## 4 Simulations
### Comparison Between the Numerical and Theoretical Results  
Tables 1 and 2 illustrate the errors between the BER performance and the theoretical results of suboptimal receiver in noncoherent PPM system. The error is defined as  
\[error=\frac{\|P_{eb,num}-P_{eb,thy}\|}{P_{eb,thy}}\,, \tag{24}\]  
where \(P_{eb,num}\) represents the numerical result, \(P_{eb,thy}\) denotes the theoretical result and \(\|\cdot\|\) is the absolute value operator. The ensemble runs obtaining \(P_{eb,num}\) is 100. In Table 1, each test uses \(1\times 10^{3}\) PPM symbols, while the number of PPM symbols used for Table 2 is \(1\times 10^{5}\). For convenience, we only analyzed the cases where the BER results are larger than \(10^{-4}\). The theoretical results are derived based on the assumption that the PPM symbols are uniformly distributed in the possible time slots. However, the numerical simulation can not accurately meet the assumption. Thus, there are certain errors between the numerical results and the theoretical results. It can be seen from Tables 1 and 2 that, as the number of PPM symbols increases, the deviation between the numerical results and the theoretical results tends to decrease. The reason for this is that the PPM symbols asymptotically obey the uniform distribution in the possible time slots when the number of PPM symbols increases. Figure 7: BER performance of noncoherent PPM optimal receiver. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \(E_{b}/N_{0}\) (dB) & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline Binary (\%) & 0.5 & 1.5 & 0.2 & 1.0 & 1.2 & 0.9 & 0.5 & 2.1 & 0.7 & 0.8 & 1.1 & 0.3 & 0.5 \\
4-ary (\%) & 1.2 & 0.6 & 0.9 & 0.5 & 1.5 & 2.1 & 1.7 & 0.9 & 1.3 & 0.8 & / & / & / \\
8-ary (\%) & 0.4 & 1.2 & 2.2 & 1.4 & 0.8 & 1.3 & 0.9 & 1.3 & 1.6 & / & / & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The errors between the BER performance and the theoretical results of optimal receiver (\(100\times 10^{5}\) PPM symbols). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \(E_{b}/N_{0}\) (dB) & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline Binary (\%) & 2.1 & 3.6 & 3.1 & 0.7 & 3.1 & 4.5 & 0.3 & 1.2 & 0.3 & 3.9 & 1.3 & 2.8 & 0.4 \\
4-ary (\%) & 0.4 & 1.7 & 1.4 & 3.8 & 0.3 & 2.7 & 2.5 & 1.8 & 1.5 & 1.3 & 1.4 & 6.8 & 7.6 \\
8-ary (\%) & 4.8 & 0.3 & 0.5 & 4.1 & 3.1 & 5.1 & 0.3 & 3.0 & 3.6 & 1.8 & 20.7 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The errors between the BER performance and the theoretical results of suboptimal receiver (\(100\times 10^{5}\) PPM symbols). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \(E_{b}/N_{0}\) (dB) & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline Binary (\%) & 1.6 & 0.9 & 2.2 & 1.1 & 1.5 & 2.3 & 2.7 & 1.3 & 3.5 & 3.7 & 1.5 & 2.3 & 5.4 \\
4-ary (\%) & 2.5 & 3.3 & 0.8 & 1.9 & 2.5 & 2.6 & 0.9 & 1.5 & 6.1 & 9.3 & / & / & / \\
8-ary (\%) & 2.3 & 1.5 & 2.3 & 0.7 & 2.5 & 1.7 & 3.5 & 4.6 & 17.9 & / & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The errors between the BER performance and the theoretical results of optimal receiver (\(100\times 10^{5}\) PPM symbols). variance \(N_{0}/2\). The values of \(E_{b}/N_{0}\) ranges from 0 dB to 12 dB. We consider the case where the BER is larger than \(10^{-4}\). The width of a time slot is 5 ns. The other parameters are set as \(L=2\), \(\alpha_{l}=\{0,-2\}\) dB and \(\tau_{l}=\{0,2\}\) ns. Figures 8 and 9 show the BER results of the suboptimal receiver and the optimal receiver when the channel is Rayleigh and AWGN. Compared with the AWGN channel, the BER performance of the noncoherent Q-ary PPM system decreases in the Rayleigh fading channel. As the focus of this paper is the AWGN channel for which the receivers are designed, it is reasonable that the BER degradation occurs in the Rayleigh fading channel.