Z-OFDM: A New High-Performance Solution for Underwater Acoustic Communication

Abstract

This paper presents Z-OFDM, a high-performance solution for underwater acoustic communication. Traditional underwater orthogonal frequency division multiplexing (OFDM) systems suffer from spectrum leakage and distortion due to the narrowband nature of underwater acoustic signals and the picket fence effect of the fast Fourier transform (FFT). Z-OFDM addresses these issues by integrating zoom-fast Fourier transform (ZoomFFT) with OFDM and redesigning the modulator and demodulator to replace the conventional FFT. This integration enhances spectral resolution, resulting in higher channel capacity, improved Signal to Interference plus Noise Ratio (SINR), and reduced Bit Error Rate (BER). Computer simulations using underwater acoustic channels from Fuxian Lake and Wuyuan Bay demonstrate that the Z-OFDM system achieves a 6 dB gain compared to conventional OFDM systems at a BER of ${10}^{-3}$. These results demonstrate the effectiveness of Z-OFDM in overcoming the limitations of traditional FFT-based OFDM systems in underwater environments.

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 [16]. 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]. Currently, there are mainly two methods for spectrum refinement in engineering: zoom-fast Fourier transform (ZoomFFT) and Chirp-Z transform [24]. Both algorithms can effectively improve the spectral resolution of the narrowband signal. Chirp-Z transform increases the number of FFT transform points by interpolating the target frequency band without expanding the data length to improve the spectral resolution of the signal [25]. ZoomFFT moves the signal spectrum to the baseband, decimates it, and outputs it for spectrum refinement [26]. For the multi-frequency harmonic components with single frequency and less severe spectral line interference, Chirp-Z can obtain higher frequency, amplitude, and phase accuracy than Zoom FFT [27]. For the dense multi-frequency harmonic components with serious interference, Chirp-Z can only amplify the frequency band but cannot eliminate the interference effect [28], while ZoomFFT can separate and correct the interference frequency components by increasing the refinement factor to obtain high-precision signal parameters [29,30,31]. Since OFDM belongs to the multi-carrier communication system, the underwater acoustic OFDM signal is a multi-frequency signal. To improve spectrum utilization and communication speed under extremely limited bandwidth resources [32], the spectrum lines of underwater acoustic OFDM signals are usually quite dense [33]. In addition, the underwater acoustic OFDM signal is vulnerable to the influence of underwater auditory channels, resulting in Inter Carrier Interference (ICI). In contrast, ZoomFFT is more suitable for underwater acoustic OFDM signal processing [34].

Considering the advantage of ZoomFFT, the UWA OFDM communication system based on ZoomFFT (Z-OFDM) is proposed in the paper, which uses ZoomFFT to replace FFT in the modulation and demodulation. This modulation method improves the channel capacity and SINR of the system and reduces the BER of the system. The Z-OFDM system is expected to become a new high-performance and low-complexity solution for UWA communication.

The rest of the paper is organized as follows. In Section 2, the principle of ZoomFFT is reviewed. In Section 3, the design and performance of the Z-OFDM system are presented. Section 4 focuses on the computer simulation experiments of the Z-OFDM system. In Section 5, the innovations and advantages of the Z-OFDM system compared to the traditional FFT-based OFDM system are discussed based on the experimental results. Finally, the conclusion is drawn in Section 6.

2. ZoomFFT Algorithm

2.1. Principle

The ZoomFFT can be summarized with the following four steps shown in Figure 1 [21]: frequency shift, low pass filtering, resampling, and FFT.

The sequence $s\left(n\right)$ is the input signal with the length of N. And $S_D\left(k\right)$ is the corresponding refinement spectrum of sequence $s\left(n\right)$, which can be expressed as [21]

$$\begin{array}{cc}\hfill S_D\left(k\right)& =\mathcal{F}(h_{LP}\left(nD\right)\ast \Re \left(s\left(nD\right)e^{-j{\omega }_{0}nD}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{D}}{H}_{LP}(e^{j{\displaystyle \frac{2\pi }{TD}}k})S(e^{j({\displaystyle \frac{2\pi }{TD}}k-{\omega }_0)})\hfill \end{array}$$

(1)

where $h_{LP}$ the impulse response of the low pass filter is used for anti-aliasing. ${\omega }_0$ is the frequency shift parameter. D is the refinement factor.

