## 2016 |

Durisi, Giuseppe; Koch, Tobias; Ostman, Johan; Polyanskiy, Yury; Yang, Wei Short-Packet Communications Over Multiple-Antenna Rayleigh-Fading Channels Journal Article IEEE Transactions on Communications, 64 (2), pp. 618–629, 2016, ISSN: 0090-6778. Abstract | Links | BibTeX | Tags: diversity branches, Encoding, ergodic capacity, Fading, fading channels, finite-blocklength information theory, finiteblocklength information theory, infinite-blocklength performance metrics, Journal, machine-type communication systems, maximum coding rate, Mission critical systems, mission-critical machine-type communications, multiple antennas, multiple-antenna Rayleigh block-fading channels, Multiplexing, optimal number, outage capacity, rate gain, Rayleigh channels, Receivers, Reliability, short-packet communications, spatial multiplexing, Throughput, Time-frequency analysis, time-frequency-spatial degrees of freedom, transmit antennas, transmit diversity, Transmitting antennas, Ultra-reliable low-latency communications @article{Durisi2016b, title = {Short-Packet Communications Over Multiple-Antenna Rayleigh-Fading Channels}, author = {Giuseppe Durisi and Tobias Koch and Johan Ostman and Yury Polyanskiy and Wei Yang}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7362178}, doi = {10.1109/TCOMM.2015.2511087}, issn = {0090-6778}, year = {2016}, date = {2016-02-01}, journal = {IEEE Transactions on Communications}, volume = {64}, number = {2}, pages = {618--629}, publisher = {IEEE}, abstract = {Motivated by the current interest in ultra-reliable, low-latency, machine-type communication systems, we investigate the tradeoff between reliability, throughput, and latency in the transmission of information over multiple-antenna Rayleigh block-fading channels. Specifically, we obtain finite-blocklength, finite-SNR upper and lower bounds on the maximum coding rate achievable over such channels for a given constraint on the packet error probability. Numerical evidence suggests that our bounds delimit tightly the maximum coding rate already for short blocklengths (packets of about 100 symbols). Furthermore, our bounds reveal the existence of a tradeoff between the rate gain obtainable by spreading each codeword over all available time-frequency-spatial degrees of freedom, and the rate loss caused by the need of estimating the fading coefficients over these degrees of freedom. In particular, our bounds allow us to determine the optimal number of transmit antennas and the optimal number of time-frequency diversity branches that maximize the rate. Finally, we show that infinite-blocklength performance metrics such as the ergodic capacity and the outage capacity yield inaccurate throughput estimates}, keywords = {diversity branches, Encoding, ergodic capacity, Fading, fading channels, finite-blocklength information theory, finiteblocklength information theory, infinite-blocklength performance metrics, Journal, machine-type communication systems, maximum coding rate, Mission critical systems, mission-critical machine-type communications, multiple antennas, multiple-antenna Rayleigh block-fading channels, Multiplexing, optimal number, outage capacity, rate gain, Rayleigh channels, Receivers, Reliability, short-packet communications, spatial multiplexing, Throughput, Time-frequency analysis, time-frequency-spatial degrees of freedom, transmit antennas, transmit diversity, Transmitting antennas, Ultra-reliable low-latency communications}, pubstate = {published}, tppubtype = {article} } Motivated by the current interest in ultra-reliable, low-latency, machine-type communication systems, we investigate the tradeoff between reliability, throughput, and latency in the transmission of information over multiple-antenna Rayleigh block-fading channels. Specifically, we obtain finite-blocklength, finite-SNR upper and lower bounds on the maximum coding rate achievable over such channels for a given constraint on the packet error probability. Numerical evidence suggests that our bounds delimit tightly the maximum coding rate already for short blocklengths (packets of about 100 symbols). Furthermore, our bounds reveal the existence of a tradeoff between the rate gain obtainable by spreading each codeword over all available time-frequency-spatial degrees of freedom, and the rate loss caused by the need of estimating the fading coefficients over these degrees of freedom. In particular, our bounds allow us to determine the optimal number of transmit antennas and the optimal number of time-frequency diversity branches that maximize the rate. Finally, we show that infinite-blocklength performance metrics such as the ergodic capacity and the outage capacity yield inaccurate throughput estimates |

