| Interference has been a fundamental performance bottleneck in wireless communication. Conventional interference control schemes, which are mainly based on channel orthogonalization, are non-capacity achieving in general. To improve the performance of wireless networks, cooperative interference control schemes are proposed. In particular, interference alignment (IA), a recently developed interference control scheme, achieves optimal capacity scaling in a wide range of wireless networks. The key idea of IA is to reduce the effect of aggregated interference by aligning interference from different transmitters into a lower dimensional subspace at each receiver. However, to achieve the optimal capacity scaling, classical IA schemes require infinite dimension of signal space, which is difficult to implement in practice. To overcome this problem, researchers have proposed IA designs with signal space dimension limited by the number of antennas.;However, despite the numerous works dedicated to IA, when the signal space has finite dimension, two fundamental questions remain open in general: 1) Under what network topology is IA feasible? 2) In a feasible network, how can we find an IA solution? The feasibility analysis of IA is difficult as the IA constraints are sets of non-linear equations, for which no systematic tool exists to characterize the feasible region. Finding solutions of IA is challenging due to the non-convex nature of the interference minimization problem.;In this thesis, by adopting tools from algebraic geometry, we establish a framework which shows the (almost surely) equivalence of the feasibility of IA problem, the algebraic independence of IA constraints, and the linear independence of the first order terms of IA constraints. This framework enables us to propose and prove a necessary and sufficient condition for IA to be feasible in MIMO interference networks with general topology. Based on this condition, we generate several insights into the relation between network topology and IA feasibility.;In addition, by exploiting the connection between algebraic independence and full rankness of Jacobian matrix, we prove that when IA is feasible, in the corresponding interference minimization problem, there is no performance gap between local and global optimums. This fact enables us to find IA solutions by adopting existing local search algorithms. Combining the results on IA feasibility analysis and algorithm design, we have established a unified algebraic framework that consolidates the theoretical basis of IA.;Further, we extend IA to networks with partial connectivity. Classical IA algorithms are designed for networks with fully connected interference graphs. We envision that in interference networks, partial connectivity can potentially allow IA algorithms to cancel interference more efficiently. We develop a new IA algorithm that dynamically adapts to partial connectivity parameters and hence achieves better performance than classical IA algorithms.;Finally, we extend IA to cellular networks. Classical IA algorithms, designed for interference networks, exploit the fact that the channel state of direct links and cross (i.e. interfering) links are statistically independent. However, in cellular networks, there is overlap between the direct and cross links. With this overlap, classical IA algorithms will cancel part of the desired signals when canceling interference. To overcome this challenge, we decompose the IA problem for cellular networks into equivalent intra- and inter-cell interference cancellation problems and develop an IA algorithm for MIMO cellular networks. |
