder, by looking for checks nodes of minimum degree to be eliminated first.}, keywords = {binary erasure channel, Channel Coding, computational complexity, Decoding, generalized peeling decoder, generalized tree-structured expectation propagatio, graphical models, Iterative decoding, LDPC codes, MAP decoding, MAP decoding algorithm, Maximum likelihood decoding, parity check codes, TEP decoder, tree graph theory, Tree graphs, tree-structured expectation propagation, trees (mathematics)}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we propose a decoding algorithm for LDPC codes that achieves the MAP solution over the BEC. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), extends the idea of our previous work, the TEP decoder. The GTEP modifies the graph by eliminating a check node of any degree and merging this information with the remaining graph. The GTEP decoder upon completion either provides the unique MAP solution or a tree graph in which the number of parent nodes indicates the multiplicity of the MAP solution. This algorithm can be easily described for the BEC, and it can be cast as a generalized peeling decoder. The GTEP naturally optimizes the complexity of the decoder, by looking for checks nodes of minimum degree to be eliminated first. |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Tree-Structured Expectation Propagation for Decoding Finite-Length LDPC Codes Journal Article IEEE Communications Letters, 15 (2), pp. 235–237, 2011, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: belief propagation decoder, BP algorithm, BP decoder, code graph, communication complexity, computational complexity, Decoding, finite-length analysis, finite-length low-density parity-check code, LDPC code, LDPC decoding, parity check codes, radiowave propagation, stopping set, TEP algorithm, TEP decoder, tree-structured expectation propagation @article{Olmos2011c, title = {Tree-Structured Expectation Propagation for Decoding Finite-Length LDPC Codes}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5682215}, issn = {1089-7798}, year = {2011}, date = {2011-01-01}, journal = {IEEE Communications Letters}, volume = {15}, number = {2}, pages = {235--237}, abstract = {In this paper, we propose Tree-structured Expectation Propagation (TEP) algorithm to decode finite-length Low-Density Parity-Check (LDPC) codes. The TEP decoder is able to continue decoding once the standard Belief Propagation (BP) decoder fails, presenting the same computational complexity as the BP decoder. The BP algorithm is dominated by the presence of stopping sets (SSs) in the code graph. We show that the TEP decoder, without previous knowledge of the graph, naturally avoids some fairly common SSs. This results in a significant improvement in the system performance.}, keywords = {belief propagation decoder, BP algorithm, BP decoder, code graph, communication complexity, computational complexity, Decoding, finite-length analysis, finite-length low-density parity-check code, LDPC code, LDPC decoding, parity check codes, radiowave propagation, stopping set, TEP algorithm, TEP decoder, tree-structured expectation propagation}, pubstate = {published}, tppubtype = {article} } In this paper, we propose Tree-structured Expectation Propagation (TEP) algorithm to decode finite-length Low-Density Parity-Check (LDPC) codes. The TEP decoder is able to continue decoding once the standard Belief Propagation (BP) decoder fails, presenting the same computational complexity as the BP decoder. The BP algorithm is dominated by the presence of stopping sets (SSs) in the code graph. We show that the TEP decoder, without previous knowledge of the graph, naturally avoids some fairly common SSs. This results in a significant improvement in the system performance. |

## 2010 |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Tree-Structure Expectation Propagation for Decoding LDPC Codes over Binary Erasure Channels Inproceedings 2010 IEEE International Symposium on Information Theory, pp. 799–803, IEEE, Austin, TX, 2010, ISBN: 978-1-4244-7892-7. Abstract | Links | BibTeX | Tags: belief propagation, binary erasure channels, Bipartite graph, BP decoder, Capacity planning, Channel Coding, codeword, computational complexity, Decoding, Finishing, graph theory, H infinity control, LDPC code decoding, LDPC Tanner graph, Maxwell decoder, parity check codes, Performance analysis, tree structure expectation propagation, trees (mathematics), Upper bound @inproceedings{Olmos2010, title = {Tree-Structure Expectation Propagation for Decoding LDPC Codes over Binary Erasure Channels}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5513636}, isbn = {978-1-4244-7892-7}, year = {2010}, date = {2010-01-01}, booktitle = {2010 IEEE International