## 2 Channel Estimation in OFDM  
The main principle of OFDM is to transmit data stream in low rate through numerous orthogonal subcarriers simultaneously [25]. Figure 1 shows a typical end-to-end configuration of OFDM communication system. The transmitted data is modulated on \(N\) subcarriers and passed through the UWA channels. The CIR of UWA channel can be expressed as [26],  
\[h(n)=\sum_{l=0}^{L-1}h_{l}\delta(n-l) \tag{1}\]  
where \(L\) is the length of channel and \(h_{l}\) is the coefficient of \(l\)th tap. The \(L\) can be calculated by \(L=\tau_{max}/T_{s}\) where \(\tau_{max}\) is the maximum delay of the channel and \(T_{s}\) is the length of an OFDM symbol. Now we suppose the guard interval (CP) length is no shorter than the maximum delay of UWA channel, which means the current received OFDM symbol is not affected by the previous symbol. Then the received signal processed by FFT can be expressed as,  
\[\mathbf{Y}=\mathbf{X}\mathbf{H}+\mathbf{V} \tag{2}\]  
where \(\mathbf{Y},\mathbf{X},\mathbf{H}\), and \(\mathbf{V}\) denote the matrix of received symbol, transmitted symbol, channel frequency response, and noise in frequency domain, respectively. It can also be presented as,  
Figure 1: Block diagram of orthogonal frequency division multiplexing (OFDM) system transceiver. \[\left[ \begin{array}{c}Y[0]\\ Y[1]\\ \vdots\\ Y[N-1]\end{array} \right]=\left[ \begin{array}{cccc}X[0]&0&\cdots&0\\ 0&X[1]&&\vdots\\ \vdots&&\ddots&0\\ 0&\cdots&0&X[N-1]\end{array} \right]\left[ \begin{array}{c}H[0]\\ H[1]\\ \vdots\\ H[N-1]\end{array} \right]+\left[ \begin{array}{c}V[0]\\ V[1]\\ \vdots\\ V[N-1]\end{array} \right] \tag{3}\]  
In the channel estimation problem of OFDM system, the matrix of transmitted pilot symbol \(\mathbf{X_{p}}\) and the matrix of received pilot symbol \(\mathbf{Y_{p}}\) are usually given and the estimate \(\hat{\mathbf{H_{p}}}\) of channel frequency response \(\mathbf{H_{p}}\) needs to be figured out. Least square (LS), minimum mean square error (MMSE), and linear minimum mean square error (LMMSE) are the most traditional algorithms used in OFDM channel estimation. LS algorithm focuses on finding the estimate \(\hat{\mathbf{H_{p}}}\) to minimize the value of following cost function,  
\[J\left(\hat{\mathbf{H_{p}}}\right)=\left\|\mathbf{Y_{p}}-\hat{\mathbf{Y_{p}}} \right\| \tag{4}\]  
The solution to the LS channel estimation is as follows [26; 27],  
\[\hat{\mathbf{H_{LS}}}=(\mathbf{X_{p}^{H}X_{p}})^{-1}\mathbf{X_{p}^{H}Y_{p}}= \mathbf{X_{p}}^{-1}\mathbf{Y_{p}} \tag{5}\]  
MMSE algorithm aims at finding the estimate \(\hat{\mathbf{H_{p}}}\) to minimize the value of mean square, and can be expressed as,  
\[J\left(\hat{\mathbf{H_{p}}}\right)=E\left\{\left\|\mathbf{H_{p}}-\hat{\mathbf{ H_{p}}}\right\|^{2}\right\} \tag{6}\]  
The solution to the MMSE channel estimation is [26; 27],  
\[\hat{\mathbf{H_{MMSE}}}=\  
into four major components. In the zone of modelling parameters, users can set the position and depth of transmitter and receiver, as well as signal frequency, number of beams, and SWH. Figures of normalized CIR and eigenrays trace are presented in the middle. The status indicator displays the consequences of script function in progress. Furthermore, the channel list is used to save or delete modelled channels. The interface of OFDM channel estimation is illustrated in Figure 2b. The parameters, such as modulation type, number of carriers, simulated channel, and algorithm, are freely set in the zone of estimation parameters. Figures of the simulation results are plotted in the lower part and can be saved with specified names if necessary. Figure 3 shows the flow chart of using the simulation platform in this paper. When conducting a simulation, first, users should enter proper channel parameters and click 'Generate Channel' button repeatedly until enough channels are saved in the list. Then, parameters of OFDM channel estimation and UWA channels need to be configured. For deterministic channels, a UWA channel needs to be selected from the channel list.