Alamouti Scheme. MISO. STBC for real constellation

Question

I have a real orthogonal design for codeword...

$$ \begin{bmatrix} s1 & s2 \\ -s2 & s1 \end{bmatrix} $$

$$ s=\begin{bmatrix} s1\\ s2 \end{bmatrix} $$

$$ Y^T=\sqrt{\frac{P}{Mt}} h^T X + n^T $$

...and receive signal:

$$ Y1=\begin{bmatrix} y1\\ y2 \end{bmatrix} $$

So for:

$$ X = \begin{bmatrix} x1 & x2 \\ -x2 & x1 \end{bmatrix} $$

and $$ Y = \begin{bmatrix} y1 & y2 \end{bmatrix} $$

...i have:

$$ Y=\sqrt{\frac{P}{Mt}} \cdot \begin{bmatrix}h1 & h2\\ h2 & -h1\end{bmatrix}\cdot \begin{bmatrix}x1 & x2\end{bmatrix}^T+N $$

...right?

Answer 1

Your analysis is correct. I would write it as follows: if you define the transmitted matrix as \begin{equation*} X = \begin{bmatrix} x_1 & -x_2 \\ x_2 & x_1 \end{bmatrix} \end{equation*} with one column transmitted at each symbol interval (that is, first you transmit $[x_1, x_2]$ and then $[-x_2, x_1]$), then the system can be written as $$Y=hX$$ with $h = [h_1, h_2]$, or equivalently it can be written as $$Y = Hx$$ with \begin{equation*} H = \begin{bmatrix} h_1 & h_2 \\ -h_2 & h_1 \end{bmatrix} \end{equation*} and $x=[x_1, x_2]^T$.

Of course, you can also transpose all vectors and matrices and arrive at equivalent results.

Answer 2

If you transmit $[x_1\,\,\,x_2]$ in the first time slot, and $[-x_2\,\,\,x_1]$ in the second time slot, the received signals will be

$$y_1=h_1x_1+h_2x_2+n_1$$

and

$$y_2=-h_1x_2+h_2x_1+n_2$$

This can be written in matrix-vector form as

$$\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}$$.

where

$$\mathbf{H}=\begin{pmatrix}h_1&h_2\\h_2&-h_1\end{pmatrix}$$

and $$\mathbf{x}=\begin{pmatrix}x_1\\x_2\end{pmatrix}$$

Note that although the channel coefficients maybe complex-valued, the only part that will affect the transmission is the real part since the signals are real. In this case

$$\mathbf{H}^T\mathbf{H}=(h_1^2+h_2^2)\mathbf{I}_{2\times 2}$$

which means that the two signals can be completely decoupled, and detected optimally separately.