## 2014 |

A, Pastore; Koch, Tobias; Fonollosa, Javier Rodriguez A Rate-Splitting Approach to Fading Channels With Imperfect Channel-State Information Journal Article IEEE Transactions on Information Theory, 60 (7), pp. 4266–4285, 2014, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: channel capacity, COMONSENS, DEIPRO, Entropy, Fading, fading channels, flat fading, imperfect channel-state information, MobileNET, Mutual information, OTOSiS, Random variables, Receivers, Signal to noise ratio, Upper bound @article{Pastore2014a, title = {A Rate-Splitting Approach to Fading Channels With Imperfect Channel-State Information}, author = {Pastore A and Tobias Koch and Javier Rodriguez Fonollosa}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6832779 http://www.tsc.uc3m.es/~koch/files/IEEE_TIT_60(7).pdf http://arxiv.org/pdf/1301.6120.pdf}, issn = {0018-9448}, year = {2014}, date = {2014-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {60}, number = {7}, pages = {4266--4285}, publisher = {IEEE}, abstract = {As shown by Médard, the capacity of fading channels with imperfect channel-state information can be lower-bounded by assuming a Gaussian channel input (X) with power (P) and by upper-bounding the conditional entropy (h(X|Y,hat Ħ)) by the entropy of a Gaussian random variable with variance equal to the linear minimum mean-square error in estimating (X) from ((Y,hat Ħ)) . We demonstrate that, using a rate-splitting approach, this lower bound can be sharpened: by expressing the Gaussian input (X) as the sum of two independent Gaussian variables (X_1) and (X_2) and by applying Médard's lower bound first to bound the mutual information between (X_1) and (Y) while treating (X_2) as noise, and by applying it a second time to the mutual information between (X_2) and (Y) while assuming (X_1) to be known, we obtain a capacity lower bound that is strictly larger than Médard's lower bound. We then generalize this approach to an arbi- rary number (L) of layers, where (X) is expressed as the sum of (L) independent Gaussian random variables of respective variances (P_ell ) , (ell = 1,dotsc ,L) summing up to (P) . Among all such rate-splitting bounds, we determine the supremum over power allocations (P_ell ) and total number of layers (L) . This supremum is achieved for (L rightarrow infty ) and gives rise to an analytically expressible capacity lower bound. For Gaussian fading, this novel bound is shown to converge to the Gaussian-input mutual information as the signal-to-noise ratio (SNR) grows, provided that the variance of the channel estimation error (H-hat Ħ) tends to zero as the SNR tends to infinity.}, keywords = {channel capacity, COMONSENS, DEIPRO, Entropy, Fading, fading channels, flat fading, imperfect channel-state information, MobileNET, Mutual information, OTOSiS, Random variables, Receivers, Signal to noise ratio, Upper bound}, pubstate = {published}, tppubtype = {article} } As shown by Médard, the capacity of fading channels with imperfect channel-state information can be lower-bounded by assuming a Gaussian channel input (X) with power (P) and by upper-bounding the conditional entropy (h(X|Y,hat Ħ)) by the entropy of a Gaussian random variable with variance equal to the linear minimum mean-square error in estimating (X) from ((Y,hat Ħ)) . We demonstrate that, using a rate-splitting approach, this lower bound can be sharpened: by expressing the Gaussian input (X) as the sum of two independent Gaussian variables (X_1) and (X_2) and by applying Médard's lower bound first to bound the mutual information between (X_1) and (Y) while treating (X_2) as noise, and by applying it a second time to the mutual information between (X_2) and (Y) while assuming (X_1) to be known, we obtain a capacity lower bound that is strictly larger than Médard's lower bound. We then generalize this approach to an arbi- rary number (L) of layers, where (X) is expressed as the sum of (L) independent Gaussian random variables of respective variances (P_ell ) , (ell = 1,dotsc ,L) summing up to (P) . Among all such rate-splitting bounds, we determine the supremum over power allocations (P_ell ) and total number of layers (L) . This supremum is achieved for (L rightarrow infty ) and gives rise to an analytically expressible capacity lower bound. For Gaussian fading, this novel bound is shown to converge to the Gaussian-input mutual information as the signal-to-noise ratio (SNR) grows, provided that the variance of the channel estimation error (H-hat Ħ) tends to zero as the SNR tends to infinity. |

