## 4 NN-Based Joint Equalization and Detection  
Unlike the frequency domain equalizers mentioned above, this paper combines equalization and symbol detection and implements a joint detector with the NN. The detected symbol can be written as  
\[S_{d}(n)=\underset{D\big{[}\hat{H}(n),R(n)\big{]}}{\arg\min}L\big{\{}D\big{[} \hat{H}(n),R(n)\big{]},S(n)\big{\}}, \tag{12}\]  
where \(L\{\cdot\}\) is the loss function and \(D[\cdot]\) represents the process of the proposed network. It has been proved that a simple network structure is enough to well solve the receiver problems [33; 42]. Inspired by this, the proposed joint detector utilizes a single-layered network. As shown in Figure 4, after channel estimation, \(R(n)\) and \(H(n)\) are sent to the network for detection. Moreover, a block-input strategy is proposed to further decrease the complexity of the network. \(N_{b}\) is the number of blocks and \(N_{\text{L}}\) denotes the number of neurons. The structure configurations of the network are described as follows. **Input:** Before input to the network, the received OFDM symbol \(R(n)\) with \(N\) subcarriers is firstly divided into \(N_{b}\) blocks. Each block contains \(N_{\text{c}}\) subcarriers. The same process is conducted with the corresponding estimated channel function \(\hat{H}(n)\). Thus, a small reusable network can be designed for each combination of the data and channel block. To further determine the optimal hyperparameters, a proper \(N_{\text{c}}\) should be defined to balance the computation and accuracy of the network, i.e., two rules are proposed:  
1. \(N_{\text{c}}\) should optimize the performance of the joint detector with estimated channel. 2. The required number of neurons is positively correlated to the input dimension, and \(N_{\text{c}}\) should be as small as possible to minimize the computation of the network. Based on the analysis above, the narrow coherence bandwidth \(W_{\text{c}}\) of the channel is considered, which always limits the performance of the underwater OFDM system. Because the subcarriers in each coherence frequency band suffer from relatively flat fading, an opportunity is found to find the optimal \(N_{\text{c}}\)[43]. By setting \(N_{\text{c}}\) in the range of coherence bandwidth, the joint detection network is able to deal with the symbol blocks separately  
Figure 4: Block diagram of the proposed scheme on joint equalization and detection based on the NN. The detection process integrates the equalization and symbol demapping, which is completed by a low-complexity network, taking the segmented subcarriers as input. in a flat fading channel. Figure 5a shows the frequency response of a simple channel with one path. \(A_{1}\) to \(D_{3}\) are values of the frequency points. According to [43], for a channel that follows the homogeneous assumption, the coherence bandwidth is inversely proportional to the maximum channel delay \(T_{\text{d}}\), which can be approximately represented as  
\[W_{\text{c}}\approx\frac{1}{T_{\text{d}}}. \tag{13}\]  
For the channel in Figure 5a, \(W_{\text{c}}\) is easily observed as the frequency range between \(B_{2}\) and \(D_{3}\), written as  
\[W_{\text{c}}=f_{D_{3}}-f_{B_{2}}. \tag{14}\]  
In practice, the UWA channel contains more paths as shown in Figure 5b, whose frequency response is the sum of more than one paths with time-varying coefficients distributions [39]. With the same estimated \(W_{\text{c}}\) as Figure 5a, different situations are listed in Table 1 by taking different sections as blocks. It should be noted that the starting point of the first block should always be \(A_{1}\), which is also the first subcarrier of an OFDM symbol. When the width of the block \(W_{b}=W_{\text{c}}\), block \([A_{1},C_{2}]\) suffers from selective fading because the section includes inflection points of the frequency response. The same result occurs when \(W_{b}=W_{\text{c}}/2\) because \([B_{1},B_{3}]\) includes an inflection point. When \(W_{b}=W_{\text{c}}/3\), no sections include an inflection point or experience flat fading.