## 3 Parameter Estimation Method for the Weak PRBC-PAM Signal
### Parameter Estimation
When the frequency to be measured is not equal to the internal driving signal, i.e., \(\omega_{0}\
eq\omega\), there is a significant difference in the time-domain plots of the output \(Sdf(t)\) and the PAM pseudorandom sequences. Figure 13a-d shows the detection system's output signal, its three-valued signal \(Sdf(t)\), the estimated pseudorandom code obtained by the AZV method, and the actual PAM pseudorandom sequence when \(\omega\) = 100 MHz, \(\omega_{0}\) = 104 MHz, and the symbol width is \(p\) = 100 ns. It is easy to know that the values of 1, \(-\)1, and 0 for the PAM pseudorandom sequence and \(Sdf(t)\) no longer exhibit a one-to-one correspondence, and the ratio of the existing times of \(-\)1 and 0 no longer conforms to the regularity of the duty cycle because of the frequency difference, and the chaotic state and the critical chaotic state only exhibit little difference. According to Equation (8), when \(\Delta\omega\) = 0, the chaotic state with an equivalent amplitude of \(\left\lvert F_{\mathrm{c}}-A\right\rvert\) is significantly different from the critical chaotic state with an equivalent amplitude of \(F_{\mathrm{c}}\), which makes it relatively easy to identify the different states of the Duffing oscillator output signal. Figure 11: Time\(-\)domain diagram of the PAM pseudorandom sequence. Figure 12: Pseudo\(-\)random code time-domain diagram: (**a**) Pseudorandom code time-domain diagram obtained by \(Sdf(t)\) with the AZV method; (**b**) Pseudorandom code time-domain diagram of the actual signal. However, when \(\Delta\omega\
eq 0\), the value of \(\cos(\Delta\omega+\varphi(t))\) changes with the change in \(\Delta\omega\), and the equivalent amplitudes of the chaotic state also change constantly between \(\left|F_{\mathrm{c}}-A\right|\) and \(F_{\mathrm{c}}\). When the equivalent amplitude of the chaotic state is close to \(F_{\mathrm{c}}\), it is difficult to distinguish between the chaotic state and the critical chaotic state. Although the equivalent amplitude of the large-scale period state also changes between \(F_{\mathrm{c}}\) and \((F_{\mathrm{c}}+A)\), due to the sensitivity of the Duffing oscillator for weak signals, as long as the equivalent amplitude \(F_{\mathrm{e}}\) is slightly larger than the critical amplitude \(F_{\mathrm{c}}\), the Duffing oscillator system is in a large-scale period state, which means that the change in \(\Delta\omega\) has little effect on the large-scale period state. Therefore, the time occupied by \(-1\) in \(Sdf(t)\) (the existence time of the critical chaotic state) and the time occupied by \(-1\) in the estimated pseudorandom code after AZV are not equal to the duty cycle of the pulse signal, but the time occupied by \(1\) (the existence time of the large-scale periodic state) is still equal to the duty cycle. The ratio of the time taken by the value of \(1\) in the actual pseudorandom code to the time taken by the value of \(1\) in the estimated pseudorandom code after AZV is denoted as \(z_{1}\), and the ratio of the time taken by the value of \(-1\) in the actual pseudorandom code to the time taken by the value of \(-1\) in the estimated pseudorandom code after the AZV is \(z_{2}\). Set \(z_{\mathrm{m}}\) as the evaluation indicator, which can be expressed as follows:  
\[(z_{2}-z_{1})/z_{1} \tag{12}\]  
Figure 13: Time-domain diagram of several signals when \(\Delta\omega\
eq 0\): (**a**) detection system’s output; (**b**) \(Sdf(t)\); (**c**) Estimated pseudorandom code after the AZV method; (**d**) Actual PAM pseudorandom sequence. To find the threshold of \(z_{\rm m}\), the values of \(z_{\rm m}\) obtained under different frequency differences \(\Delta\omega\) are shown in Figure 14 to help determine whether the frequencies are equal. Here, the signal to be detected has a frequency of 100 MHz. With the increase in \(\Delta\omega\) (the frequency of the Duffing oscillator internal periodic driving force signal changes adaptively), although the value of \(z_{\rm m}\) fluctuates to some extent, the overall trend also increases.