Yang, Wei; Durisi, Giuseppe; Koch, Tobias; Polyanskiy, Yury Dispersion of Quasi-Static MIMO Fading Channels via Stokes' Theorem Inproceedings 2014 IEEE International Symposium on Information Theory, pp. 2072–2076, IEEE, Honolulu, 2014, ISBN: 978-1-4799-5186-4. Abstract | Links | BibTeX | Tags: channel capacity, differential form integration, Dispersion, Fading, fading channels, fading distribution, integration, Manifolds, Measurement, MIMO, MIMO communication, quasistatic MIMO fading channels dispersion, quasistatic multiple-input multiple-output fading, radio transmitters, Random variables, Stoke Theorem, transmitter @inproceedings{Yang2014b, title = {Dispersion of Quasi-Static MIMO Fading Channels via Stokes' Theorem}, author = {Wei Yang and Giuseppe Durisi and Tobias Koch and Yury Polyanskiy}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=6875198}, isbn = {978-1-4799-5186-4}, year = {2014}, date = {2014-01-01}, booktitle = {2014 IEEE International Symposium on Information Theory}, pages = {2072--2076}, publisher = {IEEE}, address = {Honolulu}, abstract = {This paper analyzes the channel dispersion of quasi-static multiple-input multiple-output fading channels with no channel state information at the transmitter. We show that the channel dispersion is zero under mild conditions on the fading distribution. The proof of our result is based on Stokes' theorem, which deals with the integration of differential forms on manifolds with boundary.}, keywords = {channel capacity, differential form integration, Dispersion, Fading, fading channels, fading distribution, integration, Manifolds, Measurement, MIMO, MIMO communication, quasistatic MIMO fading channels dispersion, quasistatic multiple-input multiple-output fading, radio transmitters, Random variables, Stoke Theorem, transmitter}, pubstate = {published}, tppubtype = {inproceedings} } This paper analyzes the channel dispersion of quasi-static multiple-input multiple-output fading channels with no channel state information at the transmitter. We show that the channel dispersion is zero under mild conditions on the fading distribution. The proof of our result is based on Stokes' theorem, which deals with the integration of differential forms on manifolds with boundary. |

## 2013 |

Yang, Wei; Durisi, Giuseppe; Koch, Tobias; Polyanskiy, Yury Quasi-Static SIMO Fading Channels at Finite Blocklength Inproceedings 2013 IEEE International Symposium on Information Theory, pp. 1531–1535, IEEE, Istanbul, 2013, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: achievability bounds, AWGN channel, AWGN channels, channel capacity, channel dispersion, channel gains, Dispersion, error probability, error statistics, Fading, fading channels, fading realizations, fast convergence, finite blocklength, maximal achievable rate, numerical evaluation, outage capacity, quasistatic SIMO fading channels, Random variables, Receivers, SIMO Rician channel, single-input multiple-output, Transmitters, zero dispersion @inproceedings{Yang2013a, title = {Quasi-Static SIMO Fading Channels at Finite Blocklength}, author = {Wei Yang and Giuseppe Durisi and Tobias Koch and Yury Polyanskiy}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6620483}, issn = {2157-8095}, year = {2013}, date = {2013-01-01}, booktitle = {2013 IEEE International Symposium on Information Theory}, pages = {1531--1535}, publisher = {IEEE}, address = {Istanbul}, abstract = {We investigate the maximal achievable rate for a given blocklength and error probability over quasi-static single-input multiple-output (SIMO) fading channels. Under mild conditions on the channel gains, it is shown that the channel dispersion is zero regardless of whether the fading realizations are available at the transmitter and/or the receiver. The result follows from computationally and analytically tractable converse and achievability bounds. Through numerical evaluation, we verify that, in some scenarios, zero dispersion indeed entails fast convergence to outage capacity as the blocklength increases. In the example of a particular 1×2 SIMO Rician channel, the blocklength required to achieve 90% of capacity is about an order of magnitude smaller compared to the blocklength required for an AWGN channel with the same capacity.}, keywords = {achievability bounds, AWGN channel, AWGN channels, channel capacity, channel dispersion, channel gains, Dispersion, error probability, error statistics, Fading, fading channels, fading realizations, fast convergence, finite blocklength, maximal achievable rate, numerical evaluation, outage capacity, quasistatic SIMO fading channels, Random variables, Receivers, SIMO Rician channel, single-input multiple-output, Transmitters, zero dispersion}, pubstate = {published}, tppubtype = {inproceedings} } We investigate the maximal achievable rate for a given blocklength and error probability over quasi-static single-input multiple-output (SIMO) fading channels. Under mild conditions on the channel gains, it is shown that the channel dispersion is zero regardless of whether the fading realizations are available at the transmitter and/or the receiver. The result follows from computationally and analytically tractable converse and achievability bounds. Through numerical evaluation, we verify that, in some scenarios, zero dispersion indeed entails fast convergence to outage capacity as the blocklength increases. In the example of a particular 1×2 SIMO Rician channel, the blocklength required to achieve 90% of capacity is about an order of magnitude smaller compared to the blocklength required for an AWGN channel with the same capacity. |

