n_T < 2n_R$, the slope of the lower bound is equal to $\nicefrac{L}{2}$. In Fig. \[fig:RsAUpLowBounds\] on the top of the next page, the average rate, the upper bound  and the lower bounds from Table \[tab:LowBounds\] for $n_T=$ and $n_R=1,\dots,4$ are depicted. From the Fig., we observe that the upper bound in  and lower bounds track the average rate quite well. Only in the aforementioned case $n_R < n_T < 2n_R$, the slope of the lower bound differs from the exact performance and the upper bound. Note that for $n_R=1$, the upper bound coincides with the exact performance.

------------------------------------------------------------------------

Characterization of the absolute rate loss
------------------------------------------

In this subsection, we characterize the absolute rate loss of the stacked OSTBC to the ergodic capacity using Fischer’s inequality. First of all, we discuss the case of $n_T\geq 2n_R$. Note that the rate loss with the basic Alamouti scheme ($n_T=2$) was also analyzed in [@Sandhu; @HassibiHoch2002] using different approaches. Starting from , applying Fischer’s inequality and averaging over all channel realizations we arrive at $$\begin{aligned}
R_{sA}  \leq \mathbb{E}\Bigg[ & \frac{1}{2}\log_2\Bigg(\det\left(\mathbf{I}_{n_R}
+\frac{\rho}{n_T}\mathbf{H}\mathbf{H}^H\right) \nonumber \\
& \times \left(\mathbf{I}_{n_R} +\frac{\rho}{n_T}\mathbf{H}_e\mathbf{H}_e^H\right)\Bigg)\Bigg] = C(\rho, n_T,n_R) \label{eq:AbsRatLossNTNR} \\
& \qquad[\text{case } n_T \geq 2n_R] \nonumber,\end{aligned}$$ i.e. as long as 