$$\begin{aligned}
\frac{d({{\sf{R}}}_E(\zeta)/\zeta)}{d\zeta} &=-\frac{1}{\theta
T}\frac{\frac{p_z({\alpha_{\text{opt}}})N_0}{{\bar{P}}}\frac{2^{{r_{\text{opt}}}\zeta}({\dot{r}_{\text{opt}}}\zeta+{r_{\text{opt}}})\zeta\log_e2-(2^{{r_{\text{opt}}}\zeta}-1)}{\zeta^2}(1-e^{-\theta
T{r_{\text{opt}}}})-\theta Te^{-\theta T{r_{\text{opt}}}}{\dot{r}_{\text{opt}}}P(z>{\alpha_{\text{opt}}})}{1-P(z>{\alpha_{\text{opt}}})(1-e^{-\theta
T{r_{\text{opt}}}})}\\
&=-\frac{p_z({\alpha_{\text{opt}}})N_0}{\theta
T{\bar{P}}}\frac{2^{{r_{\text{opt}}}\zeta}{r_{\text{opt}}}\zeta\log_e2-(2^{{r_{\text{opt}}}\zeta}-1)}{\zeta^2}
\label{eq:R_Ederiv}\end{aligned}$$ where (\[eq:R\_Ederiv\]) is obtained by using the equation $\frac{2^{{r_{\text{opt}}}/B}p_z({\alpha_{\text{opt}}})\log_e2}{B{{\text{\footnotesize{SNR}}}}}(1-e^{-\theta
T{r_{\text{opt}}}})=\theta Te^{-\theta T{r_{\text{opt}}}}P\{z>{\alpha_{\text{opt}}}\}$ that needs to be satisfied by ${r_{\text{opt}}}$ and ${\alpha_{\text{opt}}}$ as shown in the proof of Theorem \[theo:wideband\] in (\[eq:conditionb\]). Note that the probability density function $p_z(z) \ge 0$ for all $z \ge 0$. Hence, if $2^{{r_{\text{opt}}}\zeta}{r_{\text{opt}}}\zeta\log_e2-(2^{{r_{\text{opt}}}\zeta}-1) \ge 0$ for all ${r_{\text{opt}}}\ge 0$ and $\zeta \ge 0$, then $\frac{d({{\sf{R}}}_E(\zeta)/\zeta)}{d\zeta}\le 0$ proving that ${{\sf{R}}}_E(\zeta)/\zeta$ is indeed a monotonically decreasing function of $\zeta$. Now, we denote $x={r_{\text{opt}}}\zeta\geq0$ and define $f(x)=2^x x\log_e2-(2^x-1)$. The first derivative of $f$ with respect to $x$ is $\dot{f}(x)=x2^x(\log_e2)^2\geq0$, implying that $f$ is a monotonically increasing function. Since $f(0) = 0$, we immediately conclude that $f(x) \ge 0$ for all $x \ge 0$. Hence, $2^{{r_{\text{opt}}}\zeta}{r_{\text{opt}}}\zeta\log_e2-(2^{{r_{\text{opt}}}\zeta}-1) \ge 0$ for all ${r_{\text{opt}}}\ge 0$ and $\zeta \ge 0$, $\frac{d({{\sf{R}}}_E(\zeta)/\zeta)}{d\zeta}\leq0$, and $\frac{E_b}{N_0}_{{\text{min}}}$ is achieved in the limit as $\zeta \to 0$. $\square$

Having analytically characterized the spectral efficiency–bit energy tradeoff in the wideband regime, we now provide numerical results to illustrate the theoretical findings. Fig. \[fig:4\] plots the spectral efficiency curves as a function of the bit energy in the Rayleigh channel. In all the curves, we have ${\bar{P}}/N_{0}=10^4$. Moreover, we 