## 2 Principles
\tag{4}\]  
The purpose of this design is to increase load capacity and stabilize output voltage. The physical equipment of the amplification matching circuit is shown in Figure 7a. The temporal waveform after the amplification matching circuit is shown in Figure 7b. The maximum output signal value is about 6.5 V. The cross-correlation (CC) function between chaotic circuit 1 and chaotic circuit 2 is shown in Figure 7c. It is indicated that outputs of  
Figure 5: Amplifying matching circuit flow chart. Figure 6: Amplifying matching circuit schematic diagram. chaotic circuit 1 and chaotic circuit 2 are well synchronized with the CC of 0.99. The CC can be expressed as:  
\[CC=\frac{\sum\limits_{i=0}^{n-1}\left[(x_{i}-\mu_{x})(y_{i}-\mu_{y})\right]}{ \sqrt{\sum\limits_{i=0}^{n-1}\left(x_{i}-\mu_{x}\right)^{2}\cdot\sum\limits_{i= 0}^{n-1}\left(y_{i}-\mu_{y}\right)^{2}}} \tag{5}\]  
where \(\mu_{x}\) and \(\mu_{y}\) denote the mean of the time series of the chaotic circuit 1 and the chaotic circuit 2, respectively. A schematic of digital domain encryption is shown in Figure 8. The 16 quadrature amplitude modulation orthogonal frequency division multiplexing (16 QAM-OFDM) modulation was adopted in our scheme. In order to reduce the complexity of the algorithm and improve the bit error performance, we designed an optimization algorithm. Firstly, as shown in Figure 8a, the sub-bands in the frequency domain are divided. As is shown in Figure 8b, the sub-bands in the frequency domain are disturbed, in order to reduce the number of disturbed objects in the frequency domain. Then, the encryption in the time domain is shown in Figure 8c. The optimization scheme of region division is not used in the time domain, so as to achieve the effect of all information scrambling. It was assumed that the process of OFDM has \(M\) subcarriers for data transmission. It can be divided into \(K\) frequency bands according to region division, and each frequency band has \(n\) sub-bands. Its relation satisfies \(M=K\times n\). The number of frequency bands \(K\) can be selected, but \(K\) must be greater than 3. A total of 512 subcarriers were used for data transmission, and the four subcarriers were divided into a frequency band, with a total of 128 frequency bands. Frequency domain encryption is to perturb the 128 frequency bands. Compared with the undivided frequency band algorithm, the computational cost of the proposed algorithm was reduced by four times, and all the data were perturbed and encrypted. Figure 8: Schematic of the proposed digital domain encryption. (**a**) the divided frequency domain, (**b**) the disturbed frequency domain, and (**c**) the disturbed time domain. Figure 7: (**a**) Physical equipment of the amplifying matching circuit, (**b**) temporal waveform after amplifying, and (**c**) cross-correlation between chaotic circuit 1 and chaotic circuit 2. The chaotic sequence \(\mathbf{y}\) was used in this encryption part, but the original chaotic sequence did not meet the requirements of signal encryption, so it needed to be optimized, as shown in the following process:  
\[\mathbf{Y}=Mat\Big{\{}mod(\mathbf{y}{\cdot}10^{2},1)\cdot[mod(\mathbf{y}{\cdot}10^{2},1)^{ \prime}]\Big{\}} \tag{6}\]  
where \(Mat\{\cdot\}\) means to set the non-integer element of the matrix to 0, and \(\mathbf{Y}\) is the generated permutation matrix. The encrypted information in frequency domain can be expressed as:  
\[\left[ \begin{array}{c}F_{e1}\\ F_{e2}\\ \vdots\\ F_{ek}\end{array} \right]=\left[ \begin{array}{c}F_{1}\\ F_{2}\\ \vdots\\ F_{k}\end{array} \right]\cdot\mathbf{Y} \tag{7}\]  
where \(F_{ek}\) is the signal of encrypted frequency band and \(F_{k}\) is the signal of original frequency band. The data of the time domain are perturbed so that all data are encrypted. The data are interleaving diffusion perturbation by the optimized chaotic sequence. Zoning perturbation was not used; otherwise, the data are only scrambled in a local area, which increases the possibility of cracking.