Interference alignment using finite and dependent channel extensions: the single beam case

Vector space interference alignment (IA) is known to achieve high degrees of freedom (DoF) with infinite independent channel extensions, but its performance is largely unknown for a finite number of possibly dependent channel extensions. In this paper, we consider a $K$-user $M_t \times M_r$ MIMO interference channel (IC) with arbitrary number of channel extensions $T$ and arbitrary channel diversity order $L$ (i.e., each channel matrix is a generic linear combination of $L$ fixed basis matrices). We study the maximum DoF achievable via vector space IA in the single beam case (i.e. each user sends one data stream). We prove that the total number of users $K$ that can communicate interference-free using linear transceivers is upper bounded by $NL+N^2/4$, where $N = \min\{M_tT, M_rT \}$. An immediate consequence of this upper bound is that for a SISO IC the DoF in the single beam case is no more than $\min\left\{\sqrt{ 5K/4}, L + T/4\right\}$. When the channel extensions are independent, i.e. $L$ achieves the maximum $M_r M_t T$, we show that this maximum DoF lies in $[M_r+M_t-1, M_r+M_t]$ regardless of $T$. Unlike the well-studied constant MIMO IC case, the main difficulty is how to deal with a hybrid system of equations (zero-forcing condition) and inequalities (full rank condition). Our approach combines algebraic tools that deal with equations with an induction analysis that indirectly considers the inequalities.