## 2015 |

## Inproceedings |

Luengo, David; Martino, Luca; Elvira, Victor; Bugallo, Monica F Efficient Linear Combination of Partial Monte Carlo Estimators Inproceedings 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4100–4104, IEEE, Brisbane, 2015, ISBN: 978-1-4673-6997-8. Abstract | Links | BibTeX | Tags: covariance matrices, efficient linear combination, Estimation, fusion, Global estimator, global estimators, least mean squares methods, linear combination, minimum mean squared error estimators, Monte Carlo estimation, Monte Carlo methods, partial estimator, partial Monte Carlo estimators, Xenon @inproceedings{Luengo2015bb, title = {Efficient Linear Combination of Partial Monte Carlo Estimators}, author = {David Luengo and Luca Martino and Victor Elvira and Monica F Bugallo}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7178742 http://www.tsc.uc3m.es/~velvira/papers/ICASSP2015_luengo.pdf}, doi = {10.1109/ICASSP.2015.7178742}, isbn = {978-1-4673-6997-8}, year = {2015}, date = {2015-04-01}, booktitle = {2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, pages = {4100--4104}, publisher = {IEEE}, address = {Brisbane}, abstract = {In many practical scenarios, including those dealing with large data sets, calculating global estimators of unknown variables of interest becomes unfeasible. A common solution is obtaining partial estimators and combining them to approximate the global one. In this paper, we focus on minimum mean squared error (MMSE) estimators, introducing two efficient linear schemes for the fusion of partial estimators. The proposed approaches are valid for any type of partial estimators, although in the simulated scenarios we concentrate on the combination of Monte Carlo estimators due to the nature of the problem addressed. Numerical results show the good performance of the novel fusion methods with only a fraction of the cost of the asymptotically optimal solution.}, keywords = {covariance matrices, efficient linear combination, Estimation, fusion, Global estimator, global estimators, least mean squares methods, linear combination, minimum mean squared error estimators, Monte Carlo estimation, Monte Carlo methods, partial estimator, partial Monte Carlo estimators, Xenon}, pubstate = {published}, tppubtype = {inproceedings} } In many practical scenarios, including those dealing with large data sets, calculating global estimators of unknown variables of interest becomes unfeasible. A common solution is obtaining partial estimators and combining them to approximate the global one. In this paper, we focus on minimum mean squared error (MMSE) estimators, introducing two efficient linear schemes for the fusion of partial estimators. The proposed approaches are valid for any type of partial estimators, although in the simulated scenarios we concentrate on the combination of Monte Carlo estimators due to the nature of the problem addressed. Numerical results show the good performance of the novel fusion methods with only a fraction of the cost of the asymptotically optimal solution. |

## 2013 |

## Inproceedings |

Alvarado, Alex; Brannstrom, Fredrik; Agrell, Erik; Koch, Tobias High-SNR Asymptotics of Mutual Information for Discrete Constellations Inproceedings 2013 IEEE International Symposium on Information Theory, pp. 2274–2278, IEEE, Istanbul, 2013, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: AWGN channels, discrete constellations, Entropy, Fading, Gaussian Q-function, high-SNR asymptotics, IP networks, least mean squares methods, minimum mean-square error, MMSE, Mutual information, scalar additive white Gaussian noise channel, Signal to noise ratio, signal-to-noise ratio, Upper bound @inproceedings{Alvarado2013b, title = {High-SNR Asymptotics of Mutual Information for Discrete Constellations}, author = {Alex Alvarado and Fredrik Brannstrom and Erik Agrell and Tobias Koch}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6620631}, issn = {2157-8095}, year = {2013}, date = {2013-01-01}, booktitle = {2013 IEEE International Symposium on Information Theory}, pages = {2274--2278}, publisher = {IEEE}, address = {Istanbul}, abstract = {The asymptotic behavior of the mutual information (MI) at high signal-to-noise ratio (SNR) for discrete constellations over the scalar additive white Gaussian noise channel is studied. Exact asymptotic expressions for the MI for arbitrary one-dimensional constellations and input distributions are presented in the limit as the SNR tends to infinity. Asymptotics of the minimum mean-square error (MMSE) are also developed. It is shown that for any input distribution, the MI and the MMSE have an asymptotic behavior proportional to a Gaussian Q-function, whose argument depends on the minimum Euclidean distance of the constellation and the SNR. Closed-form expressions for the coefficients of these Q-functions are calculated.}, keywords = {AWGN channels, discrete constellations, Entropy, Fading, Gaussian Q-function, high-SNR asymptotics, IP networks, least mean squares methods, minimum mean-square error, MMSE, Mutual information, scalar additive white Gaussian noise channel, Signal to noise ratio, signal-to-noise ratio, Upper bound}, pubstate = {published}, tppubtype = {inproceedings} } The asymptotic behavior of the mutual information (MI) at high signal-to-noise ratio (SNR) for discrete constellations over the scalar additive white Gaussian noise channel is studied. Exact asymptotic expressions for the MI for arbitrary one-dimensional constellations and input distributions are presented in the limit as the SNR tends to infinity. Asymptotics of the minimum mean-square error (MMSE) are also developed. It is shown that for any input distribution, the MI and the MMSE have an asymptotic behavior proportional to a Gaussian Q-function, whose argument depends on the minimum Euclidean distance of the constellation and the SNR. Closed-form expressions for the coefficients of these Q-functions are calculated. |

