Assuming the channel gain and the amplitude of the transmitted signal are pre-known by the receivers, the receivers can use $yl1$, $yl2$ signals to reconstruct the transmitted signals $x1$, $x2$ as $x\u02dc1$, $x\u02dc2$, where $x\u02dc1$, $x\u02dc2$ are denoted by the following formulas: Display Formula
$x\u02dc1=\u2211l=1L=2yl1hl1(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)+\u2211l=1L=2yl2hl2(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d)\u2212A\u2211l=1L=2hl1(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)hl2(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d),x\u02dc2=\u2211l=1L=2yl1hl2(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d)\u2212\u2211l=1L=2yl2hl1(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)+A\u2211l=1L=2hl12(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d).$(4)
Substituting Eq. (3) into Eq. (4), we obtain the following simplified results: Display Formula$x\u02dc1=x1\u2211l=1L=2[hl12(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)+hl22(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d)]+\u2211l=1L=2[nl1hl1(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)+nl2hl2(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d)],x\u02dc2=x2\u2211l=1L=2[hl12(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)+hl22(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d)]+\u2211l=1L=2[nl1hl2(\theta 1,\theta 2,\varphi 1,\varphi 2,rl2,d)\u2212nl2hl1(\theta 1,\theta 2,\varphi 1,\varphi 2,rl1,d)].$(5)
From Eq. (5), we can see the interference between two Tx antennas is eliminated due to the contribution of the modified Alamouti code.