Symposium on Information Theory}, pages = {799--803}, publisher = {IEEE}, address = {Austin, TX}, abstract = {Expectation Propagation is a generalization to Belief Propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal distribution constraints in some check nodes of the LDPC Tanner graph. These additional constraints allow decoding the received codeword when the BP decoder gets stuck. In this paper, we first present the new decoding algorithm, whose complexity is identical to the BP decoder, and we then prove that it is able to decode codewords with a larger fraction of erasures, as the block size tends to infinity. The proposed algorithm can be also understood as a simplification of the Maxwell decoder, but without its computational complexity. We also illustrate that the new algorithm outperforms the BP decoder for finite block-size codes.}, keywords = {belief propagation, binary erasure channels, Bipartite graph, BP decoder, Capacity planning, Channel Coding, codeword, computational complexity, Decoding, Finishing, graph theory, H infinity control, LDPC code decoding, LDPC Tanner graph, Maxwell decoder, parity check codes, Performance analysis, tree structure expectation propagation, trees (mathematics), Upper bound}, pubstate = {published}, tppubtype = {inproceedings} } Expectation Propagation is a generalization to Belief Propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal distribution constraints in some check nodes of the LDPC Tanner graph. These additional constraints allow decoding the received codeword when the BP decoder gets stuck. In this paper, we first present the new decoding algorithm, whose complexity is identical to the BP decoder, and we then prove that it is able to decode codewords with a larger fraction of erasures, as the block size tends to infinity. The proposed algorithm can be also understood as a simplification of the Maxwell decoder, but without its computational complexity. We also illustrate that the new algorithm outperforms the BP decoder for finite block-size codes. |

Salamanca, Luis; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Channel Decoding with a Bayesian Equalizer Inproceedings 2010 IEEE International Symposium on Information Theory, pp. 1998–2002, IEEE, Austin, TX, 2010, ISBN: 978-1-4244-7892-7. Abstract | Links | BibTeX | Tags: a posteriori probability, Bayesian equalizer, Bayesian methods, BER, Bit error rate, Channel Coding, channel decoding, channel estate information, Communication channels, Decoding, equalisers, Equalizers, error statistics, low-density parity-check decoders, LPDC decoders, Maximum likelihood decoding, maximum likelihood detection, maximum likelihood estimation, Noise reduction, parity check codes, Probability, Uncertainty @inproceedings{Salamanca2010a, title = {Channel Decoding with a Bayesian Equalizer}, author = {Luis Salamanca and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5513348}, isbn = {978-1-4244-7892-7}, year = {2010}, date = {2010-01-01}, booktitle = {2010 IEEE International Symposium on Information Theory}, pages = {1998--2002}, publisher = {IEEE}, address = {Austin, TX}, abstract = {Low-density parity-check (LPDC) decoders assume the channel estate information (CSI) is known and they have the true a posteriori probability (APP) for each transmitted bit. But in most cases of interest, the CSI needs to be estimated with the help of a short training sequence and the LDPC decoder has to decode the received word using faulty APP estimates. In this paper, we study the uncertainty in the CSI estimate and how it affects the bit error rate (BER) output by the LDPC decoder. To improve these APP estimates, we propose a Bayesian equalizer that takes into consideration not only the uncertainty due to the noise in the channel, but also the uncertainty in the CSI estimate, reducing the BER after the LDPC decoder.}, keywords = {a posteriori probability, Bayesian equalizer, Bayesian methods, BER, Bit error rate, Channel Coding, channel decoding, channel estate information, Communication channels, Decoding, equalisers, Equalizers, error statistics, low-density parity-check decoders, LPDC decoders, Maximum likelihood decoding, maximum likelihood detection, maximum likelihood estimation, Noise reduction, parity check codes, Probability, Uncertainty}, pubstate = {published}, tppubtype = {inproceedings} } Low-density parity-check (LPDC) decoders assume the channel estate information (CSI) is known and they have the true a posteriori probability (APP) for each transmitted bit. But in most cases of interest, the CSI needs to be estimated with the help of a short training sequence and the LDPC decoder has to decode the received word using faulty APP estimates. In this paper, we study the uncertainty in