DoF Achieving Propagation Delay Aligned Structure for $K \times 2$ X Channels

This letter investigates the implementation of perfect interference alignment (IA) based on propagation delay (PD) for <inline-formula> <tex-math notation="LaTeX">$K\times 2$ </tex-math></inline-formula> X channels where each of the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> source nodes sends an independent message to each of the two destination nodes. The key problem is to find a feasible PD alignment (PDA) structure to achieve the degrees of freedom (DoF), which is <inline-formula> <tex-math notation="LaTeX">$2K/(K+1)$ </tex-math></inline-formula>. From the simple <inline-formula> <tex-math notation="LaTeX">$K=2, 3$ </tex-math></inline-formula> to the general <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>, we show the DoF achievability of the proposed PDA structure combined with proper time slot scheduling at the transmitter side. Moreover, the feasibility condition in Euclidean space is given. The connection to cyclic IA is further built up when <inline-formula> <tex-math notation="LaTeX">$K=2$ </tex-math></inline-formula>.