## 3 UWA Receiver Structure
### Signal Detection in UWA Environments  
For the received signal described in Equation (1), using the minimum squared Euclidian distance is considered to detect the above symbol, the optimization problem can be expressed as  
\[\hat{S}(n)=\operatorname*{arg\,min}_{S^{m}(n),m\in[1,M]}\|R(n)-H(n)S^{m}(n)\|^{2}, \tag{5}\]  
where \(\|\cdot\|\) represents 2-norm. \(H(n)\) is the channel frequency response in any one OFDM symbol. \(M\) is the modulation order, and \(S^{m}(n)\) is the \(m\)-th referred symbol. Further, the squared Euclidian distance can be written as  
\[E_{S^{k}}^{2}=\left\{ \begin{array}{cc}\|(H(n)-1)S^{m}(n)+Z(n)\|^{2}&k=m\\ \left\|H(n)S^{k}(n)-S^{m}(n)+Z(n)\right\|^{2}&k\
eq m\end{array} \right.. \tag{6}\]  
Considering a system utilizing frequency domain equalization (FDE) for better performance, for the signal in Equation (6), a multiplier \(G(n)\) is included in the detector, yielding  
\[E_{S^{k}}^{2}=\left\{ \begin{array}{cc}\|(H(n)G(n)-1)S^{m}(n)+Z(n)G(n)\|^{2}&k =m\\ \left\|H(n)G(n)S^{k}(n)-S^{m}(n)+Z(n)G(n)\right\|^{2}&k\
eq m\end{array} \right.. \tag{7}\]  
Considering \(G(n)\) as the entry of multiplier matrix \(\mathbf{G}\), for ZF equalizer,  
\[\mathbf{G}=\left(\mathbf{\hat{H}^{H}\hat{H}}\right)^{-1}\mathbf{\hat{H}^{H}}, \tag{8}\]  
where \(\mathbf{\hat{H}}\) is the estimated channel matrix. Multiplying \(\mathbf{R}\) in Equation (2) with \(\mathbf{G}\), and expending the equation, the equalized symbol can be expressed as  
\[\left[ \begin{array}{c}\hat{s}_{1}\\ \hat{s}_{2}\\ \vdots\\ \hat{s}_{N}\end{array} \right]=\left[ \begin{array}{cccc}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{array} \right]\left[ \begin{array}{c}s_{1}\\ s_{2}\\ \vdots\\ s_{N}\end{array} \right]+\left[ \begin{array}{cccc}g_{11}&g_{12}&\cdots&g_{1N }\\ g_{21}&g_{22}&\cdots&g_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ g_{N1}&g_{N2}&\cdots&g_{NN}\end{array} \right]\left[ \begin{array}{c}z_{1}\\ z_{2}\\ z_{N}\end{array} \right], \tag{9}\]  
where \(g_{nn}\) is the element of \(\mathbf{G}\). It can be seen in Equation (10) that the equalized symbol includes amplified noise which will influence the detection. In the noiseless channel, the second term is zero. Figure 2: Time and frequency response of a typical UWA channel: (**a**) is the CIR of the channel, which contains many strongly reflected paths with large delays. Its frequency response in (**b**) shows non-negligible frequency selectivity. For the MMSE equalizer, there is  
\[\mathbf{G}=\mathbf{\hat{H}}^{\mathrm{H}}\bigg{(}\mathbf{\hat{H}}\mathbf{\hat{H}}^ {\mathrm{H}}+\frac{\sigma^{2}}{P}\mathbf{I}\bigg{)}^{-1}, \tag{10}\]  
where \(\sigma^{2}\) and \(P\) are powers of noise and signal, respectively, and \(\mathbf{I}\) is the identity matrix. With Equation (11), the MMSE-equalized symbol can be expressed as  
\[\left[ \begin{array}{c}\hat{s}_{1}\\ \hat{s}_{2}\\ \vdots\\ \hat{s}_{N}\end{array} \right]=\left[ \begin{array}{cccc}\frac{1}{|H_{11}|+ \frac{\sigma^{2}}{\sigma^{2}}}&0&\cdots&0\\ 0&\frac{1}{|H_{22}|+\frac{\sigma^{2}}{\sigma^{2}}}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\frac{1}{|H_{\mathrm{NN}}|+\frac{\sigma^{2}}{\sigma^{2}}}\end{array}  \right]\left[ \begin{array}{c}s_{1}\\ s_{2}\\ \vdots\\ s_{N}\end{array} \right]+\left[ \begin{array}{cccc}g_{11}&g_{12}&\cdots&g_{1N }\\ g_{21}&g_{22}&\cdots&g_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ g_{N1}&g_{N2}&\cdots&g_{NN}\end{array} \right]\left[ \begin{array}{c}n_{1}\\ n_{2}\\ \vdots\\ n_{N}\end{array} \right]. \tag{11}\]  
The first term of Equation (11) includes a factor matrix that only contains positive coefficients. As Figure 3 shows, for a single received point in the decision regions, the elements of this factor matrix linearly scale the received point to draw it closer to the reference point, which does not change its quadrant. Consequently, for the low-level constellations that can decide the symbols according to the quadrants they lie, such as BPSK and QPSK, the MMSE does not perform better than the ZF equalizer.