## 1 Introduction  
Coherent underwater acoustic communications are challenged by acoustic channels, which are often characterized as time-varying, dispersive, sparse, _etc_. [1]. Therefore, much of the recent research hasbeen focused on the development of channel equalizers to remove intersymbol interference (ISI) in multipath environments, especially for high-rate coherent communications. One receiver cannot avoid deep fading in time-varying channels, and thus the equalizers fail to remove ISI. With multiple sensors exploiting spatial diversity, [PERSON] _et al._[2] has proposed a multichannel decision feedback equalizer (McDFE). The disadvantage of McDFE is its complexity due to the computational load, which increases with the time spread of underwater channels. Therefore, it is difficult to apply McDFE in underwater channels of long time spread, especially using a large number of receiving hydrophones. Another novel method is the time reversal mirror (TRM), originally proposed by [PERSON] [3], which was later applied for underwater communications. The focusing of TRM results in a significant reduction of ISI for underwater communications, which has been demonstrated by [PERSON] _et al._[4, 5]. Two vertical hydrophone arrays and two-way transmission are required by TRM to achieve the focusing at the transmitter. During the transmission, the underwater channel is required to be constant. An alternative technique for underwater communications is proposed by [PERSON] _et al._[6] to take advantage of the focusing at the receiver, commonly referred as passive time reversal or passive-phase conjugation (PPC). This method requires only one receiving array and one-way transmission. ISI cannot be eliminated by the focusing, and thus a subsequent channel equalizer is used to remove residual ISI [6, 7, 8, 9], where a single channel decision feedback equalizer (DFE) is used. It is referred to as PPC-DFE in this paper. Spatial diversity is used by the focusing to suppress ISI. In a real oceanic environment, it is difficult to predict time variant spatial coherence [10], when interchannel correlations impact spatial focusing. [PERSON] [11] has discussed the upper bound performance of time reversal communications, but it is very difficult to predict real performance of time reversal communications, as spatial coherence is neglected in the model. By numerical simulations and experimental demonstrations, [PERSON] [12] has demonstrated that McDFE achieves superior performance over that of PPC-DFE. This leads to a receiver structure which uses adaptive multichannel combining after PPC processing in each individual channel. [PERSON] _et al._[13] have presented a receiver structure--joint PPC and McDFE (PPC-McDFE). This receiver structure involves temporal focusing (pulse compression by PPC processing) for time delayed arrivals [14], and thus the computational load of a subsequent McDFE is much reduced. It is well known that temporal focusing degrades with time evolution in time-varying channels. To counter for this degradation, the block-based approach proposed by [PERSON] [15] can be used to extend PPC-McDFE in time-varying channels. PPC processing requires information of the channel characteristics, which can be estimated using training symbols. Underwater channels are often sparse, especially at the high-frequency regime, where there are a few dominant arrivals. The dominant arrivals can be estimated using the matching pursuit (MP) algorithm [16]. [PERSON] [17] has shown that the MP algorithm exploits the channel sparseness to improve the performance of PPC-DFE. It is an open question whether the MP algorithm can improve the performance of PPC-McDFE, in comparison with the conventional channel estimation method--the least squares (LS) method. The above brief introduction shows that different approaches have been proposed and have been tested in field experiments. However, the experiments were conducted under different conditions, and it is therefore difficult to compare the performance of different receiver structures.