# Z-OFDM: A New High-Performance Solution for Underwater Acoustic Communication
## 1 Introduction  
Underwater Acoustic (UWA) communication systems play a vital role in fields such as oil exploration, oceanographic studies, and military applications [1]. Compared to optical and radio frequency (RF) communications, UWA communication is currently the dominant technology for underwater wireless communication because it can provide transmission distances of up to several kilometers. Orthogonal Frequency Division Multiplexing (OFDM) has been widely used in UWA communication systems [2; 3; 4; 5]. As a multicarrier technology, OFDM benefits from the computational efficiency of processing based on the fast Fourier transform (FFT), which allows the bandwidth to be easily divided into several orthogonal narrowband subcarriers. This simple structure enables high transmission rates and a strong ability to against the frequency-selective multipath distortion of UWA channels [6]. In traditional UWA OFDM communication systems, the design of modulators and demodulators typically employs the FFT algorithm. According to the Web of Science, between 2021 and 2024, a total of 228 studies applied OFDM in UWA communication systems, with 97.8% of them using FFT as the modulation and demodulation algorithm. These studies focus on improving various aspects of the communication system, including channel estimation (CE) [4], carrier frequency offset (CFO) estimation [7], pilot insertion [8], equalizers [9], and mappers [10], to enhance system performance. However, systems that utilize FFT obtain a rough \"panoramic spectrum\" over the entire folding frequency [11]. It is important to note that UWA OFDM communication is a narrowband communication technology. Due to the limitations of FFT size, spectral leakage and picket fence effects cause many subcarriers near interference frequencies to suffer severe signal-to-interference-plus-noise ratio (SINR) degradation. This makes UWA OFDM systems more susceptible to narrowband interference (NBI), thereby limiting the performance improvement of the systems. Specifically, UWA OFDM communication systems are extremely sensitive to Carrier Frequency Offset (CFO) caused by the Doppler effect. If the CFO exceeds the carrier spacing, due to the spectral leakage of FFT, the spectrum of NBI will spread in the entire frequency domain, thereby severely affecting adjacent sub-channels [12]. Suppressing Doppler distortion and reducing CFO are considered feasible methods to mitigate interference and have been extensively studied [7; 13; 14; 15]. For instance, Doppler shift compensation often employs resampling-based techniques. However, this significantly increases the size of the FFT, which in turn leads to higher computational complexity and reduced data rates [14]. OFDM is often used as the physical layer solution for UWA wireless sensor networks [2 ]. It is foreseeable that if the spectral leakage and picket fence effects of FFT are overlooked while continuing to optimize other modules of the communication system, it will greatly increase the computational complexity of the entire network and negatively impact energy efficiency. The fractional Fourier transform (FRFT) is another implementation method for UWA OFDM modulation and demodulation [17]. The complexity of the OFDM system based on FRFT is similar to FFT, but the performance is better in UWA channels [18]. However, FRFT is only a generalized Fourier transform, which cannot solve the spectrum leakage and picket fence effect of FFT itself. Consequently, it also limits improving UWA OFDM communication performance [19; 20]. Based on the previous analysis, improving FFT or FRFT to address their inherent issues of spectral leakage and picket fence effects would be a promising direction. To eliminate the picket fence effect, refining the spectrum of the received signal is necessary to improve its spectral resolution [21]. The basic idea of spectrum refinement is to amplify a specific frequency band in the signal spectrum [22], increasing the spectral line density near a particular frequency to achieve selectable frequency band analysis [23].