@article {637,
	title = {Union Bound Minimization Approach for Designing Grassmannian Constellations},
	journal = {IEEE Transactions on Communications},
	volume = {71},
	year = {2023},
	month = {April},
	pages = {1940-1952},
	doi = {10.1109/TCOMM.2023.3244965},
	author = {Cuevas, Diego and Alvarez-Vizoso, Javier and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@article {655,
	title = {Statistical Characterization of the Chordal Product Determinant of Grassmannian Codes},
	journal = {Information and Inference: A journal of the IMA},
	volume = {12},
	year = {2023},
	month = {September},
	doi = {10.1093/imaiai/iaad035},
	author = {Alvarez-Vizoso, Javier and Beltr{\'a}n, Carlos and Cuevas, Diego and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@article {335,
	title = {On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients},
	journal = {IEEE Transaction on Information Theory},
	volume = {61},
	year = {2015},
	month = {November},
	pages = {6028-6048},
	abstract = {In this paper, we study the number of different interference alignment (IA) solutions that exists for the K-user multiple-input multiple-output (MIMO) interference channels with constant coefficients, when the alignment is performed via beamforming and without symbol extensions. When counting the number of IA solutions for a given problem, the most interesting case happens when the number of equations and variables of the polynomial system of equations are the same and the system is feasible. In this situation, the number of IA solutions is finite and, as we show in this paper, is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of solutions is the scaled average over a subset of the solution variety (formed by all triplets of channels, precoders and decoders satisfying the IA polynomial equations) of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, the average of the determinant (properly scaled) gives us the number of solutions. Our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas and streams per user.},
	doi = {10.1109/TIT.2015.2482493},
	author = {Gonz{\'a}lez, {\'O}scar and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio}
}
@article {482,
	title = {Homotopy Continuation for Spatial Interference Alignment in Arbitrary MIMO X Networks},
	journal = {IEEE Transactions on Signal Processing},
	volume = {65},
	year = {2017},
	month = {April},
	pages = {1752-1764},
	issn = {1053-587X},
	doi = {10.1109/TSP.2016.2637310},
	author = {Fanjul, Jacobo and Gonz{\'a}lez, {\'O}scar and Santamar{\'\i}a, Ignacio and Beltr{\'a}n, Carlos}
}
@article {336,
	title = {A Feasibility Test for Linear Interference Alignment in MIMO Channels with Constant Coefficients},
	journal = {IEEE Transactions on Information Theory.},
	volume = {60},
	year = {2014},
	month = {March},
	pages = {1840-1856},
	abstract = {In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input-multiple-output (MIMO) channels with constant coefficients for any number of users, antennas, and streams per user, and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. In particular, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians), and the solution set (channels, decoders, and precoders satisfying the IA polynomial equations), not only as algebraic sets, but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class.},
	keywords = {algebraic geometry, differential topology, interference alignment, MIMO interference channel, polynomial equations},
	issn = {0018-9448},
	doi = {10.1109/TIT.2014.2301440},
	author = {Gonz{\'a}lez, {\'O}scar and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio}
}
@article {639,
	title = {Constrained Riemannian Noncoherent Constellations for the MIMO Multiple Access Channel},
	journal = {IEEE Transactions on Information Theory},
	volume = {69},
	year = {2023},
	month = {July},
	pages = {4559-4578},
	abstract = {We consider the design of multiuser constellations for a multiple access channel (MAC) with $K$ users, with M antennas each, that transmit simultaneously to a receiver equipped with N antennas through a Rayleigh block-fading channel, when no channel state information (CSI) is available to either the transmitter or the receiver. In full-diversity scenarios where the coherence time is at least T >= (K+1)M, the proposed constellation design criterion is based on the asymptotic expression of the multiuser pairwise error probability (PEP) derived by Brehler and Varanasi (2001). In non-full diversity scenarios, for which the previous PEP expression is no longer valid, the proposed design criteria are based on proxies of the PEP recently proposed by Ngo and Yang (2021). Although both the PEP expression and its bounds or proxies were previously considered intractable for optimization, in this work we derive their respective unconstrained gradients. These gradients are in turn used in the optimization of the proposed cost functions in different Riemannian manifolds representing different power constraints. In particular, in addition to the standard unitary space-time modulation (USTM) leading to optimization on the Grassmann manifold, we consider a more relaxed per-codeword power constraint leading to optimization on the so-called oblique manifold, and an average power constraint leading to optimization on the so-called trace manifold. Equipped with these theoretical tools, we design multiuser constellations for the MIMO MAC in full-diversity and non-full-diversity scenarios with state-of-the-art performance in terms of symbol error rate (SER).},
