linear approximation in (\[eq:linearapprox\]), we can easily show for fixed spectral efficiency ${{\sf{R}}}\left(\frac{E_b}{N_0}\right)$ for which the linear approximation is accurate that the increase in the bit energy in dB, when the QoS exponent increases from $\theta_1$ to $\theta_2$, is $$\begin{gathered}
\frac{E_b}{N_0}\bigg|_{dB,
\theta_2}-\frac{E_b}{N_0}\bigg|_{dB,\theta_1} =
\left(\frac{1}{\mathcal{S}_{0,\theta_2}} -
\frac{1}{\mathcal{S}_{0,\theta_1}}\right)
{{\sf{R}}}\left(\frac{E_b}{N_0}\right) 10\log_{10}2.
\label{eq:bitenergydifference}\end{gathered}$$ As observed in Fig. \[fig:2\] (and also as will be seen in Fig. \[fig:3\] below), spectral efficiency curves are almost linear in the low-power regime, validating the accuracy of the linear approximation in (\[eq:linearapprox\]) obtained through $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0}$ and $\mathcal{S}_0$.

Fig. \[fig:3\] plots the spectral efficiency curves as a function of the bit energy for Nakagami-$m$ channels for different values of $m$. $\theta$ is set to be 0.01. For $m=\{0.6,1,2,5\}$, we compute that ${{\alpha^{\ast}_{\text{opt}}}}=\{1.2764,1,0.809,0.7279\}$, $\frac{E_b}{N_0}_{\min} = \{3.099,    2.751,    2.176,    1.343\}$, and $\mathcal{S}_0=\{0.1707,0.2605,0.4349,0.7479\}$, respectively. We observe that as $m$ increases and hence the channel quality improves, lower bit energies are required. Finally, in Fig. \[fig:spec-comp\], we plot the spectral efficiency vs. $E_b/N_0$ for different transmission strategies. The variable-rate/variable-power and variable-rate/fixed-power strategies are studied in [@deli]. We immediately see that substantially more energy is required for fixed-rate/fixed-power transmission schemes considered in this paper.

The Effect of Imperfect Channel Knowledge in the Low-Power Regime 