Interference Alignment in MIMO Networks: Feasibility and Transceiver Design

Wireless communications have gone through an exponential growth in the last several years and it is forecast that this growth will be sustained for the coming decades. This ever-increasing demand for radio resources is now facing one of its main limitations: inter-user interference, arising from the fact of multiple users accessing the propagation medium simultaneously which limits the total amount of data that can be reliably communicated through the wireless links. Traditionally, interference has been dealt with by allocating disjoint channel resources to distinct users. However, the advent of a novel interference coordination technique known as interference alignment (IA) brought to the forefront the promise of a much larger spectral efficiency. This dissertation revolves around the idea of linear interference alignment for a network consisting of several mutually interfering transmitter-receiver pairs, which is com-monly known as interference channel. In particular, we consider that each of the nodes is equipped with multiple antennas and exploits the spatial dimension to perform interference alignment. This work explores the problem of linear spatial domain interference alignment in three different facets. Our first contribution is to analyze the conditions, i.e., number of antennas, users and streams, under which IA is feasible. For this task, we distinguish between systems in which each user transmits a single stream of information (single-beam systems) and those in which multiple streams per user are transmitted (multi-beam systems). For single-beam systems, the problem translates into determining the feasibility of a network flow problem. We show that this problem admits a closed-form solution with a time-complexity that is lin-ear in the number of users. For multi-beam systems, we propose a numerical feasibility test that completely settles the question of IA feasibility for arbitrary networks and is shown to belong to the bounded-error probabilistic polynomial time (BPP) complexity class. The second contribution, consists in generalizing the aforementioned feasibility results to characterize the number of existing IA solutions. We show that di?erent IA solutions can exhibit dramatically different performances and, consequently, the number of solutions turns out to be an important metric to evaluate the ability of a system to improve its performance in terms of sum-rate or robustness while maintaining perfect IA. Again, we provide a closed-form expression for the number of solutions in single-beam systems, highlighting interesting connections with classical combinatorial and graph-theoretic problems. For multi-beam systems, we approximate the number of solutions numerically by means of Monte Carlo integration. Finally, our contributions conclude with the design of two algorithms for the computation of IA solutions. The first of them, based on a numerical technique known as homotopy continuation, is theoretically guaranteed to converge to any optimum solution (provided that it exists) and can systematically compute different IA solutions in parallel. The second, essentially a Gauss-Newton method, can be used to reliably compute IA solutions with computation times that are remarkably shorter than those required by the fastest available algorithm at the time of writing. In view of the results provided by the proposed algorithms, we explore the possibility of computing a small subset of solutions and picking the best one according to a certain metric. For example, our numerical results show that the sum-rate performance obtained by picking the best out of a small number of solutions rivals that obtained by the best-performing state-of-the-art algorithm.