## 1 Introduction  
With its high-rate data transmission in marine activities, multiple-input-multiple-output (MIMO) technology has seen widespread use in underwater acoustic communications in the past few decades [1, 2, 3, 4, 5, 6, 7, 8, 9]. However, MIMO underwater acoustic channels are characterized by long multipath spreads, spatial correlation, serious noise, etc., which can lead to severe co-channel interference (CCI), intersymbol interference (ISI), and noise interference and drastically impact communication performance [10, 11]. To achieve high reliability with relatively low-complexity interference suppression, multi-carrier modulation (MCM) has traditionally been exploited in MIMO underwater acoustic communications [1, 2, 3, 4, 5] since it can reduce the range of symbols affected by multi-path spread and spatial correlation through band splitting. Of all the MCM technologies, orthogonal frequency division multiplexing (OFDM) is the most widely used owing to its simple implementation, and ability to reduce the span of multipath spread and spatial correlation to a symbol interval [1, 2, 3, 4, 5]. However, the subbands of OFDM are overlapped and have narrow bandwidth. Additionally, slight frequency offset can lead to severe intercarrier interference (ICI), while complex frequency offset compensation must be adopted in the acoustic communication using OFDM [12, 13, 14]. In recent years, filtered multitone (FMT), an attractive MCM, has been applied to MIMO underwater acoustic communications [4, 5]. Like OFDM, FMT can maintain high spectral efficiency and is easily implemented by using Fast-Fourier transform (FFT). Moreover, in contrast to OFDM, FMT is insensitive to frequency offset, because it has non-overlapping subbands with high spectral containment [15; 16]. In MIMO-FMT underwater acoustic communications, offering a compromise between bandwidth efficiency and the complexity of filter implementation, the bandwidth of each subband is still larger than the coherence bandwidth of the acoustic channel. Therefore, although the span of ISI and CCI has been obviously reduced by band splitting, there are still shortened ISI and CCI in demodulated signals of the receiver. Due to its simple implementation and good spatial-temporal compression, time reversal (TR) has been traditionally used as pre-processor to alleviate ISI and CCI in MIMO-FMT acoustic communications [4; 5]. However, residual ISI and CCI, as well as noise still exist in the output signals of TR processing; and, therefore, it is necessary to use other technologies as post-processor for TR to further eliminate interference [17]. In [4], a MIMO-FMT TR underwater acoustic communication using successive interference cancellation (SIC) is proposed to remove residual ISI and CCI after TR processing. Although the research has demonstrated that the method presented in [4] can achieve a good communication performance, the suppression of noise has not been considered. Deficiencies in existing research motivated us to explore joint suppression of residual ISI and CCI, as well as noise, in MIMO-FMT TR underwater acoustic communications. Generally, applying error control coding (ECC) directly to the method presented in [4] can achieve joint suppression of residual ISI and CCI, as well as noise. However, if interference cancellation, adaptive equalization, and decoding are all based on hard decision, the suppression performance will be limited. Turbo equalization is a powerful interference suppression technique for MIMO underwater acoustic communications [6; 7; 8; 9]. Compared with conventional equalization and decoding based on hard decision, turbo equalization can iteratively exchange soft information of symbols with soft-decision decoding; and, therefore, more information of transmitted bits can be acquired and better communication can be achieved. According to the implementation of adaptive adjustment, turbo equalization can be divided into channel estimation (CE)-based turbo equalization and direct adaptation (DA)-based turbo equalization [18; 19].