## 2016 |

## Journal Articles |

Stinner, Markus ; Olmos, Pablo M On the Waterfall Performance of Finite-Length SC-LDPC Codes Constructed From Protographs Journal Article IEEE Journal on Selected Areas in Communications, 34 (2), pp. 345–361, 2016, ISSN: 0733-8716. Abstract | Links | BibTeX | Tags: Analytical models, capacity-achieving codes, Complexity theory, Couplings, Decoding, Encoding, finite-length analysis, Iterative decoding, Low-density parity-check (LDPC) codes, spatially coupled LDPC codes constructed from prot @article{Stinner2016, title = {On the Waterfall Performance of Finite-Length SC-LDPC Codes Constructed From Protographs}, author = {Stinner, Markus and Olmos, Pablo M.}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7339427}, doi = {10.1109/JSAC.2015.2504279}, issn = {0733-8716}, year = {2016}, date = {2016-02-01}, journal = {IEEE Journal on Selected Areas in Communications}, volume = {34}, number = {2}, pages = {345--361}, abstract = {An analysis of spatially coupled low-density parity-check (SC-LDPC) codes constructed from protographs is proposed. Given the protograph used to generate the SC-LDPC code ensemble, a set of scaling parameters to characterize the average finite-length performance in the waterfall region is computed. The error performance of structured SC-LDPC code ensembles is shown to follow a scaling law similar to that of unstructured randomly constructed SC-LDPC codes. Under a finite-length perspective, some of the most relevant SC-LDPC protograph structures proposed to date are compared. The analysis reveals significant differences in their finite-length scaling behavior, which is corroborated by simulation. Spatially coupled repeat-accumulate codes present excellent finite-length performance, as they outperform in the waterfall region SC-LDPC codes of the same rate and better asymptotic thresholds.}, keywords = {Analytical models, capacity-achieving codes, Complexity theory, Couplings, Decoding, Encoding, finite-length analysis, Iterative decoding, Low-density parity-check (LDPC) codes, spatially coupled LDPC codes constructed from prot}, pubstate = {published}, tppubtype = {article} } An analysis of spatially coupled low-density parity-check (SC-LDPC) codes constructed from protographs is proposed. Given the protograph used to generate the SC-LDPC code ensemble, a set of scaling parameters to characterize the average finite-length performance in the waterfall region is computed. The error performance of structured SC-LDPC code ensembles is shown to follow a scaling law similar to that of unstructured randomly constructed SC-LDPC codes. Under a finite-length perspective, some of the most relevant SC-LDPC protograph structures proposed to date are compared. The analysis reveals significant differences in their finite-length scaling behavior, which is corroborated by simulation. Spatially coupled repeat-accumulate codes present excellent finite-length performance, as they outperform in the waterfall region SC-LDPC codes of the same rate and better asymptotic thresholds. |

## 2015 |

## Journal Articles |

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, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Olmos, Pablo M. and Urbanke, Rudiger L.}, 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} } 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 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{Olmos2015b, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Olmos, Pablo M. and Urbanke, Rudiger L.}, 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. |

