## 3 LLR Calculation under Phase Noncoherent Channel
Specifically, we consider (a) an AWGN channel with perfect CSI, i.e., perfect knowledge of the variance \(\sigma^{2}\) is available at the receiver, and (b) an AWGN channel without CSI, i.e., no knowledge of the variance \(\sigma^{2}\) is available. In the first case, we will develop the optimal LLR calculation with full complexity. Clearly, in the second case, we will be given a simplified LLR calculation form without CSI. According to (7), the LLR can be expressed as [29]:  
\[ \begin{split}\zeta_{m}&=\ln\,\frac{p(c_{m}{=}0|R)} {p(c_{m}{=}1|R)}=\ln\,\frac{\sum\limits_{S^{*}c_{m}=0}p(R|S)}{\sum\limits_{S^ {*}_{y}c_{m}=1}p(R|S)}\\ &=\ln\,\frac{\sum\limits_{S^{*}c_{m}=0}l_{0}\left(\frac{\rho}{ \sigma^{2}}\sum\limits_{k=1}^{N}\sum_{i=1}^{16}r_{k,i}s_{y,i}^{*}\right| \right)}{\sum\limits_{S^{*}_{y}c_{m}=1}l_{0}\left(\frac{\rho}{\sigma^{2}}\sum \limits_{k=1}^{N}\sum_{i=1}^{16}r_{k,i}s_{y,i}^{*}\right|\right)}\\ &=\ln\,\frac{\sum\limits_{S^{*}_{y}c_{m}=0}l_{0}\left(\frac{\rho} {\sigma^{2}}\left|\left\langle R,S^{*}\right\rangle\right|\right)}{\sum \limits_{S^{*}_{y}c_{m}=1}l_{0}\left(\frac{\rho}{\sigma^{2}}\left|\left\langle R,S^{*}\right\rangle\right|\right)},\,\,m\in\{0,\,1,\,\cdots,4N-1\}\end{split}  \tag{8}\]  
Here, we introduce a method for extracting the bit LLR under the fading channels with perfect CSI, and then soft decision decoding can be used to improve receiver performance. However, there are still shortcomings of this scheme. First, the zero-order Bessel function is involved, and the resource consumption (implementation complexity, storage space, energy consumption, and delay) is relatively large, especially when the observation interval \(N\) is large. Second, in order to obtain the bit LLR information, the receiver needs to accurately estimate the CSI. An inaccurate estimation of the CSI would cause a serious deterioration in the subsequent decoding. That is, robustness to CSI is insufficiency. Therefore, in the following, we further turn our attention towards developing a bit LLR extraction scheme under multiple observation intervals with low complexity and no CSI. First, calculate the decision metric for each symbol:  
\[V_{k,y}=\left|\sum\limits_{i=1}^{16}r_{k,i}s_{y,i}^{*}\right|^{2},\ 1\leq y\leq 16,1 \leq k\leq\ N \tag{9}\]  
Secondly, the maximum and two submaximal metrics for each symbol are recorded as follows [30]:  
\[V_{k,\hat{y}_{1}}=\max\limits_{1\leq y\leq 16}\left\{V_{k,y}\right\},\ 1\leq k\leq\ N \tag{10}\]  
\[V_{k,\hat{y}_{2}}=\max\limits_{1\leq y\leq 16,y\
eq\hat{y}_{1}}\left\{V_{k,y} \right\},\ 1\leq k\leq\ N \tag{11}\]  
\[V_{k,\hat{y}_{3}}=\max\limits_{1\leq y\leq 16,y\
eq\{\hat{y}_{1},\hat{y}_{2} \}}\left\{V_{k,y}\right\},\ 1\leq k\leq\ N \tag{12}\]  
The LLR in (8) is finally simplified as  
\[\zeta_{m}=\ln\frac{\sum\limits_{y_{i}:c_{m}=0,l}I_{0}\left(\frac{\rho}{\sigma ^{2}}\middle|\sum\limits_{k=1}^{N}w_{k,\hat{y}_{i}}\right|\right)}{\sum\limits _{y_{i}:c_{m}=1,l}I_{0}\left(\frac{\rho}{\sigma^{2}}\middle|\sum\limits_{k=1} ^{N}w_{k,\hat{y}_{i}}\right|\right)},\ l\in\{1,\ 2,\ 3\},\ m\in\{0,\ 1,\ \cdots,4N-1\} \tag{13}\]  
where \(w_{k,\hat{y}_{i}}=\sum\limits_{i=1}^{16}r_{k,i}s_{y_{i},i}^{*}\), and \(\hat{y}_{l}\) is given in (10) to (12). As shown in (13), the calculation number for \(I_{0}(x)\) is successfully reduced. However, the implementation for \(I_{0}(x)\) is also complicated for the receiver in WSN. Consequently, we will focus our attention on the simplification of \(I_{0}(x)\). As shown in Figure 3, \(I_{0}(x)\) increases rapidly with the increase in \(x\), and the main influence of \(\sum\limits_{S_{y}}I_{0}\left(\frac{\rho}{\sigma^{2}}\middle|\sum\limits_{k=1} ^{N}w_{k,\hat{y}_{i}}\right|\right)\) can be determined by \(I_{0}\left(\max\limits_{S_{y}}\left[\frac{\rho}{\sigma^{2}}\middle|\sum \limits_{k=1}^{N}w_{k,\hat{y}_{i}}\right|\right]\right)\).