Signal & System Theory Group

Dr. Christian Lameiro

Akademischer Rat auf Zeit

M.E. (University of Cantabria, Spain), 2010
Ph.D. (University of Cantabria, Spain), 2015


Room: P1.7.01.4
Phone: +49 (0)5251 60-4367
[javascript protected email address]




Christian Lameiro received the Telecommunication Engineering degree in 2010, and the Master and Ph.D. degrees in electrical engineering in 2011 and 2015, respectively, from the University of Cantabria, Spain, He was a research assistant from 2010 to 2015 at the Advanced Signal Processing Group of the University of Cantabria, Spain. He was a visiting researcher with the Associate Institute for Signal Processing, Technical University of Munich, Munich, 2013. In 2014, he was a DAAD fellow at the SST group. He finally joined the SST group in 2015, where he currently holds an Akademischer Rat auf Zeit position. His current research interests include signal processing for wireless communications, multiple-input multiple-output (MIMO) systems and communications in interference-limited networks.

List of Publications

1

Information-theoretic analysis of a family of improper discrete constellations
(Ignacio Santamaria, Pedro M. Crespo, Christian Lameiro and Peter J. Schreier)
Entropy 20 (1), 2018.
DOI:10.3390/e20010045.
[BibTeX]
@article{Lameiro:2018ad,
author = {Santamaria, Ignacio and Crespo, Pedro M. and Lameiro, Christian and Schreier, Peter J.},
title = {Information-theoretic analysis of a family of improper discrete constellations},
year = {2018},
journal = {Entropy},
number = {1},
volume = {20},
doi = {10.3390/e20010045},
}
[Abstract]

2

Improper Gaussian signaling for multiple-access channels in underlay cognitive radio
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
Submitted to IEEE Trans. Wireless Comm., 2017.
Available at https://arxiv.org/abs/1711.09768 [BibTeX]
@article{Lameiro:2017ab,
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
title = {Improper {G}aussian signaling for multiple-access channels in underlay cognitive radio},
year = {2017},
journal = {Submitted to {IEEE} {T}rans.\ Wireless {C}omm.},
note = {Available at https://arxiv.org/abs/1711.09768},
}
[Abstract]

3

Spatial interference shaping for underlay MIMO cognitive networks
(Christian Lameiro, Wolfgang Utschick and Ignacio Santamaría)
Signal Process. 134,  pp. 174–184, 2017. [BibTeX]
@article{Lameiro:2016ab,
author = {Lameiro, Christian and Utschick, Wolfgang and Santamar{\'i}a, Ignacio},
title = {Spatial interference shaping for underlay {MIMO} cognitive networks},
year = {2017},
journal = {Signal Process.},
pages = {174–184},
volume = {134},
}
[Abstract]

4

Determining the dimension of the improper signal subspace in complex-valued data
(Tanuj Hasija, Christian Lameiro and Peter J. Schreier)
IEEE Signal Process. Lett. 24, November 2017. [BibTeX]
@article{HasijaSPletter:2016aa,
abstract = {Abstract: A complex-valued signal is improper if it is correlated with its complex conjugate. The dimension of the improper signal subspace, i.e., the number of improper components in a complex-valued measurement, is an important parameter and is unknown in most of the applications. In this letter, we introduce two approaches to estimate this dimension: one based on an information-theoretic criterion and the other based on hypothesis testing. We also present reduced-rank versions of these approaches that work for scenarios where the number of observations is comparable to or even smaller than the dimension of the data. Unlike other techniques for determining model orders, our techniques also work in the presence of additive colored noise.},
author = {Hasija, Tanuj and Lameiro, Christian and Schreier, Peter J.},
month = {{N}ovember},
title = {Determining the dimension of the improper signal subspace in complex-valued data},
year = {2017},
journal = {IEEE Signal Process. Lett.},
volume = {24},
}
[Abstract]
Abstract: A complex-valued signal is improper if it is correlated with its complex conjugate. The dimension of the improper signal subspace, i.e., the number of improper components in a complex-valued measurement, is an important parameter and is unknown in most of the applications. In this letter, we introduce two approaches to estimate this dimension: one based on an information-theoretic criterion and the other based on hypothesis testing. We also present reduced-rank versions of these approaches that work for scenarios where the number of observations is comparable to or even smaller than the dimension of the data. Unlike other techniques for determining model orders, our techniques also work in the presence of additive colored noise.

5

Rate region boundary of the SISO Z-interference channel with improper signaling
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
IEEE Trans. Comm. 65 (3),  pp. 1022–1034, March 2017. [BibTeX]
@article{Lameiro:2016ac,
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
month = {March},
title = {Rate region boundary of the {SISO Z}-interference channel with improper signaling},
year = {2017},
journal = {{IEEE} {T}rans.\ {C}omm.},
number = {3},
pages = {1022–1034},
volume = {65},
}
[Abstract]

