## 3 The Existing OFDM-Based Frequency Domain Equalization Algorithm  
Based on Equation (5), if \(P_{2}=0\), that is, the length of the cyclic prefix is greater than the maximum multipath time delay, the received signals are only affected by frequency selective fading, which causes the distortion of signals at the subcarrier frequency. In a severe case, the information cannot be correctly restored, undermining the symbol error ratio. The existing equalization technique can be used to make satisfactory compensation for frequency selective fading [23]. The complex baseband sub-symbol of the periodic signal for the \(m\)th symbol of OFDM is assumed to be \(X_{m}=[X_{m}(0),\cdots X_{m}(V-1)]^{T}\). Based on Equation (5), if \(P_{2}=0\), the received periodic signal \(y_{m}(n)\) of the \(m\)th symbol is fast Fourier-transformed to obtain the restored information \(Y_{m}=[Y_{m}(0),Y_{m}(1)\cdots Y_{m}(V-1)]^{T}\) as follows:  
\[Y_{m}=HX_{m}+N \tag{6}\]  
where \(H\) stands for the frequency spectrum of the impulse response \(h(n)\) of the underwater acoustic channel, and \(N\) is the frequency spectrum of the ambient noise in the ocean. The multiplication between matrixes in Equation (6) means the multiplication of the corresponding elements in these matrixes. The existing OFDM-based frequency domain adaptive equalization algorithm is normally implemented in two phases. The first phase is training. In this initial phase, a known training sequence \(x_{i}(n)\) with the length equivalent to \(C_{t}\) symbol periods is employed to iteratively adjust the equalization filter coefficient \(w(n)\). After such adjustment, the equivalent channel impulse response \(h(n)^{\star}\)\(w(n)\) is approximate to the ideal channel impulse response \(b(n)\). Thus, the compensation for multipath effect is achieved for the underwater acoustic channel [24]. With the purpose in the initial phase, the receiving frequency domain value \(Y_{i}\) of each symbol is used to obtain the following error function \(E_{i}\) when the period training sequence of the \(i^{\text{th}}\) symbol is reached:\[E_{i}=(B_{i}-W_{i}H)X_{i}=B_{i}X_{i}-W_{i}Y_{i}\quad i=1,2\ldots C_{t} \tag{7}\]  
where \(W_{i}\) and \(B_{i}\) are the frequency spectrum of the equalization filter coefficient \(w(n)\) and the frequency spectrum of the equalized equivalent impulse response \(b(n)\), respectively, when the period training sequence of the \(i\)th symbol is reached. Following the least mean square (LMS) principles, the optimal \(W_{i}\) and \(B_{i}\) are searched to minimize \(\|E_{i}\|^{2}\). The iteration of \(W_{i}\) and \(B_{i}\) is defined by Equation (8), where \(u_{w}\) and \(u_{b}\) are the step length factor for adjusting \(W_{i}\) and \(B_{i}\), respectively. \[\left\{ \begin{array}{l}W_{i}=W_{i-1}+2u_{w}E_{i}Y_{i}^{*}\\ B_{i}=B_{i-1}-2u_{b}E_{i}X_{i}^{*}\end{array} \right. \begin{array}{l}i=1,2 \ldots C_{t}\end{array}  \tag{8}\]  
After the iterative adjustment of the training sequence with \(C_{t}\) symbol periods, \(W_{i}\) and \(B_{i}\) are updated to better overcome the frequency selective fading caused by the multipath effect of the underwater acoustic channel [25]. At this time, the second phase of the frequency domain equalization algorithm begins, that is, restoring the effective information sequence at the receiver. The frequency domain value \(Y_{i}\) of the effective information sequence \(y_{i}(n)\) for the \(i\)th (\(i>C_{t}\)) symbol period is determined, and then adjusted by Equation (9) with \(W_{i}\) and \(B_{i}\) obtained in the training phase. After that, the restored signal frequency domain value \(X^{\prime}_{i}\) is obtained, and subsequently decided according to the mode of modulation. The complex baseband sub-signal \(X^{\prime}_{i}\) is then obtained. The restoration of information is completed. In the phase of information restoration, the \(X^{\prime}_{i}\) obtained from the decision can be also utilized to iterate the equalizer coefficient, so as to maintain the minor compensation for the time-varying underwater acoustic channel.