## Inproceedings |

Santos, Irene ; Murillo-Fuentes, Juan Jose ; Olmos, Pablo M Block Expectation Propagation Equalization for ISI Channels Inproceedings 2015 23rd European Signal Processing Conference (EUSIPCO), pp. 379–383, IEEE, Nice, 2015, ISBN: 978-0-9928-6263-3. Abstract | Links | BibTeX | Tags: Approximation algorithms, Approximation methods, BCJR algorithm, channel equalization, Complexity theory, Decoding, Equalizers, expectation propagation, ISI, low complexity, Signal processing algorithms @inproceedings{Santos2015, title = {Block Expectation Propagation Equalization for ISI Channels}, author = {Santos, Irene and Murillo-Fuentes, Juan Jose and Olmos, Pablo M.}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7362409}, doi = {10.1109/EUSIPCO.2015.7362409}, isbn = {978-0-9928-6263-3}, year = {2015}, date = {2015-08-01}, booktitle = {2015 23rd European Signal Processing Conference (EUSIPCO)}, pages = {379--383}, publisher = {IEEE}, address = {Nice}, abstract = {Actual communications systems use high-order modulations and channels with memory. However, as the memory of the channels and the order of the constellations grow, optimal equalization such as BCJR algorithm is computationally intractable, as their complexity increases exponentially with the number of taps and size of modulation. In this paper, we propose a novel low-complexity hard and soft output equalizer based on the Expectation Propagation (EP) algorithm that provides high-accuracy posterior probability estimations at the input of the channel decoder with similar computational complexity than the linear MMSE. We experimentally show that this quasi-optimal solution outperforms classical solutions reducing the bit error probability with low complexity when LDPC channel decoding is used, avoiding the curse of dimensionality with channel memory and constellation size.}, keywords = {Approximation algorithms, Approximation methods, BCJR algorithm, channel equalization, Complexity theory, Decoding, Equalizers, expectation propagation, ISI, low complexity, Signal processing algorithms}, pubstate = {published}, tppubtype = {inproceedings} } Actual communications systems use high-order modulations and channels with memory. However, as the memory of the channels and the order of the constellations grow, optimal equalization such as BCJR algorithm is computationally intractable, as their complexity increases exponentially with the number of taps and size of modulation. In this paper, we propose a novel low-complexity hard and soft output equalizer based on the Expectation Propagation (EP) algorithm that provides high-accuracy posterior probability estimations at the input of the channel decoder with similar computational complexity than the linear MMSE. We experimentally show that this quasi-optimal solution outperforms classical solutions reducing the bit error probability with low complexity when LDPC channel decoding is used, avoiding the curse of dimensionality with channel memory and constellation size. |

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 = {Olmos, Pablo M. and Mitchell, David G. M. and Costello, Daniel J.}, 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. |

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 = {Stinner, Markus and Olmos, Pablo M.}, 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. |

## 2014 |

## Journal Articles |

Yang, W; Durisi, Giuseppe ; Koch, Tobias ; Polyanskiy, Yury Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength Journal Article IEEE Transactions on Information Theory, 60 (7), pp. 4232–4265, 2014, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: channel dispersion, Decoding, error probability, finite blocklength regime, MIMO, MIMO channel, outage probability, quasi-static fading channel, Rayleigh channels, Receivers, Transmitters @article{Yang2014, title = {Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength}, author = {Yang, W. and Durisi, Giuseppe and Koch, Tobias and Polyanskiy, Yury}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6802432 http://arxiv.org/abs/1311.2012}, issn = {0018-9448}, year = {2014}, date = {2014-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {60}, number = {7}, pages = {4232--4265}, publisher = {IEEE}, abstract = {This paper investigates the maximal achievable rate for a given blocklength and error probability over quasi-static multiple-input multiple-output fading channels, with and without channel state information at the transmitter and/or the receiver. The principal finding is that outage capacity, despite being an asymptotic quantity, is a sharp proxy for the finite-blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver, at only one of them, or at neither of them. These results follow from analytically tractable converse and achievability bounds. Numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases. In the example of a particular 1 $,times,$ 2 single-input multiple-output Rician fading channel, the blocklength required to achieve 90% of capacity is about an order of magnitude smaller compared with the blocklength required for an AWGN channel with the same capacity. For this specific scenario, the coding/decoding schemes adopted in the LTE-Advanced standard are benchmarked against the finite-blocklength achievability and converse bounds.}, keywords = {channel dispersion, Decoding, error probability, finite blocklength regime, MIMO, MIMO channel, outage probability, quasi-static fading channel, Rayleigh channels, Receivers, Transmitters}, pubstate = {published}, tppubtype = {article} } This paper investigates the maximal achievable rate for a given blocklength and error probability over quasi-static multiple-input multiple-output fading channels, with and without channel state information at the transmitter and/or the receiver. The principal finding is that outage capacity, despite being an asymptotic quantity, is a sharp proxy for the finite-blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver, at only one of them, or at neither of them. These results follow from analytically tractable converse and achievability bounds. Numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases. In the example of a particular 1 $,times,$ 2 single-input multiple-output Rician fading channel, the blocklength required to achieve 90% of capacity is about an order of magnitude smaller compared with the blocklength required for an AWGN channel with the same capacity. For this specific scenario, the coding/decoding schemes adopted in the LTE-Advanced standard are benchmarked against the finite-blocklength achievability and converse bounds. |

## Inproceedings |

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 & 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 = {Olmos, Pablo M. and Mitchell, David G.M. and Truhachev, Dmitry and Costello, Daniel J.}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6955088}, year = {2014}, date = {2014-01-01}, booktitle = {8th IEEE International Symposium on Turbo Codes & 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 = {Stinner, Markus and Olmos, Pablo M.}, 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. |