the CSI estimate and how it affects the bit error rate (BER) output by the LDPC decoder. To improve these APP estimates, we propose a Bayesian equalizer that takes into consideration not only the uncertainty due to the noise in the channel, but also the uncertainty in the CSI estimate, reducing the BER after the LDPC decoder. |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Joint Nonlinear Channel Equalization and Soft LDPC Decoding with Gaussian Processes Journal Article IEEE Transactions on Signal Processing, 58 (3), pp. 1183–1192, 2010, ISSN: 1053-587X. Abstract | Links | BibTeX | Tags: Bayesian nonlinear classification tool, Bit error rate, Channel Coding, channel equalizers, Channel estimation, Coding, equalisers, equalization, error statistics, Gaussian processes, GPC, joint nonlinear channel equalization, low-density parity-check (LDPC), low-density parity-check channel decoder, Machine learning, nonlinear channel, nonlinear codes, parity check codes, posterior probability estimates, soft LDPC decoding, soft-decoding, support vector machine (SVM) @article{Olmos2010a, title = {Joint Nonlinear Channel Equalization and Soft LDPC Decoding with Gaussian Processes}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5290078}, issn = {1053-587X}, year = {2010}, date = {2010-01-01}, journal = {IEEE Transactions on Signal Processing}, volume = {58}, number = {3}, pages = {1183--1192}, abstract = {In this paper, we introduce a new approach for nonlinear equalization based on Gaussian processes for classification (GPC). We propose to measure the performance of this equalizer after a low-density parity-check channel decoder has detected the received sequence. Typically, most channel equalizers concentrate on reducing the bit error rate, instead of providing accurate posterior probability estimates. We show that the accuracy of these estimates is essential for optimal performance of the channel decoder and that the error rate output by the equalizer might be irrelevant to understand the performance of the overall communication receiver. In this sense, GPC is a Bayesian nonlinear classification tool that provides accurate posterior probability estimates with short training sequences. In the experimental section, we compare the proposed GPC-based equalizer with state-of-the-art solutions to illustrate its improved performance.}, keywords = {Bayesian nonlinear classification tool, Bit error rate, Channel Coding, channel equalizers, Channel estimation, Coding, equalisers, equalization, error statistics, Gaussian processes, GPC, joint nonlinear channel equalization, low-density parity-check (LDPC), low-density parity-check channel decoder, Machine learning, nonlinear channel, nonlinear codes, parity check codes, posterior probability estimates, soft LDPC decoding, soft-decoding, support vector machine (SVM)}, pubstate = {published}, tppubtype = {article} } In this paper, we introduce a new approach for nonlinear equalization based on Gaussian processes for classification (GPC). We propose to measure the performance of this equalizer after a low-density parity-check channel decoder has detected the received sequence. Typically, most channel equalizers concentrate on reducing the bit error rate, instead of providing accurate posterior probability estimates. We show that the accuracy of these estimates is essential for optimal performance of the channel decoder and that the error rate output by the equalizer might be irrelevant to understand the performance of the overall communication receiver. In this sense, GPC is a Bayesian nonlinear classification tool that provides accurate posterior probability estimates with short training sequences. In the experimental section, we compare the proposed GPC-based equalizer with state-of-the-art solutions to illustrate its improved performance. |

Fresia, Maria; Perez-Cruz, Fernando; Poor, Vincent H; Verdu, Sergio Joint Source and Channel Coding Journal Article IEEE Signal Processing Magazine, 27 (6), pp. 104–113, 2010, ISSN: 1053-5888. Abstract | Links | BibTeX | Tags: belief propagation, Channel Coding, combined source-channel coding, Decoding, Encoding, graphical model, Hidden Markov models, Iterative decoding, joint source channel coding, JSC coding, LDPC code, low density parity check code, Markov processes, parity check codes, Slepian-Wolf problem, variable length codes @article{Fresia2010, title = {Joint Source and Channel Coding}, author = {Maria Fresia and Fernando Perez-Cruz and Vincent H Poor and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5563107}, issn = {1053-5888}, year = {2010}, date = {2010-01-01}, journal = {IEEE Signal Processing Magazine}, volume = {27}, number = {6}, pages = {104--113}, abstract = {The objectives of this article