On the Degrees of Freedom of the Compound MIMO Broadcast Channels with Finite States

TL;DR

This paper investigates the degrees of freedom in compound MIMO broadcast channels with finite states, demonstrating interference alignment techniques achieve optimal DoF under certain conditions, regardless of the number of channel states.

Contribution

It extends interference alignment methods to complex channels and shows optimal DoF can be achieved without transmitter cooperation in finite-state MIMO broadcast channels.

Findings

Interference alignment achieves DoF of MK/(M+K-1)

Optimal DoF when each receiver's channel states are at least M

System benefits from interference alignment despite high channel uncertainty

Abstract

Multiple-antenna broadcast channels with $M$ transmit antennas and $K$ single-antenna receivers is considered, where the channel of receiver $r$ takes one of the $J_r$ finite values. It is assumed that the channel states of each receiver are randomly selected from $\mathds{R}^{M\times 1}$ (or from $\mathds{C}^{M\times 1}$). It is shown that no matter what $J_r$ is, the degrees of freedom (DoF) of $\frac{MK}{M+K-1}$ is achievable. The achievable scheme relies on the idea of interference alignment at receivers, without exploiting the possibility of cooperation among transmit antennas. It is proven that if $J_r \geq M$, $r=1,...,K$, this scheme achieves the optimal DoF. This results implies that when the uncertainty of the base station about the channel realization is considerable, the system loses the gain of cooperation. However, it still benefits from the gain of interference alignment. In fact, in this case, the compound broadcast channel is treated as a compound X channel. Moreover, it is shown that when the base station knows the channel states of some of the receivers, a combination of transmit cooperation and interference alignment would achieve the optimal DoF. Like time-invariant $K$-user interference channels, the naive vector-space approaches of interference management seem insufficient to achieve the optimal DoF of this channel. In this paper, we use the Number-Theory approach of alignment, recently developed by Motahari et al.[1]. We extend the approach of [1] to complex channels as well, therefore all the results that we present are valid for both real and complex channels.