Interference Alignment and the Degrees of Freedom for the 3 User Interference Channel

We explore the degrees of freedom region for the 3 user wireless interference channel with perfect channel knowledge at all nodes. 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. We show that it is easy to construct constant K user interference channels with nonzero channel coefficients that have the full K/2 degrees of freedom. Even if channel coefficients are randomly drawn from a continuous distribution, we show that the total number of degrees of freedom for the 3 user interference channel is 3/2 with probability 1 when the channel coefficients can vary with time or frequency dimensions. If all nodes have M > 1 antennas then we show that 3M/2 degrees of freedom are achieved without the need for varying channel coefficients, i.e., with constant channel matrices. Interference alignment and zero forcing suffice to achieve all the degrees of freedom in all cases. We also investigate whether the degree of freedom D directly lead to an O(1) capacity characterization, i.e. an approximation accurate to within a bounded constant. 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. However, for the 3 user interference channel with only one antenna at each node we are only able to establish the degree of freedom but not the precise O(1) capacity. We also consider the degree of freedom benefits from cognitive sharing of messages. 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 one receiver has cognitive knowledge of all the other users' messages the degrees of …