## 1 Introduction  
Underwater acoustic (UWA) communication technique so far is the only medium to communicate over long distances in water. However, it offers more challenges than conventional terrestrial wireless communications [1]. The complicated marine environment causes various kinds of interferences in UWA channels [2; 3]. For lower frequencies, noise in the seawater is random, inconsistent and frequency varying while for higher frequencies the acoustic signals suffer high attenuation; thus, there is only a limited band of frequencies for UWA communications. Besides, the acoustic signals reach the receiver along different paths due to the interface, reflections, refractions, random scattering, etc. These factors lead to distortion of the received signals and degradation of the performance of UWA communication systems. To cope with challenging UWA channels, researchers have proposed several algorithms and communication techniques [4; 5; 6; 7]. For the multipath interference and the low signal-to-noise ratio (SNR) in UWA communications, a commonly used technique is the spread spectrum communication [8; 9]. The spread spectrum method is a type of transmission, in which the bandwidth occupied by the signal exceeds the minimum bandwidth required to transmit information [10]. Traditional spread spectrum communications include Direct Sequence Spread Spectrum (DSSS), Frequency Hopping Spread Spectrum (FHSS) and Time Hopping Spread Spectrum (THSS), where DSSS is used more commonly due to its resistance to frequency selective fading [11]. DSSS technique promises high reliability but at the cost of a lower communication rate because it uses one sequence to spread only 1 bit of information [12]. In order to improve the performance of the DSSS system, a new M-ary cyclic shift keying spread spectrum (MCSK) technique is thus proposed, which can not only guarantee the high reliability but can also improve the communication rate to a significant extent. The MCSK is the combination of two techniques that have been used in UWA communications, namely, M-ary spread spectrum modulation and cyclic shift keying (CSK) modulation [13]. In M-ary modulation, multiple spread spectrum sequences are used to represent different information, and each spread spectrum sequence carries more information [14; 15]. CSK technique uses the phases of cyclic shift sequences to transmit information. The amount of information contained in each cyclic shift sequence depends on the length of the sequence. The combination of M-ary and CSK can greatly increase the amount of information carried by one sequence, so the MCSK method can significantly improve the spread spectrum gain on communication rates. Aiming at the influence of multipath interference of UWA channels on communications, Reference [16] first applies the time-reversal mirror (TRM) technique to UWA communications. In order to meet the requirement of low power consumption, we mainly analyze single-element TRM for unidirectional transmission in this paper. By theoretical derivation and simulations, we analyze the focusing gain and applicable conditions of single-element TRM in detail. Moreover, we propose to combine virtual time-reversal mirror (VTRM) with the MCSK technique in UWA communications. Compared with the system proposed in Reference [17], the bit error rate (BER) performance of our system is much better in UWA multipath channels. This MCSK communication scheme based on VTRM (MCSK-VTRM) can significantly improve both the availability and reliability of information transmission. The content of this paper is organized as follows: Section 2 introduces the MCSK communication technique and analyses its performance. Section 3 introduces how to use TRM to equalize UWA channels and analyzes the influence of multipath structures of channels on TRM. Besides, we introduce the VTRM technique and how to estimate UWA channels with matching pursuit (MP) and base pursuit de-noising (BPDN) in VTRM. Section 4 analyzes the performance of MCSK-VTRM in UWA communications by simulations and experiments.