## 2013 |

## Journal Articles |

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 = {Olmos, Pablo M. and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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 = {Salamanca, Luis and Olmos, Pablo M. and Perez-Cruz, Fernando and Murillo-Fuentes, Juan Jose}, 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 = {Bravo-Santos, Ángel M.}, 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. |

## 2012 |

## Journal Articles |

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 = {Salamanca, Luis and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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; 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 = {Olmos, Pablo M. and Salamanca, Luis and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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. |

## Inproceedings |

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 = {Salamanca, Luis and Murillo-Fuentes, Juan Jose and Olmos, Pablo M. and Perez-Cruz, Fernando}, 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; 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 = {Olmos, Pablo M. and Perez-Cruz, Fernando and Salamanca, Luis and Murillo-Fuentes, Juan Jose}, 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} } |

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 = {Olmos, Pablo M. and Perez-Cruz, Fernando and Salamanca, Luis and Murillo-Fuentes, Juan Jose}, 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. |

Pastore, Adriano ; Koch, Tobias ; Fonollosa, Javier Rodriguez Improved Capacity Lower Bounds for Fading Channels with Imperfect CSI Using Rate Splitting Inproceedings 2012 IEEE 27th Convention of Electrical and Electronics Engineers in Israel, pp. 1–5, IEEE, Eilat, 2012, ISBN: 978-1-4673-4681-8. Abstract | Links | BibTeX | Tags: channel capacity, channel capacity lower bounds, conditional entropy, Decoding, Entropy, Fading, fading channels, Gaussian channel, Gaussian channels, Gaussian random variable, imperfect channel-state information, imperfect CSI, independent Gaussian variables, linear minimum mean-square error, mean square error methods, Medard lower bound, Mutual information, Random variables, rate splitting approach, Resource management, Upper bound, wireless communications @inproceedings{Pastore2012, title = {Improved Capacity Lower Bounds for Fading Channels with Imperfect CSI Using Rate Splitting}, author = {Pastore, Adriano and Koch, Tobias and Fonollosa, Javier Rodriguez}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6377031}, isbn = {978-1-4673-4681-8}, year = {2012}, date = {2012-01-01}, booktitle = {2012 IEEE 27th Convention of Electrical and Electronics Engineers in Israel}, pages = {1--5}, publisher = {IEEE}, address = {Eilat}, abstract = {As shown by Medard (“The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inform. Theory, May 2000), the capacity of fading channels with imperfect channel-state information (CSI) can be lower-bounded by assuming a Gaussian channel input X, and by upper-bounding the conditional entropy h(XY, Ĥ), conditioned on the channel output Y and the CSI Ĥ, by the entropy of a Gaussian random variable with variance equal to the linear minimum mean-square error in estimating X from (Y, Ĥ). We demonstrate that, by using a rate-splitting approach, this lower bound can be sharpened: we show that by expressing the Gaussian input X as as the sum of two independent Gaussian variables X(1) and X(2), and by applying Medard's lower bound first to analyze the mutual information between X(1) and Y conditioned on Ĥ while treating X(2) as noise, and by applying the lower bound then to analyze the mutual information between X(2) and Y conditioned on (X(1), Ĥ), we obtain a lower bound on the capacity that is larger than Medard's lower bound.}, keywords = {channel capacity, channel capacity lower bounds, conditional entropy, Decoding, Entropy, Fading, fading channels, Gaussian channel, Gaussian channels, Gaussian random variable, imperfect channel-state information, imperfect CSI, independent Gaussian variables, linear minimum mean-square error, mean square error methods, Medard lower bound, Mutual information, Random variables, rate splitting approach, Resource management, Upper bound, wireless communications}, pubstate = {published}, tppubtype = {inproceedings} } As shown by Medard (“The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inform. Theory, May 2000), the capacity of fading channels with imperfect channel-state information (CSI) can be lower-bounded by assuming a Gaussian channel input X, and by upper-bounding the conditional entropy h(XY, Ĥ), conditioned on the channel output Y and the CSI Ĥ, by the entropy of a Gaussian random variable with variance equal to the linear minimum mean-square error in estimating X from (Y, Ĥ). We demonstrate that, by using a rate-splitting approach, this lower bound can be sharpened: we show that by expressing the Gaussian input X as as the sum of two independent Gaussian variables X(1) and X(2), and by applying Medard's lower bound first to analyze the mutual information between X(1) and Y conditioned on Ĥ while treating X(2) as noise, and by applying the lower bound then to analyze the mutual information between X(2) and Y conditioned on (X(1), Ĥ), we obtain a lower bound on the capacity that is larger than Medard's lower bound. |