6

Experimental evaluation of interference alignment for broadband WLAN systems
(Christian Lameiro, 'Oscar González, Jose A. García-Naya, Ignacio Santamaría and Luis Castedo)
EURASIP Journal on Wireless Communications and Networking, 2015.
DOI:10.1186/s13638-015-0409-z.
[BibTeX]
@article{LameiroGonzalezGarcia-Naya:2015:Experimental-evaluation-of-interference-alignment,
abstract = {In this paper, we present an experimental study on the performance of spatial interference alignment (IA) in indoor wireless local area network scenarios that use orthogonal frequency division multiplexing (OFDM) according to the physical-layer specifications of the IEEE 802.11a standard. Experiments have been carried out using a wireless network testbed capable of implementing a 3-user MIMO interference channel. We have implemented IA decoding schemes that can be designed according to distinct criteria (e.g., zero-forcing or MaxSINR). The measurement methodology has been validated considering practical issues like the number of OFDM training symbols used for channel estimation or feedback time. In case of asynchronous users, a time-domain IA decoding filter is also compared to its frequency-domain counterpart. We also evaluated the performance of IA from bit error ratio measurement-based results in comparison to different time-division multiple access transmission schemes. The comparison includes single- and multiple-antenna systems transmitting over the dominant mode of the MIMO channel. Our results indicate that spatial IA is suitable for practical indoor scenarios in which wireless channels often exhibit relatively large coherence times.},
author = {Lameiro, Christian and Gonz{\'a}lez, \'Oscar and Garc{\'i}a-Naya, Jose A. and Santamar{\'i}a, Ignacio and Castedo, Luis},
title = {Experimental evaluation of interference alignment for broadband {WLAN} systems},
year = {2015},
journal = {EURASIP Journal on Wireless Communications and Networking},
doi = {10.1186/s13638-015-0409-z},
}
[Abstract]
In this paper, we present an experimental study on the performance of spatial interference alignment (IA) in indoor wireless local area network scenarios that use orthogonal frequency division multiplexing (OFDM) according to the physical-layer specifications of the IEEE 802.11a standard. Experiments have been carried out using a wireless network testbed capable of implementing a 3-user MIMO interference channel. We have implemented IA decoding schemes that can be designed according to distinct criteria (e.g., zero-forcing or MaxSINR). The measurement methodology has been validated considering practical issues like the number of OFDM training symbols used for channel estimation or feedback time. In case of asynchronous users, a time-domain IA decoding filter is also compared to its frequency-domain counterpart. We also evaluated the performance of IA from bit error ratio measurement-based results in comparison to different time-division multiple access transmission schemes. The comparison includes single- and multiple-antenna systems transmitting over the dominant mode of the MIMO channel. Our results indicate that spatial IA is suitable for practical indoor scenarios in which wireless channels often exhibit relatively large coherence times.

7

Benefits of improper signaling for underlay cognitive radio
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
IEEE Wireless Comm. Lett. 4,  pp. 22–25, February 2015.
DOI:10.1109/LWC.2014.2360179.
[BibTeX]
@article{LameiroSantamariaSchreier:2015:Benefits-of-Improper-Signaling-for-Underlay,
abstract = {In this letter we study the potential benefits of improper signaling for a secondary user (SU) in underlay cognitive radio networks. We consider a basic yet illustrative scenario in which the primary user (PU) always transmit proper Gaussian signals and has a minimum rate constraint. After parameterizing the SU transmit signal in terms of its power and circularity coefficient (which measures the degree of impropriety), we prove that the SU improves its rate by transmitting improper signals only when the ratio of the squared modulus between the SU-PU interference link and the SU direct link exceeds a given threshold. As a by-product of this analysis, we obtain the optimal circularity coefficient that must be used by the SU depending on its power budget. Some simulation results show that the SU benefits from the transmission of improper signals especially when the PU is not highly loaded.},
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
month = {{F}ebruary},
title = {Benefits of improper signaling for underlay cognitive radio},
year = {2015},
journal = {IEEE Wireless Comm. Lett.},
pages = {22–25},
volume = {4},
doi = {10.1109/LWC.2014.2360179},
}
[Abstract]
In this letter we study the potential benefits of improper signaling for a secondary user (SU) in underlay cognitive radio networks. We consider a basic yet illustrative scenario in which the primary user (PU) always transmit proper Gaussian signals and has a minimum rate constraint. After parameterizing the SU transmit signal in terms of its power and circularity coefficient (which measures the degree of impropriety), we prove that the SU improves its rate by transmitting improper signals only when the ratio of the squared modulus between the SU-PU interference link and the SU direct link exceeds a given threshold. As a by-product of this analysis, we obtain the optimal circularity coefficient that must be used by the SU depending on its power budget. Some simulation results show that the SU benefits from the transmission of improper signals especially when the PU is not highly loaded.

8

A quadratically convergent method for interference alignment in MIMO interference channels
(Óscar González, Christian Lameiro and Ignacio Santamaría)
IEEE Signal Process. Lett. 21,  pp. 1423–1427, November 2014.
DOI:10.1109/LSP.2014.2338132.
[BibTeX]
@article{GonzalezLameiroSantamaria:2014:A-Quadratically-Convergent-Method-for-Interference,
abstract = {Alternating minimization and steepest descent are commonly used strategies to obtain interference alignment (IA) solutions in the $K$-user multiple-input multiple-output (MIMO) interference channel (IC). Although these algorithms are shown to converge monotonically, they experience a poor convergence rate, requiring an enormous amount of iterations which substantially increases with the size of the scenario. To alleviate this drawback, in this letter we resort to the Gauss-Newton (GN) method, which is well-known to experience quadratic convergence when the iterates are sufficiently close to the optimum. We discuss the convergence properties of the proposed GN algorithm and provide several numerical examples showing that it always converges to the optimum with quadratic rate, reducing dramatically the required computation time in comparison to other algorithms, hence paving a new way for the design of IA algorithms.},
author = {Gonz{\'a}lez, {\'O}scar and Lameiro, Christian and Santamar{\'i}a, Ignacio},
month = {{N}ovember},
title = {A quadratically convergent method for interference alignment in {MIMO} interference channels},
year = {2014},
journal = {IEEE Signal Process. Lett.},
pages = {1423–1427},
volume = {21},
doi = {10.1109/LSP.2014.2338132},
}
[Abstract]
Alternating minimization and steepest descent are commonly used strategies to obtain interference alignment (IA) solutions in the K-user multiple-input multiple-output (MIMO) interference channel (IC). Although these algorithms are shown to converge monotonically, they experience a poor convergence rate, requiring an enormous amount of iterations which substantially increases with the size of the scenario. To alleviate this drawback, in this letter we resort to the Gauss-Newton (GN) method, which is well-known to experience quadratic convergence when the iterates are sufficiently close to the optimum. We discuss the convergence properties of the proposed GN algorithm and provide several numerical examples showing that it always converges to the optimum with quadratic rate, reducing dramatically the required computation time in comparison to other algorithms, hence paving a new way for the design of IA algorithms.

