method of using a Nyquist pulse in the transmitter and a rectangular filter in the receiver, other scenarios can be designed that generate Nyquist pulses at the input $r(t)$ of the sampling unit. In one of these methods, the transmitted pulse is a root-Nyquist pulse, and the receiver contains a filter matched to the transmitted pulse . Consequently, the output of the matched filter will be ISI-free if for any integer $k$ $$\intop_{-\infty}^{\infty}q(t)q(t-k{T_\mathrm{s}})dt=\begin{cases}
E_{{q}} & k=0\\
0 & k\neq0
\end{cases}
\label{eq:Root_Nyquist Criteria},$$ where ${E_q}=\intop_{-\infty}^{\infty}q^{2}(t)dt.$ Tables \[pulse-def\] and \[tab:All\_Info\_Nyq\] also includes two root-Nyquist pulses that have been previously used for conventional coherent channels, where again $0\leq \alpha \leq 1$. These are the root raised cosine (RRC) pulse and the first-order Xia pulse [@xia1997family]. Both have the lowpass bandwidth $B=(1+\alpha)/(2{T_\mathrm{s}})$.

Although the output of the matched filter for both the first order Xia pulse and the RRC pulse are similar ($r(t)$ consists of RC pulses in both cases), the RRC is symmetric in time, whereas the Xia pulse has more energy in the precursor (i.e., the part of the pulse before the peak) [@tan2004transmission]. Moreover, the maximum of Xia pulse does not happen at the origin. The important point with the Xia 