## 4 Quadrature Chaos Shift Keying Communication System
### Result Analyses  
The QCSK communication system was considered in four configurations: three based on the Vilnius chaos oscillator, corresponding to the configurations described in Table 1, and one based on the Colpitts chaos oscillator with \(v_{\text{C1}}\) as the synchronization signal. As demonstrated in our previous study, the remaining two configurations of the Colpitts chaos oscillator were excluded from further analysis due to their low correlation coefficient values. These four QCSK configurations were tested in an AWGN channel. Table 2 presents an example of falsely detected bit counts and BER values under different SNR conditions, comparing the performance with and without the added detection threshold. These results are provided for the QCSK configuration using \(v_{R1}\) as the synchronization signal.  
As shown in Table 2, an imbalance emerges at higher SNR values, where correlation coefficient levels become more distinct if the detection threshold is not applied. This imbalance leads to detection errors, as one signal may dominate, increasing the probability of misdetection of '1' or '0' bits. When the threshold is introduced, it compensates for this imbalance, resulting in improved BER performance and a more stable data detection process.  
The noise immunity study results for all four configurations are presented in Figure 10, where the \(y\)-axis represents the BER and the x-axis shows bit energy and the noise power spectral density ratio (\(E_{b}/N_{0}\)).  
\begin{table}
\begin{tabular}{c c c c|c c c} \hline \hline  & \multicolumn{3}{c|}{**Without Threshold**} & \multicolumn{3}{c}{**With Threshold**} \\
**SNR, dB** & \multirow{2}{*}{**BER**} & **False ‘0’** & **False ‘1’** & \multirow{2}{*}{**BER**} & **False ‘0’** & **False ‘1’** \\  & & **Count** & **Count** & & **Count** & **Count** \\ \hline \(-20\) & 0.4835 & 2414 & 2421 & 0.4810 & 2416 & 2394 \\ \(-15\) & 0.4407 & 2287 & 2120 & 0.4370 & 2168 & 2202 \\ \(-10\) & 0.3249 & 1761 & 1488 & 0.3195 & 1598 & 1597 \\ \(-5\) & 0.1134 & 702 & 432 & 0.1053 & 519 & 534 \\ \(0\) & 0.0121 & 79 & 42 & 0.0107 & 57 & 50 \\ \(2\) & 0.0048 & 34 & 14 & 0.0043 & 21 & 22 \\ \(4\) & 0.0019 & 12 & 7 & 0.0014 & 7 & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Error count for the QCSK configuration without and with the detection threshold in a system based on the Vilnius chaos oscillator using \(v_{R1}\) as the synchronization signal.  
Figure 10: BER versus \(E_{b}/N_{0}\) curves of the QCSK data transmission system.  
The best performance among the tested QCSK configurations was observed in the Vilnius-chaos-oscillator-based system when \(v_{\mathrm{C2}}\) was used as the synchronization signal. In the chaotic synchronization noise immunity study, this configuration demonstrated the highest correlation values under noisy conditions, ensuring that \(X\) and \(Y\) correlations are of the same level. Additionally, the cross-correlation between chaotic signals forming the information-carrying signals was minimal, and it remained relatively unaffected by added noise, further enhancing system's resilience.  
The QCSK system based on the Colpitts chaos oscillator achieved the second-best performance using \(v_{\mathrm{C1}}\) for synchronization. In the previous study, this configuration displayed similar synchronization characteristics to the \(v_{\mathrm{C2}}\) synchronized Vilnius chaos oscillator. However, in the Colpitts oscillator, \(X^{\prime}\) and \(Y^{\prime}\) exhibited different correlation coefficient values, leading to a system imbalance, which required compensation during detection.  
The remaining two configurations of the Vilnius chaos oscillator exhibited similar performance trends, aligning with findings from the synchronization noise immunity study. However, the worst BER performance was observed in the Vilnius-chaos-oscillator-based QCSK system when \(v_{\mathrm{R1}}\) was used as the synchronization signal. This configuration was more noise-sensitive since spectral components of the information-carrying signal occupied a wider band [37].