## 1 Introduction  
Reliable underwater acoustic communications are challenging because of the limited bandwidths, time-varying multipath delay, double-selective channel fading, and strong background noise [1]. For example, the intersymbol interference (ISI) could span tens or even hundreds of symbol durations. Many approaches have been proposed for underwater acoustic (UWA) communications over the last two decades, including the single carrier with time-domain decision feedback equalizer (SC-TDE) [2; 3], orthogonal frequency-division multiplexing (OFDM) [4], single-carrier with frequency domain equalization (SC-FDE) [5; 6; 7; 8], and time reversal communications [9; 10]. SC-FDE is a promising low-complexity approach for mitigating intersymbol interference (ISI). The main advantage of SC-FDE is its ability to cope with long delay spread channels with single-tap equalizers. In addition, the estimated channels can be updated easily via pilot signals on a block-by-blocks. Compared to time-domain equalization, SC-FDE has lower computational complexity [11]. Compared to OFDM, SC-FDE exhibits similar or better performance with similar computational complexity while avoiding OFDM's high peak-to-average power ratio (PAPR) and sensitivity to carrier frequency offset (CFO) [11]. Therefore, SC-FDE has been proposed for IEEE 802.16e [11; 12; 13] and high-rate underwater acoustic communications [5; 6; 7; 8]. In our previous work, a hybrid time frequency domain equalizer (HT-FDE) is proposed to combat the residual ISI in single carrier UWA communication systems [7]. In phase-coherent UWA communication, multichannel receivers are usually employed to enhance signal-to-noise ratios, mitigate channel fading effects, and reduce the ISI. Spatial diversity assumes the received multichannel signals are independent and the probability that all of them fade at the same time is small. To obtain spatial diversity gain, the optimal element spacing for the receiving array is required on the order of three to four wavelengths for vertical arrays and on the order of 30 to 60 wavelengths for horizontal arrays [14]. Recently, phase-coherent UWA communication has been considered for some practical applications, such as underwater unmanned vehicles (UUV) and gliders. The aforementioned optimal element space requirements may be unsuitable for these small underwater platforms. Several approaches to achieve phase coherent UWA communication with a single receiver have been developed in recent years [15; 16; 17; 18; 19; 20; 21]. In synthetic aperture communications, only one or a few hydrophones are employed on the underwater platforms. A virtual array is generated by the relative motion between the transmitter and the receiver to achieve spatial diversity gain with reduced size and cost. Non-coherent synthetic aperture time-reversal communications in shallow water was demonstrated at sea where only one transmitter and one receiver were required [15]. The benefit of this system is that it requires only two transducers, significantly decreasing the technical cost compared to typical time reversal communication schemes. The use of a single receiver that combined three consecutive receptions indicated that the synthetic aperture communication is feasible at 9200 km range [16]. High-rate coherent synthetic aperture communications in shallow water at two different bandwidths was presented in [17]. The authors combined two to five consecutive transmissions successfully. They provided an effective data rate of 30/M kbits/s (M is the transmission number) with 8-phase shift keying (PSK) modulations at about a 3.5 km range in shallow water. A glider with a single hydrophone is used for long-range acoustic communication in deep water [18]. This system improved the performance even when individual receptions had low signal-to-noise ratios (SNR). The authors used a binary phase shift keying (BPSK) modulation and time reversal with decision feedback equalization (TR-DFE) with a matched pursuit sparse channel estimation algorithm.