## 2009 |

## Journal Articles |

Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Gaussian Process Regressors for Multiuser Detection in DS-CDMA Systems Journal Article IEEE Transactions on Communications, 57 (8), pp. 2339–2347, 2009, ISSN: 0090-6778. Abstract | Links | BibTeX | Tags: analytical nonlinear multiuser detectors, code division multiple access, communication systems, Detectors, digital communication, digital communications, DS-CDMA systems, Gaussian process for regressi, Gaussian process regressors, Gaussian processes, GPR, Ground penetrating radar, least mean squares methods, maximum likelihood, maximum likelihood detection, maximum likelihood estimation, mean square error methods, minimum mean square error, MMSE, Multiaccess communication, Multiuser detection, nonlinear estimator, nonlinear state-ofthe- art solutions, radio receivers, Receivers, regression analysis, Support vector machines @article{Murillo-Fuentes2009, title = {Gaussian Process Regressors for Multiuser Detection in DS-CDMA Systems}, author = {Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5201027}, issn = {0090-6778}, year = {2009}, date = {2009-01-01}, journal = {IEEE Transactions on Communications}, volume = {57}, number = {8}, pages = {2339--2347}, abstract = {In this paper we present Gaussian processes for Regression (GPR) as a novel detector for CDMA digital communications. Particularly, we propose GPR for constructing analytical nonlinear multiuser detectors in CDMA communication systems. GPR can easily compute the parameters that describe its nonlinearities by maximum likelihood. Thereby, no cross-validation is needed, as it is typically used in nonlinear estimation procedures. The GPR solution is analytical, given its parameters, and it does not need to solve an optimization problem for building the nonlinear estimator. These properties provide fast and accurate learning, two major issues in digital communications. The GPR with a linear decision function can be understood as a regularized MMSE detector, in which the regularization parameter is optimally set. We also show the GPR receiver to be a straightforward nonlinear extension of the linear minimum mean square error (MMSE) criterion, widely used in the design of these receivers. We argue the benefits of this new approach in short codes CDMA systems where little information on the users' codes, users' amplitudes or the channel is available. The paper includes some experiments to show that GPR outperforms linear (MMSE) and nonlinear (SVM) state-ofthe- art solutions.}, keywords = {analytical nonlinear multiuser detectors, code division multiple access, communication systems, Detectors, digital communication, digital communications, DS-CDMA systems, Gaussian process for regressi, Gaussian process regressors, Gaussian processes, GPR, Ground penetrating radar, least mean squares methods, maximum likelihood, maximum likelihood detection, maximum likelihood estimation, mean square error methods, minimum mean square error, MMSE, Multiaccess communication, Multiuser detection, nonlinear estimator, nonlinear state-ofthe- art solutions, radio receivers, Receivers, regression analysis, Support vector machines}, pubstate = {published}, tppubtype = {article} } In this paper we present Gaussian processes for Regression (GPR) as a novel detector for CDMA digital communications. Particularly, we propose GPR for constructing analytical nonlinear multiuser detectors in CDMA communication systems. GPR can easily compute the parameters that describe its nonlinearities by maximum likelihood. Thereby, no cross-validation is needed, as it is typically used in nonlinear estimation procedures. The GPR solution is analytical, given its parameters, and it does not need to solve an optimization problem for building the nonlinear estimator. These properties provide fast and accurate learning, two major issues in digital communications. The GPR with a linear decision function can be understood as a regularized MMSE detector, in which the regularization parameter is optimally set. We also show the GPR receiver to be a straightforward nonlinear extension of the linear minimum mean square error (MMSE) criterion, widely used in the design of these receivers. We argue the benefits of this new approach in short codes CDMA systems where little information on the users' codes, users' amplitudes or the channel is available. The paper includes some experiments to show that GPR outperforms linear (MMSE) and nonlinear (SVM) state-ofthe- art solutions. |