The number of FFT points is N. The spectral resolution of the Zoom-FFT algorithm is [35]

$$\begin{array}{c}\hfill \Delta f_N^{\prime }={\displaystyle \frac{f_s^{\prime }}{N}}={\displaystyle \frac{f_s/D}{N}}={\displaystyle \frac{\Delta f_N}{D}}\end{array}$$

(2)

where the sampling frequencies for the FFT and Zoom-FFT algorithms are, respectively, $f_s$ and $f_s^{\prime }$, and the spectral resolutions are $\Delta f_N$ and $\Delta f_N^{\prime }$. The resolution of the spectrum after refinement is increased D-fold compared with the original range.

2.2. Energy Efficiency

When the number of points in the ZoomFFT is N, the total number of sampling points is $ND$, the order of the low-pass filter is R, and the FFT requires ${\displaystyle \frac{N}{2}}{log}_{2}N$ complex multiplications, the total number of complex multiplications (NCM) for the ZoomFFT algorithm, including frequency shifting, low-pass filtering, and FFT, can be expressed as [21,24]

$$\begin{array}{cc}\hfill T_{ZoomFFT}& =T_{shift}\left(n\right)+T_{LP}\left(n\right)+T_{FFT}\left(n\right)\hfill \\ \hfill & =\mathsf{\Theta }((R+1)ND+{\displaystyle \frac{N}{2}}lo{g}_{2}N)\hfill \end{array}$$

(3)

If the traditional FFT algorithm is required to achieve the same frequency resolution as the ZoomFFT, the number of points needs to be increased by D. Therefore, the NCM for the traditional FFT algorithm is

$$\begin{array}{cc}\hfill T_{FFT}& =\mathsf{\Theta }({\displaystyle \frac{ND}{2}}lo{g}_{2}ND)\hfill \end{array}$$

(4)

It is shown from Equation (3) and (4) that the spectrum resolution with ZoomFFT is the same as the spectrum of the original signal with FFT, but the computational complexity has been reduced. When the value of D is relatively small (e.g., $D=2,3$), the computational complexity of the ZoomFFT is not significantly different from that of the FFT due to the frequency shift and filtering steps involved in the ZoomFFT. However, as D increases, in order to achieve the same frequency resolution, the FFT requires a much larger size, specifically D times that of the ZoomFFT. Given that the growth rate of $N{log}_{2}N$ is greater than that of N, this results in a substantial increase in the complexity of the FFT, far exceeding that of the ZoomFFT. Therefore, the larger the value of D, the more pronounced the advantage of the ZoomFFT algorithm becomes.

Using a benchmark signal with a center frequency of 31 kHz, a bandwidth of approximately 2 kHz, and a sampling rate of 144 kHz, both FFT and ZoomFFT are implemented on the same hardware platform [36]. Under the condition of the refinement factor, D is 3, and the execution time of ZoomFFT is 9.528 ms, while that of FFT is 9.722 ms, showing little difference between the two. When the refinement factor D is increased to 12, the execution time of ZoomFFT is 8.578 ms, while that of FFT is 32.604 ms. To achieve the same resolution, the FFT algorithm requires an increase in the number of points, resulting in a much longer execution time compared to ZoomFFT. It is evident that for a specific frequency band, ZoomFFT exhibits a shorter execution time, thereby reducing overall energy consumption.

3. Communication System Model

3.1. Theory

The composition of the Z-OFDM system, which belongs to single input-single output (SISO) communication system, is shown in Figure 2 [35]. The input binary data are mapped into a symbol constellation point $X_p$ by Quadrature Phase Shift Keying (QPSK) [37] and transferred to the OFDM time domain signal $x_p\left(t\right)$ by the inverse fast Fourier transform (IFFT) algorithm. Up-sampling the OFDM signal $x_p\left(t\right)$ [38], and then the low-pass filter $h_l\left(t\right)$ is added to the system to reduce the aliasing effect. Furthermore, the cyclic prefix (CP) is inserted at the head of the transmitting sequence to minimize the impact of inter-symbol interference (ISI). The signal is then up-converted to the carrier frequency. Subsequently, the signal is transmitted through the underwater acoustic channel, and the received signal $y\left(t\right)$ is obtained at the receiver. Then, the CP is deleted and the time-domain signal is down-converted, passed through a low-pass filter, and then down-sampled by decimation. Finally, passing the down-sampled signal through FFT and performing channel equalization to reduce channel effects, then demodulating it by QPSK to get the message sequence [39]. The following will introduce the components of the Z-OFDM system from the perspectives of the transmitter model and the receiver model.

