of SER for the sampling receiver, we first apply a maximum likelihood detector to (\[eq:RecSamplesEQ\]), assuming a special case in which $\mathcal{C}$ is an $M$-PAM constellation, which yields the SER $$P_{\mathrm{err}}=2\frac{M-1}{M}Q\left(\frac{AG(0)\Delta a q(0)}{2\sqrt{G(0)^2N_{0}B}}\right)$$where $$Q(x)=\frac{1}{\sqrt{2\pi}}\intop_{x}^{\infty}\exp\left(\frac{-x^{2}}{2}\right)dx$$ is the Gaussian Q-function. As a result, $$A=\frac{2}{\Delta a q(0)}Q^{-1}\left(P_{\mathrm{err}}\frac{M}{2\left(M-1\right)}\right)\sqrt{N_{0}B}$$ and $$\frac{A_{\mathrm{ref}}}{A}=\frac{\Delta a q(0)}{\Delta a_{\mathrm{ref}}}\frac{Q^{-1}\left(P_{\mathrm{err}}\right)}{Q^{-1}\left(P_{\mathrm{err}}\frac{M}{2\left(M-1\right)}\right)}\sqrt{\frac{B_{\mathrm{ref}}}{B}},\label{eq:Ratio_Sampl}$$ where $B_{\mathrm{ref}}=1/T_\mathrm{b}$ is the bandwidth of the reference pulse. The optical power gain now follows from (\[eq:OptGainGeneral\]).

For the matched filter receiver, by applying the maximum likelihood detector to (\[eq:Rec\_MF\]), the SER will be $$\begin{aligned}
P_{\mathrm{err}}=2\frac{M-1}{M}Q\left(\frac{A\Delta a{E_q}\zeta}{2\sqrt{\frac{\zeta^2N_{0}{E_q}}{2}}}\right)\nonumber\\
\qquad=2\frac{M-1}{M}Q\left(A\Delta a\sqrt{\frac{{E_q}}{2N_{0}}}\right).\nonumber\end{aligned}$$ As a result, $$A=\frac{1}{\Delta a}Q^{-1}\left(P_{\mathrm{err}}\frac{M}{2\left(M-1\right)}\right)\sqrt{\frac{2N_{0}}{{E_q}}}$$and $$\frac{A_{\mathrm{ref}}}{A}=\frac{\Delta a}{\Delta a_{\mathrm{ref}}}\frac{\sqrt{2}Q^{-1}\left(P_{\mathrm{err}}\right)}{Q^{-1}\left(P_{\mathrm{err}}\frac{M}{2\left(M-1\right)}\right)}\sqrt{{E_q}B_{\mathrm{ref}}}.\label{eq:Ratio_MFRx}$$

In contrast to the case with equal eye openings (see Fig. \[fig:OptPower\_Eye\]), Nyquist and root-Nyquist pulses can be compared with each other when the SER is kept constant, since neither (\[eq:Ratio\_Sampl\]) nor (\[eq:Ratio\_MFRx\]) depend on the filter gains $G(0)$ and $\zeta$.

By increasing the bandwidth, the gain for SRC decreases slightly, whereas it increases for SDJ, where $\mu=0$ for both cases. The reason is that for these pulses by increasing $\alpha$, both $\overline{q}$ and the ratio $A_{\mathrm{ref}}/A$ decreases. We observe that for the regular Nyquist pulses in Sec. \[sec:Pulse-shaping\], the gain increases by increasing the bandwidth. The reason is that by increasing the roll-off 