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Compared to conventional time-domain equalization, frequency-domain equalization (FDE) presents a computationally efficient alternative for the reception of single carrier (SC) transmissions. In this paper, we consider iterative FDE (IFDE) with explicit frequency-domain channel estimation (FDCE) for non-cyclic-prefixed SC systems. First, an improved IFDE algorithm is presented based on soft iterative interference cancellation. Second, a new adaptive FDCE (AFDCE) algorithm based on per-tone Kalman filtering is proposed to track and predict the frequency-domain channel coefficients. The AFDCE algorithm employs across-tone noise reduction, exploits temporal correlation between successive blocks, and adaptively updates the auto-regressive model coefficients, bypassing the need for prior knowledge of channel statistics. Finally, block-overlapping is used to facilitate the joint operation of IFDE and AFDCE. Simulation results show that, compared to related IFDE and adaptive channel estimation schemes, the proposed schemes offer lower mean-square error (MSE) in channel prediction, lower bit error rate (BER) after decoding, and robustness to non-stationary channels.