Campo, Adria Tauste ; Vazquez-Vilar, Gonzalo ; Guillen i Fabregas, Albert ; Koch, Tobias ; Martinez, Alfonso Achieving Csiszár's Source-Channel Coding Exponent with Product Distributions Inproceedings 2012 IEEE International Symposium on Information Theory Proceedings, pp. 1548–1552, IEEE, Cambridge, MA, 2012, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: average probability of error, Channel Coding, code construction, codewords, Csiszár's source-channel coding, Decoding, Encoding, error probability, error statistics, Joints, Manganese, product distributions, random codes, random-coding upper bound, source coding, source messages, Upper bound @inproceedings{Campo2012a, title = {Achieving Csiszár's Source-Channel Coding Exponent with Product Distributions}, author = {Campo, Adria Tauste and Vazquez-Vilar, Gonzalo and Guillen i Fabregas, Albert and Koch, Tobias and Martinez, Alfonso}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6283524}, issn = {2157-8095}, year = {2012}, date = {2012-01-01}, booktitle = {2012 IEEE International Symposium on Information Theory Proceedings}, pages = {1548--1552}, publisher = {IEEE}, address = {Cambridge, MA}, abstract = {We derive a random-coding upper bound on the average probability of error of joint source-channel coding that recovers Csiszár's error exponent when used with product distributions over the channel inputs. Our proof technique for the error probability analysis employs a code construction for which source messages are assigned to subsets and codewords are generated with a distribution that depends on the subset.}, keywords = {average probability of error, Channel Coding, code construction, codewords, Csiszár's source-channel coding, Decoding, Encoding, error probability, error statistics, Joints, Manganese, product distributions, random codes, random-coding upper bound, source coding, source messages, Upper bound}, pubstate = {published}, tppubtype = {inproceedings} } We derive a random-coding upper bound on the average probability of error of joint source-channel coding that recovers Csiszár's error exponent when used with product distributions over the channel inputs. Our proof technique for the error probability analysis employs a code construction for which source messages are assigned to subsets and codewords are generated with a distribution that depends on the subset. |

## 2011 |

## Journal Articles |

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 = {Olmos, Pablo M. and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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. |

## Inproceedings |

Koch, Tobias ; Lapidoth, Amos Asymmetric Quantizers are Better at Low SNR Inproceedings 2011 IEEE International Symposium on Information Theory Proceedings, pp. 2592–2596, IEEE, St. Petersburg, 2011, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: asymmetric one-bit quantizer, asymmetric signal constellations, channel capacity, Channel Coding, Constellation diagram, Decoding, discrete-time average-power-limited Gaussian chann, Gaussian channels, quantization, Signal to noise ratio, signal-to-noise ratio, SNR, spread spectrum communication, spread-spectrum communications, ultra wideband communication, ultrawideband communications, Upper bound @inproceedings{Koch2011, title = {Asymmetric Quantizers are Better at Low SNR}, author = {Koch, Tobias and Lapidoth, Amos}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6034037}, issn = {2157-8095}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE International Symposium on Information Theory Proceedings}, pages = {2592--2596}, publisher = {IEEE}, address = {St. Petersburg}, abstract = {We study the behavior of channel capacity when a one-bit quantizer is employed at the output of the discrete-time average-power-limited Gaussian channel. We focus on the low signal-to-noise ratio regime, where communication at very low spectral efficiencies takes place, as in Spread-Spectrum and Ultra-Wideband communications. It is well known that, in this regime, a symmetric one-bit quantizer reduces capacity by 2/$pi$, which translates to a power loss of approximately two decibels. Here we show that if an asymmetric one-bit quantizer is employed, and if asymmetric signal constellations are used, then these two decibels can be recovered in full.}, keywords = {asymmetric one-bit quantizer, asymmetric signal constellations, channel capacity, Channel Coding, Constellation diagram, Decoding, discrete-time average-power-limited Gaussian chann, Gaussian channels, quantization, Signal to noise ratio, signal-to-noise ratio, SNR, spread spectrum communication, spread-spectrum communications, ultra wideband communication, ultrawideband communications, Upper bound}, pubstate = {published}, tppubtype = {inproceedings} } We study the behavior of channel capacity when a one-bit quantizer is employed at the output of the discrete-time average-power-limited Gaussian channel. We focus on the low signal-to-noise ratio regime, where communication at very low spectral efficiencies takes place, as in Spread-Spectrum and Ultra-Wideband communications. It is well known that, in this regime, a symmetric one-bit quantizer reduces capacity by 2/$pi$, which translates to a power loss of approximately two decibels. Here we show that if an asymmetric one-bit quantizer is employed, and if asymmetric signal constellations are used, then these two decibels can be recovered in full. |

