set $T = 2$ ms in the numerical results throughout the paper. As predicted by the result of Theorem \[theo:minenergywideband\], $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0} =
\frac{E_b}{N_0}_{{\text{min}}}$ in all cases in Fig. \[fig:4\]. It can be found that ${{\alpha^{\ast}_{\text{opt}}}}=\{1, 0.9858,0.8786,0.4704,0.1177\}$ from which we obtain $\frac{E_b}{N_0}_{{\text{min}}} = \{2.75, 2.79,    3.114,     5.061,   10.087\}$dB for $\theta = \{0, 0.001,0.01,0.1,1\}$, respectively. For the same set of $\theta$ values in the same sequence, we compute the wideband slope values as $\mathcal{S}_0=\{0.7358,
0.7463,0.8345,1.4073,3.1509\}$. We immediately observe that more stringent QoS constraints and hence higher values of $\theta$ lead to higher minimum bit energy values and also higher energy requirements at other nonzero spectral efficiencies. Fig. \[fig:wbm\] provides the spectral efficiency curves for Nakagami-$m$ fading channels for different values of $m$. In this figure, we set $\theta = 0.01$. For $m = 0.6, 1, 2, 5$, we find that ${{\alpha^{\ast}_{\text{opt}}}}= \{1.0567,  0.8786    0.7476    0.6974\}$, $\frac{E_b}{N_0}_{{\text{min}}} =\{3.618,    3.114,    2.407,    1.477\}$, and $\mathcal{S}_0 = \{0.6382,    0.8345,    1.1220,    1.4583\}$, respectively. Note that as $m$ increases and hence the channel conditions improve, the minimum bit energy decreases and the wideband slope increases, improving the energy efficiency both at zero spectral efficiency and at nonzero but small spectral efficiency values. As $m \to \infty$, the 