3.1.1. Transmitter Model

The transmitter sends data in blocks with a duration of $T_D$. The input bit stream for each block is mapped into M symbols using a QPSK constellation scheme (in this study, $M=4$). The data stream is then converted from serial to parallel, forming parallel symbol streams, which are transmitted on orthogonal carriers centered at different subcarrier frequencies. Suppose the original data rate is R; the parallel stream rate is ${\displaystyle \frac{R}{K}}$, where K is the number of parallel streams.

The data streams are loaded onto the subcarriers using IFFT. The symbol duration on each subcarrier increases by a factor of K. The number of subcarriers K is chosen such that the symbol time on each subcarrier is much greater than the channel delay spread, or the subcarrier bandwidth is less than the channel coherence bandwidth. The modulation scheme modulates only the odd subcarriers in the first half of the N subcarriers to meet the Hermitian symmetry requirement.

Let $X_p=[X_0,X_1,\dots ,X_{N-1}]$ be the QPSK modulated information sequence, where N is the number of subcarriers. Zeros are inserted at the carrier positions, and the modulated information sequence is loaded onto the corresponding subcarriers. Suppose the sequence after zero insertion is $X_i\left[k\right]$; this step can be expressed by Equation (5).

$$X_i\left[k\right]=\left\{\begin{array}{cc}{X}_p\left[k\right],\hfill & \mathrm{if}k\in C_D\hfill \\ 0,\hfill & \mathrm{else}\hfill \end{array}\right.$$

(5)

where $C_D$ is the data set composed of the positions of the information sequence.

The sequence after zero insertion is transformed into a time-domain signal through IFFT, which can be expressed as

$$\begin{array}{cc}\hfill x_p\left(t\right)& ={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{X}_i\left[k\right]e^{j2\pi {\displaystyle \frac{k}{T}}t}\hfill \end{array}$$

(6)

where $x_p\left(t\right)$ is the time-domain signal, and N is the number of IFFT points.

The baseband signal $x_p\left(t\right)$ is upsampled (using an upsampling factor D), resulting in the refined frequency bands. These bands are then frequency-shifted to their actual frequencies and passed through a bandpass filter to prevent aliasing, and that can be expressed as

$$x_D\left(t\right)=x_p({\displaystyle \frac{t}{D}})e^{-j{\omega }_{0}t}\ast h_t\left(t\right)$$

(7)

where $h_t\left(t\right)$ is the impulse response of the low-pass filter, and ${\omega }_0$ is the frequency shift parameter.

A CP is inserted at the front end of the signal to eliminate ISI. The signal with the CP added is then converted from parallel to serial, and upconverted to the carrier frequency $f_c$. After up-conversion, the transmitted signal $x\left(t\right)$ and its spectrum $X\left(\omega \right)$ can be described as follows:

$$\begin{array}{c}\hfill x\left(t\right)=2\mathcal{R}((x_p({\displaystyle \frac{t}{D}})e^{-j{\omega }_{0}t}\ast h_t\left(t\right))e^{-j2\pi f_{c}t})\end{array}$$

(8)

$$\begin{array}{c}\hfill X\left(\omega \right)=X_p((\omega +{\omega }_0)D+2\pi f_c)H_T(\omega +2\pi f_c)\end{array}$$

(9)

3.1.2. Receiver Model

The signal is subject to multipath propagation within the limited bandwidth through the underwater channel. Multipath propagation causes ISI, which restricts the data through put and degrades the system’s performance. For this reason, the UWA channel impulse response is represented by [40]

$$\begin{array}{c}\hfill h_c\left(t\right)=\sum _{l=0}^N{a}_l\delta (t-{\tau }_l)\end{array}$$

(10)

in which both $a_l$ and ${\tau }_l$ are the amplitude gain and Delay on the l path.

