$\mathcal{S}_0$ provide a linear approximation of the spectral efficiency curve at low spectral efficiencies, i.e., $$\begin{gathered}
\label{eq:linearapprox}
{{\sf{R}}}_E\left(\frac{E_b}{N_0}\right) = \frac{\mathcal{S}_0}{10\log_{10}
2} \left( \frac{E_b}{N_0}\bigg|_{dB}-\frac{E_b}{N_0}\bigg|_{{{\sf{R}}}= 0,
dB}\right) + \epsilon\end{gathered}$$ where $\frac{E_b}{N_0}\Big|_{dB} = 10\log_{10}\frac{E_b}{N_0}$ and $\epsilon = o\left( \frac{E_b}{N_0}-\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=
0}\right)$. Moreover, $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=0}$ is the minimum bit energy $\frac{E_b}{N_0}_{{\text{min}}}$ when the spectral efficiency ${{\sf{R}}}_E$ is a non-decreasing concave function of ${{\text{\footnotesize{SNR}}}}$. Indeed, we show that when the channel is perfectly known at the receiver, $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=0} = \frac{E_b}{N_0}_{{\text{min}}}$ in the wideband regime as $B \to \infty$. Moreover, we demonstrate that $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=0} = \frac{E_b}{N_0}_{{\text{min}}}$ for Rayleigh and Nakagami fading channels (with integer fading parameter $m$) in the low-power regime as ${\bar{P}}\to 0$. On the other hand, for general treatment, we refer to the bit energy required as ${{\text{\footnotesize{SNR}}}}$ vanishes as $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=0}$ throughout the paper. As we shall see in Section \[sec:imperfect\] that $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}=0}$ is not necessarily the minimum bit energy in a certain scenario of the imperfectly-known channel.

Energy Efficiency in the Wideband Regime {#sec:wideband}
========================================

In this section, we consider the wideband regime in which the bandwidth is large. We assume that the average power ${\bar{P}}$ is kept constant. Note that as the bandwidth $B$ increases, ${{\text{\footnotesize{SNR}}}}=
\frac{{\bar{P}}}{N_0B}$ approaches zero and we operate in the 