# Chaos-Based Underwater Communication With Arbitrary Transducers and Bandwidth
## 2 DCSK Based on First Order Hybrid System
### The Proposed DCSK Scheme  
The block diagram of the proposed DCSK is shown in Figure 4. In our proposed method, the chaotic reference signal and the information bearing signal are modulated by a sinusoidal signal and a cosine signal with the same frequency, respectively, and then the two modulated signals are added together to form one frame to transmit 1 bit information. In each frame, the separation of the reference and the information bearing signals is performed by orthogonal sinusoidal carriers rather than by the time delay in the conventional DCSK, as done in PS-DCSK. Assume that the bit transmission rate is \(R_{b}\) bits per one second, the corresponding bit duration is \(T_{b}\), and the symbol rate in the chaotic signal is \(R_{c}\) symbols per one second, then the modulated signal for the \(k\)th bit is given by  
\[S_{k}\left(t\right)=u\left(t\right)\sin\left(2\pi ft\right)+b_{k}u\left(t \right)\cos\left(2\pi ft\right)\quad\Delta t\left(k-1\right)L\leq t<\Delta tkL \enspace, \tag{5}\]  
where \(u\left(t\right)\) is the chaotic state generated by Equation (1), \(b_{k}\) is the \(k\)th binary information, \(\Delta t=T_{b}/L\) is the digital-to-analog (D/A) sampling time, \(f\) is the frequency of the sinusoidal signal satisfying \(f\gg 1/\Delta t\), and the spreading gain \(L=n_{samp}N_{s}\), \(n_{samp}\) is the number of over sampling points for one symbol \(s\), \(N_{s}\) is the number of symbols in the modulated signal used for encoding a bit. Hence, the symbol rate in the chaotic signal is \(R_{c}=R_{b}N_{s}\) symbols per one second. The \(k\)th frame modulated signal \(S_{k}\left(t\right)\) is transmitted over the communication channel. At the receiver end, the received signal \(v_{k}\left(t\right)\) is demodulated by the corresponding sinusoidal and its orthogonal signal, respectively. The demodulated signals \(v_{1k}\left(t\right)\) and \(v_{2k}\left(t\right)\) are given by  
\[v_{1k}\left(t\right)=v_{k}\left(t\right)\cdot\sin\left(2\pi ft \right)=\left(S_{k}\left(t\right)+n\left(t\right)\right)\cdot\sin\left(2\pi ft \right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad  
Equation (3) is the optimal coherent matched filter corresponding to the chaotic signal in Equation (1). Equation (3) maximizes the signal to noise ratio at the receiver, thus achieving the lowest BER. The proposed method removes the delay and switch components in both the transmitter and the receiver as compared with the conventional DCSK. Instead it uses the orthogonal signal pair to modulate the reference and the information bearing signals. It helps thus to achieve higher data transmission rate and it does not require the radio-frequency delay equipment in the communication system, which is rather difficult to implement in the current CMOS technology [35]. As a result, the proposed method is easy to be implemented. Moreover, the frequency modulation moves the chaotic signal spectrum to the bandwidth of the communication equipment, which is compatible with the conventional transducer or antenna.