9

Amplify-and-forward strategies in the two-way relay channel with analog Tx-Rx beamforming
(Christian Lameiro, Javier Vía and Ignacio Santamaría)
IEEE Trans. Veh. Techn. 62,  pp. 642–654, February 2013.
DOI:10.1109/TVT.2012.2226612.
[BibTeX]
@article{LameiroViaSantamaria:2013:Amplify-and-Forward-Strategies-in-the-Two-Way-Relay,
abstract = {In this paper, we study the multiple-input multiple-output two-way relay channel (MIMO-TWRC) when the nodes and relay use analog beamforming. Following the amplify-andforward (AF) strategy, the problem consists of finding the transmit and receive beamformers of the nodes and relay (as well as the power allocated to each one) that achieve the boundary of the optimal rate region. To solve it, we first express the optimal node beamformers in terms of relay beamformers and then show that the optimal rate region can efficiently be characterized using convex optimization techniques. We also extend our study to the multiple-relay scenario when the source nodes are single antenna and propose a distributed algorithm to compute the relay beamforming matrices. The proposed algorithm consists of solving two different subproblems. First, each individual TWRC is optimized independently. Next, a distributed beamforming is applied to make the signals from the relays add up coherently at the source nodes. Numerical examples are provided to illustrate the performance of the proposed techniques and to compare the performance of analog beamforming architectures against conventional MIMO schemes that operate at the baseband.},
author = {Lameiro, Christian and V{\'i}a, Javier and Santamar{\'i}a, Ignacio},
month = {{F}ebruary},
title = {Amplify-and-forward strategies in the two-way relay channel with analog {Tx-Rx} beamforming},
year = {2013},
journal = {IEEE Trans. Veh. Techn.},
pages = {642–654},
volume = {62},
doi = {10.1109/TVT.2012.2226612},
}
[Abstract]
In this paper, we study the multiple-input multiple-output two-way relay channel (MIMO-TWRC) when the nodes and relay use analog beamforming. Following the amplify-andforward (AF) strategy, the problem consists of finding the transmit and receive beamformers of the nodes and relay (as well as the power allocated to each one) that achieve the boundary of the optimal rate region. To solve it, we first express the optimal node beamformers in terms of relay beamformers and then show that the optimal rate region can efficiently be characterized using convex optimization techniques. We also extend our study to the multiple-relay scenario when the source nodes are single antenna and propose a distributed algorithm to compute the relay beamforming matrices. The proposed algorithm consists of solving two different subproblems. First, each individual TWRC is optimized independently. Next, a distributed beamforming is applied to make the signals from the relays add up coherently at the source nodes. Numerical examples are provided to illustrate the performance of the proposed techniques and to compare the performance of analog beamforming architectures against conventional MIMO schemes that operate at the baseband.

10

Improper signaling for OFDM underlay cognitive radio systems
(Mohammad Soleymani, Christian Lameiro, Peter J. Schreier and Ignacio Santamaría)
Proc. IEEE Statistical Signal Processing Workshop (SSP), Freiburg, Germany, June 2018. [BibTeX]
@inproceedings{SoleymaniSSP18,
address = {Freiburg, Germany},
author = {Mohammad Soleymani and Lameiro, Christian and Schreier, Peter J. and Santamar{\'\i}a, Ignacio},
booktitle = {Proc. IEEE Statistical Signal Processing Workshop (SSP)},
month = {June},
title = {Improper signaling for {OFDM} underlay cognitive radio systems},
year = {2018},
}
[Abstract]

11

Performance analysis of maximally improper signaling for multiple-antenna systems
(Christian Lameiro, Ignacio Santamaria and Peter J. Schreier)
Proc. IEEE Wireless Comm. Networking Conf. (WCNC), Barcelona, Spain, April 2018. [BibTeX]
@inproceedings{Lameiro:2018aa,
address = {Barcelona, Spain},
author = {Lameiro, Christian and Santamaria, Ignacio and Schreier, Peter J.},
booktitle = {Proc. IEEE Wireless Comm. Networking Conf. (WCNC)},
month = {April},
title = {Performance analysis of maximally improper signaling for multiple-antenna systems},
year = {2018},
}
[Abstract]

12

A sparse CCA algorithm with application to model-order selection for small sample support
(C. Lameiro and P. J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 4721–4725, New Orleans, USA, March 2017.
DOI:10.1109/ICASSP.2017.7953052.
[BibTeX]
@inproceedings{Lameiro:2017df,
address = {New Orleans, USA},
author = {C. Lameiro and P. J. Schreier},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {March},
title = {A sparse {CCA} algorithm with application to model-order selection for small sample support},
year = {2017},
pages = {4721–4725},
doi = {10.1109/ICASSP.2017.7953052},
}
[Abstract]

13

Cross-validation techniques for determining the number of correlated components between two data sets when the number of samples is very small
(Christian Lameiro and Peter J. Schreier)
Proc. Asilomar Conf. Signals Syst. Computers, Pacific Grove, CA, USA, November 2016. [BibTeX]
@inproceedings{Lameiro:2016aa,
address = {Pacific Grove, CA, USA},
author = {Lameiro, Christian and Schreier, Peter J.},
booktitle = {{P}roc.\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {November},
title = {Cross-validation techniques for determining the number of correlated components between two data sets when the number of samples is very small},
year = {2016},
}
[Abstract]

14

Maximally improper interference in underlay cognitive radio networks
(Christian Lameiro, I. Santamaría, W. Utschick and Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Shanghai, China, March 2016. [BibTeX]
@inproceedings{Lameiro:2016aa,
address = {Shanghai, China},
author = {Lameiro, Christian and Santamar{\'i}a, I. and Utschick, W. and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}arch},
title = {Maximally improper interference in underlay cognitive radio networks},
year = {2016},
}
[Abstract]