## 2008 |

## Journal Articles |

Perez-Cruz, Fernando; Murillo-Fuentes, Juan Jose; Caro, S Nonlinear Channel Equalization With Gaussian Processes for Regression Journal Article IEEE Transactions on Signal Processing, 56 (10), pp. 5283–5286, 2008, ISSN: 1053-587X. Abstract | Links | BibTeX | Tags: Channel estimation, digital communications receivers, equalisers, equalization, Gaussian processes, kernel adaline, least mean squares methods, maximum likelihood estimation, nonlinear channel equalization, nonlinear equalization, nonlinear minimum mean square error estimator, regression, regression analysis, short training sequences, Support vector machines @article{Perez-Cruz2008c, title = {Nonlinear Channel Equalization With Gaussian Processes for Regression}, author = {Fernando Perez-Cruz and Juan Jose Murillo-Fuentes and S Caro}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4563433}, issn = {1053-587X}, year = {2008}, date = {2008-01-01}, journal = {IEEE Transactions on Signal Processing}, volume = {56}, number = {10}, pages = {5283--5286}, abstract = {We propose Gaussian processes for regression (GPR) as a novel nonlinear equalizer for digital communications receivers. GPR's main advantage, compared to previous nonlinear estimation approaches, lies on their capability to optimize the kernel hyperparameters by maximum likelihood, which improves its performance significantly for short training sequences. Besides, GPR can be understood as a nonlinear minimum mean square error estimator, a standard criterion for training equalizers that trades off the inversion of the channel and the amplification of the noise. In the experiment section, we show that the GPR-based equalizer clearly outperforms support vector machine and kernel adaline approaches, exhibiting outstanding results for short training sequences.}, keywords = {Channel estimation, digital communications receivers, equalisers, equalization, Gaussian processes, kernel adaline, least mean squares methods, maximum likelihood estimation, nonlinear channel equalization, nonlinear equalization, nonlinear minimum mean square error estimator, regression, regression analysis, short training sequences, Support vector machines}, pubstate = {published}, tppubtype = {article} } We propose Gaussian processes for regression (GPR) as a novel nonlinear equalizer for digital communications receivers. GPR's main advantage, compared to previous nonlinear estimation approaches, lies on their capability to optimize the kernel hyperparameters by maximum likelihood, which improves its performance significantly for short training sequences. Besides, GPR can be understood as a nonlinear minimum mean square error estimator, a standard criterion for training equalizers that trades off the inversion of the channel and the amplification of the noise. In the experiment section, we show that the GPR-based equalizer clearly outperforms support vector machine and kernel adaline approaches, exhibiting outstanding results for short training sequences. |

## Inproceedings |

Perez-Cruz, Fernando; Rodrigues, Miguel R D; Verdu, Sergio Optimal Precoding for Digital Subscriber Lines Inproceedings 2008 IEEE International Conference on Communications, pp. 1200–1204, IEEE, Beijing, 2008, ISBN: 978-1-4244-2075-9. Abstract | Links | BibTeX | Tags: Bit error rate, channel matrix diagonalization, Communications Society, Computer science, digital subscriber lines, DSL, Equations, fixed-point equation, Gaussian channels, least mean squares methods, linear codes, matrix algebra, MIMO, MIMO communication, MIMO Gaussian channel, minimum mean squared error method, MMSE, multiple-input multiple-output communication, Mutual information, optimal linear precoder, precoding, Telecommunications, Telephony @inproceedings{Perez-Cruz2008a, title = {Optimal Precoding for Digital Subscriber Lines}, author = {Fernando Perez-Cruz and Miguel R D Rodrigues and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4533270}, isbn = {978-1-4244-2075-9}, year = {2008}, date = {2008-01-01}, booktitle = {2008 IEEE International Conference on Communications}, pages = {1200--1204}, publisher = {IEEE}, address = {Beijing}, abstract = {We determine the linear precoding policy that maximizes the mutual information for general multiple-input multiple-output (MIMO) Gaussian channels with arbitrary input distributions, by capitalizing on the relationship between mutual information and minimum mean squared error (MMSE). The optimal linear precoder can be computed by means of a fixed- point equation as a function of the channel and the input constellation. We show that diagonalizing the channel matrix does not maximize the information transmission rate for nonGaussian inputs. A full precoding matrix may significantly increase the information transmission rate, even for parallel non-interacting channels. We illustrate the application of our results to typical Gigabit DSL systems.}, keywords = {Bit error rate, channel matrix diagonalization, Communications Society, Computer science, digital subscriber lines, DSL, Equations, fixed-point equation, Gaussian channels, least mean squares methods, linear codes, matrix algebra, MIMO, MIMO communication, MIMO Gaussian channel, minimum mean squared error method, MMSE, multiple-input multiple-output communication, Mutual information, optimal linear precoder, precoding, Telecommunications, Telephony}, pubstate = {published}, tppubtype = {inproceedings} } We determine the linear precoding policy that maximizes the mutual information for general multiple-input multiple-output (MIMO) Gaussian channels with arbitrary input distributions, by capitalizing on the relationship between mutual information and minimum mean squared error (MMSE). The optimal linear precoder can be computed by means of a fixed- point equation as a function of the channel and the input constellation. We show that diagonalizing the channel matrix does not maximize the information transmission rate for nonGaussian inputs. A full precoding matrix may significantly increase the information transmission rate, even for parallel non-interacting channels. We illustrate the application of our results to typical Gigabit DSL systems. |