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 = {Olmos, Pablo M. and Urbanke, Rudiger}, 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 = {Olmos, Pablo M. and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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. |

Asyhari, Taufiq A; Koch, Tobias ; Guillén i Fàbregas, Albert Nearest Neighbour Decoding and Pilot-Aided Channel Estimation in Stationary Gaussian Flat-Fading Channels Inproceedings 2011 IEEE International Symposium on Information Theory Proceedings, pp. 2786–2790, IEEE, St. Petersburg, 2011, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: Channel estimation, Decoding, Fading, fading channels, Gaussian channels, MIMO, MIMO communication, MISO, multiple-input multiple-output, nearest neighbour decoding, noncoherent multiple-input single-output, pilot-aided channel estimation, Receiving antennas, Signal to noise ratio, signal-to-noise ratio, SNR, stationary Gaussian flat-fading channels, Wireless communication @inproceedings{Asyhari2011, title = {Nearest Neighbour Decoding and Pilot-Aided Channel Estimation in Stationary Gaussian Flat-Fading Channels}, author = {Asyhari, A. Taufiq and Koch, Tobias and Guillén i Fàbregas, Albert}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6034081}, issn = {2157-8095}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE International Symposium on Information Theory Proceedings}, pages = {2786--2790}, publisher = {IEEE}, address = {St. Petersburg}, abstract = {We study the information rates of non-coherent, stationary, Gaussian, multiple-input multiple-output (MIMO) flat-fading channels that are achievable with nearest neighbour decoding and pilot-aided channel estimation. In particular, we analyse the behaviour of these achievable rates in the limit as the signal-to-noise ratio (SNR) tends to infinity. We demonstrate that nearest neighbour decoding and pilot-aided channel estimation achieves the capacity pre-log-which is defined as the limiting ratio of the capacity to the logarithm of SNR as the SNR tends to infinity-of non-coherent multiple-input single-output (MISO) flat-fading channels, and it achieves the best so far known lower bound on the capacity pre-log of non-coherent MIMO flat-fading channels.}, keywords = {Channel estimation, Decoding, Fading, fading channels, Gaussian channels, MIMO, MIMO communication, MISO, multiple-input multiple-output, nearest neighbour decoding, noncoherent multiple-input single-output, pilot-aided channel estimation, Receiving antennas, Signal to noise ratio, signal-to-noise ratio, SNR, stationary Gaussian flat-fading channels, Wireless communication}, pubstate = {published}, tppubtype = {inproceedings} } We study the information rates of non-coherent, stationary, Gaussian, multiple-input multiple-output (MIMO) flat-fading channels that are achievable with nearest neighbour decoding and pilot-aided channel estimation. In particular, we analyse the behaviour of these achievable rates in the limit as the signal-to-noise ratio (SNR) tends to infinity. We demonstrate that nearest neighbour decoding and pilot-aided channel estimation achieves the capacity pre-log-which is defined as the limiting ratio of the capacity to the logarithm of SNR as the SNR tends to infinity-of non-coherent multiple-input single-output (MISO) flat-fading channels, and it achieves the best so far known lower bound on the capacity pre-log of non-coherent MIMO flat-fading channels. |

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 = {Salamanca, Luis and Olmos, Pablo M. and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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. |

