performance approaches that of the unfaded additive Gaussian noise channel (AWGN) for which we have $\frac{E_b}{N_0}_{{\text{min}}} = -1.59$ dB and $\mathcal{S}_0 = 2$ [@sergio].

Energy Efficiency in the Low-Power Regime {#sec:lowpower}
=========================================

In this section, we investigate the spectral efficiency–bit energy tradeoff as the average power ${\bar{P}}$ diminishes. We assume that the bandwidth allocated to the channel is fixed. Note that ${{\text{\footnotesize{SNR}}}}=
{\bar{P}}/(N_0B)$ vanishes with decreasing ${\bar{P}}$, and we again operate in the low-${{\text{\footnotesize{SNR}}}}$ regime similarly as in Section \[sec:wideband\]. However, energy requirements in the low-power regime will be different from those in the wideband regime, because the arrival rates that can be supported get smaller with decreasing power in this regime.

The following result provides the expressions for the bit energy at zero spectral efficiency and the wideband slope.

\[theo:lowpower\] In the low-power regime, the bit energy at zero spectral efficiency and wideband slope are given by $$\begin{gathered}
\frac{E_b}{N_0}\bigg|_{{{\sf{R}}}= 0}=\frac{\log_e2}{{{\alpha^{\ast}_{\text{opt}}}}P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}} \quad \text{and} \label{eq:ebminlp}\\
\mathcal{S}_0=\frac{2 P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}}{1+\beta
(1-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\})},\label{eq:s0lp}\end{gathered}$$ respectively, where $\beta=\frac{\theta TB}{\log_e2}$ is normalized QoS constraint. In the above formulation, ${{\alpha^{\ast}_{\text{opt}}}}$ is again defined as $
{{\alpha^{\ast}_{\text{opt}}}}= \lim_{{{\text{\footnotesize{SNR}}}}\to 0} {\alpha_{\text{opt}}}, $ and ${{\alpha^{\ast}_{\text{opt}}}}$ satisfies $$\begin{gathered}
 \label{eq:stacond}
{{\alpha^{\ast}_{\text{opt}}}}p_z({{\alpha^{\ast}_{\text{opt}}}})=P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}.\end{gathered}$$

*Proof:* We first consider the Taylor series expansion of ${r_{\text{opt}}}$ in the low-${{\text{\footnotesize{SNR}}}}$ regime: 