After passing through the UWA channel, the received signal can be written as [41]

$$\begin{array}{cc}\hfill y\left(t\right)& =h_c\left(t\right)\ast x\left(t\right)+w\left(t\right)\hfill \\ \hfill & =\sum _{l=0}^L{a}_{l}x(t-{\tau }_l)+w\left(t\right)\hfill \end{array}$$

(11)

where $w\left(t\right)$ is the additive white Gaussian noise (AWGN) process.

At the receiver, the first step is frequency down-conversion, which translates the received signal from its carrier frequency to baseband. This is represented by

$$y_m\left(t\right)=y\left(t\right)e^{j2\pi f_{c}t}$$

(12)

where $f_c$ is the carrier frequency.

After frequency down-conversion, the signal $y_m\left(t\right)$ undergoes serial-to-parallel conversion. This process converts the entire signal into multiple OFDM symbols. Each OFDM symbol contains a CP that must be removed for restoring the original symbol. Once the CP is removed, the signal is frequency-shifted to near zero frequency and then undergoes to low-pass filtering using a Kaiser window function to prevent spectral aliasing during the downsampling. The process is represented as

$$y_f\left(t\right)=y_m\left(t\right)e^{j{\omega }_{0}t}\ast h_r\left(t\right)$$

(13)

where $h_r\left(t\right)$ is the impulse response of the low-pass filter.

After low-pass filtering, the signal is down-sampled to enhance its resolution. Down-sampling involves selecting every D-th sample, and the process is described as

$$y_D\left(t\right)=y_f\left(tD\right)$$

(14)

where D is the down-sampling factor.

This step ensures that the sampling rate of the received signal returns to the original baseband sampling rate.

The down-sampled signal is then transformed to the frequency domain using FFT. At this step, the refined spectrum of the received signal is obtained. The FFT process is represented as

$$Y_z\left(\omega \right)=\sum _{t=0}^{N-1}{y}_D\left(t\right)e^{-j\omega t}$$

(15)

where $Y_z\left(\omega \right)$ is the frequency-domain representation, and N is the FFT size.

Let ${\omega }_c=2\pi f_c$. The complete calculation of the ZoomFFT output at the l subcarrier is as follows:

$$\begin{array}{cc}\hfill Y_z\left(\omega \right)& =\mathcal{F}(y\left(tD\right)e^{j({\omega }_c+{\omega }_0)tD}\ast h_r\left(tD\right))\hfill \\ \hfill & =H_C({\displaystyle \frac{\omega -({\omega }_c+{\omega }_0)}{D}})X({\displaystyle \frac{\omega -({\omega }_c+{\omega }_0)}{D}})H_r({\displaystyle \frac{\omega }{D}})+W\left(\omega \right)\hfill \end{array}$$

(16)

Subsequently, the refined spectrum of the received signal is subjected to channel equalization. This process eliminates the impact of the channel on the data symbols, thereby reducing the BER. The channel’s target frequency band spectrum is refined using ZoomFFT. By combining the refined spectrum with the Minimum Mean Square Error (MMSE) equalizer, the influence of the channel on the received signal can be more effectively mitigated. The principle of the equalizer $G\left(k\right)$, obtained through the methods in references [42,43,44], is given as shown in Equation (17).

$$\begin{array}{cc}\hfill \widehat{S}\left(k\right)& =G\left(k\right)d\left(k\right),\hfill \\ \hfill G\left(k\right)& ={\left(Y_z{Y}_z^H+{\sigma }^{2}I\right)}^{-1}{Y}_z{e}_i,\hfill \\ \hfill Y_z{e}_i& =E\left[S_i{R}_z^{\left(i\right)}\right]\hfill \end{array}$$

(17)

Then, by performing channel equalization on $Y_z\left(\omega \right)$, the effects of channel transmission are eliminated. Finally, the modulated information $X_p$ is extracted from the selected refined spectrum $Y_z\left(\omega \right)$, and by performing QPSK demapping on $X_p$, the binary data are obtained.

3.2. Performance Analysis

3.2.1. Channel Capacity

According to the definition of spectral resolution, an increase in resolution implies a decrease in the spacing of subcarriers, which means that a constant bandwidth can be synthesized by more subcarriers. For the same bandwidth, the channel capacity of the Z-OFDM system increases with the spectral refinement factor D [45].

