Interference Alignment and the Degrees of Freedom for the K User Interference Channel

While the best known outerbound for the K user interference channel states that there cannot be more than K/2 degrees of freedom, it has been conjectured that in general the constant interference channel with any number of users has only one degree of freedom. In this paper, we explore the spatial degrees of freedom per orthogonal time and frequency dimension for the K user wireless interference channel where the channel coefficients take distinct values across frequency slots but are fixed in time. We answer five closely related questions. First, we show that K/2 degrees of freedom can be achieved by channel design, i.e. if the nodes are allowed to choose the best constant, finite and nonzero channel coefficient values. Second, we show that if channel coefficients can not be controlled by the nodes but are selected by nature, i.e., randomly drawn from a continuous distribution, the total number of spatial degrees of freedom for the K user interference channel is almost surely K/2 per orthogonal time and frequency dimension. Thus, only half the spatial degrees of freedom are lost due to distributed processing of transmitted and received signals on the interference channel. Third, we show that interference alignment and zero forcing suffice to achieve all the degrees of freedom in all cases. Fourth, we show that the degrees of freedom D directly lead to an O(1) capacity characterization of the form C(SN R) = D log(1 + SN R) + O(1) for the multiple access channel, the broadcast channel, the 2 user interference channel, the 2 user MIMO X channel and the 3 user interference channel with M > 1 antennas at each node. It is not known if this relationship is true for all networks in general, and the K user interference channel with a single antenna at all nodes in particular. Fifth, we consider the degree of freedom benefits from cognitive sharing of messages on the 3 user interference channel. If only one of the three messages is made available non-causally to all the nodes except its intended receiver the degrees of freedom are not increased. However, if two messages are shared among all nodes (except their intended receivers) then there are two degrees of freedom. We find that unlike the 2 user interference channel, on the 3 user interference channel a cognitive transmitter is not equivalent to a cognitive receiver from a degrees of freedom perspective. If …