## 2 Principles
The results indicate the typical characteristics of chaotic signal, including random like-noise in the time domain and the wide bandwidth in the frequency domain. There are the Lyapunov exponent of the signal greater than 0, which exhibits the chaotic properties of the \(\mathbf{x}\) signal. The autocorrelation function (ACF) can prove the high randomness. Figure 4f shows the autocorrelation of the \(\mathbf{x}\) chaotic signal. As can be seen from the results in Figure 4f, there are no similar fragments in the \(\mathbf{x}\) chaotic signal. In this scheme, the \(\mathbf{x}\) signal generated by the chaotic circuit is used to encrypt the data, and the \(\mathbf{x}\) chaotic signal is injected into PM as the driving signal to disturb the phase. The amplitude of the driving signal of PM 1 is opposite to that of the driving signal of PM 2. The chaotic circuits at the transmitter and receiver are controlled by the same power supply, which ensures the synchronization of chaotic signals. When the chaotic signal is injected into the PM, the adapter needs to be connected, which has a 50 Ohms impedance. This affects the chaotic waveform and the nonlinear dynamic characteristics of chaos disappear. Therefore, in this scheme, the chaotic circuit module needs to be connected in series with the amplifying matching module to realize the matching between modules, and the amplification of the chaotic signal is conducive to signal encryption. The specific flow chart is shown in Figure 5. The voltage is adjusted according to the feedback signal of the common emitter amplifier circuit. Figure 4: (**a**) Physical equipment of the analog circuit, (**b**) \(\mathbf{x}\)-\(\mathbf{y}\) phase trajectory, (**c**) \(\mathbf{y}\)-\(\mathbf{z}\) phase trajectory, (**d**) temporal waveform, (**e**) spectrum, and (**f**) autocorrelation. The main function of this module is realized by a triode. Due to the randomness and non-periodicity of chaotic signals, it is more challenging to amplify and match chaotic signals. If the chaotic signal is directly connected to the triode, waveform distortion or signal self-excitation may occur. Therefore, it is necessary to add a voltage follower module, low-pass filter module and current series negative feedback module to match and adjust. The specific circuit schematic diagram is shown in Figure 6.  
In Figure 6, the amplification matching circuit is mainly composed of resistance-capacitance parallel \(R_{15}\) and \(C_{4}\), \(R_{16}\) and \(C_{5}\), operational amplifiers \(U_{9}\) and \(U_{10}\), current-limiting resistance \(R_{2}\) and \(R_{13}\), triode Q1 BUV26G, current feedback sampling resistance \(R_{18}\), load resistance \(R_{19}\) and \(Vp\) power supply. The adapted voltage follower, which is made up of the operational amplifier \(U_{9}\), is designed to match the chaotic signal source of the output to that of the input and perform impedance matching, allowing the signal of the amplifier circuit to operate within its normal range. At the same time, it has an isolation effect and can cut off the interference effect of the back electromotive force on the front stage. The low frequency signal can thus flow normally through the low pass filter module, which is made up of inductance and capacitance. The output signal is injected into the voltage follower composed of operational amplifiers \(U_{9}\) and \(U_{10}\), where resistors \(R_{17}\) and \(R_{18}\) are used for limiting the current. Finally, the signal is injected from the base of the triode, and the emitter outputs the amplified chaotic signal. Since the signal is input from the base and output from the emitter, the common emitter amplifier circuit is formed. The signal is transmitted back to a low-pass filter module made up of an inductor and capacitor through a feedback loop. The corresponding circuit components can be selected as:  
\[\left\{ \begin{array}{l}C_{4}=C_{5}=22\;\mathrm{nF},\;R_{15}=R_{16}=1000\; \mathrm{K}\Omega\\ R_{17}=R_{18}=100\;\mathrm{K}\Omega,\;R_{19}=3\;\Omega,\;R_{20}=2\;\Omega\\ \end{array} \right.