15

An experimental evaluation of broadband spatial IA for uncoordinated MIMO-OFDM systems
(Jacobo Fanjul, Christian Lameiro, Ignacio Santamaría, Jose A. García-Naya and Luis Castedo)
Proc. IEEE Int. Conf. Digital Signal Process., Singapore, July 2015. [BibTeX]
@inproceedings{FanjulLameiroSantamaria:2015:An-Experimental-Evaluation-of-Broadband-Spatial,
abstract = {In this paper we present an experimental study on the performance of spatial Interference Alignment (IA) in broadband indoor wireless local area network scenarios that use Orthogonal Frequency Division Multiplexing (OFDM) according to the IEEE 802.11a physical-layer specifications. Experiments have been carried out using a wireless network testbed made up of six nodes equipped with Multiple-Input Multiple-Output (MIMO) radio interfaces. This setup allows the implementation of a 3-user MIMO interference channel. We have implemented different IA decoding schemes that operate either before or after the Fast Fourier Transform block. IA has been experimentally evaluated comparing both approaches to analyze its performance in synchronous and asynchronous transmissions. Our results indicate that spatial IA performs satisfactorily in practical broadband indoor scenarios in which wireless channels often exhibit relatively large coherence times.},
address = {Singapore},
author = {Fanjul, Jacobo and Lameiro, Christian and Santamar{\'i}a, Ignacio and Garc{\'i}a-Naya, Jose A. and Castedo, Luis},
booktitle = {Proc. IEEE Int. Conf. Digital Signal Process.},
month = {July},
title = {An experimental evaluation of broadband spatial {IA} for uncoordinated {MIMO-OFDM} systems},
year = {2015},
}
[Abstract]
In this paper we present an experimental study on the performance of spatial Interference Alignment (IA) in broadband indoor wireless local area network scenarios that use Orthogonal Frequency Division Multiplexing (OFDM) according to the IEEE 802.11a physical-layer specifications. Experiments have been carried out using a wireless network testbed made up of six nodes equipped with Multiple-Input Multiple-Output (MIMO) radio interfaces. This setup allows the implementation of a 3-user MIMO interference channel. We have implemented different IA decoding schemes that operate either before or after the Fast Fourier Transform block. IA has been experimentally evaluated comparing both approaches to analyze its performance in synchronous and asynchronous transmissions. Our results indicate that spatial IA performs satisfactorily in practical broadband indoor scenarios in which wireless channels often exhibit relatively large coherence times.

16

Analysis of maximally improper signalling schemes for underlay cognitive radio
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
Proc. IEEE Int. Conf. Comm., London, UK, June 2015. [BibTeX]
@inproceedings{LameiroSantamariaSchreier:2015:Analysis-of-maximally-improper-signalling,
abstract = {In this paper, the impact of improper Gaussian signaling is studied for an underlay cognitive radio (CR) scenario comprised of a primary user (PU), which has a rate constraint, and a secondary user (SU), both single-antenna. We first derive expressions for the achievable rate of the SU when it transmits proper and maximally improper Gaussian signals (assuming that the SU is solely limited by the CR constraint). These expressions depend on the channel gains to and from the SU through a single variable. Thereby, we observe that improper signaling is beneficial whenever the SU rate is below a threshold, which depends on the signal-to-noise ratio (SNR) and rate requirement of the PU. Furthermore, we provide bounds on the achievable gain that also depend only on the PU parameters. Then, the achievable rate is studied from a statistical viewpoint by deriving its cumulative distribution function considering a constant received SNR at the PU. In addition, we specialize this expression for the Z interference channel, for which the expected achievable rate is also derived. Numerical examples illustrate our claims and show that the SU may significantly benefit from using improper signaling.},
address = {London, UK},
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {C}omm.},
month = {{J}une},
title = {Analysis of maximally improper signalling schemes for underlay cognitive radio},
year = {2015},
}
[Abstract]
In this paper, the impact of improper Gaussian signaling is studied for an underlay cognitive radio (CR) scenario comprised of a primary user (PU), which has a rate constraint, and a secondary user (SU), both single-antenna. We first derive expressions for the achievable rate of the SU when it transmits proper and maximally improper Gaussian signals (assuming that the SU is solely limited by the CR constraint). These expressions depend on the channel gains to and from the SU through a single variable. Thereby, we observe that improper signaling is beneficial whenever the SU rate is below a threshold, which depends on the signal-to-noise ratio (SNR) and rate requirement of the PU. Furthermore, we provide bounds on the achievable gain that also depend only on the PU parameters. Then, the achievable rate is studied from a statistical viewpoint by deriving its cumulative distribution function considering a constant received SNR at the PU. In addition, we specialize this expression for the Z interference channel, for which the expected achievable rate is also derived. Numerical examples illustrate our claims and show that the SU may significantly benefit from using improper signaling.

17

Interference-temperature limit for cognitive radio networks with MIMO primary users
(Christian Lameiro, Wolfgang Utschick and Ignacio Santamaría)
Proc. Asilomar Conf. Signals Syst. Computers, Pacific Grove, CA, USA, November 2014.
DOI:10.1109/ACSSC.2014.7094625.
[BibTeX]
@inproceedings{LameiroUtschickSantamaria:2014:Interference-Temperature-Limit-for-Cognitive-Radio,
abstract = {This paper derives the interference-temperature (IT) limit for a multi-antenna primary user (PU) with a rate constraint. While in the case of a single-antenna PU there is a one-to-one mapping between IT and achievable rate, this correspondence does not hold anymore when a multiple-input multiple-output (MIMO) system is considered. In such cases, the spatial distribution of the interference must be taken into account, since it strongly affects the PU performance. To this end, we derive a closed-form expression for the maximum IT that can be tolerated by identifying the worst-case interference covariance matrix, which results in a multilevel waterfilling problem.},
address = {Pacific Grove, CA, USA},
author = {Lameiro, Christian and Utschick, Wolfgang and Santamar{\'i}a, Ignacio},
booktitle = {{P}roc.\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Interference-temperature limit for cognitive radio networks with {MIMO} primary users},
year = {2014},
doi = {10.1109/ACSSC.2014.7094625},
}
[Abstract]
This paper derives the interference-temperature (IT) limit for a multi-antenna primary user (PU) with a rate constraint. While in the case of a single-antenna PU there is a one-to-one mapping between IT and achievable rate, this correspondence does not hold anymore when a multiple-input multiple-output (MIMO) system is considered. In such cases, the spatial distribution of the interference must be taken into account, since it strongly affects the PU performance. To this end, we derive a closed-form expression for the maximum IT that can be tolerated by identifying the worst-case interference covariance matrix, which results in a multilevel waterfilling problem.

