Degrees of Freedom of MIMO $X$ Networks: Spatial Scale Invariance and One-Sided Decomposability

We show that an M×N user MIMO X network with A antennas at each node has A(MN/(M+N-1)) degrees of freedom (DoF), thus resolving in this case a discrepancy between the spatial scale invariance conjecture (scaling the number of antennas at each node by a constant factor will scale the total DoF by the same factor) and a decomposability property of overconstrained wireless networks. While the best previously known general DoF outer bound is consistent with the spatial invariance conjecture, the best previously known general DoF inner bound, inspired by the K user MIMO interference channel, was based on the decomposition of every transmitter and receiver into multiple single antenna nodes, transforming the network into an AM×AN user SISO X network. While such a decomposition is DoF optimal for the K user MIMO interference channel, a gap remained between the best inner and outer bounds for the MIMO X channel. Here we close this gap with the new insight that the MIMO X network is only one-sided decomposable, i.e., either all the transmitters or all the receivers (but not both) can be decomposed by splitting multiple antenna nodes into multiple single antenna nodes without loss of DoF. The result is extended to SIMO and MISO X networks as well and in each case the DoF results satisfy the spatial scale invariance property.