Rodrigues, Miguel R D; Perez-Cruz, Fernando; Verdu, Sergio Multiple-Input Multiple-Output Gaussian Channels: Optimal Covariance for Non-Gaussian Inputs Inproceedings 2008 IEEE Information Theory Workshop, pp. 445–449, IEEE, Porto, 2008, ISBN: 978-1-4244-2269-2. Abstract | Links | BibTeX | Tags: Binary phase shift keying, covariance matrices, Covariance matrix, deterministic MIMO Gaussian channel, fixed-point equation, Gaussian channels, Gaussian noise, Information rates, intersymbol interference, least mean squares methods, Magnetic recording, mercury-waterfilling power allocation policy, MIMO, MIMO communication, minimum mean-squared error, MMSE, MMSE matrix, multiple-input multiple-output system, Multiple-Input Multiple-Output Systems, Mutual information, Optimal Input Covariance, Optimization, Telecommunications @inproceedings{Rodrigues2008, title = {Multiple-Input Multiple-Output Gaussian Channels: Optimal Covariance for Non-Gaussian Inputs}, author = {Miguel R D Rodrigues and Fernando Perez-Cruz and Sergio Verdu}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4578704}, isbn = {978-1-4244-2269-2}, year = {2008}, date = {2008-01-01}, booktitle = {2008 IEEE Information Theory Workshop}, pages = {445--449}, publisher = {IEEE}, address = {Porto}, abstract = {We investigate the input covariance that maximizes the mutual information of deterministic multiple-input multipleo-utput (MIMO) Gaussian channels with arbitrary (not necessarily Gaussian) input distributions, by capitalizing on the relationship between the gradient of the mutual information and the minimum mean-squared error (MMSE) matrix. We show that the optimal input covariance satisfies a simple fixed-point equation involving key system quantities, including the MMSE matrix. We also specialize the form of the optimal input covariance to the asymptotic regimes of low and high snr. We demonstrate that in the low-snr regime the optimal covariance fully correlates the inputs to better combat noise. In contrast, in the high-snr regime the optimal covariance is diagonal with diagonal elements obeying the generalized mercury/waterfilling power allocation policy. Numerical results illustrate that covariance optimization may lead to significant gains with respect to conventional strategies based on channel diagonalization followed by mercury/waterfilling or waterfilling power allocation, particularly in the regimes of medium and high snr.}, keywords = {Binary phase shift keying, covariance matrices, Covariance matrix, deterministic MIMO Gaussian channel, fixed-point equation, Gaussian channels, Gaussian noise, Information rates, intersymbol interference, least mean squares methods, Magnetic recording, mercury-waterfilling power allocation policy, MIMO, MIMO communication, minimum mean-squared error, MMSE, MMSE matrix, multiple-input multiple-output system, Multiple-Input Multiple-Output Systems, Mutual information, Optimal Input Covariance, Optimization, Telecommunications}, pubstate = {published}, tppubtype = {inproceedings} } We investigate the input covariance that maximizes the mutual information of deterministic multiple-input multipleo-utput (MIMO) Gaussian channels with arbitrary (not necessarily Gaussian) input distributions, by capitalizing on the relationship between the gradient of the mutual information and the minimum mean-squared error (MMSE) matrix. We show that the optimal input covariance satisfies a simple fixed-point equation involving key system quantities, including the MMSE matrix. We also specialize the form of the optimal input covariance to the asymptotic regimes of low and high snr. We demonstrate that in the low-snr regime the optimal covariance fully correlates the inputs to better combat noise. In contrast, in the high-snr regime the optimal covariance is diagonal with diagonal elements obeying the generalized mercury/waterfilling power allocation policy. Numerical results illustrate that covariance optimization may lead to significant gains with respect to conventional strategies based on channel diagonalization followed by mercury/waterfilling or waterfilling power allocation, particularly in the regimes of medium and high snr. |