18

Interference shaping constraints for underlay MIMO interference channels
(Christian Lameiro, Ignacio Santamaría and Wolfgang Utschick)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Florence, Italy, May 2014.
DOI:10.1109/ICASSP.2014.6855020.
[BibTeX]
@inproceedings{LameiroSantamariaUtschick:2014:Interference-Shaping-Constraints-for-Underlay,
abstract = {In this paper, a cognitive radio (CR) scenario comprised of a secondary interference channel (IC) and a primary point-to-point link (PPL) is studied, when the former interferes the latter. In order to satisfy a given rate requirement at the PPL, typical approaches impose an interference temperature constraint (IT). When the PPL transmits multiple streams, however, the spatial structure of the interference comes into play. In such cases, we show that spatial interference shaping constraints can provide higher sum-rate performance to the IC while ensuring the required rate at the PPL. Then, we extend the interference leakage minimization algorithm (MinIL) to incorporate such constraints. An additional power control step is included in the optimization procedure to improve the sum-rate when the interference alignment (IA) problem becomes infeasible due to the additional constraint. Numerical examples are provided to illustrate the effectiveness of the spatial shaping constraint in comparison to IT when the PPL transmits multiple data streams.},
address = {Florence, Italy},
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Utschick, Wolfgang},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {Interference shaping constraints for underlay {MIMO} interference channels},
year = {2014},
doi = {10.1109/ICASSP.2014.6855020},
}
[Abstract]
In this paper, a cognitive radio (CR) scenario comprised of a secondary interference channel (IC) and a primary point-to-point link (PPL) is studied, when the former interferes the latter. In order to satisfy a given rate requirement at the PPL, typical approaches impose an interference temperature constraint (IT). When the PPL transmits multiple streams, however, the spatial structure of the interference comes into play. In such cases, we show that spatial interference shaping constraints can provide higher sum-rate performance to the IC while ensuring the required rate at the PPL. Then, we extend the interference leakage minimization algorithm (MinIL) to incorporate such constraints. An additional power control step is included in the optimization procedure to improve the sum-rate when the interference alignment (IA) problem becomes infeasible due to the additional constraint. Numerical examples are provided to illustrate the effectiveness of the spatial shaping constraint in comparison to IT when the PPL transmits multiple data streams.

19

Spatial shaping and precoding design for underlay MIMO interference channels
(Christian Lameiro, Wolfgang Utschick and Ignacio Santamaría)
Proc.Int. ITG Work. Smart Antennas, Erlangen, Germany, March 2014. [BibTeX]
@inproceedings{LameiroUtschickSantamaria:2014:Spatial-Shaping-and-Precoding-Design,
abstract = {In this paper, we study coexistence issues between an underlay single-beam interference channel (IC) and a primary point-to-point link (PPL) that has a rate constraint. We derive spatial shaping constraints at the secondary transmitters and show that they generalize the so-called interference temperature (IT) when the PPL transmits multiple streams. We propose a successive convex approximation algorithm to compute the spatial shaping matrices that maximize the allowed transmit power of the IC while ensuring the rate constraint of the PPL. Then, we provide general interference leakage minimization (MinIL) and maximum signal-to-interference-plus-noise ratio (MaxSINR) algorithms that can incorporate both types of constraint. An additional power control step is included in the optimization procedure to further improve the sum-rate of the IC. Different numerical examples are provided to illustrate the effectiveness of the proposed techniques and to compare the performance improvement when the IC is subject to spatial shaping constraints in comparison to IT.},
address = {Erlangen, Germany},
author = {Lameiro, Christian and Utschick, Wolfgang and Santamar{\'i}a, Ignacio},
booktitle = {Proc.Int. ITG Work. Smart Antennas},
month = {{M}arch},
title = {Spatial shaping and precoding design for underlay {MIMO} interference channels},
year = {2014},
}
[Abstract]
In this paper, we study coexistence issues between an underlay single-beam interference channel (IC) and a primary point-to-point link (PPL) that has a rate constraint. We derive spatial shaping constraints at the secondary transmitters and show that they generalize the so-called interference temperature (IT) when the PPL transmits multiple streams. We propose a successive convex approximation algorithm to compute the spatial shaping matrices that maximize the allowed transmit power of the IC while ensuring the rate constraint of the PPL. Then, we provide general interference leakage minimization (MinIL) and maximum signal-to-interference-plus-noise ratio (MaxSINR) algorithms that can incorporate both types of constraint. An additional power control step is included in the optimization procedure to further improve the sum-rate of the IC. Different numerical examples are provided to illustrate the effectiveness of the proposed techniques and to compare the performance improvement when the IC is subject to spatial shaping constraints in comparison to IT.

20

An interference alignment algorithm for structured channels
(Christian Lameiro, Óscar González and Ignacio Santamaría)
Proc. IEEE Work. Signal Process. Advances in Wireless Comm., Darmstadt, Germany, June 2013.
DOI:10.1109/SPAWC.2013.6612059.
[BibTeX]
@inproceedings{LameiroGonzalezSantamaria:2013:An-Interference-Alignment-Algorithm-for-Structured,
abstract = {In this paper we propose a new interference alignment (IA) algorithm specifically designed to work with structured channels (e.g., diagonal or block-diagonal). Multiple-input multiple-output (MIMO) structured channels arise when symbol extensions -either in time or frequency- are employed jointly with the spatial dimension in the design of the precoders. In this case, the rank constraint in the direct channels must explicitly be taken into account into the optimization problem to ensure that there is no degrees-of-freedom (DoF) loss. To this end, we propose an algorithm that minimizes the interference leakage while ensuring that the direct links are full rank and the transmitters satisfy a power constraint. The algorithm is based upon an alternating optimization procedure, which solves a generalized eigenvalue problem at each step. We show through simulations the advantages of the proposed algorithm in several scenarios that use symbols extensions or improper (a.k.a. asymmetric) signalling.},
address = {Darmstadt, Germany},
author = {Lameiro, Christian and Gonz{\'a}lez, {\'O}scar and Santamar{\'i}a, Ignacio},
booktitle = {Proc. IEEE Work. Signal Process. Advances in Wireless Comm.},
month = {{J}une},
title = {An interference alignment algorithm for structured channels},
year = {2013},
doi = {10.1109/SPAWC.2013.6612059},
}
[Abstract]
In this paper we propose a new interference alignment (IA) algorithm specifically designed to work with structured channels (e.g., diagonal or block-diagonal). Multiple-input multiple-output (MIMO) structured channels arise when symbol extensions -either in time or frequency- are employed jointly with the spatial dimension in the design of the precoders. In this case, the rank constraint in the direct channels must explicitly be taken into account into the optimization problem to ensure that there is no degrees-of-freedom (DoF) loss. To this end, we propose an algorithm that minimizes the interference leakage while ensuring that the direct links are full rank and the transmitters satisfy a power constraint. The algorithm is based upon an alternating optimization procedure, which solves a generalized eigenvalue problem at each step. We show through simulations the advantages of the proposed algorithm in several scenarios that use symbols extensions or improper (a.k.a. asymmetric) signalling.

