## 3 Iterative Multistep mDDP-Based ICI Canceling UWA OFDM Communication System
1 is performed sequentially, but since hard-determined data can be used, the CTF can be obtained with higher accuracy  
Figure 6: Flowchart of iterative multistep modified delay and Doppler profiler. and the four parameters for each path, the relative delay time \(\tau_{p}\), attenuation \(r_{p}\), Doppler-shift \(\alpha_{p}\), and sampling clock error \(\beta_{p}\), can be obtained. The details of the iterative multistep mDDP are explained below using equations. The FFT output of the OFDM demodulation generates the data symbol \(\hat{d}(k,l)\) as expressed in Equation (1), which is referred from the original mDDP in paper [18]'s Equation (6) and the mDDP and DDP in papers [17; 19; 20; 21]. \[\hat{d}(k,l)=h(k,l,l)d(k,l)+\sum_{ \begin{subarray}{c}n=0\\ n\
eq l\end{subarray} }^{N-1}h(k,l,n)d(k,n)+w(k,l), \tag{1}\]  
where \(d(k,n)\) is the sending data symbol of the subcarrier index \(n\) at the \(k\)th OFDM symbol, \(w(k,l)\) is additive noise that corresponds to the \(l\)th subcarrier in the \(k\)th OFDM symbol, and \(h(k,l,n)\) is the transfer function of the \(k\)th OFDM symbol with a sending subcarrier index \(n\) to receiving subcarrier \(l\). \(N\) is the number of subcarriers. The magnitude of the \(k\)th OFDM \(h(k,l,n)\) is at a maximum at \(n=l\) and is getting smaller as \(|n-l|\) is getting larger. Then, the range of the sum in Equation (1) can be decreased from \(l-NumCOL/2\) to \(l+NumCOL/2\) for approximation, where \(NumCOL\) is the total number of sums. Assuming \(NumCOL=3\), Equation (1) can be shown as Equation (2) in matrix form. Non-diagonal components in the \(h(k,*,*)\) matrix correspond to ICI. \[ \begin{bmatrix}\hat{d}(k,l-1)\\ \hat{d}(k,l)\\ \hat{d}(k,l+1)\end{bmatrix} = \begin{bmatrix}h(k,l-1,l-1)&h(k,l,l-1)&h(k,l+1,l- 1)\\ h(k,l-1,l)&h(k,l,l)&h(k,l+1,l)\\ h(k,l-1,l+1)&h(k,l,l+1)&h(k,l+1,l+1)\end{bmatrix}\begin{bmatrix}d(k,l-1)\\ d(k,l)\\ d(k,l+1)\end{bmatrix} + \begin{bmatrix}w(k,l-1)\\ w(k,l)\\ w(k,l+1)\end{bmatrix}  \tag{2}\]  
By neglecting the additive noise component, sending data symbol \(d(k,n)\) can be estimated from the receiver FFT outputs \(\hat{d}(k,l-1)\) to \(\hat{d}(k,l+1)\), as shown in Equation (3). \[d(k,l)\cong \begin{bmatrix}0&1&0\end{bmatrix}\begin{bmatrix}h(k,l-1,l-1)&h(k,l,l-1)&h(k,l+1,l-1)\\ h(k,l-1,l)&h(k,l,l)&h(k,l+1,l)\\ h(k,l-1,l+1)&h(k,l,l+1)&h(k,l+1,l+1)\end{bmatrix} ^{-1} \begin{bmatrix}\hat{d}(k,l -1)\\ \hat{d}(k,l)\\ \hat{d}(k,l+1)\end{bmatrix} , \tag{3}\]  
Equation (3) computation is implemented in a multitap ICI cancel equalizer in Figure 5. Once all \(h(k,l,n)\) values are available, ICI cancelling is possible. \(h(k,l,n)\) can be expressed as Equations (4)-(6) according to the original mDDP in paper [18]. \[h(k,l,n)=\sum_{p=1}^{Np}h_{p}(k,l,n) \tag{4}\]  
\[h_{p}(k,l,n)=\frac{sinc(A)}{sinc(\frac{A}{N})}\cdot e^{i\frac{n(N-1)A}{N}} \cdot r_{p}\cdot e^{-i\frac{2n(n+q)r_{p}^{\prime}}{N}}\cdot e^{i\frac{2n(N+C )(n+\beta_{p})}{N}}, \tag{5}\]  
\[A = n-l+n\beta_{p}+\alpha_{p}\big{(}1+\beta_{p}\big{)} \tag{6}\] \[\cong n-l+n\beta_{p}+\alpha_{p},\]  
where \(\alpha_{p}=\Delta f_{p}/f_{0}\) represents the normalized frequency offset of the \(p\)th path, \(\beta_{p}\) represents the sampling clock error of the \(p\)th path and \(\tau_{p}^{\prime}\) denotes relative delay time in sampling points. Here, mDDP is applied repeatedly, but in the first mDDP, only the CTF of the SP position within one symbol is available for ICI cancellation, but in the second and subsequent mDDPs, the path information obtained in the first mDDP and the hard-decision results of the data carriers are used, and ICI cancellation can be performed. We defined the first mDDP as mode 1 and the second and subsequent mDDPs as mode 2. From Equation (1), the inputs such as \(h(k,l,l)\) and \(h(k-2,l,l)\) of mDDP mode 1 in Figure 6 can be approximated as Equations (7) and (8), where the subcarrier index of \(l\) is only the SP positions.