Bravo-Santos, Ángel M Polar Codes for the Rayleigh Fading Channel Journal Article IEEE Communications Letters, PP (99), pp. 1–4, 2013, ISSN: 1089-7798. Abstract | Links | BibTeX | Tags: fading channels, polar codes, Rayleigh channels @article{Bravo-Santos2013a, title = {Polar Codes for the Rayleigh Fading Channel}, author = {Ángel M Bravo-Santos}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6663750}, issn = {1089-7798}, year = {2013}, date = {2013-01-01}, journal = {IEEE Communications Letters}, volume = {PP}, number = {99}, pages = {1--4}, abstract = {The application of polar codes for the Rayleigh fading channel is considered. We construct polar codes for the block Rayleigh fading channel with known channel side information (CSI) and for the Rayleigh channel with known channel distribution information (CDI). The construction of polar codes for the Rayleigh fading with known CSI allows them to work with any signal noise ratio (SNR). The rate of the codeword is adapted correspondingly. Polar codes for Rayleigh fading with known CDI suffer a penalty for not having complete information about the channel. The penalty, however, is small, about 1.3 dB. We perform simulations and compare the obtained results with the theoretical limits. We show that they are close to the theoretical limit. We compare polar codes with other good codes and the results show that long polar codes are closer to the limit.}, keywords = {fading channels, polar codes, Rayleigh channels}, pubstate = {published}, tppubtype = {article} } The application of polar codes for the Rayleigh fading channel is considered. We construct polar codes for the block Rayleigh fading channel with known channel side information (CSI) and for the Rayleigh channel with known channel distribution information (CDI). The construction of polar codes for the Rayleigh fading with known CSI allows them to work with any signal noise ratio (SNR). The rate of the codeword is adapted correspondingly. Polar codes for Rayleigh fading with known CDI suffer a penalty for not having complete information about the channel. The penalty, however, is small, about 1.3 dB. We perform simulations and compare the obtained results with the theoretical limits. We show that they are close to the theoretical limit. We compare polar codes with other good codes and the results show that long polar codes are closer to the limit. |

