## 3 System Architecture and Channel Models
### Noise-Resistant Channel Measurement Model  
Considering the OFDM system with cyclic prefix (CP) and \(K\) subcarriers, the non-overlapping sets of \(K_{d}\) data subcarriers \(\mathbf{x_{d}}\), \(K_{p}\) pilot subcarriers \(\mathbf{x_{p}}\) and \(K_{n}\) null subcarriers \(\mathbf{x_{n}}\) satisfy \(K=K_{d}+K_{p}+K_{n}\). Let \(T\) denotes the OFDM symbol period without CP, \(T_{cp}\) denotes the length of CP and \(T_{s}=T+T_{cp}\) is the whole OFDM symbol period. The \(k\)th subcarrier is at the frequency:  
\[f_{k}=f_{c}+k/T,\qquad k=-K/2,\ldots,K/2-1, \tag{1}\]  
where \(f_{c}\) is the carrier frequency. Assuming that there are \(M\) OFDM blocks in one frame, a passband waveform of the \(m\)th OFDM block at time \(t^{\prime}\in[-T_{cp}+mT_{s},T+mT_{s}]\) is then given by  
\[\tilde{x}^{(m)}(t)=2 Re\big{\{}\frac{1}{\sqrt{K}}\sum_{k=-K/2}^{K/2-1}x_{k}^{(m )}e^{2\pi f_{k}t}q(t)\big{\}}, \tag{2}\]  
where \(m=1,2,\ldots,M\), \(t=t^{\prime}-mT_{s}\), \(x_{k}^{(m)}\) is defined as a transmitted symbol on the \(k\)th subcarrier at the \(m\)th OFDM block and q(t) is the pulse shaping filter:  
\[q(t)= \begin{cases}1,&t\in[-T_{cp},T],\\ 0,&\text{otherwise}.\end{cases}  \tag{3}\]  
The UWA channel is a typical sparse multipath channel. For horizontal shallow water multi-path transmission with a range much greater than depth, each path has almost the same Doppler factor [38]. Thus, we assume that all the paths have the equal Doppler scale factor \(a\), and the Doppler effect can be almost eliminated through effective Doppler estimation and compensation. Furthermore, the path delay remains stable across several consecutive OFDM blocks, and the path gains and Doppler scale factors are constant during  
Figure 2: Architecture of Bayesian learning-based channel estimation in UWA-OFDM communication. one OFDM block, but vary from block to block. The time-varying channel response during the \(m\)th OFDM block can be expressed as  
\[h^{(m)}(\tau,t)=\sum_{l=0}^{L-1}A_{l}^{(m)}\delta\Big{[}\tau-\Big{(}\tau_{l}-a^{( m)}t\Big{)}\Big{]}, \tag{4}\]  
where \(L\) is the path number of the multipath channel, which may be distributed in more than one cluster. \(A_{l}\) and \(\tau_{l}\) denote the gain and delay of the \(l\)th path. Through the time-varying channel, the received passband signal during the \(m\)th OFDM block can be expressed as  
\[\tilde{y}^{(m)}(t)=\sum_{l=0}^{L-1}A_{l}^{(m)}\tilde{x}^{(m)}\Big{[}\Big{(}1+a ^{(m)}\Big{)}t-\tau_{l}\Big{]}+\tilde{w}^{(m)}(t), \tag{5}\]  
where \(\tilde{w}(t)\) is the additive noise. After synchronization, a popular two-step approach [39] can be adopted to mitigate the Doppler effect, which takes a coarse Doppler estimation and resampling as the first step and then performs the fine Doppler shift compensation using the null subcarriers. Moreover, CP can also be used for Doppler shift estimation [40]. Performing CP-OFDM demodulation, the output \(K\times 1\) vector of the \(m\)th OFDM block can be expressed as  
\[\mathbf{y}^{(m)}=\mathbf{X}^{(m)}\mathbf{F}\mathbf{h}^{(m)}+\mathbf{w}^{(m)}, \tag{6}\]  
where \(\mathbf{X}^{(m)}\) is an \(K\times K\) diagonal matrix whose diagonal entries are the \(K\) transmitted symbols, \(\mathbf{F}\) is an \(K\times L\) discrete Fourier transform (DFT) matrix and \(\mathbf{w}^{(m)}\) is an \(K\times 1\) additive noise vector. \(\mathbf{h}^{(m)}\) is the discretized CIR vector as \(\mathbf{h}^{(m)}=[h_{1}^{(m)},h_{2}^{(m)},\cdots,h_{L}^{(m)}]\). For \(M\) consecutive OFDM blocks, the transmission model of all \(K\times M\) symbols can be expressed as  
\[\mathbf{Y}=\Big{[}\mathbf{\Phi}^{(1)}\mathbf{h}^{(1)},\mathbf{\Phi}^{(2)} \mathbf{h}^{(2)},\cdots,\mathbf{\Phi}^{(M)}\mathbf{h}^{(M)}\Big{]}+\mathbf{W}, \tag{7}\]  
where \(\mathbf{Y}=[\mathbf{y}^{(1)},\mathbf{y}^{(2)},\cdots,\mathbf{y}^{(M)}]\), \(\mathbf{W}=[\mathbf{w}^{(1)},\mathbf{w}^{(2)},\cdots,\mathbf{w}^{(M)}]\), and \(\mathbf{H}=[\mathbf{h}^{(1)},\mathbf{h}^{(2)},\cdots,\mathbf{h}^{(M)}]\).