given by (\[eq:gammadensity0\]), then ${{\alpha^{\ast}_{\text{opt}}}}$ is unique and hence is the same for all $\theta \ge 0$. We immediately conclude from these results that $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0}$ also has the same value for all $\theta \ge 0$ and therefore does not depend on $\theta$ when $z$ has the distribution given in (\[eq:gammadensity0\]).

Moreover, using the results of Theorem \[theo:stacond\] above and Theorem \[theo:minenergywideband\] in Section \[sec:wideband\], we can further show that $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0}$ is the minimum bit energy. Note that this implies that the same minimum bit energy can be attained regardless of how strict the QoS constraint is. On the other hand, we note that the wideband slope $\mathcal{S}_0$ in general varies with $\theta$.

In the low-power regime, when $\theta = 0$, the minimum bit energy is achieved as ${\bar{P}}\to 0$, i.e., $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0} =
\frac{E_b}{N_0}_{\min}$. Moreover, if the probability density function of $z$ is in the form given in (\[eq:gammadensity0\]) then the minimum bit energy is achieved as ${\bar{P}}\to 0$, i.e. $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=
0} = \frac{E_b}{N_0}_{\min}$, for all $\theta \ge 0$.

*Proof:* Recall from (\[eq:R\_Etheta0\]) that in the limit as $\theta \to 0$, $$\begin{gathered}
{{\sf{R}}}_E({{\text{\footnotesize{SNR}}}},0) = \lim_{\theta \to 0} {{\sf{R}}}_E({{\text{\footnotesize{SNR}}}},\theta) = \max_{r\ge
0} \,\,\frac{r}{B} \, P\left\{z >
\frac{2^{\frac{r}{B}}-1}{{{\text{\footnotesize{SNR}}}}}\right\}.\end{gathered}$$ Since the 