## 3 Parameter Estimation Method for the Weak PRBC-PAM Signal  
### System Output Symbolization  
Driven by the PRBC-PAM signal, the transition of Equation (6)'s output between three states, namely, chaotic, critical chaotic, and large-scale periodic state, is caused by the gradual variation in the driving force. So, it is natural to infer PRBC-PAM parameters from the information brought by the output sequences using the concept of SSS (Symbol Sequence Statistics) [27; 28; 29]. According to the basic qualitative analysis of Section 2, it is possible to establish a quantitative relationship between the detection system's output and input. Inspired by idea of the symbol sequence statistic, the output can be roughened into a new sequence composed of 1, 0, \(-1\) to distinguish chaotic states, critical chaotic states, and large-scale periodic states, to highlight input features, so that the processed output sequence only contains the output state type rather than the real value, highlighting the relationship between the input and output of the detection system, and further obtaining estimated parameters. The difference between the critical chaotic state and chaotic state is small and hard to distinguish by general signal processing methods. Due to the fact that the three states are in different time positions for the same signal and their frequencies are different, the time-frequency processing technique, short-time Fourier transform (STFT), can be used for state recognition to realize three-valued processing [30; 31]. The STFT is a kind of local time-frequency analysis technique. It uses the window function to intercept the signal for a short period and obtains the frequency of the intercepted segment through traditional Fourier transform; then, the time-frequency distribution can be obtained by moving carefully designed window functions along the time axis. The three-dimensional time-frequency distribution of the three states obtained by the STFT is shown in Figure 7, from which it is evident that although the critical chaotic state and the chaotic state are similar, the three states can be effectively distinguished by the STFT method. The amplitude spectrum peaks of the chaotic state fluctuate violently and irregularly, which is relatively chaotic. Although the amplitude spectrum of the critical chaotic state is also messy, the peak value's distribution basically follows a horizontal line. The amplitude spectral peaks of large-scale periodic states are very clean and flat. To further analyze the impact of the internal driving signal's frequency on the STFT results, time-domain diagrams of three states with different internal driving frequencies are given in Figure 8. For one given state, the time-amplitude curves of the different frequencies vary slightly, but the overall trend stays the same. For these three states, they are obviously different at different frequencies. Firstly, the maximum amplitude of the chaotic state is far from the other two. Secondly, the maximum amplitudes in Figure 8b,c are nearly the same. But in terms of the average value, the mean amplitude in Figure 8b is smaller than in Figure 8c because of the fluctuation in the maximum amplitude curve. Figure 7: Three-dimensional time–frequency distribution diagram obtained by the STFT in three states: (**a**) Chaotic state; (**b**) Critical chaotic state; (**c**) Large-scale periodic state. In summary, the time-domain diagram is obtained from the frequency point corresponding to the maximum amplitude in the three-dimensional time-frequency distribution graph obtained by the STFT method, and then, the mean value of the highest amplitude curve is used as the decision parameter of the system state, which is expressed as follows:  
\[ \begin{array}{l}E_{\mathrm{m}}=\frac{1}{N}\sum\limits_{i=1}^{N}\left(\max \left|\sum\limits_{i=0}^{N-1}x(k)g_{\mathrm{Ham}}(k-i)e^{-/2\pi r/i}\right| \right)\\ g_{\mathrm{Ham}}(n)=0.54-0.46\cos\left(2\pi\frac{n}{N-1}\right),0\leq n\leq N- 1\end{array}