receiver, for which no bias is needed. Using (\[eq:OpticalPower\]), $P_{\mathrm{opt}}^{\mathrm{ref}}=A_{\mathrm{ref}}\mathbb{E}_{\mathrm{ref}}\left\{a_k\right\}$ and $$\Upsilon=10\log_{10}\left(\frac{A_{\mathrm{ref}}\mathbb{E}_{\mathrm{ref}}\left\{a_k\right\}}{A\left(\mu+\mathbb{E}\left\{a_k\right\}\overline{q}\right)}\right)\label{eq:OptGainGeneral}$$ where $A_{\mathrm{ref}}$ and $\mathbb{E}_{\mathrm{ref}}\left\{a_{k}\right\}$ are the scaling factor and the symbol average for the reference system, respectively. Defining $$\Delta a=\min_{a,a'\in\mathcal{C},a\neq a'}\left|a-a'\right|\label{eq:MinDistance}$$ as the minimum distance between any two constellation points $a$ and $a'$, $\mathbb{E}_{\mathrm{ref}}\left\{a_k\right\}=\Delta a_{\mathrm{ref}}/2$, where $\Delta a_{\mathrm{ref}}$ is the minimum distance for the reference system. The expressions in (\[eq:OptGainGeneral\]) and (\[eq:MinDistance\]) hold in general for all finite set of constellation points $\mathcal{C}$.

Initially, we compare the pulses in a noise-free setting. For any Nyquist pulse with a sampling receiver, the minimum eye opening after filtering is given by (\[eq:RecSamplesEQ\]) as $$\min_{ a,a'\in \mathcal{C}, a\neq a'} \left|AG(0)\left(\mu+aq(0)\right)-AG(0)\left(\mu+a'q(0)\right)\right|=AG(0)\Delta a q(0). \label{eq:MinEye_Sample}$$ As a result, to have equal eye opening we require ${A_{\mathrm{ref}}}/{A}={\Delta a q(0)}/{\Delta a_{\mathrm{ref}}}$, which substituted into yields $$\Upsilon=10\log_{10}\left(\frac{\Delta a q(0)}{\mu+\mathbb{E}\left\{a_k\right\}\overline{q}}\right).\label{eq:OptGainEye}$$

Fig. \[fig:OptPower\_Eye\] demonstrates the comparison of the optical power gain for various pulses defined in Sec. \[sec:Pulse-shaping\] for both OOK and 4-PAM formats, where the signals are scaled to have equal eye opening. The S2 pulse with OOK modulation, which is used as a baseline for comparison, is shown in the figure with an arrow. The results for SRC and SDJ have been derived before in [@4132995 