conclude that $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0} = \frac{E_b}{N_0}_{\min}$ for all $\theta \ge 0$ when $z$ has a Gamma distribution. $\square$

Next, we provide numerical results which confirm the theoretical conclusions and illustrate the impact of QoS constraints on the energy efficiency. We set $B=10^5$ Hz in the computations. Fig. \[fig:2\] plots the spectral efficiency as a function of the bit energy for different values of $\theta$ in the Rayleigh fading channel (or equivalently Nakagami-$m$ fading channel with $m
= 1$) for which ${\mathbb{E}}\{|h|^2\}={\mathbb{E}}\{z\}=1$. In all cases in Fig. \[fig:2\], we readily note that $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0} = \frac{E_b}{N_0}_{{\text{min}}}$. Moreover, as predicted, the minimum bit energy is the same and is equal to the one achieved when there are no QoS constraints (i.e., when $\theta =
0$). From the equation ${{\alpha^{\ast}_{\text{opt}}}}p_z({{\alpha^{\ast}_{\text{opt}}}})=P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}$, we can find that ${{\alpha^{\ast}_{\text{opt}}}}=1$ in the Rayleigh channel for which $p_z({{\alpha^{\ast}_{\text{opt}}}}) = P\{z
> {{\alpha^{\ast}_{\text{opt}}}}\} = e^{-{{\alpha^{\ast}_{\text{opt}}}}}$. Hence, the minimum bit energy is $\frac{E_b}{N_0}_{\min} = 2.75$ dB. On the other hand, the wideband slopes are $\mathcal{S}_0=\{0.7358, 0.6223,0.2605,0.0382,0.0040\}$ for $\theta=\{0, 0.001,0.01,0.1,1\}$, respectively. Hence, $\mathcal{S}_0$ decreases with increasing $\theta$ and consequently more bit energy is required at a fixed nonzero spectral efficiency. Assuming that the minimum bit energies are the same and considering the 