## 1 Introduction  
The rapid rise of scientific and commercial activities in underwater environments has led to an increasing demand for high-rate and wide-band underwater acoustic (UWA) communications to sustain a mass of data transmission in UWA sensor networks. Multipath spread of UWA channelsoften is on the order of ten milliseconds [1; 2; 3], and therefore the high-rate and wide-band UWA communications using single-carrier (SC) modulation often have to adopt adaptive equalization with high computation complexity to suppress the intersymbol interference (ISI) spanning several tens or hundreds of symbols [3; 4; 5; 6]. To reduce the span of ISI, multicarrier (MC) modulation has been used in UWA communications [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Based on whether the divided subbands are overlapping or not, the MC modulation used in UWA communications falls into two types. Orthogonal frequency division multiplexing (OFDM) is representative of the type with overlapping subbands [6; 7; 8; 9; 10; 11; 12]. In OFDM, the communication band is divided into hundreds of overlapping subbands with very narrow bandwidth that makes communication performance sensitive to frequency offset. Since the Doppler spread of UWA channels can lead to massive intercarrier interference (ICI), complicated techniques must be exploited in OFDM modulation UWA communications to suppress ICI. On the contrary, the MC modulation with non-overlapping subbands can avoid the problem of frequency offset sensitivity. According to inserting guard bands or not, the MC modulation with non-overlapping subbands can be further classified into two types. Compared to the type with guard bands, the type without guard bands can keep good balance between ICI suppression performance and spectral efficiency. Therefore, as the representation of MC modulation without overlapping subbands and guard bands, filtered multitone (FMT) modulation has attracted research interest and been applied to UWA communications in recent years [13; 14; 15; 16; 17]. In FMT modulation UWA communications, the span of ISI is obviously shortened through band splitting, but there is always some ISI due to the multipath spread of subchannels. To suppress the ISI, traditional minimum mean square error (MMSE) adaptive equalization is usually used in FMT modulation UWA communications [13; 17]. For adaptive equalization, when the transmitted symbols are perfectly known in advance, the ISI can be completely eliminated. In practice, the transmitted symbols cannot be known beforehand, and therefore the performance of adaptive equalization is limited. Moreover, the UWA channel is a noisy channel, and the performance of adaptive equalization can also be degraded by noise. Turbo equalization is a technique that processes the received signal using adaptive equalization and channel decoding in an iterative fashion. In turbo equalization, adaptive equalization can exploit the last extrinsic information of channel decoding as a priori information to improve ISI suppression performance; channel decoding also can use the last extrinsic information of adaptive equalization as a priori information to further reduce noise. Since the ISI and noise are jointly suppressed, turbo equalization can overcome the shortcomings of adaptive equalization and has been used in UWA communications in recent decades [19; 20; 21; 22; 23; 24; 25; 26]. Ideally, when adaptive equalization and channel decoding are both based on the maximum-likelihood (ML) or maximum a priori probability (MAP) algorithm, turbo equalization can achieve the best performance. However, the complexity of the ML/MAP algorithm can increase exponentially with the span of ISI, and therefore high computation complexity prohibits the application of ML/MAP-based turbo equalization to UWA communications. In practical UWA communications, turbo equalization based on the minimum mean square error (MMSE) algorithm is generally used in consideration of communication performance and computation complexity.