Koch, Tobias; Kramer, Gerhard On Noncoherent Fading Relay Channels at High Signal-to-Noise Ratio Journal Article IEEE Transactions on Information Theory, 59 (4), pp. 2221–2241, 2013, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: channel capacity, Channel models, Fading, fading channels, MIMO communication, multiple-input single-output fading channel statis, noncoherent, noncoherent fading relay channel capacity, radio receiver, radio receivers, radio transmitter, radio transmitters, Receivers, relay channels, relay networks (telecommunication), Relays, Signal to noise ratio, signal-to-noise ratio, SNR, statistics, time selective, Transmitters, Upper bound @article{Koch2013a, title = {On Noncoherent Fading Relay Channels at High Signal-to-Noise Ratio}, author = {Tobias Koch and Gerhard Kramer}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6378474}, issn = {0018-9448}, year = {2013}, date = {2013-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {59}, number = {4}, pages = {2221--2241}, abstract = {The capacity of noncoherent regular-fading relay channels is studied where all terminals are aware of the fading statistics but not of their realizations. It is shown that if the fading coefficient of the channel between the transmitter and the receiver can be predicted more accurately from its infinite past than the fading coefficient of the channel between the relay and the receiver, then at high signal-to-noise ratio (SNR), the relay does not increase capacity. It is further shown that if the fading coefficient of the channel between the transmitter and the relay can be predicted more accurately from its infinite past than the fading coefficient of the channel between the relay and the receiver, then at high SNR, one can achieve communication rates that are within one bit of the capacity of the multiple-input single-output fading channel that results when the transmitter and the relay can cooperate.}, keywords = {channel capacity, Channel models, Fading, fading channels, MIMO communication, multiple-input single-output fading channel statis, noncoherent, noncoherent fading relay channel capacity, radio receiver, radio receivers, radio transmitter, radio transmitters, Receivers, relay channels, relay networks (telecommunication), Relays, Signal to noise ratio, signal-to-noise ratio, SNR, statistics, time selective, Transmitters, Upper bound}, pubstate = {published}, tppubtype = {article} } The capacity of noncoherent regular-fading relay channels is studied where all terminals are aware of the fading statistics but not of their realizations. It is shown that if the fading coefficient of the channel between the transmitter and the receiver can be predicted more accurately from its infinite past than the fading coefficient of the channel between the relay and the receiver, then at high signal-to-noise ratio (SNR), the relay does not increase capacity. It is further shown that if the fading coefficient of the channel between the transmitter and the relay can be predicted more accurately from its infinite past than the fading coefficient of the channel between the relay and the receiver, then at high SNR, one can achieve communication rates that are within one bit of the capacity of the multiple-input single-output fading channel that results when the transmitter and the relay can cooperate. |

## 2012 |

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 = {Adriano Pastore and Tobias Koch and Javier Rodriguez Fonollosa}, 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. |

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

## 2011 |

Asyhari, Taufiq A; Koch, Tobias; i Fàbregas, Albert Guillén 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 = {Taufiq A Asyhari and Tobias Koch and Albert Guillén i Fàbregas}, 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. |

Asyhari, Taufiq A; Koch, Tobias; i Fabregas, Albert Guillen 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 = {Taufiq A Asyhari and Tobias Koch and Albert Guillen i Fabregas}, 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 |

Koch, Tobias; Lapidoth, Amos Gaussian Fading Is the Worst Fading Journal Article IEEE Transactions on Information Theory, 56 (3), pp. 1158–1165, 2010, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: Additive noise, channel capacity, channels with memory, Distribution functions, ergodic fading processes, Fading, fading channels, flat fading, flat-fading channel capacity, Gaussian channels, Gaussian fading, Gaussian processes, H infinity control, high signal-to-noise ratio (SNR), Information technology, information theory, multiple-input single-output fading channels, multiplexing gain, noncoherent, noncoherent channel capacity, peak-power limited channel capacity, Signal to noise ratio, signal-to-noise ratio, single-antenna channel capacity, spectral distribution function, time-selective, Transmitters @article{Koch2010a, title = {Gaussian Fading Is the Worst Fading}, author = {Tobias Koch and Amos Lapidoth}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5429105}, issn = {0018-9448}, year = {2010}, date = {2010-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {56}, number = {3}, pages = {1158--1165}, abstract = {The capacity of peak-power limited, single-antenna, noncoherent, flat-fading channels with memory is considered. The emphasis is on the capacity pre-log, i.e., on the limiting ratio of channel capacity to the logarithm of the signal-to-noise ratio (SNR), as the SNR tends to infinity. It is shown that, among all stationary and ergodic fading processes of a given spectral distribution function and whose law has no mass point at zero, the Gaussian process gives rise to the smallest pre-log. The assumption that the law of the fading process has no mass point at zero is essential in the sense that there exist stationary and ergodic fading processes whose law has a mass point at zero and that give rise to a smaller pre-log than the Gaussian process of equal spectral distribution function. An extension of these results to multiple-input single-output (MISO) fading channels with memory is also presented.}, keywords = {Additive noise, channel capacity, channels with memory, Distribution functions, ergodic fading processes, Fading, fading channels, flat fading, flat-fading channel capacity, Gaussian channels, Gaussian fading, Gaussian processes, H infinity control, high signal-to-noise ratio (SNR), Information technology, information theory, multiple-input single-output fading channels, multiplexing gain, noncoherent, noncoherent channel capacity, peak-power limited channel capacity, Signal to noise ratio, signal-to-noise ratio, single-antenna channel capacity, spectral distribution function, time-selective, Transmitters}, pubstate = {published}, tppubtype = {article} } The capacity of peak-power limited, single-antenna, noncoherent, flat-fading channels with memory is considered. The emphasis is on the capacity pre-log, i.e., on the limiting ratio of channel capacity to the logarithm of the signal-to-noise ratio (SNR), as the SNR tends to infinity. It is shown that, among all stationary and ergodic fading processes of a given spectral distribution function and whose law has no mass point at zero, the Gaussian process gives rise to the smallest pre-log. The assumption that the law of the fading process has no mass point at zero is essential in the sense that there exist stationary and ergodic fading processes whose law has a mass point at zero and that give rise to a smaller pre-log than the Gaussian process of equal spectral distribution function. An extension of these results to multiple-input single-output (MISO) fading channels with memory is also presented. |

