On the number of interference alignment solutions for the K-user MIMO channel with constant coefficients

In this paper, we study the number of different interference alignment (IA) solutions in a K-user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in this paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average 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, its average (properly scaled) gives the number of solutions. For single-beam systems, the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas, and streams per user.The work of Ó. González and I. Santamaría was supported by MICINN (Spanish Ministry for Science and Innovation) under grants TEC2013-47141-C4-3-R (RACHEL), TEC2010-19545-C04-03 (COSIMA), CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS) and FPU grant AP2009-1105. Carlos Beltrán was partially supported by MICINN grant MTM2010-16051

Homotopy continuation for spatial interference alignment in arbitrary MIMO X networks

In this paper, we propose an algorithm to design interference alignment (IA) precoding and decoding matrices for arbitrary MIMO X networks. The proposed algorithm is rooted in the homotopy continuation techniques commonly used to solve systems of nonlinear equations. Homotopy methods find the solution of a target system by smoothly deforming the solution of a start system which can be trivially solved. Unlike previously proposed IA algorithms, the homotopy continuation technique allows us to solve the IA problem for both unstructured (i.e., generic) and structured channels such as those that arise when time or frequency symbol extensions are jointly employed with the spatial dimension. To this end, we consider an extended system of bilinear equations that include the standard alignment equations to cancel the interference, and a new set of bilinear equations that preserve the desired dimensionality of the signal spaces at the intended receivers. We propose a simple method to obtain the start system by randomly choosing a set of precoders and decoders, and then finding a set of channels satisfying the system equations, which is a linear problem. Once the start system is available, standard prediction and correction techniques are applied to track the solution all the way to the target system. We analyze the convergence of the proposed algorithm and prove that, for many feasible systems and a sufficiently small continuation parameter, the algorithm converges with probability one to a perfect IA solution. The simulation results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom in a variety of MIMO X networks with or without symbol extensions. Further, the algorithm provides insights into the feasibility of IA in MIMO X networks for which theoretical results are scarce.This work has been supported by the Ministerio de Economía y Competitividad (MINECO) of Spain, under grants TEC2013-47141-C4-R (RACHEL), TEC2016-75067-C4-4-R (CARMEN), MTM2014-57590-P, and FPI grant BES-2014-069786

Constrained riemannian noncoherent constellations for the MIMO multiple access channel

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).This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/AEI/10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033

Noncoherent multiuser Grassmannian constellations for the MIMO multiple access channel

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 in [1]. Although this PEP expression was previously considered intractable for optimization, in this work we derive a closed-form formula for its unconstrained gradient and perform Riemannian optimization in the Grassmannian manifold to design multiuser constellations for the MIMO MAC with state-of-the-art performance in terms of symbol error rate (SER).This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltran was also partly supported under grant ´PID2020-113887GB-I00 funded by MCIN/AEI /10.13039/501100011033.The work of I. Santamaría was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/AEI /10.13039/501100011033

Union bound minimization approach for designing grassmannian constellations

In this paper, we propose an algorithm for designing unstructured Grassmannian constellations for noncoherent multiple-input multiple-output (MIMO) communications over Rayleigh block-fading channels. Unlike the majority of existing unitary space-time or Grassmannian constellations, which are typically designed to maximize the minimum distance between codewords, in this work we employ the asymptotic pairwise error probability (PEP) union bound (UB) of the constellation as the design criterion. In addition, the proposed criterion allows the design of MIMO Grassmannian constellations specifically optimized for a given number of receiving antennas. A rigorous derivation of the gradient of the asymptotic UB on a Cartesian product of Grassmann manifolds, is the main technical ingredient of the proposed gradient descent algorithm. A simple modification of the proposed cost function, which weighs each pairwise error term in the UB according to the Hamming distance between the binary labels assigned to the respective codewords, allows us to jointly solve the constellation design and the bit labeling problem. Our simulation results show that the constellations designed with the proposed method outperform other structured and unstructured Grassmannian designs in terms of symbol error rate (SER) and bit error rate (BER), for a wide range of scenarios.This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033

Constellations on the sphere with efficient encoding-decoding for noncoherent communications

In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for onthe-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray-like coding. Simulation results show that Grass-Lattice has symbol and bit error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature and a coherent pilot-based scheme.This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltran was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019- 104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033. A short preliminary version of this paper was presented at the 2022 IEEE Global Communications Conference: Signal Processing for Communications (Globecom 2022 SPC)

A measure preserving mapping for structured Grassmannian constellations in SIMO channels

