## 2 Detection Model and Mechanism
### Detection System Excited by the PRBC-PAM Signal  
The signal generation process of the PRBC-PAM signal usually includes directional coupling, pseudorandom code generation, synchronous pulse circuit combination, and phase modulation, as shown in Figure 4.  
Equation (2) shows the PRBC signal generated by the m-sequence generator:  
\[p(t)=\frac{1}{\sqrt{NP}}\sum_{m=0}^{N-1}\sum_{n=0}^{P-1}C_{i}\mu(t-nT_{\rm c}-mT _{\rm r}) \tag{2}\]  
where \(\mu(t)=1/\sqrt{T_{\rm c}}rect(t/T_{\rm c})\), \(rect(t/T_{\rm c})= \begin{cases}1&0\leq t<T_{\rm c}\\ 0&\text{else}\end{cases} \) is the unit pulse with width \(T_{\rm c}\), \(P\) is the pulse number, \(T_{\rm r}=PT_{\rm c}\) is the pseudorandom code repetition period, \(N\) is the modulation period number, and \(C_{\rm i}=\{+1,-1\}\). Here, we express the pulse signal as follows:  
\[u(t)=\frac{1}{\sqrt{M}}\sum_{i=0}^{M-1}\mu\big{(}t-T_{\rm p}-iT_{\rm m}\big{)} \tag{3}\]  
Figure 4: Generation process of the PRBC–PAM signal. Figure 3: Time–output diagram of the intermittent chaotic state. Figure 2: Phase diagrams of chaotic and large\(-\)scale periodic states. (**a**) Chaotic state, \(F=0.82<0.826\) (**b**) large-scale periodic state, \(F=0.83>0.826\). where \(T_{\rm P}\) is the pulse width, and \(T_{\rm m}\) is the pulse repetition period. In general, the code width of the pseudorandom code in the PRBC-PAM signal is equal to the repetition period of the modulated pulse, namely, \(T_{\rm c}=T_{\rm m}\). Hence, the PAM pseudorandom sequence can be expressed as a product of the PRBC signal and the pulse signal as Equation (4):  
\[s_{\rm P}(t)=\frac{1}{\sqrt{NP}}\sum_{m=0}^{N-1}\sum_{n=0}^{P-1}C_{i}\mu\left( t-T_{\rm P}-nT_{\rm m}-mT_{\rm r}\right) \tag{4}\]  
Figure 5 shows the time-domain graph of the PRBC signal, the high-frequency pulse signal, and the PAM pseudorandom sequence in the case of \(T_{\rm c}=T_{\rm m}\). The transmitted PRBC-PAM signal is obtained by modulating the oscillator signal from the PAM pseudorandom sequence, expressed as Equation (5):  
\[ \begin{array}{l}s(t)=As_{\rm p}(t)\cos(\omega_{0}t+\varphi_{0})\\ =Au(t)\cos(\omega_{0}t+\varphi_{0}+\varphi_{1})\end{array}  \tag{5}\]  
where \(A\) is the amplitude of the weak signal, \(s_{\rm p}(t)\) is the PAM pseudorandom sequence, \(u(t)\) is the pulse signal, \(\omega_{0}\) is the carrier frequency, \(\varphi_{0}\) is the initial phase, and \(\varphi_{\rm i}\) is the phase code \(\varphi_{i}=\left\{ \begin{array}{l}0,p(t)=1\\ \pi,p(t)=-1\end{array} \right.\), where \(p(t)\) is the pseudorandom code shown in Equation (2). For simplicity, \(s(t)\) is obtained by combining \(s_{\rm p}(t)\) with cosine modulation, or \(u(t)\) with pseudocode cosine modulation. When the weak signal detection system is excited by a weak PRBC-PAM signal, i.e., the simple sinusoidal wave to be measured in the original Duffing weak signal detection system is replaced by a PRBC-PAM signal, then we can write the detection model as follows:  
\[ \begin{cases}\dot{x}=gy\\ \dot{y}=g\big{(}-0.5y+x-x^{3}+F\cos(\omega t)+\\ Au(t)\cos(\omega_{0}t+\varphi_{0}+\varphi_{\rm i})+n(t)\big{)}\end{cases}  \tag{6}\]  
Figure 5: Waveform evolution process of the PAM pseudorandom sequence: (**a**) PRBC signal; (**b**) Pulse signal; (**c**) PAM pseudorandom sequence. Similar to the simple sinusoidal case, set \(F=F_{c}\), so that the system is in a critical chaotic state. If the noise \(n(t)\) is ignored, the total driving force is as follows:  
\[F_{\rm c}\cos(\omega t)+Au(t)\cos(\omega_{0}t+\varphi_{0}+\varphi_{\rm i}) \tag{7}\] \[=F_{\rm e}(t)\cos(\omega t+\Phi(t))\]  
where  
\[ \begin{array}{l}F_{\rm e}(t)=\sqrt{F_{\rm c}^{2}+2F_{\rm c}Au(t)\cos(\Delta \omega t+\varphi(t))+A^{2}u^{2}(t)}\\ =\left\{\begin{array}{cc}F_{\rm c}&u(t)=0\\ \sqrt{F_{\rm c}+2F_{\rm c}A\cos(\Delta\omega t+\varphi(t))+A^{2}}&u(t)=1\end{array} \right.\\ \Phi(t)=\arctan\frac{Au(t)\sin(\Delta\omega t+\varphi(t))}{F_{\rm c}+Au(t)\cos( \Delta\omega t+\varphi(t))}\end{array}