## 5 Simulation and Analysis
Consequently, the compensation made by the existing frequency domain equalization algorithm for the frequency selective fading of the underwater acoustic channel is unsatisfactory at some frequencies especially when the frequency selective fading is deep (see Figure 4). The compensation is made to some extent but not entirely in an acceptable way within the OFDM effective communication frequency bands. When the compensation effect of the equalization algorithm is good, the product of the underwater acoustic channel spectrum and the equalizer spectrum should be 1. According to this standard, Figure 3 shows a good balance. However, in Figure 4, the equalizer does not effectively compensate at the frequency points with large fading of the underwater acoustic channel spectrum, and the compensation ability is insufficient. In the simulation of the proposed algorithm, the length \(\Delta\) of cyclic prefix of the OFDM signals was set to 0 ms. It was intended to fully verify how well the proposed OFDM-based frequency domain equalization algorithm prevented the influence of redundant cyclic prefixes. In this case, the proposed algorithm was explored without any cyclic prefix, while the existing frequency domain equalization algorithm still contained a 5.12 ms cyclic prefix. In the initial adjustment, the step length of the proposed algorithm \(\mu_{0}(s)\) and \(\rho_{0}(t)\) was set to 0.01, and the number of orders for the frequency domain decision feedback equalizers was \(t\_i=4\) and \(t\_b=2\), respectively. Figure 5 presents the convergence curve of the compensation for multipath effect for Channel 2 with such two equalization algorithms. Based on their different structures, the existing frequency domain equalization algorithm and the frequency domain decision feedback equalization algorithm have the error of compensation for multipath effect calculated by Equations (15) and (16), respectively. In the equations, \(H_{i}(k)\) is the frequency spectrum of the underwater acoustic channel on the \(k\)th subcarrier; \(B_{i}(k)\) is the frequency spectrum of the equivalent channel on the \(k\)th subcarrier after being equalized by the existing frequency domain algorithm; \(W_{i}(k)\), \(B_{fi}(s,\,k)\), and \(Wb_{j}(t,k)\) are the frequency spectrum of the equalization filters, respectively, on the \(k\)th subcarrier; \(m_{0}\) is the initial carrier serial number of an OFDM communication signal; \(N\) is the number of subcarriers for the OFDM signal [29,30]. \[E_{tr}=20\lg[\frac{1}{N}\cdot\sum_{k=m_{0}}^{m_{0}+N-1}\left|\frac{H_{i}(k)W _{i}(k)}{B_{i}(k)}-1\right|] \tag{15}\]  
Figure 4: Compensation for Channel 2 with the existing frequency domain equalization algorithm. As shown in Figure 5, for OFDM signals, one iteration is to use one OFDM symbol for equalization processing. The reason the peak value appears near 120 is caused by the inconsistency between the statistical characteristics of training sequence and the statistical characteristics of actual transmission information. Evidently, the proposed frequency domain decision feedback equalization algorithm can better reflect the relationship between symbols since it adopts a multi-symbol periodic joint equalization mechanism. Without a cyclic prefix, it can still make good compensation for the frequency selective fading of multi-path channels. Differently, the existing algorithm uses only the information of the received symbol during each iteration, so that it cannot overcome the inter-symbol interference and inter-subcarrier interference caused by the multipath effect. Therefore, the existing algorithm provides limited compensation for the frequency selective fading of multipath channels (see Figure 4), and its error curve of compensation has poor convergence. Figure 6 shows the communication bit error rate under the same conditions as Figures 4 and 5. Due to the large noise introduced by SNR reduction, the equalization effect will be affected, resulting in the failure of the equalizer, and a significant increase in bit error rate.