## 4 Experiments and Analysis
### Comparison of Carrier Frequency Estimation Under Different SNRs  
Under different levels of SNR, the PRBC-PAM signal's carrier frequency is estimated according to the AZV and CSR methods based on the Duffing oscillator. The highest frequency of the Duffing oscillator when \(z_{\text{m}}<0.6\), the evaluation indicator \(z_{\text{m}}\), and the estimated sensitivity \(r_{\text{m}}\) are shown in Table 3. The proposed method can achieve an effective estimation of the carrier frequency under different SNRs. When \(z_{\text{m}}<0.6\), \(r_{\text{m}}\) is generally slightly greater than 0.03%. Therefore, the threshold value of the common ratio of 0.03% can meet the estimation accuracy, and it is still effective under low SNRs. Under an SNR of \(-30\) dB, \(r_{\text{m}}\) decreases because it is hard to distinguish Duffing's three-phase states under this situation, and the three-valued processing has a large error. In other words, both the values of \(1\) and \(-1\) are disordered and cannot meet the requirements of the duty cycle characteristics. The Duffing equation's output and its three-valued results are shown in Section 4.2.  
To further evaluate the proposed method in carrier frequency estimation of the PRBC-PAM signal, several related state-of-the-art methods are compared with the proposed method, including the SR algorithm in references [8; 9], the search algorithm in reference [10], the smoothed pseudo-Wigner-Ville distribution (SPWVD) method proposed in reference [14], the WRFFFT method proposed in reference [15], and the PCFCRD method  
Figure 15: The experimental layout and scenario: (**a**) The layout of the experimental setup; (**b**) The experimental scenario.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**SNR** & **Highest Frequency \(\omega_{\text{max}}\)** & \(z_{\text{m}}\) & \(r_{\text{m}}\) \\ \hline \(0\) dB & \(100.032\) MHz & \(0.5771\) & \(0.032\%\) \\ \hline \(-10\) dB & \(100.033\) MHz & \(0.5959\) & \(0.033\%\) \\ \hline \(-20\) dB & \(100.031\) MHz & \(0.5897\) & \(0.031\%\) \\ \hline \(-25\) dB & \(100.031\) MHz & \(0.5870\) & \(0.031\%\) \\ \hline \(-30\) dB & \(100.15\) MHz & \(0.5944\) & \(0.15\%\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Carrier frequency estimation results under different SNRs.  
proposed in reference [17]. The estimated carrier frequency based on the proposed Duffing oscillator and those methods are shown in Figure 13. The normalized root mean square error (NRMSE) results of each method under different SNRs are shown in Figure 16; here, we define the NRMSE as follows [27]:  
\[\text{NRMSE}:=\frac{\sqrt{\frac{1}{N}\sum\limits_{k=1}^{N}{{(x_{k}-x_{0})}^{2}} }}{x_{0}} \tag{13}\]  
where \(x_{0}\) is the parameter's true value, \(x_{k}\) is what we estimated, and \(N\) represents the number of simulations in a given experiment.  
It can be seen from Figure 16 that the carrier frequency estimation results based on the Duffing system proposed in this paper are stable at a higher level under different SNRs. The NRMSE still reaches 0.01 at \(-\)30 dB SNR, which is superior to the other methods. Although the NRMSE value of the proposed method is slightly higher when the SNR is 0 dB, its estimation accuracy is sufficiently low, and it has almost no effect on the estimation of the PAM pseudorandom sequence.