## 4 The Improved Frequency Domain Equalization Algorithm with Reduced Cyclic Prefix  
As discussed in the previous section, the existing frequency domain equalization algorithm is applicable only if the maximum time delay does not exceed the length of the cyclic prefix. It depends on a longer cyclic prefix to \"evade\" the inter-symbol interference and inter-subcarrier interference resulting from multipath effect [28]. Therefore, it is expected to develop a novel frequency domain equalization algorithm in this paper. The novel algorithm will be able to make compensation for frequency selective fading while using fewer redundant cyclic prefixes. In this way, it can \"proactively\" overcome the adverse effect of inter-symbol interference and inter-subcarrier interference. After observing the iteration defined by Equation (8), it was found that the existing frequency domain equalization algorithm used only the received symbol for operation during each iteration, but never made use of the information in the other symbols. The dependence of an OFDM-based communication system on a cyclic prefix would be effectively lowered if the symbols could be taken into account comprehensively during each iteration [29]. Enlightened by this observation, a multi-symbol buffer structure was added together with a feedback mechanism to the existing equalizer. This could make use of the feedback mechanism to resist the strong minimum phase component of a channel. Moreover, the mutual relationship between symbols could be better reflected to reduce the computation of the algorithm and improve its tracking capability. The complete structure of the equalizer as presented in Figure 2 contains a feed-forward equalization filter, a feedback equalization filter and a decision device. The frequency domain value \(Y_{i}\) was obtained after \(V\)-point inverse fast Fourier-transform for the received signal \(y_{i}(n)\) in the \(i\)th symbol period. Then \(Y_{i}\) was sent to the feed-forward equalization filter for processing to get the output \(pf_{i}\). The frequency domain coefficient \(W_{i}\) of the feed-forward equalization filter was \([Wf_{i}(1),Wf_{i}(2)\ldots Wf_{i}(t\_i)]\), and the number of orders in the buffer area was denoted by \(t\_i\). The coefficient at the \(j^{\text{th}}\) order, that is, \(Wf_{i}(j)=[Wf_{i}(j,0),Wf_{i}(j,1)\ldots Wf_{i}(j,V-1)]^{T}\) was a \(V\)-point frequency domain column vector. The frequency domain coefficient of the feedback equalization filter \(Wb_{i}=[Wb_{i}(1),Wb_{i}(2)\ldots Wb_{i}(t\_b)]\) was structurally similar to that of the feed-forward equalization filter. The number of orders was represented by \(t\_b\). The output value of the decision device was \(X_{i}\), and taken as the input of the feedback equalization filter. It had different implications in different phases [30]. When the equalization algorithm was in the training phase, the output value \(X_{i}\) of the decision device represented the frequency domain value of the known training symbol. When the \(i\)th symbol was an effective information symbol, the decision device made a decision on the information of each subcarrier based on the mode of modulation implemented. Subsequently, it used the collected information of 0 and 1 to restore the corresponding frequency domain value of the \(i\)th symbol, which was taken as the output of the decision device \(X_{i}\) in this case. The value was also sent to and transmitted in the feedback equalization filter. The output value of the feedback equalization filter was denoted by \(qb_{i}\). The output value \(pf_{i}\) of the feed-forward equalization filter was combined with the output value of the feedback equalization filter \(qb_{i}\). This gave the estimated value \(X^{\prime}_{i}\) of the complex baseband sub-signal at the receiver. The estimated value \(X^{\prime}_{i}\) was further taken as the input of the decision device. In this way, a closed loop was established for the good tracking and compensation of the underwater acoustic channel [31].