derive lower bounds for the ergodic capacity and the average rate of the stacked OSTBC. Due to the peculiar property of stacked OSTBC, lower bounds are obtained in the procedure for the following cases: (i) $n_T\leq n_R$, (ii) $n_R < n_T < 2n_R$, (iii) $2n_R\leq n_T \leq 4n_R$, and (iv) $4n_R < n_T$.

First of all, from [@OymanNBPaulraj] we obtain the following lower bound on the ergodic capacity $$\label{eq:BoundOyman}
    C \geq C_{lb}= \sum_{j=1}^{L}\log_2\left(1+\frac{\rho}{n_T}\exp\left(\sum_{p=1}^{K-j}\frac{1}{p}-\gamma\right)\right),$$ where $\gamma\approx 0.57721566$ is Euler’s constant.

In order to derive an upper bound on the ratio of the ergodic capacity to the average rate achieved with the stacked scheme, we need a lower bound for the average rate of the stacked scheme. To this end, we rewrite  as follows $$\begin{aligned}
\label{eq:MutInfoStAlReW}
I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T}
+\frac{\rho}{n_T}(\mathbf{H})^H\mathbf{H}+\frac{\rho}{n_T}(\mathbf{H}'_e)^H\mathbf{H}'_e\right),\end{aligned}$$ where $\mathbf{H}$ is the actual MIMO channel, which is obtained by taking the odd rows of the equivalent channel $\mathbf{H}'$ and $\mathbf{H}_e$ is obtained by taking the even rows of $\mathbf{H}'$. The relation between the actual channel $\mathbf{H}$ and $\mathbf{H}_e$ is described in the following proposition.

\[prop:RelHandHe\] Let $\mathbf{H}_e$ be the even and $\mathbf{H}$ the odd rows of $\mathbf{H}'$ given in , respectively. Then the following holds

1.  $\nonumber
        \mathbf{H}_e=\mathbf{H}^*\mathbf{J}
    $, where[^4] 