## 4 Quadrature Chaos Shift Keying Communication System
### System Design Particularities  
The QCSK communication system is based on chaos shift keying and chaotic synchronization. The core elements of the transmitter and receiver are chaos oscillators operating in a drive-response configuration. Figure 9 presents the block diagram of the QCSK data transmission system. At the transmitter, the state variables \(X\), \(Y\), and \(Z\) represent the output signals of the drive chaos oscillator. The \(X\) and \(Y\) signals are used for CSK modulation, forming the information-carrying signal by switching these chaotic signals in correspondence to data values, while the \(Z\) signal is used as a synchronization signal. Before CSK and quadrature modulation, a DC offset is canceled in all signals. The synchronization signal and keyed data signals are then used for quadrature modulation and transmitted through the AWGN channel. Figure 9: Quadrature chaos shift keying data transmission system block scheme. At the receiver, the input signals are quadrature demodulated and filtered using the LPF, which is tuned to preserve the main spectral components of the chaotic signal. The LPF bandwidth is selected to ensure the MSE for signals before and after filtering is \(-25\) dB. The synchronization signal is then used to synchronize the response chaos oscillator via the substitution method, as described in the previous section. The received information-carrying signals, along with the response chaos oscillator signals (\(X^{\prime}\) and \(Y^{\prime}\)), are then used for data detection. Information detection is based on correlation coefficient estimations between the received information-carrying signal and the response chaos oscillator signals \(X^{\prime}\) and \(Y^{\prime}\). After computing the correlation, a comparator makes a decision based on the correlation coefficient value. Section 3 showed different correlation values for the oscillator's state variable as well as the nonzero cross-correlation for the signals used for the information-carrying signal forming. Therefore, a nonzero threshold utilization is required for decision making. For threshold calculation, the likelihood ratio approach is utilized. Given that white Gaussian noise is added to the signal, the likelihood that the received information-carrying signal \(s(t)\) was induced by the signal \(X\) can be expressed in the form:  
\[\Lambda[X|s(t)]=\frac{1}{(\sigma\sqrt{2\pi})^{K}}\exp\left(-\frac{1}{2}\sum_{ k=1}^{K}\frac{(s_{k}-x_{k})^{2}}{\sigma^{2}}\right), \tag{7}\]  
where \(\sigma\) denotes the AWGN standard deviation and \(s_{k}\) and \(x_{k}\) denote \(k\)th sample of CSK signal \(s(t)\) and chaotic signal \(X\), respectively. Then, the logarithm of the likelihood ratio can be expressed as:  
\[\ln\Lambda[s(t)]=\frac{\Lambda[X|s(t)]}{\Lambda[Y|s(t)]}=\sum_{k=1}^{K}\frac{s _{k}}{\sigma^{2}}(x_{k}-y_{k})-\frac{1}{2}\sum_{k=1}^{K}\frac{(x_{k}^{2}-y_{k }^{2})}{\sigma^{2}}. \tag{8}\]  
In this case, the decision-making regions are split by the point where the likelihood ratio changes sign, which is given by:  
\[\sum_{k=1}^{K}s_{k}(x_{k}-y_{k})-\frac{1}{2}\sum_{k=1}^{K}(x_{k}^{2}-y_{k}^{2} )\geqslant 0. \tag{9}\]  
Assuming that the information-carrying signal \(s(t)\) is a sum of \(X\) state variable and noise signal \(n(t)\), one can express the mathematical expectation \(E[]\) of the first sum as:  
\[E[(x_{k}+n_{k})(x_{k}-y_{k})]=\sum_{k=1}^{K}x_{k}^{2}-\sum_{k=1}^{K}x_{k}y_{k}, \tag{10}\]  
where \(n_{k}\) is the \(k\)th sample of the noise signal. By substituting the calculated mathematical expectation in Equation (9), one can conclude that a high cross-correlation between the response oscillator's state variables \(X^{\prime}\) and \(Y^{\prime}\) decreases correct detection probability.