## 2 FM-CSK and QCPSK Communication Systems
### Conceptual Models  
Figures 1 and 2 show the block diagrams of the QCPSK and FM-CSK. The transmitter and the receiver of the two systems contain chaos oscillators. The \(X_{1}\), \(X_{2}\), and \(X_{3}\) depict the chaos signals (state variables) of the transmitter's chaos oscillator, while \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) depict the chaos signals (state variables) of the receiver's chaos oscillator. Both systems are based on coherent chaos signal detection. The chaos oscillators are synchronized using the Pecora-Carrol method [55]. The \(X_{2}\) signal is transmitted, and the received \(X^{\prime}_{2}\) signal substitutes the \(Y_{2}\) state variable of the receiver's chaos oscillator. As a result, \(X_{1}\) repeats the behavior of \(Y_{1}\), and \(X_{3}\) repeats the behavior of \(Y_{3}\). The main difference between the  
systems is how the binary information signal \(b(t)\) and the synchronization signal \(X_{2}\) are used for modulation and demodulation, as well as how the received information signal \(b^{\prime}(t)\) is detected.  
The QCPSK data transmission system in Figure 1 is based on chaos phase-shift keying and quadrature transceiver. The binary message signal \(b(t)\) controls a switch that toggles between \(X_{1}\) and \(-X_{1}\). Bit \"1\" shifts to \(X_{1}\), and bit \"0\" shifts to \(-X_{1}\), thus forming the phase-manipulated chaos signal labeled as \(X_{1}PM\). The synchronization \(X_{2}\) is the in-phase component, and the information-carrying \(X_{1}PM\) is the quadrature component of the transmitted signal. After propagating through the channel, the received quadrature signal is passed to the quadrature demodulator, which separates the synchronization \(X^{\prime}{}_{2}\) and information-carrying \(X^{\prime}{}_{1}PM\) signals with the use of low-pass filters (LPF). In the QCPSK system, the received information signal \(b^{\prime}(t)\) is detected by first calculating the correlation \(\beta\) between the \(X^{\prime}{}_{1}PM\) and \(Y_{1}\) (\(Y_{1}\approx X_{1}\) when synchronized). When detecting \"1\", \(X^{\prime}{}_{1}PM\) positively correlates with \(Y_{1}\) and \(\beta\) > 0. When detecting \"0\", \(X^{\prime}{}_{1}PM\) negatively correlates with \(Y_{1}\) and \(\beta\) < 0. Then the result of a correlator is processed by a comparator that outputs \"1\" if \(\beta\) > 0 and \"0\" if \(\beta\) < 0, thus acquiring the received information signal \(b^{\prime}(t)\).  
The FM-CSK data transmission system in Figure 2 is based on chaos shift keying and frequency modulation. The binary message signal \(b(t)\) controls a switch that toggles between chaos signals \(X_{1}\) and \(X_{3}\) of equal powers. Bit \"1\" shifts to \(X_{1}\), and bit \"0\" shifts to \(X_{3}\), thus forming the information-carrying signal \(X_{13}\). The \(X_{2}\) and \(X_{13}\) signals are passed through pre-emphasis filters, and each is used for frequency modulation. Both signals are then transmitted in parallel, propagating through the channel. In the receiver, both signals are passed through a bandpass filter (BPF), frequency-demodulated, and passed through a de-emphasis filter. Using pre-emphasis and de-emphasis filters eliminates noise power spectral density increase with frequency caused by frequency demodulation. The received information signal \(b^{\prime}(t)\) is detected by estimating the correlation between the received \(X^{\prime}{}_{13}\) signal and \(Y_{1}\) and \(Y_{3}\). If \(X^{\prime}{}_{13}\) correlates with \(Y_{1}\), then the \"1\" bit is received, and if \(X^{\prime}{}_{13}\) correlates with \(Y_{3}\), then the \"0\" bit is received. As demonstrated in work [53], an appropriate bit length is selected to minimize the correlation between the modulated bits.