21

Degrees-of-freedom for the 4-user SISO interference channel with improper signaling
(Christian Lameiro and Ignacio Santamaría)
Proc. IEEE Int. Conf. Comm., Budapest, Hungary, June 2013.
DOI:10.1109/ICC.2013.6655009.
[BibTeX]
@inproceedings{LameiroSantamaria:2013:Degrees-of-Freedom-for-the-4-User-SISO-Interference,
abstract = {It has been recently shown that for the 3-user single-input single-output (SISO) interference channel with constant channel coefficients, a maximum of 1.2 degrees-of-freedom (DoF) are achievable using linear interference alignment schemes when improper (a.k.a. asymmetric) Gaussian signaling is applied. In this paper, we study the 4-user SISO interference channel and provide inner and outer bounds for the total number of DoF achievable for this channel. We prove that at least 4/3 DoF are achievable for the 4-user channel using also linear interference alignment techniques and improper signaling. A simple converse proof shows that no more than 8/5 DoF are achievable for this scheme. Simulation results seem to indicate that the inner bound is in fact tight for this channel, and serve to illustrate the sum-rate improvement with respect to time division multiple access (TDMA) techniques.},
address = {Budapest, Hungary},
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {C}omm.},
month = {{J}une},
title = {Degrees-of-freedom for the 4-user {SISO} interference channel with improper signaling},
year = {2013},
doi = {10.1109/ICC.2013.6655009},
}
[Abstract]
It has been recently shown that for the 3-user single-input single-output (SISO) interference channel with constant channel coefficients, a maximum of 1.2 degrees-of-freedom (DoF) are achievable using linear interference alignment schemes when improper (a.k.a. asymmetric) Gaussian signaling is applied. In this paper, we study the 4-user SISO interference channel and provide inner and outer bounds for the total number of DoF achievable for this channel. We prove that at least 4/3 DoF are achievable for the 4-user channel using also linear interference alignment techniques and improper signaling. A simple converse proof shows that no more than 8/5 DoF are achievable for this scheme. Simulation results seem to indicate that the inner bound is in fact tight for this channel, and serve to illustrate the sum-rate improvement with respect to time division multiple access (TDMA) techniques.

22

Computing the degrees of freedom for arbitrary MIMO interference channels
(Óscar González, Christian Lameiro, Javier Vía, Carlos Beltrán and Ignacio Santamaría)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Vancouver, Canada, May 2013.
DOI:10.1109/ICASSP.2013.6638491.
[BibTeX]
@inproceedings{GonzalezLameiroVia:2013:Computing-the-Degrees-of-Freedom-for-Arbitrary,
abstract = {In this paper we provide an efficient procedure to compute the total number of degrees of freedom (DoF), achievable by linear beamforming, of the $K$-user multiple-input multiple-output (MIMO) interference channel with an arbitrary number of Tx-Rx antennas at each link. Firstly, we derive an analytical outer bound that generalizes the results that exist for the symmetric $K$-user $M\timesN$ interference channel. Secondly, we obtain a tighter bound by solving a convex optimization problem that includes as constraints the DoF characterizations for point-to-point MIMO links and for 2-user interference channels. The solution to this convex problem admits an interesting waterfilling interpretation. Finally, exploiting this outer bound and using a recently proposed feasibility test, we show that it is possible to obtain the DoF for any interference channel in an efficient way. Some simulations results are included to illustrate the tightness of the derived bounds, as well as to study the DoF achievable for the 4-user channel when we distribute the total number of antennas among users and between transmitters and receivers in different ways.},
address = {Vancouver, Canada},
author = {Gonz{\'a}lez, {\'O}scar and Lameiro, Christian and V{\'i}a, Javier and Beltr{\'a}n, Carlos and Santamar{\'i}a, Ignacio},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {Computing the degrees of freedom for arbitrary {MIMO} interference channels},
year = {2013},
doi = {10.1109/ICASSP.2013.6638491},
}
[Abstract]
In this paper we provide an efficient procedure to compute the total number of degrees of freedom (DoF), achievable by linear beamforming, of the K-user multiple-input multiple-output (MIMO) interference channel with an arbitrary number of Tx-Rx antennas at each link. Firstly, we derive an analytical outer bound that generalizes the results that exist for the symmetric K-user MtimesN interference channel. Secondly, we obtain a tighter bound by solving a convex optimization problem that includes as constraints the DoF characterizations for point-to-point MIMO links and for 2-user interference channels. The solution to this convex problem admits an interesting waterfilling interpretation. Finally, exploiting this outer bound and using a recently proposed feasibility test, we show that it is possible to obtain the DoF for any interference channel in an efficient way. Some simulations results are included to illustrate the tightness of the derived bounds, as well as to study the DoF achievable for the 4-user channel when we distribute the total number of antennas among users and between transmitters and receivers in different ways.

