$\lambda_c = \frac{2 \hbar }{{mu}}$. Non-vanishing value of Eq. (\[wignerbeta3\]) requires ${\bf{x}}_1 = \sum\limits_{j = 2}^N
{n_j {\bf{Q}}_j } T_{j,1}$ to be on the order of $\lambda_c $. Combining with Eq. (\[qconserve\]), we have: $${\bf{x}}_1  = \sum\limits_{j = 2}^N {n_j {\bf{Q}}_j } T_{j,1} \sim
o(\lambda_c ). \label{xconserve1}$$

![Echo line in Fig. \[npulserulefig\]. Here the black, blue and red vertical bars indicate the arriving time of the first pulse $T_1$, the echo time $T^{echo}$, and the arrival time of the $N^{th}$ pulse $T_N$ respectively.[]{data-label="echofig"}](npulseruleswithechofig.eps "fig:"){width="5"}\

Equations (\[qconserve\]) and (\[xconserve1\]) require the input displacement line in Fig. \[npulserulefig\] to start with $({\bf{q}}_1=0 ,{\bf{x}}_1 \sim o(\lambda_c) )$ in reciprocal phase space such that $ W({\bf{q}},{\bf{x}}, {T_1^-})$ is non-zero and can be scattered toward $({\bf{Q}}_N ,0)$ at time ${T_N }$. This is summarized in Fig. \[iniconditionfig\] as a supplementary diagram to Fig. \[npulserulefig\]. With the basic diagrams in Fig. \[proprulefig\] and with the input displacement line specified in Fig. \[iniconditionfig\], output of an arbitrary N-pulse TLI can be evaluated using the displacement diagram in Fig. \[npulserulefig\].

The condition of ${\bf{x}}_1 \sim o(\lambda_c) $ in Eq. (\[xconserve1\]) is different for specific initial condition with different temperatures, or initial atom velocity spreadings. In the next subsection, 