{erf}^{-1}\left(1-\frac{2}{e\Psi}\right)\right),\end{aligned}$$ where $\beta\simeq0.5772$ denotes the Euler’s constant.

See Appendix \[App:proofaveGlsn\]

Based on , we further obtain the limiting behavior of the average throughput gain when $\Psi$ becomes large in the following corollary.

\[Cor:aveGlsngrO\] As $\Psi\rightarrow\infty$, $G_{\bar{R}}(\Psi)$ is asymptotically equivalent to $$\label{eq:galarpsiasy}
G_{\bar{R}}(\Psi)\sim\frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\sqrt{\ln(\Psi)}.
$$

See Appendix \[App:proofCor:aveGlsngrO\]

From we find that $G_{\bar{R}}(\Psi)=O\left(\sqrt{\ln(\Psi)}\right)$ as $\Psi\rightarrow\infty$. Thus, the growth of the average throughput from adding reconfiguration states becomes small when the number of reconfiguration states is already large, although having more distinct reconfiguration states always benefits the average throughput.

Outage Throughput Gain
----------------------

In the above analysis, we focused on the performance gain in terms of the average throughput. However, it is insufficient to use the average throughput as the sole measure of the rate performance of the systems with multiple antennas. For scenarios where the channel remains (quasi) static during the transmission, it is appropriate to evaluate the system performance by the outage throughput, since every possible target transmission rate is associated with an unavoidable probability of outage. In the following, we analyze the performance gain of employing the reconfigurable antennas in terms of the outage throughput.

For a given target rate $R$, an outage event happens when the maximum achievable throughput is less than the target rate, and the outage probabilities for the systems without and with the reconfigurable antennas are given by ${\mathbb{P}}(R_{{\widehat{\psi}}}<R)$ and ${\mathbb{P}}(R_{\psi}<R)$, respectively. At a required outage level $0<\epsilon<1$, the outage throughputs for the systems with and without the reconfigurable antennas are given by [@Tse_05_Fundamentals] $