# Chaos-Based Underwater Communication With Arbitrary Transducers and Bandwidth
## 1 Introduction  
The underwater communication has numerous applications for the marine researchers, oceanographers, marine commercial operators, and defense organizations. Underwater acoustic communication is the main medium for such applications [1; 2; 3; 4; 5; 6]. However, the reliability of underwater acoustic communication is restricted by several physical channel constraints, such as multipath propagation, amplitude damping, time-varying characteristics, and ambient noise [7; 8; 9]. Chaotic signal has been applied in digital communication systems for the past two decades and chaotic modulation schemes have attracted considerable interest [10; 11; 12; 13]. It is mainly due to the fact that chaotic signals demonstrate some attractive nature features, including a delta-function-like autocorrelation property required by spread spectrum systems [14], immunity to multipath propagation [15] and jamming signal [16], a large channel capacity due to broadband property [17], and increased transmission security due to a noise-like waveform [18; 19; 20]. Chaos-based communication systems can be classified into two categories, i.e., coherent and non-coherent communication. In coherent communication systems, transmitter and receiver need to be synchronized, being highly susceptible to channel constraints, in particular to multipath propagation [21]. In this case, robust chaotic synchronization in a complicated wireless channel is rather difficult to be achieved. To avoid the chaos synchronization demand, non-coherent communication systems have been proposed, which do not require chaos synchronization and channel estimation, offering a more suitable approach for this complicated communication channel. In non-coherent communication systems, such as differential chaos shift keying (DCSK) [22], one bit duration is divided into two equal time slots, where the reference signal is transmitted in the first slot, and its repeated or its inverted copy (according to the bit to be transmitted) is sent in the second slot. The DCSK and its modified versions exhibit both low bit error rate (BER) and robustness against channel distortion [23; 24; 25; 26]. Most of the previous researcher focused on the problem of low data transmission rate [25; 26; 27; 28; 29; 30; 31], in which half of bit energy is used to transmit the reference signal. In Reference [27], [PERSON] and [PERSON] proposed a Reference-Modulated DCSK (RM-DCSK) system; the first information bit is sent in the first time slot, and the waveform in the second slot carries not only the second information bit but also the reference signal of the next frame. In Reference [28], the High Efficiency DCSK (HE-DCSK) system was proposed, where the information bearing signal carries two-bit information using the superposition of one information bit multiplied by the current reference and another information bit multiplied by the last reference in the second slot. The two methods achieve double data transmission rate and double bandwidth efficiency compared with the conventional DCSK. High data rate Code Shift DCSK (HCS-DCSK) system [29] uses the sum of multiple bits multiplied by different (uncorrelated) chaotic references to form the transmitting signal for just one frame to achieve high bit rate, but it needs the synchronized chaotic references at the receiver, which is difficult to implement. Phase-Separate DCSK (PS-DCSK) system [30] uses the sinusoidal and its orthogonal signals to separate the reference signals and the information bearing signals, respectively. It not only avoids radio frequency delay lines, which is difficult to integrate in Complementary Metal Oxide Semiconductor (CMOS) technology, but it also achieves twice as much data rate as the original DCSK. Those methods utilize the orthogonality property of the chaotic signal and the combined information signals of multi-bits to form one frame to be transmitted. They achieve higher transmission rates and BER as comparable to the conventional DCSK in presence of additive Gaussian white noise (AWGN) channel.