# Synchronization of Analog-Discrete Chaotic Systems for Wireless Sensor Network Design
## 3 Study on Synchronization Noise Immunity
### Analog-Discrete Synchronization  
The first part of this study is evaluating the analog-discrete synchronization noise immunity. Figure 12 outlines the experiment in block diagrams. The drive chaos oscillator is the LTspice model in one case and the hardware model in the other case, while the response oscillator is the discrete model. The band-limited white Gaussian noise (B.L. AWGN) is added to the chaotic signals of the drive chaos oscillator. Each noise signal is formed individually for each chaotic signal by applying the low-pass filters with the cut-off frequency matched to the bandwidth of the chaotic signals. The assumption that the noise signal's presence in the same bandwidth as the signal is made. Table 1 compiles cut-off frequencies of filters for all signals. The table also shows that chaotic signals of the different models have mostly similar bandwidths. More information on how the cut-off frequency was selected is available in Appendix A.  
Figure 11: Pecora–Carroll synchronization technique [37]. Figure 12: Chaos oscillator’s discrete model synchronization with LTspice model **(a)** and hardware prototype **(b)**. The quality of synchronization is evaluated using the correlation coefficient. The correlation is computed using (13). Here the example correlation is computed for arbitrary \(A\) and \(B\). \[\beta=\frac{\sum\limits_{m}\left(A_{m}-\overline{A}\right)\cdot\left(B_{m}- \overline{B}\right)}{\sqrt{\frac{\sum\limits_{m}\left(A_{m}-\overline{A} \right)^{2}\cdot\sum\limits_{m}\left(B_{m}-\overline{B}\right)^{2}}}}, \tag{13}\]  
where \(\beta\)--correlation coefficient and \(m\)--array index. Correlation equals 1 for the identical signal when synchronization is achieved and 0 when there is no similarity between the signals (no synchronization). The case of correlation equal \(-1\) is for the case of opposite signals; however, the negative values of the correlation should not occur in this study. The work estimates the 1000 \(\theta\) correlation time variations applying N = 1000 long sliding window. The final result is obtained as an average of 100 Monte-Carlo iterations. The added noise signal will influence the level of synchronization, decreasing the quality of synchronization with the increase of the noise power relative to signal power. The signal-to-noise ratio (SNR) formula used in the work is given in (14). \[SNR(dB)=10\cdot\log_{10}\left(\frac{P_{signal}}{P_{noise}}\right). \tag{14}\]  
Figure 13 demonstrates the implementation of the block diagrams in Figure 12. Figure 13 shows the discrete model synchronization with LTspice noise immunity. The LTspice signals are exported to MATLAB, where they are resampled and scaled. Then, the noise signals of a defined SNR are added to the signals, after which the \(y_{1}+n_{2}\) is passed to the discrete model. Finally, the correlation is calculated for \(x_{2}\), \(x_{2}+n_{1}\) and \(z_{2}\), \(z_{1}+n_{3}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Chaos Oscillator**} & \multirow{2}{*}{**Model**} & **Cut-Off Frequency of the Filter for Forming** & **Cut-Off Frequency of the Filter for Forming** & **Cut-Off Frequency of the Filter for Forming** \\  & & \(n_{1}\) **(Noise for \(x\))** & \(n_{2}\) **(Noise for \(y\))** & \(n_{3}\) **(Noise for \(z\))** \\ \hline \multirow{3}{*}{Vilnius} & Discrete & 0.25 & 0.40 & 0.69 \\  & LTspice & 0.40 & 0.40 & 0.76 \\  & Hardware & 0.29 & 0.37 & 0.58 \\  & Discrete & 0.60 & 0.67 & 1.00 \\  & LTspice & 0.50 & 0.83 & 0.93 \\  & Hardware & 0.60 & 0.67 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cut-off frequencies for forming the B.L. AWGN for each chaotic signal. Figure 13: Implementation of the discrete model synchronization with LTspice model (**a**) and implementation of the discrete model synchronization with hardware prototype (**b**). Figure 13b demonstrates the discrete model synchronization with hardware noise immunity.