## 1 Introduction
MMSE turbo equalization can be classified into two types based on whether the estimated channel response is used in computing equalizer coefficients or not. Compared with the MMSE turbo equalization without channel estimation (CE), the CE-based MMSE turbo equalization requires a shorter training sequence and has faster convergence, but a slightly higher computation complexity due to the large-dimension matrix inversion in computing equalizer coefficients. To further reduce the computation complexity, the low-complexity CE-based MMSE turbo equalization is proposed and analyzed based on the terrestrial wireless channel model in [19]. Additionally, simulation analysis shows that this performance is nearly as good as that of the traditional CE-based MMSE turbo equalization [19]. At present, in FMT modulation UWA communications, the traditional MMSE adaptive equalization is a conventional approach used to suppress the ISI [13; 17]. However, when the effect of ISI spans tens of symbols or large constellation sizes are used, the performance of FMT modulation UWA communications using traditional MMSE adaptive equalization is unacceptable. Considering the aforementioned advantages of turbo equalization and disadvantages of traditional MMSE adaptive equalization, the FMT modulation UWA communication using low-complexity CE-based MMSE turbo equalization is proposed in this paper. The proposed method applies turbo equalization to FMT modulation UWA communications to improve the ISI suppression performance. In the proposed method, error control coding (ECC) is used to encode the information bit subsequence before FMT modulation, and the low-complexity CE-based MMSE turbo equalization is adopted at the receiver to suppress ISI in each subsequence after FMT demodulation. The proposed method obtains good performance with relatively low computation complexity because the equalizer in turbo equalization is based on the low-complexity CE-based MMSE algorithm. The proposed method is analyzed in theory and verified by simulation results and data collected from a real experiment. The proposed method is compared with the FMT modulation UWA communications using traditional MMSE adaptive equalization, and the results show the validity of the proposed method. The contribution of this paper is threefold: (1) turbo equalization is first applied to FMT modulation UWA communications to overcome the shortcomings of adaptive equalization; (2) the low-complexity CE-based MMSE algorithm is exploited to adjust the equalizer coefficients of turbo equalization to suppress ISI with relatively low computation complexity; (3) simulation and real experiments are designed to verify the proposed method. This paper is organized as follows: in Section 2, the transmitter structure of the proposed method is briefly described. In Section 3, the receiver structure of the proposed method is presented and the principle of the low-complexity CE-based MMSE turbo equalization is analyzed in detail. In Section 4, the simulation and experiment results of the proposed method are presented, and the proposed method is compared with the results of the FMT modulation UWA communication using traditional MMSE adaptive equalization. Finally, the conclusion and summary is in Section 5.