Interference Alignment with Diversity for the $2 \times 2$ $X$ Network with four antennas

A transmission scheme based on the Alamouti code, which we call the Li-Jafarkhani-Jafar (LJJ) scheme, was recently proposed for the $2 \times 2$ $X$ Network (i.e., two-transmitter (Tx) two-receiver (Rx) $X$ Network) with two antennas at each node. This scheme was claimed to achieve a sum degrees of freedom (DoF) of $\frac{8}{3}$ and also a diversity gain of two when fixed finite constellations are employed at each Tx. Furthermore, each Tx required the knowledge of only its own channel unlike the Jafar-Shamai scheme which required global CSIT to achieve the maximum possible sum DoF of $\frac{8}{3}$. In this paper, we extend the LJJ scheme to the $2 \times 2$ $X$ Network with four antennas at each node. The proposed scheme also assumes only local channel knowledge at each Tx. We prove that the proposed scheme achieves the maximum possible sum DoF of $\frac{16}{3}$. In addition, we also prove that, using any fixed finite constellation with appropriate rotation at each Tx, the proposed scheme achieves a diversity gain of at least four.