$$\label{eq:subsR}
{r_{\text{opt}}}=a{{\text{\footnotesize{SNR}}}}+b{{\text{\footnotesize{SNR}}}}^2+o({{\text{\footnotesize{SNR}}}}^2)$$ where $a$ and $b$ are real-valued constants. Substituting (\[eq:subsR\]) into (\[eq:thresh\]), we obtain the Taylor series expansion for ${\alpha_{\text{opt}}}$: $$\label{eq:subsalpha}
{\alpha_{\text{opt}}}=
\frac{a\log_e2}{B}+\left(\frac{b\log_e2}{B}+\frac{a^2\log_e^2
2}{2B^2}\right){{\text{\footnotesize{SNR}}}}+o({{\text{\footnotesize{SNR}}}}).$$ &gt;From (\[eq:subsalpha\]), we note that in the limit as ${{\text{\footnotesize{SNR}}}}\to
0$, we have $$\label{eq:coeffRa}
{{\alpha^{\ast}_{\text{opt}}}}=\frac{a\log_e2}{B}.$$ Next, we obtain the Taylor series expansion with respect to ${{\text{\footnotesize{SNR}}}}$ for $P\{z>{\alpha_{\text{opt}}}\}$ using the Leibniz Integral Rule [@Protter]: $$\begin{aligned}
\label{eq:probalpha}
P\{z>{\alpha_{\text{opt}}}\}
=P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}-\left(\frac{b\log_e2}{B}+\frac{a^2\log_e^2
2}{2B^2}\right)p_z({{\alpha^{\ast}_{\text{opt}}}}){{\text{\footnotesize{SNR}}}}+o({{\text{\footnotesize{SNR}}}}).\end{aligned}$$ Using (\[eq:subsR\]), (\[eq:subsalpha\]), and (\[eq:probalpha\]), we find the following series expansion for ${{\sf{R}}}_E$ given in (\[eq:spececopt\]): $$\begin{aligned}
\label{eq:reducespec}
\hspace{-0.2cm}{{\sf{R}}}_E({{\text{\footnotesize{SNR}}}})&=-\frac{1}{\theta
TB}\log_e\Bigg[1-\bigg(P\{z>
{{\alpha^{\ast}_{\text{opt}}}}\}-\bigg(\frac{b\log_e2}{B}+\frac{a^2\log_e^2
2}{2B^2}\bigg)p_z({{\alpha^{\ast}_{\text{opt}}}}){{\text{\footnotesize{SNR}}}}+o({{\text{\footnotesize{SNR}}}})\bigg)\nonumber\\
&\phantom{-\frac{1}{\theta T}-1-\bigg(P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}}\times\big(\theta
Ta{{\text{\footnotesize{SNR}}}}+(\theta
Tb-\frac{(\theta Ta)^2}{2}){{\text{\footnotesize{SNR}}}}^2+o({{\text{\footnotesize{SNR}}}}^2)\big)\Bigg]\nonumber\\
&=\frac{a P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}}{B}{{\text{\footnotesize{SNR}}}}+\frac{1}{B}\Big(-\frac{\theta
Ta^2}{2}P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}-\frac{a^3p_z({{\alpha^{\ast}_{\text{opt}}}})\log_e^2
2}{2B^2}+\frac{\theta T(P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}a)^2}{2}\Big){{\text{\footnotesize{SNR}}}}^2
+o({{\text{\footnotesize{SNR}}}}^2). $$ Then, using (\[eq:coeffRa\]), we immediately derive from (\[eq:reducespec\]) that $$\begin{aligned}
\dot{{{\sf{R}}}}_E(0)&=\frac{{{\alpha^{\ast}_{\text{opt}}}}P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}}{\log_e2},\label{eq:dotec}\\
\ddot{{{\sf{R}}}}_E(0)&=-\frac{{{\alpha^{\ast}_{\text{opt}}}}^3 p_z\{{{\alpha^{\ast}_{\text{opt}}}}\}}{\log_e2}-\frac{\theta TB{{\alpha^{\ast}_{\text{opt}}}}^2}{\log_e^2
2}P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}(1-P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}).\label{eq:ddotec}\end{aligned}$$ Similarly as in the discussion in the proof of Theorem \[theo:wideband\] in Section \[sec:wideband\], the optimal fixed-rate ${r_{\text{opt}}}$, akin to (\[eq:conditionb\]), should satisfy $$\label{eq:constthr}
\frac{2^{{r_{\text{opt}}}/B}p_z({\alpha_{\text{opt}}})\log_e2}{B{{\text{\footnotesize{SNR}}}}}(1-e^{-\theta
T{r_{\text{opt}}}})=\theta Te^{-\theta T{r_{\text{opt}}}}P\{z>{\alpha_{\text{opt}}}\}.$$ Taking the limits of both sides of (\[eq:constthr\]) as ${{\text{\footnotesize{SNR}}}}\to
0$ and employing (\[eq:subsR\]), we obtain $$\label{eq:condition}
\frac{a p_z({{\alpha^{\ast}_{\text{opt}}}})\log_e2}{B}=P\{z>{{\alpha^{\ast}_{\text{opt}}}}\}.$$ &gt;From (\[eq:coeffRa\]), (\[eq:condition\]) simplifies to $$\label{eq:reducedalpha}
{{\alpha^{\ast}_{\text{opt}}}}p_z({{\alpha^{\ast}_{\text{opt}}}})=P\{z>{{\alpha^{\ast}_{\text{opt}}}}\},$$ proving the condition in (\[eq:stacond\]). Moreover, using (\[eq:reducedalpha\]), the first term in the expression for $\ddot{{{\sf{R}}}}_E(0)$ in (\[eq:ddotec\]) becomes $-\frac{{{\alpha^{\ast}_{\text{opt}}}}^2
P\{z\geq{{\alpha^{\ast}_{\text{opt}}}}\}}{\log_e2}$. Together with this change, evaluating the expressions in (\[eq:ebresult\]) with the results 