	doi = {10.1109/TIT.2023.3249903},
	author = {Alvarez-Vizoso, Javier and Cuevas, Diego and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@article {650,
	title = {Constellations on the Sphere with Efficient Encoding-Decoding for Noncoherent Communications},
	journal = {IEEE Transactions on Wireless Communications},
	year = {2023},
	author = {Cuevas, Diego and Alvarez-Vizoso, Javier and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@conference {642,
	title = {Noncoherent Multiuser Grassmannian Constellations for the MIMO Multiple Access Channel},
	booktitle = {IEEE International Conference on Acoustics Speech and Signal Processing},
	year = {2023},
	month = {June},
	address = {Rhodes, Greece},
	author = {Alvarez-Vizoso, Javier and Cuevas, Diego and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@conference {629,
	title = {A measure preserving mapping for structured Grassmannian constellations in SIMO channels},
	booktitle = {IEEE Global Communications Conference (GLOBECOM)},
	year = {2022},
	month = {Dec.},
	address = {Rio de Janeiro, Brazil},
	author = {Cuevas, Diego and Alvarez-Vizoso, Javier and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@conference {278,
	title = {A general test to check the feasibility of linear interference alignment},
	booktitle = {IEEE International Symposium on Information Theory (ISIT 2012), Cambridge, MA, USA},
	year = {2012},
	month = {July, 2012},
	keywords = {feasibility, IA},
	author = {Gonz{\'a}lez, {\'O}scar and Santamar{\'\i}a, Ignacio and Beltr{\'a}n, Carlos}
}
@conference {359,
	title = {Finding the Number of Feasible Solutions for Linear Interference Alignment Problems},
	booktitle = {2013 IEEE International Symposium on Information Theory},
	year = {2013},
	month = {July},
	pages = {384--388},
	address = {Istanbul, Turkey},
	abstract = {In this paper, we study how many different solutions exist for a feasible interference alignment (IA) problem. We focus on linear IA schemes without symbol extensions for the K-user multiple-input multiple-output (MIMO) interference channel. When the IA problem is feasible and the number of variables matches the number of equations in the polynomial system, the number of solutions is known to be finite. Unfortunately, the exact number of solutions is only known for a few particular cases, mainly single-beam MIMO networks. In this paper, we prove that the number of IA solutions is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of solutions is the scaled average over a subset of the solution variety (formed by all triplets of channels, precoders and decoders satisfying the IA polynomial equations) of the determinant of certain Hermitian matrix related to the geometry of the problem. Our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas and streams per user.},
	isbn = {978-1-4799-0446-4},
	author = {Gonz{\'a}lez, {\'O}scar and Santamar{\'\i}a, Ignacio and Beltr{\'a}n, Carlos}
}
@conference {618,
	title = {A fast algorithm for designing Grassmannian constellations},
	booktitle = {25th ITG Workshop on Smart Antennas},
	year = {2021},
	month = {November},
	address = {EURECOM, French Riviera, France},
	author = {Cuevas, Diego and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
@conference {354,
	title = {Computing the Degrees of Freedom for Arbitrary MIMO Interference Channels},
	booktitle = {IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013)},
	year = {2013},
	month = {May},
	address = {Vancouver, Canada},
	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 x N 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.},
	keywords = {convex optimization, degrees of freedom, interference alignment, multiple-input multiple-output},
	isbn = {978-1-4799-0356-6},
	author = {Gonz{\'a}lez, {\'O}scar and Lameiro, Christian and V{\'\i}a, Javier and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio}
}
@conference {627,
	title = {Coherence-based subspace packings for MIMO noncoherent communications},
	booktitle = {30th European Signal Processing Conference (EUSIPCO 2022)},
	year = {2022},
	month = {August},
	address = {Belgrade, Serbia},
	author = {Alvarez-Vizoso, Javier and Cuevas, Diego and Beltr{\'a}n, Carlos and Santamar{\'\i}a, Ignacio and Tucek, V{\'\i}t and Peters, Gunnar}
}