According to the definition given in the literature [46], the OFDM channel capacity can be expressed as

$$\begin{array}{cc}\hfill C& =\sum _{l=0}^k{C}_k\hfill \\ \hfill & =\Delta f\sum _{k=1}^{N}lo{g}_2(1+{\displaystyle \frac{P_k}{N_0}})\hfill \end{array}$$

(18)

where $C_k$ is the information capacity of the k-th sub-channel, $P_k$ is the average power of the sub-carriers under the first sub-channel.

Assuming that the bandwidth of the UWA signal is constant, the relationship between the bandwidth, B, the number of sub-carriers, and the sub-carrier interval should satisfy the description given in Equation (19).

$$\begin{array}{c}\hfill B=N\Delta f\end{array}$$

(19)

Combining Equations (2), (18), and (19), it is derived that the channel capacity of the Z-OFDM system can be described as

$$\begin{array}{cc}\hfill C_Z& =\sum _{k=1}^{ND}{\displaystyle \frac{\Delta f}{D}}lo{g}_2(1+{\displaystyle \frac{P_k}{N_0}})\hfill \end{array}$$

(20)

The power of the channel noise in the k-th sub-channel is denoted as $N_0$. Figure 3 shows the channel capacity of the Z-OFDM system over AWGN channel. From Figure 3, it can be concluded that the channel capacity of the Z-OFDM system increases as the spectral refinement factor D increases for the same ${\displaystyle \frac{E_b}{N_0}}$. In contrast, the channel capacity of the FFT-based OFDM system is smaller than that of the Z-OFDM system.

3.2.2. Signal Interference plus Noise Ratio

This section analyzes the Signal Interference plus Noise Ratio (SINR) of the Z-OFDM communication system [47,48,49,50].

According to the literature, the SINR of the Z-OFDM system can be defined as [51]

$$\begin{array}{cc}\hfill SIN{R}_i& ={\displaystyle \frac{E\left[R\right(\tau \left)\right]}{Var\left[I_Q\right]}}\hfill \end{array}$$

(21)

where $I_Q$ denoted the total interference.

This section $I_Q$ was only considered as AWGN to simplify the research, $R_i\left(\tau \right)$ which was given in

$$\begin{array}{cc}\hfill R_i\left(\tau \right)& ={\int }_{\tau }^{T_c}{P}_{Zj}\left(t\right)P_{Zj}(t-\tau )\hfill \end{array}$$

(22)

where $P_{Zj}$ is the average power of the Z-OFDM signal, which is given in

$$\begin{array}{cc}\hfill P_{Zj}& ={\displaystyle \frac{1}{T_c}}{\int }_{\tau }^{T_c}{y}_z^2\left(t\right)dt\hfill \\ \hfill & ={\displaystyle \frac{D}{T_c}}{\int }_{\tau }^{T_c}|X({\displaystyle \frac{2\pi f}{D}}){|}^{2}d{\displaystyle \frac{f}{D}}\hfill \\ \hfill & =P_{i}D\hfill \end{array}$$

(23)

Combining Equations (21), (22), and (23), they can be written as Equation (23) in another way [52].

$$\begin{array}{cc}\hfill SIN{R}_{Zj}& ={\displaystyle \frac{E\left[R_{Zj}\left(\tau \right)\right]}{Var\left[I_Q\right]}}\hfill \\ \hfill & ={\displaystyle \frac{E\left[R_i\left(\tau \right)D^2\right]}{Var\left[I_Q\right]}}\hfill \end{array}$$

(24)

The comparison between the Z-OFDM system and the FFT-based OFDM system over different values of factor D is shown in Figure 4. The SINR of the Z-OFDM system increased with the increase in factor D and is higher than the SINR of the FFT-based OFDM system.