23

Pre- and post-FFT Interference Leakage Minimization for MIMO OFDM networks
(Christian Lameiro, Óscar González, Javier Vía, Ignacio Santamaría and Robert W. Heath Jr.)
Proc. Int. Symp. Wireless Comm. Syst., Paris, France, August 2012.
DOI:10.1109/ISWCS.2012.6328429.
[BibTeX]
@inproceedings{LameiroGonzalezVia:2012:Pre–and-Post-FFT-Interference-Leakage,
abstract = {Interference alignment (IA) has been shown to achieve the maximum degrees of freedom in the multiple-input multiple-output (MIMO) K-user interference channel (IFC). In the presence of frequency-selective channels, orthogonal frequency-division multiplexing (OFDM) is typically used to deal with the multipath nature of the channel. While IA techniques can be applied in a per-subcarrier basis (post-FFT), the existence of symbol timing offsets (STOs) between the desired and the interfering OFDM symbols decreases the system performance dramatically. To solve this problem, we design pre-FFT precoders and decoders for single-beam MIMO IFCs for OFDM transmissions. Since the IA decoders operate before the FFT, they mitigate the interference before synchronization takes place. We show that our proposed scheme improves the system performance when STOs occur, in comparison with traditional post-FFT IA techniques. We provide simulation results to compare post- and pre-FFT beamforming techniques and to illustrate the performance of the proposed method.},
address = {Paris, France},
author = {Lameiro, Christian and Gonz{\'a}lez, {\'O}scar and V{\'i}a, Javier and Santamar{\'i}a, Ignacio and Heath Jr., Robert W.},
booktitle = {Proc. Int. Symp. Wireless Comm. Syst.},
month = {{A}ugust},
title = {Pre- and post-{FFT} Interference Leakage Minimization for {MIMO OFDM} networks},
year = {2012},
doi = {10.1109/ISWCS.2012.6328429},
}
[Abstract]
Interference alignment (IA) has been shown to achieve the maximum degrees of freedom in the multiple-input multiple-output (MIMO) K-user interference channel (IFC). In the presence of frequency-selective channels, orthogonal frequency-division multiplexing (OFDM) is typically used to deal with the multipath nature of the channel. While IA techniques can be applied in a per-subcarrier basis (post-FFT), the existence of symbol timing offsets (STOs) between the desired and the interfering OFDM symbols decreases the system performance dramatically. To solve this problem, we design pre-FFT precoders and decoders for single-beam MIMO IFCs for OFDM transmissions. Since the IA decoders operate before the FFT, they mitigate the interference before synchronization takes place. We show that our proposed scheme improves the system performance when STOs occur, in comparison with traditional post-FFT IA techniques. We provide simulation results to compare post- and pre-FFT beamforming techniques and to illustrate the performance of the proposed method.

24

A distributed algorithm for two-way multiple-relay networks
(Christian Lameiro, Javier Vía and Ignacio Santamaría)
Proc. IEEE Sensor Array and Multichannel Signal Process. Work., Hoboken, NJ, USA, June 2012.
DOI:10.1109/SAM.2012.6250440.
[BibTeX]
@inproceedings{LameiroViaSantamaria:2012:A-Distributed-Algorithm-for-Two-Way-Multiple-Relay,
abstract = {In this paper we propose an efficient transmission strategy for the two-way relay channel (TWRC) with multiple relays, when these are multiple-input multiple-output (MIMO) transceivers that apply the amplify-and-forward (AF) protocol. Although the optimal beamforming strategy is known, it requires a central node, with channel state information (CSI) of the entire network, to compute all the beamforming matrices, which is impractical. To reduce the overhead, in this paper we present a distributed algorithm for the computation of the relay beamforming matrices. The proposed algorithm divides the problem in two stages. First, each relay computes its own beamforming matrix in parallel using only local CSI. Next, a distributed beamforming is applied to make the signals add up coherently at the nodes. Although the proposed algorithm is suboptimal, we show through simulations that it performs very close to the optimal achievable rate region.},
address = {Hoboken, NJ, USA},
author = {Lameiro, Christian and V{\'i}a, Javier and Santamar{\'i}a, Ignacio},
booktitle = {{P}roc.\ {IEEE} {S}ensor {A}rray and {M}ultichannel {S}ignal {P}rocess. {W}ork.},
month = {{J}une},
title = {A distributed algorithm for two-way multiple-relay networks},
year = {2012},
doi = {10.1109/SAM.2012.6250440},
}
[Abstract]
In this paper we propose an efficient transmission strategy for the two-way relay channel (TWRC) with multiple relays, when these are multiple-input multiple-output (MIMO) transceivers that apply the amplify-and-forward (AF) protocol. Although the optimal beamforming strategy is known, it requires a central node, with channel state information (CSI) of the entire network, to compute all the beamforming matrices, which is impractical. To reduce the overhead, in this paper we present a distributed algorithm for the computation of the relay beamforming matrices. The proposed algorithm divides the problem in two stages. First, each relay computes its own beamforming matrix in parallel using only local CSI. Next, a distributed beamforming is applied to make the signals add up coherently at the nodes. Although the proposed algorithm is suboptimal, we show through simulations that it performs very close to the optimal achievable rate region.

25

Interference leakage minimization for convolutive MIMO interference channels
(Óscar González, Christian Lameiro, Javier Vía, Ignacio Santamaría and Robert W. Heath Jr.)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Kyoto, Japan, March 2012.
DOI:10.1109/ICASSP.2012.6288506.
[BibTeX]
@inproceedings{GonzalezLameiroVia:2012:Interference-Leakage-Minimization-for-Convolutive,
abstract = {An alternating optimization algorithm was recently proposed for the K-user multiple-input multiple-output (MIMO) interference channel. For flat-fading channels and feasible problems, this algorithm successfully aligns the interfering signals exploiting the spatial dimensions. In this paper, we consider the case in which all pairwise MIMO channels are frequency-selective (convolutive), and the users transmit broadband signals using a single-carrier scheme. Unlike the flat-fading case, for frequency-selective channels it is necessary to add a spectral mask in the frequency response of the precoders and decoders to avoid trivial solutions. We show in the paper that each step of the alternating minimization algorithm can be reformulated as a convex optimization problem in which the autocorrelation function of the precoders or decoders is obtained. Upon convergence, a final spectral factorization stage must be applied to obtain the precoders and decoders from their autocorrelation functions. Simulation results are provided to illustrate the performance of the proposed algorithm.},
address = {Kyoto, Japan},
author = {Gonz{\'a}lez, {\'O}scar and Lameiro, Christian and V{\'i}a, Javier and Santamar{\'i}a, Ignacio and Heath Jr., Robert W.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}arch},
title = {Interference leakage minimization for convolutive {MIMO} interference channels},
year = {2012},
doi = {10.1109/ICASSP.2012.6288506},
}
[Abstract]
An alternating optimization algorithm was recently proposed for the K-user multiple-input multiple-output (MIMO) interference channel. For flat-fading channels and feasible problems, this algorithm successfully aligns the interfering signals exploiting the spatial dimensions. In this paper, we consider the case in which all pairwise MIMO channels are frequency-selective (convolutive), and the users transmit broadband signals using a single-carrier scheme. Unlike the flat-fading case, for frequency-selective channels it is necessary to add a spectral mask in the frequency response of the precoders and decoders to avoid trivial solutions. We show in the paper that each step of the alternating minimization algorithm can be reformulated as a convex optimization problem in which the autocorrelation function of the precoders or decoders is obtained. Upon convergence, a final spectral factorization stage must be applied to obtain the precoders and decoders from their autocorrelation functions. Simulation results are provided to illustrate the performance of the proposed algorithm.