Asyhari, Taufiq A; Koch, Tobias ; Guillen i Fabregas, Albert Nearest Neighbour Decoding with Pilot-Assisted Channel Estimation for Fading Multiple-Access Channels Inproceedings 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1686–1693, IEEE, Allerton, 2011, ISBN: 978-1-4577-1818-2. Abstract | Links | BibTeX | Tags: Channel estimation, Decoding, Fading, fading channels, fading multiple-access channels, MIMO, MIMO communication, multi-access systems, multiple-input multiple-output channel, nearest-neighbour decoding, noncoherent MIMO fading MAC channel, pilot-assisted channel estimation, Receiving antennas, Signal to noise ratio, signal-to-noise ratio, Time division multiple access, Vectors @inproceedings{Asyhari2011a, title = {Nearest Neighbour Decoding with Pilot-Assisted Channel Estimation for Fading Multiple-Access Channels}, author = {Asyhari, A. Taufiq and Koch, Tobias and Guillen i Fabregas, Albert}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6120371}, isbn = {978-1-4577-1818-2}, year = {2011}, date = {2011-01-01}, booktitle = {2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)}, pages = {1686--1693}, publisher = {IEEE}, address = {Allerton}, abstract = {This paper studies a noncoherent multiple-input multiple-output (MIMO) fading multiple-access channel (MAC). The rate region that is achievable with nearest neighbour decoding and pilot-assisted channel estimation is analysed and the corresponding pre-log region, defined as the limiting ratio of the rate region to the logarithm of the signal-to-noise ratio (SNR) as the SNR tends to infinity, is determined.}, keywords = {Channel estimation, Decoding, Fading, fading channels, fading multiple-access channels, MIMO, MIMO communication, multi-access systems, multiple-input multiple-output channel, nearest-neighbour decoding, noncoherent MIMO fading MAC channel, pilot-assisted channel estimation, Receiving antennas, Signal to noise ratio, signal-to-noise ratio, Time division multiple access, Vectors}, pubstate = {published}, tppubtype = {inproceedings} } This paper studies a noncoherent multiple-input multiple-output (MIMO) fading multiple-access channel (MAC). The rate region that is achievable with nearest neighbour decoding and pilot-assisted channel estimation is analysed and the corresponding pre-log region, defined as the limiting ratio of the rate region to the logarithm of the signal-to-noise ratio (SNR) as the SNR tends to infinity, is determined. |

## 2010 |

## Journal Articles |

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 = {Fresia, Maria and Perez-Cruz, Fernando and Poor, H. Vincent and Verdu, Sergio}, 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. |

## Inproceedings |

Martinez Ruiz, Manuel ; Artés-Rodríguez, Antonio ; Diaz-Rico, Jose Antonio ; Fuentes, Jose Blanco 2010 - MILCOM 2010 MILITARY COMMUNICATIONS CONFERENCE, pp. 1163–1168, IEEE, San Jose, 2010, ISSN: 2155-7578. Abstract | Links | BibTeX | Tags: Bit rate, code stream, data stream, Decoding, discrete wavelet transforms, Image coding, image compression, imagery transmission, JPEG-2000, JPEG2000 compressed images, link-16, Link-16 Enhance Throughput, Link-16 tactical network, MIDS-LVT, military communication, multirresolution, operational requirement, packing limit, PSNR, Security, Streaming media, tactical data link, time slot, Transform coding, wavelet discrete transforms, wavelets @inproceedings{Martinez-Ruiz2010, title = {New Initiatives for Imagery Transmission over a Tactical Data Link. A Case Study: JPEG2000 Compressed Images Transmitted in a Link-16 Network. Method and Results}, author = {Martinez Ruiz, Manuel and Artés-Rodríguez, Antonio and Diaz-Rico, Jose Antonio and Fuentes, Jose Blanco}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5680102}, issn = {2155-7578}, year = {2010}, date = {2010-01-01}, booktitle = {2010 - MILCOM 2010 MILITARY COMMUNICATIONS CONFERENCE}, pages = {1163--1168}, publisher = {IEEE}, address = {San Jose}, abstract = {This paper presents the results of an initiative to transmit imagery content through a Link-16 tactical network using a multirresolution approach based on wavelets to compress images. Firstly, we identify the operational requirements. Secondly, we justify why JPEG2000 is our choice for coding still images. Thirdly, we propose a method to map the JPEG2000 code-stream into Link-16 free-text messages. We propose to send the most important part of the JPEG2000 compressed image in a more error resistant Link-16 packed structure and the remaining of the image in less robust data structures but at higher data rates. Finally, we present our results based on software simulations and laboratory tests with real Link-16 terminals including a comparative analysis with Link-16 enhance throughput. A configuration using two MIDS-LVTs has being set up, along with JPEG2000 coding and decoding software tools.}, keywords = {Bit rate, code stream, data stream, Decoding, discrete wavelet transforms, Image coding, image compression, imagery transmission, JPEG-2000, JPEG2000 compressed images, link-16, Link-16 Enhance Throughput, Link-16 tactical network, MIDS-LVT, military communication, multirresolution, operational requirement, packing limit, PSNR, Security, Streaming media, tactical data link, time slot, Transform coding, wavelet discrete transforms, wavelets}, pubstate = {published}, tppubtype = {inproceedings} } This paper presents the results of an initiative to transmit imagery content through a Link-16 tactical network using a multirresolution approach based on wavelets to compress images. Firstly, we identify the operational requirements. Secondly, we justify why JPEG2000 is our choice for coding still images. Thirdly, we propose a method to map the JPEG2000 code-stream into Link-16 free-text messages. We propose to send the most important part of the JPEG2000 compressed image in a more error resistant Link-16 packed structure and the remaining of the image in less robust data structures but at higher data rates. Finally, we present our results based on software simulations and laboratory tests with real Link-16 terminals including a comparative analysis with Link-16 enhance throughput. A configuration using two MIDS-LVTs has being set up, along with JPEG2000 coding and decoding software tools. |