3.2.3. Bit Error Rate

In this paper, it is assumed that the modulation is QPSK. According to the definition given in the literature [53], the BER of the OFDM can be expressed as

$$\begin{array}{cc}\hfill P_{BER}& =Q(\sqrt{{\displaystyle \frac{2\sum |H_c{|}^2{\epsilon }_Z}{N_0}}})\hfill \end{array}$$

(25)

where $N_0$ is the noise spectral density of the AWGN channel, ${\epsilon }_Z$ is the total transmitted signal energy for the Z-OFDM signals, which was given by

$$\begin{array}{cc}\hfill {\epsilon }_Z& ={\int }_{-\infty }^{+\infty }{\left|Y_z\left(f\right)\right|}^{2}df\hfill \\ \hfill & ={\int }_{-\infty }^{+\infty }|Y_z({\displaystyle \frac{f}{D}}){|}^{2}df\hfill \\ \hfill & =D{\int }_{-\infty }^{+\infty }|Y_z({\displaystyle \frac{f}{D}}){|}^{2}d{\displaystyle \frac{f}{D}}\hfill \\ \hfill & =\epsilon D\hfill \end{array}$$

(26)

Combining Equations (25) and (26), the probability of error of the Z-OFDM system is [46,54]

$$\begin{array}{cc}\hfill P_{BER}& =Q(\sqrt{{\displaystyle \frac{2\sum |H_c{|}^2\epsilon D}{N_0}}})\hfill \end{array}$$

(27)

Figure 5 shows the BER of Z-OFDM at different values of factor D over the AWGN channel. At a BER of ${10}^{-3}$, Z-OFDM achieved a 3 dB gain compared to FFT-based OFDM when $D=2$. It also achieved a 6 dB gain when $D=4$ and an 8 dB gain when $D=6$. It also shows that the BER of the Z-OFDM system decreased as the factor D increased.

4. Computer Simulation Experiments

This section primarily introduces the computer simulation experiments and experimental results of the Z-OFDM system. The parameter of the Z-OFDM system is given in

Table 1. Parameter of communication model.

Parameter	Value
DFT Point	1024
Number of Sub-carrier	512
Number of Data Sub-carrier	324
Number of Pilot Sub-carrier	103
Sampling Rate (kHz)	144
Bandwidth (kHz)	3.6
Band (kHz)	29.1∼32.7
Guard Interval	0.125 × Symbol Length
Sub-carrier Spacing (Hz)	3.516
Refinement Factor D	[2 3 4]

. This simulation aims to get the BER of the Z-OFDM system under a natural underwater channel.

In this section, the channel data are obtained from the natural underwater channel in an experiment held by the Key Laboratory of Underwater Acoustic Communication and Marine Information Technology at Xiamen university. The experiment aimed to measure the time evolution of the magnitude impulse response at two different locations. The experiments are held in Fuxian Lake in Yunnan [7,55,56] (Figure 6a) and Wuyuan Bay in Xiamen [57,58,59] (Figure 6b).

4.1. Underwater Acoustic Channels

Figure 7 shows the two impulse responses of underwater channels at (a) Fuxian Lake and (b) Wuyuan Bay. The environmental characteristics of these locations are not similar and can represent different underwater environments.

4.1.1. Fuxian Lake

Fuxian Lake is a freshwater lake, and its underwater acoustic environment is relatively pure due to the lack of human-made noise such as boat activity. From Figure 7a, at least three transmission paths can be observed that are clear and constant. There are also several transmission paths that are not clear enough and can be ignored. The maximum Delay of these clear paths is 3 ms.

4.1.2. Wuyuan Bay

Wuyuan Bay is a semi-enclosed shallow bay. The data were collected in a shallow water channel with a depth of approximately 10 m in the bay. As seen in Figure 6b, there is a large bridge near the channel. The passage of vehicles on the bridge causes vibrations, which are transmitted to the sea through the bridge piers. This results in a complex underwater noise environment with significant human-made noise. In Figure 7b, six paths can be observed. The maximum delay of the paths is 11 ms. The transmission loss in Wuyuan Bay is too serious to keep the other transmitting path.