In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray coding. Simulation results show that Grass-Lattice has symbol error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature.This work was supported by Huawei Technologies Sweden, under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/AEI /10.13039/501100011033. The work ofI. Santamaria was also partly supported under grant PID2019-104958RB-C43(ADELE) funded by MCIN/ AEI /10.13039/501100011033

Desigualdad de género en procesos educativos incidente en violencia hacia la mujer

La igualdad de género es considerada un asunto de derecho y condición para la salud y el desarrollo de las familias y las sociedades; muchos autores consideran que la educación es clave para reducir las desigualdades de grupo. El objetivo de esta investigación es indagar sobre el impacto de la desigualdad de género en los procesos educativos y su incidencia en la violencia hacia la mujer en Ecuador. La revisión de literatura, el debate y discusión sobre el tema se ha ampliado y profundizado en los últimos años; evidenciando, una mayor sensibilización y concientización social. Esto, ha hecho posible visualizar estudios de casos relacionados con la participación de la mujer en espacios educativos, y la violencia de género en general. Aunque el desarrollo de estudios de género en Ecuador no ha sido prolífero, se espera que estableciendo un marco legal y teórico, esta situación cambie en los próximos años

Desigualdad de género en procesos educativos incidente en violencia hacia la mujer

Gender equality is considered a matter of law and condition for the health and development of families and societies; many authors believe that education is key to reducing group inequalities. The aim of this research is to investigate the impact of gender inequality in educational processes and its incidence on violence against women in Ecuador. The review of literature, debate and discussion on the subject has expanded and deepened in recent years; evidencing, greater and social awareness. This has made it possible to visualize case studies related to the participation of women in educational spaces, and gender violence in general. Although the development of gender studies in Ecuador has not been prolific, it is expected that by establishing a legal and theoretical framework, this situation will change in the coming years.La igualdad de género es considerada un asunto de derecho y condición para la salud y el desarrollo de las familias y las sociedades; muchos autores consideran que la educación es clave para reducir las desigualdades de grupo. El objetivo de esta investigación es indagar sobre el impacto de la desigualdad de género en los procesos educativos y su incidencia en la violencia hacia la mujer en Ecuador. La revisión de literatura, el debate y discusión sobre el tema se ha ampliado y profundizado en los últimos años; evidenciando, una mayor sensibilización y concientización social. Esto, ha hecho posible visualizar estudios de casos relacionados con la participación de la mujer en espacios educativos, y la violencia de género en general. Aunque el desarrollo de estudios de género en Ecuador no ha sido prolífero, se espera que estableciendo un marco legal y teórico, esta situación cambie en los próximos años

Recensiones [Revista de Historia Económica Año X Invierno 1992 n. 1 pp. 147-189]

Editada en la Universidad Carlos IIICarlos Llorca Baus. La Compañía Trasatlántica en las Campañas de Ultramar. (Por Eugenio Torres Villanueva).-- Stephan Haggard. Pathways from the Periphery. The Politics of Growth in the Newly Industrializing Countries. (Por Pablo Bustelo).-- Joám Carmona Badía. El atraso industrial de Galicia. Auge y liquidación de las manufacturas textiles (1750-1900). (Por Concepción de Castro).-- Francisco Comín Comín. Hacienda y Economía en la España Contemporánea (1800-1936). (Por Juan Hernández Andreu).-- Anthony de Jasay. Social Contract, Free Ride. (Por Carlos Rodríguez Braun.).-- José Luis Cardoso. O pensamiento económico em Portugal nos fináis do século XVIII, 1780-1808. (Por Luis Perdices Blas).-- Gottfried Plumpe. Die I. G. Farbenindustrie AG. Wirtschaft, Technik und Politik, 1904-1945. (Por Nuria Puig).-- Alfonso Sánchez Hormigo. Valentín Andrés Álvarez (Un economista del 27). (Por Juan Velarde Fuertes).-- Alfred W. Crosby. Imperialismo ecológico. La expansión biológica de Europa, 900-1900. (Por José Domingo Sánchez Martínez).-- Julián Casanova. La historia social y los historiadores. (Por Eloy Fernández Clemente).-- Manuel Lucena Salmoral. Características del comercio exterior de la provincia de Caracas durante el sexenio revolucionario (1807-1812). (Por Emiliano Gil Blanco.).-- Michael Kitson y Solomos Solomou. Protectionism and Economic Revival: the British Interwar Economy. (Por Concha Betrán Pérez)Publicad