are two-fold: First, to present the problem of joint source and channel (JSC) coding from a graphical model perspective and second, to propose a structure that uses a new graphical model for jointly encoding and decoding a redundant source. In the first part of the article, relevant contributions to JSC coding, ranging from the Slepian-Wolf problem to joint decoding of variable length codes with state-of-the-art source codes, are reviewed and summarized. In the second part, a double low-density parity-check (LDPC) code for JSC coding is proposed. The double LDPC code can be decoded as a single bipartite graph using standard belief propagation (BP) and its limiting performance is analyzed by using extrinsic information transfer (EXIT) chart approximations.}, keywords = {belief propagation, Channel Coding, combined source-channel coding, Decoding, Encoding, graphical model, Hidden Markov models, Iterative decoding, joint source channel coding, JSC coding, LDPC code, low density parity check code, Markov processes, parity check codes, Slepian-Wolf problem, variable length codes}, pubstate = {published}, tppubtype = {article} } The objectives of this article are two-fold: First, to present the problem of joint source and channel (JSC) coding from a graphical model perspective and second, to propose a structure that uses a new graphical model for jointly encoding and decoding a redundant source. In the first part of the article, relevant contributions to JSC coding, ranging from the Slepian-Wolf problem to joint decoding of variable length codes with state-of-the-art source codes, are reviewed and summarized. In the second part, a double low-density parity-check (LDPC) code for JSC coding is proposed. The double LDPC code can be decoded as a single bipartite graph using standard belief propagation (BP) and its limiting performance is analyzed by using extrinsic information transfer (EXIT) chart approximations. |

## 2009 |

Fresia, Maria; Perez-Cruz, Fernando; Poor, Vincent H Optimized Concatenated LDPC Codes for Joint Source-Channel Coding Inproceedings 2009 IEEE International Symposium on Information Theory, pp. 2131–2135, IEEE, Seoul, 2009, ISBN: 978-1-4244-4312-3. Abstract | Links | BibTeX | Tags: approximation theory, asymptotic behavior analysis, Channel Coding, combined source-channel coding, Concatenated codes, Decoding, Entropy, EXIT chart, extrinsic information transfer, H infinity control, Information analysis, joint belief propagation decoder, joint source-channel coding, low-density-parity-check code, optimized concatenated independent LDPC codes, parity check codes, Redundancy, source coding, transmitter, Transmitters @inproceedings{Fresia2009, title = {Optimized Concatenated LDPC Codes for Joint Source-Channel Coding}, author = {Maria Fresia and Fernando Perez-Cruz and Vincent H Poor}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5205766}, isbn = {978-1-4244-4312-3}, year = {2009}, date = {2009-01-01}, booktitle = {2009 IEEE International Symposium on Information Theory}, pages = {2131--2135}, publisher = {IEEE}, address = {Seoul}, abstract = {In this paper a scheme for joint source-channel coding based on low-density-parity-check (LDPC) codes is investigated. Two concatenated independent LDPC codes are used in the transmitter: one for source coding and the other for channel coding, with a joint belief propagation decoder. The asymptotic behavior is analyzed using EXtrinsic Information Transfer (EXIT) charts and this approximation is corroborated with illustrative experiments. The optimization of the degree distributions for our sparse code to maximize the information transmission rate is also considered.}, keywords = {approximation theory, asymptotic behavior analysis, Channel Coding, combined source-channel coding, Concatenated codes, Decoding, Entropy, EXIT chart, extrinsic information transfer, H infinity control, Information analysis, joint belief propagation decoder, joint source-channel coding, low-density-parity-check code, optimized concatenated independent LDPC codes, parity check codes, Redundancy, source coding, transmitter, Transmitters}, pubstate = {published}, tppubtype = {inproceedings} } In this paper a scheme for joint source-channel coding based on low-density-parity-check (LDPC) codes is investigated. Two concatenated independent LDPC codes are used in the transmitter: one for source coding and the other for channel coding, with a joint belief propagation decoder. The asymptotic behavior is analyzed using EXtrinsic Information Transfer (EXIT) charts and this approximation is corroborated with illustrative experiments. The optimization of the degree distributions for our sparse code to maximize the information transmission rate is also considered. |