## 3 Parameter Estimation Method for the Weak PRBC-PAM Signal
### Parameter Estimation
When \(z_{\rm m}\) reaches a certain value (approximately \(z_{\rm m}\) = 1.3), it decreases slightly, but it always remains above 0.9. Therefore, a certain threshold can be set when \(z_{\rm m}\) is less than the threshold, and the frequency difference is \(\Delta\omega\) = 0, which means that the carrier frequency of the signal in the test is equal to the internal driving force signal of the Duffing oscillator. When \(\Delta\omega\) = 0.03 MHz and \(\omega\) = 100.03 MHz, we can obtain \(z_{\rm m}\) = 0.5805. To verify the accuracy of the method and estimate its sensitivity, the parameters of the PRBC-PAM signals at different carrier frequencies are estimated. By adjusting the frequency of the Duffing oscillator's internal periodic driving signal, the maximum frequency of the Duffing oscillator is taken when \(z_{\rm m}\) < 0.6, nearly \(z_{\rm m}\) = 0.5805, and the estimated sensitivity can be obtained. Taking the carrier frequency of \(\omega_{0}\) = 100 MHz for the PRBC-PAM signal as an example, let the frequency \(\omega\) of the Duffing oscillator increase continuously and adaptively from 100 MHz; when \(\omega\) = 100.03 MHz and \(z_{\rm m}\) = 0.5805, the estimated sensitivity is \(r_{\rm m}\) = (100.03 \(-\) 100)/100 \(\times\) 100% = 0.03%. In addition, simulations are also carried out for the cases of \(\omega_{0}\) = 10 kHz, 100 kHz, 10 MHz, 1 GHz, and 10 GHz. The highest frequency of the Duffing oscillator when \(z_{\rm m}\) < 0.6, the evaluation indicator \(z_{\rm m}\), and the estimated sensitivity \(r_{\rm m}\) are shown in Table 1. In this paper, the carrier frequency estimation method based on the ratio of the time of the critical chaotic state and large-scale periodic state to the corresponding time of the pseudorandom code obtained by the AZV method is referred to as the CSR method. In practical applications, receivers can roughly measure the signal frequency. Therefore, the carrier frequency range of the PRBC-PAM signal, (\(\omega_{\rm start}\), \(\omega_{\rm end}\)), can be obtained based on prior knowledge. According to the estimated sensitivity above, the frequency of the internal driving force starts from \(\omega_{\rm start}\) and increases adaptively with \(q\) = 1 + 0.03%, which is the common ratio. The Duffing oscillator system is run once at each value, and the \(z_{\rm m}\) value can be obtained. When \(z_{\rm m}\) < 0.6, the PRBC-PAM signal's carrier frequency is considered equal to the internal driving force signal of the Duffing oscillator. As described before, when the frequency of the to-be-measured weak signal equals that of the Duffing oscillator's internal driving force signal, _Sdf_(_t_) has a one-to-one corre  
\begin{table}
\begin{tabular}{c c c c} \hline
**Carrier Frequency \(\omega_{\rm 0}\)** & **Highest Frequency \(\omega_{\rm max}\)** & \(z_{\rm m}\) & \(r_{\rm m}\) \\ \hline
10 kHz & 10.0034 kHz & 0.5991 & 0.034\% \\ \hline
100 kHz & 100.032 kHz & 0.5701 & 0.032\% \\ \hline
10 MHz & 10.0032 MHz & 0.5771 & 0.032\% \\ \hline
100 MHz & 100.03 MHz & 0.5805 & 0.03\% \\ \hline
1 GHz & 1.0003 GHz & 0.5978 & 0.03\% \\spondence to the PAM pseudorandom sequence, and the pseudorandom code estimated by adjusting \(Sdf(t)\) with the AZV method also exhibits one-to-one correspondence with the actual pseudorandom code. However, because the measured carrier frequency is not guaranteed to be the same as the actual carrier frequency and because it is easy for errors to be introduced during three-valued processing, adjusting the zero value can eliminate the influence of burrs or tiny errors, and thus, the estimation result of the pseudorandom code is superior to that of \(Sdf(t)\). Therefore, the PAM pseudorandom sequence estimation is completed by obtaining the duty cycle of the pulse signal and the estimated pseudorandom code based on the AZV method rather than directly by \(Sdf(t)\).