## 1 Introduction  
Underwater wireless communications are used in many applications to reduce cable costs and deployment time, such as harbor and shoreline monitoring, fish monitoring, and the monitoring of drilling sites such as oil wells and trenches [1]. The use of single-carrier transmission techniques has been the subject of study in the field of underwater acoustic communications [2; 3; 4; 5; 6]. In radio frequency wireless communications, orthogonal frequency division multiplexing (OFDM) technology, a type of multicarrier transmission, is frequently employed. OFDM technology has a long history, initially proposed in the 1950s [7] and subjected to theoretical analysis in the 1960s [8]. In the 1970s, a methodology based on the discrete Fourier transform (DFT) was proposed and regarded as a promising avenue of inquiry [9]. OFDM was subsequently adopted for digital broadcasting in 1987 [10] and continues to be employed in this capacity to the present day. To increase data bandwidth, OFDM-based wireless communication systems have received widespread attention for their high transmission rates and high spectrum efficiency, even for underwater applications. Since the Doppler shift occurs in underwater wireless communications due to the movement of equipment and vessels, signal processing is required to compensate for the Doppler shift. The performance of the OFDM system is seriously deteriorated by the inter-carrier interference (ICI) caused by the transmitter/receiver motion and ocean waves. Then, ICI mitigation in the OFDM system is required. Instead of conventional fast Fourier transform (FFT), fractional fast Fourier transform (F-FFT) is utilized with adaptive way for ICI mitigation in paper [11]. However, if it assumes differential OFDM modulation, then high-order modulation such as 64 QAM is difficult. In paper [12], ICI self-cancellation is demonstrated by inactivating partial subcarriers and utilizing opposite polarity on two adjacent active subcarriers for ICI self-cancellation. Then, the effective subcarrier utilization in OFDM used to convoy information lowers. Another approach to mitigate ICI with an iterative receiver based on the turbo principle is also proposed in paper [13]. So far, our research team has proposed a time-domain signal processing algorithm for the OFDM Doppler compensation of the desired propagation path [14; 15; 16]. But this approach cannot mitigate ICI caused by a multipath channel with a different Doppler shift. In 2023, we proposed a modified delay and Doppler profiler (mDDP) as a frequency domain signal processer to combat ICI in underwater acoustic OFDM mobile communications in multipath environments [17; 18]. However, the performance improvement in computer simulations and pool experiments was not remarkable. In a subsequent study, the accuracy of the channel transfer function (CTF), which is the input for the mDDP channel parameter estimation, was considered insufficient. In prior research [17; 18], multitap ICI cancellation was performed by obtaining channel estimates from the scattered pilots (SPs) signal. However, this was an ICI cancellation for data carriers, and ICI cancellation for pilot signals was not performed. Therefore, this paper aims to further improve the BER by using a multistep mDDP, which denoises the SPs in the first-stage mDDP to cancel ICI-derived noise components, and then improves the accuracy of the input SPs in the second and subsequent stages of the mDDP. To further improve the accuracy of channel estimation, an mDDP was devised to sequentially cancel the ICI. In this paper, the performance of underwater OFDM communications using an iterative one- to three-step mDDP are compared using computer simulation, assuming a two-wave multipath environment subject to the Doppler shift in the opposite direction as shown in Figure 1. Section 2 describes the system parameters of the underwater acoustic (UWA) OFDM modulation scheme used in underwater acoustic communication and the conventional Doppler shift cancellation scheme.