## 4 Experiments and Analysis
### Comparison of PAM Pseudorandom Sequence Estimation Under Different SNRs  
As seen in Section 4.1, when the value of \(z_{\text{m}}\) is less than 0.6 at different SNRs, the values of \(r_{\text{m}}\) for the carrier frequency are all above 0.03%, so a common ratio of 0.03% can be used to effectively estimate the carrier frequency. To verify that the carrier frequency estimation sensitivity has little influence on the PAM pseudorandom sequence estimation, the analysis is performed on the measured PAM pseudorandom sequence when \(\omega_{0}=\) 100.03 MHz. The time-domain diagram of each estimated parameter under different SNRs are shown in Figures 17-21. The figures show that with a reduced SNR, the output of the Duffing equation gradually becomes inconsistent, the boundary between the three states gradually blurs, and the difficulty of three-valued processing increases accordingly. Therefore, the deviations between the estimated pseudorandom code or the PAM pseudorandom sequence and the actual signals also increase. When the SNR is \(-\)25 dB, the similarity between the estimated results and the actual signals is still very high. However, when the SNR is \(-\)30 dB, it is difficult to distinguish the different states of the Duffing equation's output. Furthermore, the estimation of the pseudorandom code and PAM pseudorandom sequence are also quite different from the actual signals. Figure 16: NRMSE of the carrier frequency estimation of different methods under different SNRs [8; 10; 14; 15; 17]. Figure 19: Time-domain diagram of several signals under an SNR of \(-20\) dB when \(\omega_{0}=100.03\) MHz: (a) Output signal of the Duffing oscillator \(y(t)\); (b) Binarized signal \(Sdf(t)\); (c) Estimated pseudorandom code; (d) Estimated PAM pseudorandom sequence. Figure 17: Time\(-\)domain diagram of several signals under an SNR of \(0\) dB when \(\omega_{0}=100.03\) MHz: (a) Output signal of the Duffing oscillator \(y(t)\); (b) Binarized signal \(Sdf(t)\); (c) Estimated pseudorandom code; (d) Estimated PAM pseudorandom sequence. Figure 18: Time-domain diagram of several signals under an SNR of \(-10\) dB when \(\omega_{0}=100.03\) MHz: (a) Output signal of the Duffing oscillator \(y(t)\); (b) Binarized signal \(Sdf(t)\); (c) Estimated pseudorandom code; (d) Estimated PAM pseudorandom sequence. Under different SNRs, the pulse signal repetition period, duty cycle, and duty cycle evaluation indicator \(z_{\text{m}}\) are estimated and compared. In addition, the Pearson correlation coefficient is used to compare the estimated pseudorandom code and PAM pseudorandom sequence with the actual signal. The estimated results are shown in Table 4. The Pearson correlation coefficient is defined as follows:  
\[\rho_{y}=\frac{\text{cov}(x,y)}{\sigma_{x}\sigma_{y}} \tag{14}\]  
where cov(\(x\), \(y\)) is the covariance of variable \(x\) and \(y\), and \(\sigma_{x}\) and \(\sigma_{y}\) are the indicator deviations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**SNR**} & \multirow{2}{*}{**Estimated Parameters**} & \multirow{2}{*}{**Duty Cycle**} & \multirow{2}{*}{\(z_{\text{m}}\)} & \multicolumn{2}{c}{**Pearson Correlation Coefficient**} \\ \cline{5-6}  & & & & & **Pseudorandom Code** & **PAM Pseudorandom Sequence** \\ \hline
0 dB & 99.999 ns & 20.39\% & 0.5805 & 0.9700 & 0.9107 \\ \hline \(-\)10 dB & 100.665 ns & 20.78\% & 0.5953 & 0.9254 & 0.8906 \\ \hline \(-\)20 dB & 101.999 ns & 20.18\% & 0.5556 & 0.8684 & 0.7826 \\ \hline \(-\)25 dB & 100.832 ns & 19.10\% & 0.5944 & 0.7934 & 0.7587 \\ \hline \(-\)30 dB & 102.499 ns & 29.56\% & 0.4501 & 0.4118 & 0.3990 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimation results under different SNRs. Figure 21: Time-domain diagram of several signals under an SNR of \(-\)30 dB when \(\omega_{0}\) = 100.03 MHz: (**a**) Output signal of the Duffing oscillator \(y(t)\); (**b**) Binarized signal \(Sdf(t)\); (**c**) Estimated pseudorandom code; (**d**) Estimated PAM pseudorandom sequence.