$\mathbf{y}'$, $\mathbf{n}'\in \mathbb{C}^{2n_R}$ and $\mathbf{H}' \in \mathbb{C}^{2n_R\times n_T}$. The equivalent channel equals $$\begin{aligned}
\mathbf{H}'=[(\mathbf{H}_{1}')^T,\dots,(\mathbf{H}_{i}')^T,\dots,(\mathbf{H}_{n_R}')^T]^T, \nonumber\end{aligned}$$ where $\mathbf{H}_i'$ is given as $$\label{eq:neuer_Kanal}
\mathbf{H}_i'=  \left[ \mathbf{H}_{i,1}',\mathbf{H}_{i,3}',\dots,\mathbf{H}_{i,n_T-1}' \right],$$ where $$\nonumber
 \mathbf{H}_{i,j}' = \left[
\begin{array}{*{2}{c}}
          h_{ij} & h_{i(j+1)}  \\
        -h_{i(j+1)}^* & h_{ij}^*  \\
        \end{array}
\right]\;.$$

Mutual Information {#sec:MutInfo}
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The instantaneous capacity $I$ of a MIMO system with $n_T$ transmit and $n_R$ receive antennas is given as [@Telatar99; @FoschiniGans98] $$\label{eq:MutInfoNrGeneral}
I = \log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\mathbf{H}^H\mathbf{H}\right)\;.$$

In the following two subsections, we derive lower and upper bounds for both the ergodic capacity and the average rate achievable with the proposed stacked scheme in order to yield lower and upper bounds on the ratio of the ergodic capacity to the average rate of the stacked OSTBC. In the third subsection, we characterize the absolute loss of the average rate of the stacked OSTBC to the ergodic capacity.

Upper bounds on the ergodic capacity and the average rate of stacked OSTBC {#sec:MutInfoUB}
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By applying the trace-determinant inequality $\det(\mathbf{A})^{1/n}\leq \frac{1}{n}\mathrm{tr}(\mathbf{A})$, we arrive at a simple upper bound on the instantaneous capacity given as $$\label{eq:UppBoundCapa}
    I \leq I_{ub}= L \log_2
    \bigg(1+\frac{\rho}{n_TL}\underbrace{\sum_{j=1}^{n_T}\sum_{i=1}^{n_R}|h_{ji}|^2}_{\lambda}\bigg)\;,$$ where $L$ is equal to $L=\min(n_T,n_R)$. Averaging the upper bound in  over all channel realizations results in [@Gradshteyn] 