is $\rho$. It is further assumed that the transmitter has no channel state information (CSI) and the receiver has perfect CSI.

Code construction {#sec:Diversity_analy}
=================

A space time block code is defined by its transmit matrix $\mathbf{G}_{n_T}$ with entries $\{x_j\}_{j=1}^{p}$, which are elements of the vector $\mathbf{x}=[x_1,\dots,x_p]^T$ with $x_1,\dots,x_{p} \in \mathcal{C}$, where $\mathcal{C}
\subseteq \mathbb{C}$ denotes a complex modulation signal set with unit average power, e.g. $M$-PSK.. The rate $R$ of a space-time code is defined as $R=p/T$. In this paper, we focus on the rate $n_T/2$ stacked Alamouti scheme. Starting with the well known (basic) Alamouti scheme [@Alamouti] for $n_T=2$ transmit antennas $$\nonumber
 \mathbf{G}_{2}(x_1,x_2) = \left[
\begin{array}{*{2}{r}}
          x_1 & x_2  \\
        x_2^* & -x_1^*  \\
        \end{array}
\right]\;,$$ the transmit matrix of the stacked Alamouti scheme with $n_T=2n$ is constructed in the following way $$\begin{aligned}
\nonumber
&\mathbf{G}_{n_T}\left(\{x_j\}_{j=1}^{n_T}\right)  \\
& =\left[\mathbf{G}_{2}(x_1,x_2), \mathbf{G}_{2}(x_3,x_4),\dots,\mathbf{G}_{2}(x_{n_T-1},x_{n_T})\right].\nonumber\end{aligned}$$

For the case of $n=2$, i.e. $n_T=4$ transmit antennas we have $$\nonumber
\mathbf{G}_{4}(\{x_j\}_{j=1}^{4}) = \left[
\begin{array}{*{4}{r}}
          x_1 & x_2 & x_3 & x_4 \\
        x_2^* & -x_1^* & x_4^* & -x_3^* \\
        \end{array}
\right]\;,$$ which is also referred to as DSTTD[@TexasInstrum].

After some manipulations (particularly complex-conjugating) the system model in (\[eq:System\]) can be rewritten as $$\label{eq:system_H_vorne}
    \mathbf{y}'=\mathbf{H}'\mathbf{x}+ \mathbf{n}'\;,$$ where 