Two-Hop Interference Channels: Impact of Linear Schemes

We consider the two-hop interference channel (IC), which consists of two source-destination pairs communicating with each other via two relays. We analyze the degrees of freedom (DoF) of this network when the relays are restricted to perform linear schemes, and the channel gains are constant (i.e., slow fading). We show that, somewhat surprisingly, by using vector-linear strategies at the relays, it is possible to achieve 4/3 sum-DoF when the channel gains are real. The key achievability idea is to alternate relaying coefficients across time, to create different end-to-end interference structures (or topologies) at different times. Although each of these topologies has only 1 sum-DoF, we manage to achieve 4/3 by coding across them. Furthermore, we develop a novel outer bound that matches our achievability, hence characterizing the sum-DoF of two-hop ICs with linear schemes. We also generalize the result to the multi-antenna setting, where each node has M antennas, and the relays are restricted to one-shot linear schemes. We further extend the result to the case of complex channel gains, by intuitively viewing each complex node with M antennas as a real node with 2M antennas, and characterize the sum-DoF to be 2M - 1/3.