4.2. Experimental Results

In this section, the experimental results of the Z-OFDM system in two different environments (Fuxian Lake and Wuyuan Bay) are presented by illustrating the BER and its fluctuations as the refinement factor D increases. To ensure a fair comparison of performance, the experiments are conducted at the same transmission rate, which can be calculated using the following equation:

$$\begin{array}{cc}\hfill R_{\text{Z-OFDM}}& =N_s·R_b\hfill \\ \hfill & =D{N}_s·{\displaystyle \frac{1}{T^{\prime }}}·{log}_{2}M\hfill \\ \hfill & =D{N}_s·{\displaystyle \frac{B}{\left(1+\frac{T_G}{T_S}\right)D{N}_{DFT}}}·{log}_{2}M\hfill \\ \hfill & ={\displaystyle \frac{B{N}_S{log}_{2}M}{\left(1+\frac{T_G}{T_S}\right)N_{DFT}}}\hfill \end{array}$$

(28)

where $N_s$ is the number of subcarriers, $R_b$ is the transmission rate of a single subcarrier, $T^{\prime }$ is the period of a single subcarrier, B is the bandwidth, $T_G$ is the guard interval, $T_S$ is the symbol length, and $N_{DFT}$ is the number of DFT points. Since QPSK modulation is employed in this communication system, $M=4$.

In this experiment, regardless of the value of D, the transmission rate is consistently maintained at $2.025\mathrm{kbps}$.

4.2.1. Fuxian Lake

As shown in Figure 8a, the BER decreased with increasing Signal Noise Ratio (SNR) of the communication system. However, with a higher factor D, the BER of the communication system decreased more significantly. In other words, a higher factor D provided higher gain for the communication system. At a BER of ${10}^{-3}$, the Z-OFDM system achieved a 3 dB gain compared to the FFT-based OFDM system when the refinement factor $D=2$. When $D=3$, the communication system achieved a gain of 5 dB. When $D=4$, the communication system achieved a gain of 6 dB.

Figure 9 shows the BER of the Z-OFDM system with varying parameters of the underwater acoustic channel of Fuxian Lake. At the same ${\displaystyle \frac{E_b}{N_0}}$, the BER of OFDM is inconsistent with the varying parameter of the underwater acoustic channel. When the refinement factor $D=2$, the BER is still variable at the same ${\displaystyle \frac{E_b}{N_0}}$, but the variability of the BER of Z-OFDM becomes weaker compared to FFT-based OFDM. When $D=4$, the BER decreases, and the variability of the BER becomes weaker. When $D=6$, the BER decreases even more, and the variability of the BER cannot be observed clearly.

4.2.2. Wuyuan Bay

It can be observed from Figure 8b that the BER of the communication system decreases significantly with the increase in the refinement factor. At a BER of ${10}^{-2}$, when the refinement factor $D=2$, the Z-OFDM system achieved about 3 dB more gain than the FFT-based demodulation algorithm. When $D=3$, the gain obtained by the communication system was close to 5 dB. Compared with traditional FFT, the maximum gain obtained by the underwater acoustic OFDM demodulation algorithm based on ZoomFFT can reach 6 dB ($D=4$).

Figure 10 shows the BER of the Z-OFDM system with varying parameters of the underwater acoustic channel of Wuyuan Bay. In Figure 7, the variability of BER is the same as in Figure 6. With the increase in factor D, the variability of BER of the ZoomFFT decreased. However, due to geological factors, the BER of the communication system under Wuyuan Bay was higher than that under Fuxian Lake, and the variability of BER was more complex.

5. Discussion

In the Z-OFDM system, innovative designs have been implemented from both the transmitter and receiver. On the transmitter side, the modulator generates OFDM signals by loading data streams onto subcarriers and mapping them to corresponding carrier phases via ZoomIFFT. During this process, ZoomIFFT processes the spectrum with higher resolution, allowing for more subcarriers to be accommodated within the same bandwidth, thereby increasing the channel capacity. On the receiver side, the demodulator employs ZoomFFT to refine the spectrum of the received OFDM signal, which enables the receiver to more accurately recover the original signal. As a result, the SINR is improved, and the BER is reduced.