26

Capacity region of the two-way multi-antenna relay channel with analog Tx-Rx beamforming
(Christian Lameiro, Alfredo Nazábal, Fouad Gholam, Javier Vía and Ignacio Santamaría)
Proc. Int. ICST Conf. Mobile Lightweight Wireless Syst., Bilbao, Spain, May 2011.
DOI:10.1007/978-3-642-29479-2_1.
[BibTeX]
@inproceedings{LameiroNazabalGholam:2011:Capacity-Region-of-the-Two-Way-Multi-Antenna,
abstract = {In this paper we study the multiple-input multiple-output two-way relay channel (MIMO-TWRC) when the nodes use analog beamforming. Following the amplify-and-forward (AF) strategy, the problem consists of finding the transmit and receive beamformers of the nodes and the relay, and the power allocated to each one, that achieve the boundary of the capacity region. We express the optimal node beamformers in terms of the relay beamformers, and show that the capacity region can be efficiently characterized using convex optimization techniques. Numerical examples are provided to illustrate the results of this paper, and to compare the capacity region achieved by analog beamforming against the conventional MIMO schemes that operate at the baseband.},
address = {Bilbao, Spain},
author = {Lameiro, Christian and Naz{\'a}bal, Alfredo and Gholam, Fouad and V{\'i}a, Javier and Santamar{\'i}a, Ignacio},
booktitle = {Proc. Int. ICST Conf. Mobile Lightweight Wireless Syst.},
month = {{M}ay},
title = {Capacity region of the two-way multi-antenna relay channel with analog {Tx-Rx} beamforming},
year = {2011},
doi = {10.1007/978-3-642-29479-2_1},
}
[Abstract]
In this paper we study the multiple-input multiple-output two-way relay channel (MIMO-TWRC) when the nodes use analog beamforming. Following the amplify-and-forward (AF) strategy, the problem consists of finding the transmit and receive beamformers of the nodes and the relay, and the power allocated to each one, that achieve the boundary of the capacity region. We express the optimal node beamformers in terms of the relay beamformers, and show that the capacity region can be efficiently characterized using convex optimization techniques. Numerical examples are provided to illustrate the results of this paper, and to compare the capacity region achieved by analog beamforming against the conventional MIMO schemes that operate at the baseband.

27

Capacity region of the multiantenna gaussian broadcast channel with analog TX-RX beamforming
(Ignacio Santamaría, Javier Vía, Alfredo Nazábal and Christian Lameiro)
Proc. Int. ICST Conf. Comm. and Networking in China, Beijing, China, August 2010. [BibTeX]
@inproceedings{SantamariaViaNazabal:2010:Capacity-region-of-the-multiantenna-Gaussian,
abstract = {In this paper we characterize the capacity boundary of the two-user multiple-input multiple-output (MIMO) broadcast channel (BC) when the multiantenna terminals use analog beamforming at both sides of the link. Basically, the problem reduces to finding the optimal transmit direction and the optimal power allocation between users to operate at any point on the boundary. We show that both optimization problems can be solved in closed form. First, exploiting the fact that any Pareto optimal pair of rates must also be Pareto optimal of a convex region defined by the channel energies, the optimal transmit direction is shown to be the principal eigenvector of a matrix. Second, the optimal power allocation is obtained by exploiting again the Pareto optimality of the sought pair of rates. Although this paper focuses on the BC, the obtained results also serve to characterize the dual multiple-access channel (MAC). Moreover, this explicit characterization of the boundary can be extended to a $K$-user system for $K>2$. Numerical examples are provided to illustrate the results of the paper, and to show the enlargement of the capacity region achieved by analog beamforming in comparison to a single-input single-output (SISO) BC.},
address = {Beijing, China},
author = {Santamar{\'i}a, Ignacio and V{\'i}a, Javier and Naz{\'a}bal, Alfredo and Lameiro, Christian},
booktitle = {Proc. Int. ICST Conf. Comm. and Networking in China},
month = {{A}ugust},
title = {Capacity region of the multiantenna gaussian broadcast channel with analog {TX-RX} beamforming},
year = {2010},
}
[Abstract]
In this paper we characterize the capacity boundary of the two-user multiple-input multiple-output (MIMO) broadcast channel (BC) when the multiantenna terminals use analog beamforming at both sides of the link. Basically, the problem reduces to finding the optimal transmit direction and the optimal power allocation between users to operate at any point on the boundary. We show that both optimization problems can be solved in closed form. First, exploiting the fact that any Pareto optimal pair of rates must also be Pareto optimal of a convex region defined by the channel energies, the optimal transmit direction is shown to be the principal eigenvector of a matrix. Second, the optimal power allocation is obtained by exploiting again the Pareto optimality of the sought pair of rates. Although this paper focuses on the BC, the obtained results also serve to characterize the dual multiple-access channel (MAC). Moreover, this explicit characterization of the boundary can be extended to a K-user system for K>2. Numerical examples are provided to illustrate the results of the paper, and to show the enlargement of the capacity region achieved by analog beamforming in comparison to a single-input single-output (SISO) BC.