$n_T\geq 2n_R$, the average rate of the stacked OSTBC is only upper bounded by the ergodic capacity. Proceeding similarly for the case $n_T < 2n_R$ results in $$\begin{aligned}
R_{sA}  & =\frac{1}{2}\mathbb{E}\left[\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\left[%
\begin{array}{cc}
  \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}^H\mathbf{\tilde{H}}_e \\
  \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\
\end{array}%
\right]\right)\right] \nonumber\\
& \leq \mathbb{E}\left[\log_2\left(\det\left(\mathbf{I}_{\frac{n_T}{2}}
+\frac{\rho}{n_T}\mathbf{\tilde{H}}^H\mathbf{\tilde{H}}\right)\right)\right]\nonumber \\
&=C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)<C\left(\rho,n_T,n_R\right) \;\;\;[\text{case } n_T <
2n_R],\label{eq:AbsRatLossNT22NR}\end{aligned}$$ where $\mathbf{\tilde{H}}$ is obtained by taking the odd columns of the equivalent channel $\mathbf{H}'$ and $\mathbf{\tilde{H}}_e$ is obtained by taking the even columns of $\mathbf{H}'$. From , we observe that for $n_T < 2n_R$ the average rate of the stacked OSTBC is upper bounded by the ergodic capacity of a system with $\frac{n_T}{2}$ transmit and $2n_R$ receive antennas with a power penalty of $3$ dB.

We can characterize the gap in  and  due to the application of Fischer’s inequality. For $n_T < 2n_R$, we have then $$\begin{aligned}
\Delta & = C\left(\frac{\rho}{2},\frac{n_T}{2},2n_R\right)-R_{sA}  \nonumber\\
& =\frac{1}{2}
\mathbb{E}\left[\log_2\left(\frac{\det\left(\mathbf{I}_{n_T}
+\frac{\rho}{n_T}\mathbf{W}_D\right)}{\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}\left[
  \begin{array}{cc}
     \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}^H\mathbf{\tilde{H}}_e \\
     \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\
  \end{array}%
    \right]\right)}\right)\right] \nonumber,\end{aligned}$$ where $$\begin{aligned}
\nonumber
\mathbf{W}_D=\left[%
\begin{array}{cc}
  \mathbf{\tilde{H}}^H\mathbf{\tilde{H}} & \mathbf{0} \\
  \mathbf{0} & \mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}_e \\
\end{array}%
\right].\end{aligned}$$ Since the events of $\mathbf{\tilde{H}}_e^H\mathbf{\tilde{H}}$ having not full rank are of measure zero the strict form of Fischer’s inequality stated in 