Koch, Tobias; Lapidoth, Amos On Multipath Fading Channels at High SNR Journal Article IEEE Transactions on Information Theory, 56 (12), pp. 5945–5957, 2010, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: approximation theory, capacity pre-loglog, capacity to loglog, channel capacity, channels with memory, Delay, Fading, fading channels, frequency-selective fading, high signal-to-noise ratio, high SNR, Limiting, multipath, multipath channels, noncoherent, noncoherent multipath fading channel, Receivers, Signal to noise ratio, signal-to-noise ratio, Transmitters @article{Koch2010b, title = {On Multipath Fading Channels at High SNR}, author = {Tobias Koch and Amos Lapidoth}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5625630}, issn = {0018-9448}, year = {2010}, date = {2010-01-01}, journal = {IEEE Transactions on Information Theory}, volume = {56}, number = {12}, pages = {5945--5957}, abstract = {A noncoherent multipath fading channel is considered, where neither the transmitter nor the receiver is cognizant of the realization of the path gains, but both are cognizant of their statistics. It is shown that if the delay spread is large in the sense that the variances of the path gains decay exponentially or slower, then capacity is bounded in the signal-to-noise ratio (SNR). For such channels, capacity does not tend to infinity as the SNR tends to infinity. In contrast, if the variances of the path gains decay faster than exponentially, then capacity is unbounded in the SNR. It is further demonstrated that if the number of paths is finite, then at high SNR capacity grows double-logarithmically with the SNR, and the capacity pre-loglog-defined as the limiting ratio of capacity to loglog(SNR) as the SNR tends to infinity-is 1 irrespective of the number of paths. The results demonstrate that at high SNR multipath fading channels with an infinite number of paths cannot be approximated by multipath fading channels with only a finite number of paths. The number of paths that are needed to approximate a multipath fading channel typically depends on the SNR and may grow to infinity as the SNR tends to infinity.}, keywords = {approximation theory, capacity pre-loglog, capacity to loglog, channel capacity, channels with memory, Delay, Fading, fading channels, frequency-selective fading, high signal-to-noise ratio, high SNR, Limiting, multipath, multipath channels, noncoherent, noncoherent multipath fading channel, Receivers, Signal to noise ratio, signal-to-noise ratio, Transmitters}, pubstate = {published}, tppubtype = {article} } A noncoherent multipath fading channel is considered, where neither the transmitter nor the receiver is cognizant of the realization of the path gains, but both are cognizant of their statistics. It is shown that if the delay spread is large in the sense that the variances of the path gains decay exponentially or slower, then capacity is bounded in the signal-to-noise ratio (SNR). For such channels, capacity does not tend to infinity as the SNR tends to infinity. In contrast, if the variances of the path gains decay faster than exponentially, then capacity is unbounded in the SNR. It is further demonstrated that if the number of paths is finite, then at high SNR capacity grows double-logarithmically with the SNR, and the capacity pre-loglog-defined as the limiting ratio of capacity to loglog(SNR) as the SNR tends to infinity-is 1 irrespective of the number of paths. The results demonstrate that at high SNR multipath fading channels with an infinite number of paths cannot be approximated by multipath fading channels with only a finite number of paths. The number of paths that are needed to approximate a multipath fading channel typically depends on the SNR and may grow to infinity as the SNR tends to infinity. |

