Layered Interference Alignment: Achieving the Total DoF of MIMO X   Channels
  The $K\times 2$ and $2\times K$, Multiple-Input Multiple-Output (MIMO) X channel with constant channel coefficients available at all transmitters and receivers is considered. A new alignment scheme, named \emph{layered interference alignment}, is proposed in which both vector and real interference alignment are exploited, in conjunction with joint processing at receiver sides. Data streams with fractional multiplexing gains are sent in the desired directions to align the interfering signals at receivers. To decode the intended messages at receivers, a joint processing/simultaneous decoding technique, which exploits the availability of several receive antennas, is proposed. This analysis is subsequently backed up by metrical results for systems of linear forms. In particular, for such linear forms, Khintchine--Groshev type theorems are proved over real and complex numbers. It is observed that $K\times 2$ and $2\times K$, X channels with $M$ antennas at all transmitters/receivers enjoy duality in Degrees of Freedom (DoF). It is shown that incorporating the layered interference alignment is essential to characterize the total DoF of $\frac{2KM}{K+1}$ in the $K\times 2$ and $2\times K$, $M$ antenna X channels. 
Seyyed Hassan Mahboubi, Mumtaz Hussain, Abolfazl Seyed Motahari and   Amir Keyvan Khandani
cs.IT math.IT math.NT