On Achievable Degrees of Freedom for MIMO X Channels

In this paper, the achievable DoF of MIMO X channels for constant channel coefficients with $M_t$ antennas at transmitter $t$ and $N_r$ antennas at receiver $r$ ($t,r=1,2$) is studied. A spatial interference alignment and cancelation scheme is proposed to achieve the maximum DoF of the MIMO X channels. The scenario of $M_1\ge M_2\ge N_1\ge N_2$ is first considered and divided into 3 cases, $3N_2<M_1+M_2<2N_1+N_2$ (Case $A$), $M_1+M_2\ge 2N_1+N_2$ (Case $B$), and $M_1+M_2\le 3N_2$ (Case $C$). With the proposed scheme, it is shown that in Case $A$, the outer-bound $\frac{M_1+M_2+N_2}{2}$ is achievable; in Case $B$, the achievable DoF equals the outer-bound $N_1+N_2$ if $M_2>N_1$, otherwise it is 1/2 or 1 less than the outer-bound; in Case $C$, the achievable DoF is equal to the outer-bound $2\mathrm{/}3(M_1+M_2)$ if $(3N_2-M_1-M_2)\mathrm{m}\mathrm{o}\mathrm{d}3=0$, and it is 1/3 or 1/6 less than the outer-bound if $(3N_2-M_1-M_2)\mod 3=1 $\mathrm{o}\mathrm{r}$ 2$\mathrm{.}Inthescenarioof$M_t\leq N_r$, the exact symmetrical results of DoF can be obtained.