($C=\mathbb{E}[I]$ denotes ergodic capacity) $$\begin{aligned}
\label{eq:UppBoundAnaly}
    C \leq C_{ub} =\mathbb{E}\left[I_{ub}\right] = & \frac{L}{\ln(2)}\sum\limits_{k=0}^{n_Tn_R-1}
    \left(\frac{n_TL}{\rho}\right)^{n_Tn_R-k-1}\\
    & e^{\frac{n_TL}{\rho}} \Gamma\left(1-(n_Tn_R-k),\frac{n_TL}{\rho}\right) \nonumber.\end{aligned}$$ Note that for high SNR, the slope of the upper bound is equal to $L$. In addition to this upper bound, we compare the rate achieved with the stacked scheme with the following upper bound $$\label{eq:BoundGrant}
    C \leq C_{\mathrm{Jen}} =\log_2\left(\sum_{i=0}^L \binom{L}{i}\frac{K!}{(K-i)!}\left(\frac{\rho}{n_T}\right)^i\right),$$ derived in [@GrantUppBound] by using Jensen’s inequality, where $K=\max(n_T,n_R)$.

In the following, we analyze the performance of the stacked scheme with respect to mutual information and derive upper bounds for the average rate of the stacked scheme. We first analyze the case of $n_R=1$ receive antennas and then generalize the analysis to the case of arbitrary number of receive antennas.

### Case $n_R=1$

In case of $n_R=1$, the achievable rate of the stacked Alamouti scheme is $$\nonumber
I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H}_1')^H\mathbf{H}_1'\right)\;.$$ Using the determinant equality $\det(\mathbf{I}+\mathbf{AB})=\det(\mathbf{I}+\mathbf{BA})$, after some manipulations we arrive at $$\label{eq:MutInfoNReq1}
I_{sA} = \log_2\left(1 +\frac{\rho}{n_T} \sum\limits_{j=1}^{n_T}|h_{j1}|^2\right)\;,$$ which equals the capacity of a MIMO system with $n_T$ transmit and $n_R=1$ receive antennas [@Telatar99], i.e. as long as $n_R=1$, the capacity is achieved for arbitrary $n=n_T/2$. Note that in [@TirkHotWichBuch p.199] a Taylor series expansion is performed for the capacity and 