## 2008 |

Koch, Tobias; Lapidoth, Amos On Multipath Fading Channels at High SNR Inproceedings 2008 IEEE International Symposium on Information Theory, pp. 1572–1576, IEEE, Toronto, 2008, ISBN: 978-1-4244-2256-2. Abstract | Links | BibTeX | Tags: channel capacity, Delay, discrete time systems, discrete-time channels, Entropy, Fading, fading channels, Frequency, Mathematical model, multipath channels, multipath fading channels, noncoherent channel model, Random variables, Signal to noise ratio, signal-to-noise ratios, SNR, statistics, Transmitters @inproceedings{Koch2008, title = {On Multipath Fading Channels at High SNR}, author = {Tobias Koch and Amos Lapidoth}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=4595252}, isbn = {978-1-4244-2256-2}, year = {2008}, date = {2008-01-01}, booktitle = {2008 IEEE International Symposium on Information Theory}, pages = {1572--1576}, publisher = {IEEE}, address = {Toronto}, abstract = {This paper studies the capacity of discrete-time multipath fading channels. It is assumed that the number of paths is finite, i.e., that the channel output is influenced by the present and by the L previous channel inputs. A noncoherent channel model is considered where neither transmitter nor receiver are cognizant of the fading's realization, but both are aware of its statistic. The focus is on capacity at high signal-to-noise ratios (SNR). In particular, the capacity pre-loglog-defined as the limiting ratio of the capacity to loglog(SNR) as SNR tends to infinity-is studied. It is shown that, irrespective of the number of paths L, the capacity pre-loglog is 1.}, keywords = {channel capacity, Delay, discrete time systems, discrete-time channels, Entropy, Fading, fading channels, Frequency, Mathematical model, multipath channels, multipath fading channels, noncoherent channel model, Random variables, Signal to noise ratio, signal-to-noise ratios, SNR, statistics, Transmitters}, pubstate = {published}, tppubtype = {inproceedings} } This paper studies the capacity of discrete-time multipath fading channels. It is assumed that the number of paths is finite, i.e., that the channel output is influenced by the present and by the L previous channel inputs. A noncoherent channel model is considered where neither transmitter nor receiver are cognizant of the fading's realization, but both are aware of its statistic. The focus is on capacity at high signal-to-noise ratios (SNR). In particular, the capacity pre-loglog-defined as the limiting ratio of the capacity to loglog(SNR) as SNR tends to infinity-is studied. It is shown that, irrespective of the number of paths L, the capacity pre-loglog is 1. |

Koch, Tobias; Lapidoth, Amos Multipath Channels of Unbounded Capacity Inproceedings 2008 IEEE 25th Convention of Electrical and Electronics Engineers in Israel, pp. 640–644, IEEE, Eilat, 2008, ISBN: 978-1-4244-2481-8. Abstract | Links | BibTeX | Tags: channel capacity, discrete-time capacity, Entropy, Fading, fading channels, Frequency, H infinity control, Information rates, multipath channels, multipath fading channels, noncoherent, noncoherent capacity, path gains decay, Signal to noise ratio, statistics, Transmitters, unbounded capacity @inproceedings{Koch2008b, title = {Multipath Channels of Unbounded Capacity}, author = {Tobias Koch and Amos Lapidoth}, url = {http://ieeexplore.ieee.org/articleDetails.jsp?arnumber=4736611}, isbn = {978-1-4244-2481-8}, year = {2008}, date = {2008-01-01}, booktitle = {2008 IEEE 25th Convention of Electrical and Electronics Engineers in Israel}, pages = {640--644}, publisher = {IEEE}, address = {Eilat}, abstract = {The capacity of discrete-time, noncoherent, multipath fading channels is considered. It is shown that if the variances of the path gains decay faster than exponentially, then capacity is unbounded in the transmit power.}, keywords = {channel capacity, discrete-time capacity, Entropy, Fading, fading channels, Frequency, H infinity control, Information rates, multipath channels, multipath fading channels, noncoherent, noncoherent capacity, path gains decay, Signal to noise ratio, statistics, Transmitters, unbounded capacity}, pubstate = {published}, tppubtype = {inproceedings} } The capacity of discrete-time, noncoherent, multipath fading channels is considered. It is shown that if the variances of the path gains decay faster than exponentially, then capacity is unbounded in the transmit power. |