# Efficient On-Off Keying Underwater Acoustic Communication for Seafloor Observation Networks
## 3 Receiver Design Adapted to Channel  
### Iterative Threshold Estimation (ITE) Algorithm  
Based on the Max-Log-APP algorithm, the a posteriori symbol LLR of the soft-input soft-output (SISO) ACC-OOK demapper can be expressed as  
\[L\left(u_{i}\right)=L_{\mathrm{c}}\left(u_{i}\right)+L_{\mathrm{a}}\left(u_{i} \right)+L_{\mathrm{e}}\left(u_{i}\right),1\leq i\leq K \tag{4}\]  
where \(L_{\mathrm{c}}\left(u_{i}\right)\) is the channel observation LLR, \(L_{\mathrm{a}}\left(u_{i}\right)\) is the a priori LLR, \(L_{\mathrm{e}}\left(u_{i}\right)\) is the extrinsic LLR from the decoder. The \(K\) denotes the length of an encoded sequence. The \(L_{\mathrm{c}}\left(u_{i}\right)\) is computed as  
\[ \begin{split} L_{\mathrm{c}}\left(u_{i}\right)&= \ln\frac{p\left(Y_{i}|u_{i}=1\right)}{p\left(Y_{i}|u_{i}=0\right)}\\ &=\ln\left[\frac{N_{0}}{N_{0}+E_{\mathrm{c}}}\mathrm{exp}\left( \frac{Y_{i}^{2}}{N_{0}}-\frac{Y_{i}^{2}}{N_{0}+E_{\mathrm{c}}}\right)\right] \\ &=\ln\frac{N_{0}}{N_{0}+E_{\mathrm{c}}}+\frac{E_{\mathrm{c}}Y_{i} ^{2}}{N_{0}\left(N_{0}+E_{\mathrm{c}}\right)}\\ &=A\left(Y_{i}^{2}-B\right)\end{split}  \tag{5}\]  
where \(A=E_{\mathrm{c}}/[N_{0}(E_{\mathrm{c}}+N_{0})]\) and \(B=-\ln\frac{N_{0}}{N_{0}+E_{\mathrm{c}}}/A\) is an LLR bias. Due to the unknown SNR, the \(L_{\mathrm{c}}(u_{i})\) cannot be directly computed based on Equation (5). To solve this dilemma, the ITE algorithm is proposed to determine \(B\). At the initial turbo-iteration, the channel observation LLR \(L_{\mathrm{c}}^{0}\left(u_{i}\right)\) is set as  
\[L_{\mathrm{c}}^{0}\left(u_{i}\right)=Y_{i}^{2}-B^{0} \tag{6}\]  
Figure 2: Capacity of four noncoherent modulations under flat Rayleigh fading channel without CSI on the receiver side. where the \(B^{0}\) is chosen as the median of \(Y_{i}^{2}\), i.e., \(B^{0}=\text{median}\{Y_{i}^{2},1\leq i\leq K\}\). For the \(n\)-th turbo-iteration of the decoding, the channel observation LLR \(L_{\text{c}}^{n}\left(u_{i}\right)\) is computed as  
\[L_{\text{c}}^{n}\left(u_{i}\right)=L_{\text{c}}^{n-1}\left(u_{i}\right)-B^{n}. \tag{7}\]  
where \(B^{n}\) is obtained by minimizing the hamming distance \(D_{H}\) between the biased \(\{L_{\text{c}}^{n}\left(u_{i}\right)\}_{\text{DE}}\) and \(\{L_{\text{ACC}}^{n-1}\left(u_{i}\right)\}_{\text{DE}}\). The \(L_{\text{ACC}}^{n-1}\left(u_{i}\right)\) is the output LLR of the ACC-OOK demapper from the previous iteration and \(\{\cdot\}_{DE}\) represents the decision operation. Mathematically, \(B^{n}\) can be obtained as  
\[B^{n}=\arg\max_{B_{j}^{n}}\sum_{i=1}^{K}\text{sgn}\left[L_{\text{ACC}}^{n-1} \left(u_{i}\right)\left(L_{\text{c}}^{n-1}\left(u_{i}\right)-B_{j}^{n}\right)\right] \tag{8}\]  
The search scope of \(B^{n}\) depends on \(\{L_{\text{c}}^{n-1}\left(u_{i}\right),\,1\leq i\leq K\}\) as  
\[B_{1}^{n},B_{j}^{n},...,B_{K}^{n}\in\left[\min\{L_{\text{c}}^{n-1}\left(u_{i} \right)\},\max\{L_{\text{c}}^{n-1}\left(u_{i}\right)\}\right]\]  
Intuitively, we can find an optimal \(B^{n}\) from the whole search scope by sampling at equally intervals. For each sample \(B_{j}^{n}\), the calculation of \(\sum_{i=1}^{K}\text{sgn}\left[L_{\text{ACC}}^{n-1}\left(u_{i}\right)\left(L_ {\text{c}}^{n-1}\left(u_{i}\right)-B_{j}^{n}\right)\right]\) is performed once. However, it is inefficient and has a total complexity of \(\mathcal{O}(NMK)\) with \(M\) being the number of samples (\(M\leq K\)). Instead, a low-complexity search algorithm is used. First, \(\{L_{\text{c}}^{n-1}\left(u_{i}\right)\}\) is sorted into the ascending order yielding the sorted sequence \(\{L_{\text{c}}^{n-1}\left(u_{i}\right)^{\prime}\}\) which corresponds to a permutation sequence \(P\). Based on \(P\), the sequence \(\{L_{\text{ACC}}^{n-1}\left(u_{i}\right)\}\) is also permutated into \(\{L_{\text{ACC}}^{n-1}\left(u_{i}\right)^{\prime}\}\). Then, an appropriate \(B^{n}\) making the \(D_{H}\) minimum can be directly chosen from the \(\{L_{\text{c}}^{n-1}\left(u_{i}\right)^{\prime}\}\). Details of the search algorithm are summarized in Algorithm 1.