Feasibility Conditions for Interference Alignment (0904.4526v1)

Abstract: The degrees of freedom of MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment solution is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem as a system of bilinear equations and determine its solvability by comparing the number of equations and the number of variables. To this end, we divide interference alignment problems into two classes - proper and improper. An interference alignment problem is called proper if the number of equations does not exceed the number of variables. Otherwise, it is called improper. Examples are presented to support the intuition that for generic channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible.

• The paper introduces a novel classification of IA problems into proper and improper systems to determine feasibility.
• An analytical framework using bilinear equations is developed to assess if the number of variables meets the equation requirements.
• For symmetric systems, the condition M + N ≥ (K + 1)d is established, guiding efficient resource allocation in MIMO networks.

Feasibility Conditions for Interference Alignment

The paper "Feasibility Conditions for Interference Alignment" by Cenk M. Yetis, Syed A. Jafar, and Ahmet H. Kayran, addresses a critical problem in wireless communications: determining the feasibility of linear interference alignment (IA) solutions for Multiple Input Multiple Output (MIMO) networks with constant channel coefficients. The concept hinges on the degrees of freedom (DoF) in interference networks, which represent the number of interference-free signaling dimensions.

Key Contributions

1. Classification of IA Problems:
   • The paper introduces a novel classification system for IA problems divided into two categories: proper and improper. A system is termed proper if the number of equations posed by the alignment problem does not exceed the number of variables. Conversely, a system is improper if the number of equations exceeds the number of variables. This classification is pivotal since proper systems are almost surely feasible, while improper systems are almost surely infeasible.
2. Analytical Framework:
   • The authors present an analytical approach to determine the solvability of the IA problem by modeling the problem as a system of bilinear equations. They then assess the solvability by counting and comparing the number of equations to the number of variables.
3. Symmetric Systems Analysis:
   • For symmetric systems of the form $(M \times N, d)^K$, the paper establishes a simple condition. The system is proper if and only if $M + N \geq (K + 1)d$. This finding means that, for each user to achieve $d$ DoF in a $K$-user interference channel, the total number of antennas $M + N$ must be at least $(K + 1)d$.
4. Insightful Examples:
   • Various examples are provided to illustrate the theory. Notably, the authors demonstrate the classification with symmetric and asymmetric systems and validate the analytical predictions with numerical evidence.

Implications

The research has significant implications for the design and analysis of MIMO networks. By providing a systematic way to predict the feasibility of linear IA, it helps in resource allocation and network planning. The specific conditions for symmetric systems simplify feasibility checks, making it easier for engineers to design networks that can support efficient IA.

Future Directions

• Extended Models:

Future research could incorporate more complex channel models, such as those with time-varying or frequency-selective characteristics.

• Robustness Analysis:

Investigating the robustness of IA under real-world conditions, including hardware imperfections and non-idealities, could add practical insights.

• Algorithm Development:

Developing algorithms to synthesize IA solutions based on the proper/improper system classification could streamline the design process.

Numerical Validation

The authors substantiate their theoretical claims with numerical experiments. The leakage interference metric, which measures the fraction of interference power in the desired signal subspace, reaffirms the feasibility conditions posited. Proper systems show negligible interference, proving the viability of the alignment, whereas improper systems exhibit significant residual interference.

Conclusion

This paper rigorously addresses the feasibility of IA in constant MIMO interference networks by developing a classification system grounded in the number of variables and equations. The clear distinction between proper and improper systems provides a robust framework for evaluating the potential of IA solutions in wireless networks, thus contributing substantially to the field's theoretical and practical understanding.