by increasing the roll-off factor (see Fig. \[fig:RequiredDC\]), whereas the required bandwidth increases. The highest optical power gain for all pulses will be achieved when the roll-off factor $\alpha$ is one. The reason is that by increasing the roll-off factor, the required bias, which is the only parameter in (\[eq:OptGainEye\]) that depends on $\alpha$, decreases. The BTN and the PL pulses have approximately similar optical power gain, and the Poly and RC pulses have smaller gains, due to higher $\mu$, which is also visible in the eye diagrams of Fig. \[eye-diagrams\]. Comparing the binary and 4-PAM cases for the same $\alpha$ and $\Delta a$, we can see in Fig. \[fig:OptPower\_Eye\] that by using higher-order modulation formats, the optical power gain for all pulses decreases, since in (\[eq:OptGainEye\]), $\mathbb{E}\left\{a_{k}\right\}$ and $\mu$ will increase. For $0.5<BT_{\mathrm{b}}<1$, the optical power gain for the best 4-PAM system with nonnegative Nyquist pulses is up to 2.39 dB less than the gain of the best OOK system with regular Nyquist pulses.

For any root-Nyquist pulse with a matched filter receiver, the minimum eye opening after filtering is given by (\[eq:Rec\_MF\]) as $$\min_{a,a'\in \mathcal{C}, a\neq a'} \left|A \zeta\left(\mu Q(0)+a{E_q}\right)-A\zeta\left(\mu Q(0)+a'{E_q}\right)\right|=A\zeta\Delta a {E_q}.\label{eq:MinEye_MFR}$$ Since the eye openings in 