Recently, FFT-based OFDM communication systems continue to be the focus of numerous studies [7,60,61]. In [60], an FFT size of 2048 is used, resulting in a spectral resolution of 46.875 Hz. The study mitigates ICI by improving Doppler shift compensation through the use of a non-uniform fast Fourier transform matrix (NFFT). Experimental results demonstrate that the proposed method achieves at least an 8 dB gain in system performance at a BER of ${10}^{-1}$ compared to systems without Doppler compensation. In [61], the FFT size is 256, yielding a spectral resolution of 390.625 Hz. The study designs a preamble using a CAZAC sequence and develops a new time-frequency synchronization method for the OFDM system. Experimental results show that the proposed improvements lead to a significantly lower symbol error rate (SER) on the order of ${10}^{-3}$. In [7], an FFT size of 1024 is employed, providing a spectral resolution of 7.8125 Hz. The study enhances system performance by introducing a novel CE technique and an intra-block CFO estimation method. Experimental results indicate that, after compensation, the system’s BER is $3.20\times {10}^{-3}$ at an SNR of 30 dB. These studies, while achieving specific gains in BER, SER, and other metrics, still face limitations due to the inherent constraints of the FFT, such as increased computational complexity with larger FFT sizes.

As shown in

Table 2. Comparison between Z-OFDM (D = 2) and FFT-OFDM systems.

Feature/Metric	Z-OFDM	FFT-OFDM
Spectral Resolution	High (refined with Zoom-FFT)	Moderate (limited by FFT size)
Channel Capacity	High	Low
BER	Low, especially under interference	High
SINR	High	Low
Computational Complexity	Moderate (efficient spectral refinement)	High (requires larger FFT size for similar resolution)

, Z-OFDM significantly outperforms traditional FFT-based OFDM in several key areas. On the transmitter side, the channel capacity is notably increased. Experiments conducted in an AWGN channel indicate that under the same condition of ${\displaystyle \frac{E_b}{N_0}}=10$ dB, the Z-OFDM system (with a refinement factor $D=2$) achieves an increase in channel capacity by 0.2 bps/Hz compared to the FFT-based OFDM system. Moreover, when channel capacity remains constant, the Z-OFDM system can achieve up to a 3 dB gain over the FFT-based OFDM system. On the receiver side, the increased spectral resolution of Z-OFDM allows it to handle NBI more effectively, leading to lower BER and improved signal quality. Experiments show that under the same condition of ${\displaystyle \frac{E_b}{N_0}}=10$ dB, the Z-OFDM system with a refinement factor $D=2$ reduces the BER by 0.0683 and improves the SINR by 0.576 compared to the FFT-based OFDM system. When BER and SINR remain constant, the Z-OFDM system can also achieve up to a 3 dB gain over the FFT-based OFDM system. Additionally, Z-OFDM maintains computational efficiency, ensuring that these benefits are achieved without excessive processing overhead. In terms of computational complexity, as the refinement factor D in the Z-OFDM system increases, the spectral resolution becomes higher. To achieve the same resolution, the FFT-based OFDM system would be required to increase the FFT size, leading to a significant increase in computational complexity. However, the Z-OFDM system takes advantage of ZoomFFT, allowing its complexity to remain almost unchanged.

The above comparison is made with the refinement factor $D=2$. However, increasing the factor D can increase both the capacity and the SINR of the Z-OFDM system. It can also decrease the BER of the system over the AWGN channel and reduce the variability of the BER of the system over time in varying underwater acoustic channels. However, the factor D can not increase without the limitation of Shannon’s sampling law. The maximum factor D can be computed by Equation (29).

$$\begin{array}{cc}\hfill D& ={\displaystyle \frac{f_s}{2(f_H-f_L)}}\hfill \end{array}$$

(29)

The limitation of Shannon’s sampling law complicates the problem of how to choose factor D to make the Z-OFDM systematically obtain the best performance. This problem is not discussed in detail in this paper. However, it will be our key concern in the next phase of research work.

6. Conclusions

To eliminate the adverse effects caused by the picket fence effect of FFT and improve the performance of underwater acoustic OFDM systems, this paper proposed using ZoomFFT to replace FFT in the modulator and demodulator of the OFDM system. Performance analysis through simulations with underwater acoustic channel models from Xiamen University, specifically Fuxian Lake and Wuyuan Bay, demonstrated that the Z-OFDM system significantly outperforms traditional FFT-based OFDM systems. Future research will focus on optimizing the selection of the refinement factor D and exploring the application of spectrum refinement to other advanced Fourier Transform methods. Overall, Z-OFDM represents an advancement in underwater acoustic communication technology, offering improved performance and reliability.