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 = {Olmos, Pablo M. and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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 ; Jose Murillo-Fuentes, Juan ; Perez-Cruz, Fernando Bayesian BCJR for Channel Equalization and Decoding Inproceedings 2010 IEEE International Workshop on Machine Learning for Signal Processing, pp. 53–58, IEEE, Kittila, 2010, ISSN: 1551-2541. Abstract | Links | BibTeX | Tags: a posteriori probability, Bayes methods, Bayesian BCJR, Bayesian methods, Bit error rate, channel decoding, channel estate information, Channel estimation, Decoding, digital communication, digital communications, equalisers, Equalizers, error statistics, Markov processes, Maximum likelihood decoding, maximum likelihood estimation, multipath channel, probabilistic channel equalization, Probability, single input single output model, SISO model, statistical information, Training @inproceedings{Salamanca2010, title = {Bayesian BCJR for Channel Equalization and Decoding}, author = {Salamanca, Luis and Jose Murillo-Fuentes, Juan and Perez-Cruz, Fernando}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5589201}, issn = {1551-2541}, year = {2010}, date = {2010-01-01}, booktitle = {2010 IEEE International Workshop on Machine Learning for Signal Processing}, pages = {53--58}, publisher = {IEEE}, address = {Kittila}, abstract = {In this paper we focus on the probabilistic channel equalization in digital communications. We face the single input single output (SISO) model to show how the statistical information about the multipath channel can be exploited to further improve our estimation of the a posteriori probabilities (APP) during the equalization process. We consider not only the uncertainty due to the noise in the channel, but also in the estimate of the channel estate information (CSI). Thus, we resort to a Bayesian approach for the computation of the APP. This novel algorithm has the same complexity as the BCJR, exhibiting lower bit error rate at the output of the channel decoder than the standard BCJR that considers maximum likelihood (ML) to estimate the CSI.}, keywords = {a posteriori probability, Bayes methods, Bayesian BCJR, Bayesian methods, Bit error rate, channel decoding, channel estate information, Channel estimation, Decoding, digital communication, digital communications, equalisers, Equalizers, error statistics, Markov processes, Maximum likelihood decoding, maximum likelihood estimation, multipath channel, probabilistic channel equalization, Probability, single input single output model, SISO model, statistical information, Training}, pubstate = {published}, tppubtype = {inproceedings} } In this paper we focus on the probabilistic channel equalization in digital communications. We face the single input single output (SISO) model to show how the statistical information about the multipath channel can be exploited to further improve our estimation of the a posteriori probabilities (APP) during the equalization process. We consider not only the uncertainty due to the noise in the channel, but also in the estimate of the channel estate information (CSI). Thus, we resort to a Bayesian approach for the computation of the APP. This novel algorithm has the same complexity as the BCJR, exhibiting lower bit error rate at the output of the channel decoder than the standard BCJR that considers maximum likelihood (ML) to estimate the CSI. |

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 = {Salamanca, Luis and Murillo-Fuentes, Juan Jose and Perez-Cruz, Fernando}, 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. |

## 2009 |

## Inproceedings |

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 = {Fresia, Maria and Perez-Cruz, Fernando and Poor, H. Vincent}, 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. |