## 3 Parameter Estimation Method for the Weak PRBC-PAM Signal
### Parameter Estimation  
According to Equation (11), the PAM pseudorandom sequence has three values, \(1\), \(-1\), and \(0\), and the pseudorandom code has two values, \(1\) and \(-1\). By comparing the pseudorandom code and the PAM pseudorandom sequence, it is evident that the PAM pseudorandom sequence is the result of adding values of \(0\) based on the duty cycle characteristics of the pulse signal. Since the duty cycle of the pulse signal is constant, the parameters of the pulse signal can be estimated according to the ratio between the existence times of \(1\) or \(-1\) and \(0\) in the PAM pseudorandom sequence. Figure 10 shows the process for estimating the PRBC-PAM signal parameters based on the AZV and CSR methods of the Duffing oscillator. Firstly, the critical amplitude and frequency of the Duffing oscillator are preset. Secondly, the states of the Duffing oscillator are identified based on STFT; Then, the \(0\) value of the ternary signal \(Sdf(t)\) of the output signal of the Duffing oscillator is adjusted to the nearest non-zero value of the signal \(p(t)\), which is a pseudorandom sequence of the pseudo-code pulse Doppler fuze signal. The ratio \(z1\) of the time taken by value \(1\) in \(Sdf(t)\) is the duty cycle of the pulse signal. Next, by averaging the smallest numbers that are close to each other in the duration of \(1\) or \(-1\) in \(p(t)\), the code width Tc of the pseudorandom sequence is obtained, which is also equal to the repetition period Tm of the pulse signal. Finally, the pseudorandom sequence is multiplied by the pulse signal with a known duty cycle, and the PAM pseudorandom sequence is obtained after removing burrs. Figure 9: Time\(-\)domain diagrams of the detection system driven by the PRBC-PAM signal: (**a**) Time-domain diagram of the input PAM’s sequence; (**b**) Time-domain diagram of the system output; (**c**) Time-domain diagram of \(Sdf(t)\). When the frequency of the to-be-measured PRBC-PAM signal equals the internal driving signal of the Duffing oscillator, i.e., \(\omega_{0}=\omega\), the PAM pseudorandom \(\{1,-1,0\}\) code sequence expression of the PRBC-PAM signal totally equals our processed three-valued sequence \(\mathit{Sdf}(t)\), as we subtract both of them, and the result is \(0\). This can be explained by the critical threshold characteristics of the system. The set detection system in the critical chaotic state and \(\omega_{0}=\omega\) time-domain position of the positive PRBC-PAM pulses will make the equivalent threshold of the system equal to \(\mid\!F_{\mathrm{c}}+A\!\mid\), the position without pulses will still be in the critical state, and the time-domain position of the negative PRBC-PAM pulses will make the equivalent threshold of the system equal to \(\mid\!F_{\mathrm{c}}-A\!\mid\). Based on the system state, the further processed \(\mathit{Sdf}(t)\) can fully reflect the input PRBC-PAM pulse characteristics. So if we know the frequency, the PAM pseudorandom \(\{1,-1,0\}\) code sequence can be directly estimated by the three-valued Duffing oscillator's output signal, as can be seen in Figures 8(b) and 11. Figure 10: Flow chart for the PRBC-PAM signal parameter estimation based on the Duffing oscillator. In addition, the ratio of the time lengths of 1 and \(-\)1 of the PAM pseudorandom \(\{1,-1,0\}\) code sequence to the total time length is equal to the duty cycle of the pulse signal. Therefore, the pseudorandom code of the PRBC-PAM signal can be obtained by adjusting the value of 0 of the PAM pseudorandom \(\{1,-1,0\}\) code sequence to the nearest non-zero value before it. In this paper, this method is called adjusting the zero value (AZV) method. Figure 12a shows the pseudorandom code obtained after adjusting the 0 value, and Figure 12b presents the actual pseudorandom code of the signal, which shows that these two figures exhibit a one-to-one correspondence.