modify module 07

morea/07-antenna-diversity/reading-07-space-time-codes.md

@@ -15,7 +15,64 @@ In [the previous discussion](reading-07-transmit-diversity.html), we achieved th
 
 ## Alamouti scheme
 
-To illustrate the Alamouti scheme, we consider the case of two transmit antennas and single receive antenna.
+### Signaling scheme
+
+To illustrate the Alamouti scheme, we consider the case of two transmit antennas and single receive antenna. In the Alamouti scheme, we transmit two independent *complex* symbols *simultaneously* from two antennas, denoted by $x_1[1] = u_1$ and $x_2[1] = u_2$. Then in the next time slot, we still transmit these two symbols but rearrange them so that $x_1[2] = -u_2^\ast$ and $x_2[2] = u_1^\ast$. Assuming that the channel remain constant over the two consecutive time slots, the overall signal model is
+
+$$
+    \left[ \begin{array}{cc} y[1] & y[2] \end{array} \right] = 
+    \left[ \begin{array}{cc} h_1 & h_2 \end{array} \right] 
+    \left[ \begin{array}{lr} u_1 & -u_2^\ast \\ u_2 & u_1^\ast \end{array} \right]  + 
+    \left[ \begin{array}{cc} w[1] & w[2] \end{array} \right].
+$$
+
+Why we arrange the two symbols over the two time slots like this? This is because from the above signal model, we get two orthogonal vectors
+
+$$
+\left[ \begin{array}{c} u_1 \\ u_2 \end{array} \right] \quad \text{and} \quad \left[ \begin{array}{r} -u_2^\ast \\ u_1^\ast \end{array} \right].
+$$
+
+Each vector represents the symbols transmitted from the two transmit antennas. The two vectors over two time slots are orthogonal to each other. This orthogonality enables us to remove the interference between the two symbols at the receiver. 
+
+### Detection in the Alamouti scheme
+To see how to detect the symbols in the Alamouti scheme, we can rewrite the signal model as
+
+$$
+    \left[ \begin{array}{l} y[1] \\ y[2]^\ast \end{array} \right] = 
+    \left[ \begin{array}{lr} h_1 & h_2 \\ h_2^\ast & -h_1^\ast \end{array} \right] 
+    \left[ \begin{array}{c} u_1 \\ u_2 \end{array} \right]  + 
+    \left[ \begin{array}{c} w[1] \\ w[2]^* \end{array} \right] = 
+    \left[ \begin{array}{c} h_1 \\ h_2^\ast \end{array} \right] u_1 + 
+    \left[ \begin{array}{r} h_2 \\ -h_1^\ast \end{array} \right] u_2  + 
+    \left[ \begin{array}{c} w[1] \\ w[2]^* \end{array} \right].
+$$
+
+From the rewritten signal model above, it is clear that the two symbols $u_1$ and $u_2$ goes through two vector channels. More importantly, the two channels
+
+$$
+\left[ \begin{array}{c} h_1 \\ h_2^\ast \end{array} \right] \quad \text{and} \quad \left[ \begin{array}{r} h_2 \\ -h_1^\ast \end{array} \right]
+$$
+
+are also *orthogonal!*
+
+Due to orthogonality, we can easily eliminate the interference between the two symbols. For example, we can eliminate the second symbol by
+
+$$
+r_1 = \left[ \begin{array}{cc} h_1^\ast & h_2 \end{array} \right] \left[ \begin{array}{l} y[1] \\ y[2]^\ast \end{array} \right] = \Vert \mathbf{h} \Vert^2 u_1 + w_1,
+$$
+
+where $\mathbf{h} = [h_1, h_2]^T$ and $w_1 \sim \mathcal{CN}\left(0, \Vert \mathbf{h} \Vert^2 N_0 \right)$.
+
+Similarly, we can eliminate the first symbol by
+
+$$
+r_2 = \left[ \begin{array}{cc} h_2^\ast & -h_1 \end{array} \right] \left[ \begin{array}{l} y[1] \\ y[2]^\ast \end{array} \right] = \Vert \mathbf{h} \Vert^2 u_2 + w_2.
+$$
+
+The two scalars $r_1$ and $r_2$ can be used to detect symbols $u_1$ and $u_2$, respectively. Since each symbol is multiplied by $\Vert \mathbf{h} \Vert^2$, the diversity gain is 2.
+
+In conclusion, the Alamouti scheme achieves the diversity gain of 2 while transmitting one *complex* symbol per time slot, which amounts to two *real* symbols per time slot.
 
 ## General space-time codes
 
+