## 2 System Model  
We considered a UWA OFDM with \(N\) subcarriers, where the \(N_{U}\) subcarriers carry data and pilot symbols, and the remaining \(N_{G}/2\) subcarriers at each edge of the spectrum are used for guard bands. In the time and frequency domain, the pilot symbols are allocated in a comb-type fashion with the spacing of \(D_{f}\) subcarriers and \(D_{t}\) OFDM symbols. Let \(I_{f}\) and \(N_{f}=|I_{f}|\) denote the subcarrier indices and the number of pilot symbols in the frequency domain, respectively. Similarly, let \(I_{t}\) and \(N_{t}=|I_{t}|\) denote the OFDM symbol indices and the number of pilot symbols in the time domain, respectively.  
The time domain sample of the \(i\)-th OFDM symbol \(s_{i}[n]\) is represented as  
\[s_{i}[n]=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}S_{i}(k)e^{j\frac{2\pi}{N}ln},\ n=0,\ \cdots,\ N-1. \tag{1}\]  
where \(S_{i}(k)\) is the symbol on the \(k\)-th subcarrier of the \(i\)-th OFDM symbol. In order to maintain orthogonality between subcarriers under a frequency selective channel, a CP (of \(N_{cp}\) samples) is appended to the beginning of each OFDM symbol. After the CP insertion, the time domain samples are then pulse-shaped and up-converted to the carrier frequency \(f_{c}\). In the passband, the resulting transmitted signal is given by  
\[\bar{s}(t)=2\text{Re}\left(\sum_{i=0}^{N_{sum}-1}\sum_{n=-N_{cp}}^{N-1}s_{i}[n ]u(t-(iN_{s}+n)t_{s})\right)e^{j2\pi f_{c}t} \tag{2}\]  
where \(u(t)\) is a pulse-shaped filter, \(N_{s}=N+N_{cp}\), and \(N_{sym}\) and \(t_{s}\) denote the number of transmitted OFDM symbols and the sample duration, respectively.  
The time-varying UWA channel is often modeled as [5; 14; 15]  
\[h(\tau,\ t)=\sum_{l=0}^{N_{path}-1}c_{l}(t)\,\delta(\tau-\tau_{l}(t)) \tag{3}\]  
where \(N_{path}\) is the number of channel taps, and \(c_{l}(t)\) and \(\tau_{l}(t)\) denote the time-varying path gain and delay of the \(l\)-th channel tap, respectively. It can be assumed that during an OFDM symbol, the path gain of each tap is time-invariant, while its delay is time-varying by the Doppler effect [5]:  
\[c_{l}(t)=c_{l}, \tag{4}\] \[\tau_{l}(t)=\tau_{l}-a_{l}t,\]  
where \(a_{l}\) is the Doppler rate of the \(l\)-th channel tap. The channel impulse response (CIR) can then be rewritten as \[h(\tau,\,t)=\sum_{l=0}^{N_{path}-1}c_{l}\,\delta(\tau-\tau_{l}+\alpha_{l}t). \tag{5}\]  
The passband representation of the received signal is given as  
\[y(t)=\sum_{l=0}^{N_{path}-1}c_{l}\,\tilde{s}(t-\tau_{l}+\alpha_{l}t)+w(t) \tag{6}\]  
where \(w(t)\) is the additive white Gaussian noise (AWGN) with a zero mean and variance \(\sigma_{n}^{2}\).  
At the receiver, it is assumed that a two-step approach is applied to the received signal in (6) to mitigate the Doppler effect, which performs resampling in the passband and subsequently compensates the Doppler shift [5; 16]. For ease of derivation, we assume no residual Doppler rate, i.e., \(\alpha_{l}=0,\ \forall l\) with the aid of resampling and Doppler shift compensation. Then, the CIR in (5) can be rewritten to a time-invariant tapped-delay-line model as follows:  
\[h(\tau)=\sum_{l=0}^{N_{path}-1}c_{l}\,\delta(\tau-\tau_{l}) \tag{7}\]  
After resampling, the received pilot symbols carried by the \(i\)-th OFDM symbol in the baseband is represented as  
\[\mathbf{y}_{i}=\mathbf{X}_{i}\mathbf{A}\mathbf{h}+\mathbf{A}\mathbf{w},\ i\in I _{t} \tag{8}\]  
where \(\mathbf{X}_{i}\) is an \(N_{f}\times N_{f}\) diagonal matrix composed of the pilot symbols in the \(i\)-th OFDM symbol, \(\mathbf{A}\) is an \(N_{f}\times N_{path}\) DFT matrix with entries \([\mathbf{A}]_{m,\,n}=\frac{1}{\sqrt{N}}e^{-j\frac{2\pi}{N}I_{f}(m)n}\), \(\mathbf{h}=[h(0),\ h(1),\ \cdots,\ h(N_{path}-1)]^{T}\) is a time-invariant \(N_{path}\times 1\) CIR vector, and \(\mathbf{w}\) is an AWGN vector.