zero mean additive white Gaussian noise with variance $\sigma_{u}^{2}=\zeta^2N_{0}{E_q}/2$. Applying the root-Nyquist criterion given in (\[eq:Root\_Nyquist Criteria\]) to the sampled version of (\[eq:MF\_Output\]), the $i$-th filtered sample will be, for any constellation $\mathcal{C}$, $$r(i{T_\mathrm{s}})=A\zeta \left(\mu Q(0)+a_{i}{E_q}\right)+u(i{T_\mathrm{s}}).\label{eq:Rec_MF}$$

Comparison Between Pulses
-------------------------

As mentioned in Sec. \[sec:SYSTEM-MODEL\], it may be desirable to minimize the average or peak optical power. The next theorem shows that these two criteria are equivalent for narrow-band pulses ($B{T_\mathrm{s}}< 1$) and symmetric constellations ($\mathbb{E}\{a_k\} = L$).

\[th:pmax\] If $B{T_\mathrm{s}}< 1$ and $\mathbb{E}\{a_k\} = L$, then $P_\mathrm{max} = 2 P_\mathrm{opt}$.

From and Corollary \[CorrAvval\], $$\begin{aligned}
P_{\mathrm{max}} &=JA\left(\mu+\max_{\forall a,-\infty< t < \infty}\sum_{k=-\infty}^{\infty}\left[\left(a_{k}-L\right)q(t-k{T_\mathrm{s}})+Lq(t-k{T_\mathrm{s}})\right]\right)\\
&=JA\left(\mu+\max_{\forall a,-\infty< t < \infty}\left[\sum_{k=-\infty}^{\infty}\left(a_{k}-L\right)q(t-k{T_\mathrm{s}})+\frac{LQ(0)}{{T_\mathrm{s}}}\right]\right).\end{aligned}$$ In analogy to , the maximum is $$\begin{aligned}
P_{\mathrm{max}} &=JA\left(\mu+(\hat{a}-L)\max_{0\leq t<{T_\mathrm{s}}} \sum_{k=-\infty}^{\infty}|q(t-k{T_\mathrm{s}})|+\frac{LQ(0)}{{T_\mathrm{s}}}\right) =JA\left(2\mu+2\frac{LQ(0)}{{T_\mathrm{s}}}\right)\end{aligned}$$ which compared with completes the proof.

To compare the optical power of various pulses, a criterion called optical power gain is used, which is defined as[@4132995] $$\Upsilon=10\log_{10}\left(\frac{P_{\mathrm{opt}}^{\mathrm{ref}}}{P_{\mathrm{opt}}}\right),\nonumber$$ where $P_{\mathrm{opt}}^{\mathrm{ref}}$ is the average optical power for a reference system. (According to Theorem \[th:pmax\], $\Upsilon$ would be the same if defined in terms of $P_\mathrm{max}$, for all pulses in our study except SRC and SDJ.) Similarly to [@1577875], this reference is chosen to be the S2 pulse with OOK modulation and sampling 