## 1 Introduction
CE-based turbo equalization can converge quickly, but the computational complexity is relatively high [18]. DA-based turbo equalization needs a longer time to converge, but it can adjust coefficients based on the received signal with low computational complexity; and, therefore, it is more suitable for underwater acoustic communications [20; 21; 22]. In general, DA-based turbo equalization exploits the decoder soft information as an input to the feedback of adaptive decision feedback equalization (ADFE), rather than using it in adaptive coefficients adjustment. In [23], the DA-based turbo equalization using soft information is presented to improve communication performance, where the soft information is not only exploited in the feedback of ADFE, but also used adaptive coefficients adjustment. Therefore, based on the advantages of the DA-based turbo equalization using soft information, a MIMO-FMT TR underwater acoustic communication using DA-based turbo equalization is proposed, and the adopted DA-based turbo equalization mainly includes interference cancellation, soft information-based ADFE adjusted according to recursive expected least squares(RELS) algorithm and decoding. In the proposed method, soft information is exchanged between soft symbols with soft decisions of iterative decoding, and the interference suppression proceeds successively and iteratively until the performance is stable. The principle of the proposed method is analyzed, and based on the acoustic channel responses measured in a real experiment, the performance of the proposed method is assessed. Two methods are compared for the performance assessment. The first is the method proposed in [4]. In order to clearly demonstrate the difference from the proposed method, it is referred as the MIMO-FMT TR underwater acoustic communication using interference suppression without error control coding (ECC) in the paper. The second is the MIMO-FMT TR underwater acoustic communication using interference suppression based on hard-decision equalization and decoding, where ECC is introduced to suppress noise, but the equalization and decoding are all based on hard decision without exploiting soft information. The contribution of the paper is threefold: (1) first, DA-based turbo equalization is first used in MIMO-FMT TR acoustic communications to deal with residual ISI and CCI, as well as noise after TR process; (2) soft information-based ADFE adjusted according to the RELS algorithm along with interference cancellation and decoding is used to construct the adopted DA-based turbo equalization; and (3) based on the acoustic channel responses measured in a real experiment, the performance of the proposed method is compared with that of two other methods to demonstrate its validity. This paper is organized as follows. Section 2 describes the system model of the proposed method. Section 3 describes the principle of the DA-based turbo equalization, where the interference cancellation, soft information-based ADFE, and decoding is discussed in detail. Section 4 presents the assessment of the proposed method. Section 5 summarizes the conclusions. In this work, vectors and matrices are denoted in bold lowercase and uppercase, respectively. Scalars are represented by normal symbols. Conjugate operation, matrix transpose, and Hermitian are denoted by \((\cdot)^{*}\), \((\cdot)^{\mathrm{T}}\), and \((\cdot)^{\mathrm{H}}\), respectively. \(p(x|y,z)\) stands for the probability distribution of \(x\) conditioned on \(y\), which depends on the value of the deterministic quantity \(z\).