## 3 LLR Calculation under Phase Noncoherent Channel  
Assume that the fading phase is not known at the receiver, and the statistical average of this random phase is considered to eliminate the impact on our detection. Before launching the LLR calculation for decoding, it is instructive to point out the detection when no coding is considered. In this context, only the likelihood probability is required where the \(i\)th received chip sample of the \(k\)th symbol period is given by  
\[r_{k,i}=\rho s_{y,i}e^{i\theta}+\eta_{k,i}. \tag{2}\]  
Then, the likelihood probability in \(N\) symbol periods can be expressed as [28]:  
\[p(R|S,\theta) = \prod_{k=1}^{N}\prod_{i=1}^{16}p\big{(}r_{k,i}|s_{y,i},\theta\big{)} = \frac{1}{\left(\sqrt{2\pi}\sigma\right)^{16N}}\exp\left(-\frac{\parallel R- \rho Se^{i\theta}\parallel^{2}}{2\sigma^{2}}\right),\ 1\leq y\leq 16 \tag{3}\]  
where  
\[\parallel R-\rho Se^{i\theta}\parallel^{2}=\sum_{k=1}^{N}\sum_{i=1}^{16}\left| r_{k,i}-\rho s_{y,i}e^{i\theta}\right|^{2} \tag{4}\]  
After a simple analysis, we can express (4) in a more detailed form as follows [28]:  
\[ \begin{split}\parallel R-\rho Se^{i\theta}\parallel^{2}& =\parallel R\parallel^{2}+\rho^{2}\parallel S\parallel^{2}-2\rho \text{Re}\Big{\{}R^{T}S^{*}\Big{\}}e^{i(\theta-\beta)}\\ &=\sum_{k=1}^{N}\sum_{i=1}^{16}\left[\left|r_{k,i}\right|^{2}+ \rho^{2}\Big{|}s_{y,i}\right|^{2}\right]-2\rho\text{Re}\Bigg{\{}\sum_{k=1}^{N }\sum_{i=1}^{16}r_{k,i}s_{y,i}^{*}\Bigg{\}}\cos\theta\\ &-2\rho\text{Im}\Bigg{\{}\sum_{k=1}^{N}\sum_{i=1}^{16}r_{k,i}s_ {y,i}^{*}\Bigg{\}}\sin\theta\\ &=\sum_{k=1}^{N}\sum_{i=1}^{16}\left[\left|r_{k,i}\right|^{2}+ \rho^{2}\Big{|}s_{y,i}\right|^{2}\right]-2\rho\Bigg{|}\sum_{k=1}^{N}\sum_{i=1}^ {16}r_{k,i}s_{y,i}^{*}\Bigg{|}\cos(\theta-\beta)\end{split}  \tag{5}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Spreading Sequence \(s_{y}\)** & \(b_{m},0\leq m\leq 3\) & **Chip Phases for \(s_{y}\)=\((s_{y,1},s_{y,2},\ldots,s_{y,15},s_{y,16})\)** \\ \hline \(\mathbf{s}_{1}\) & 0000 & \(0,\frac{\pi}{16},\frac{\pi}{4},\frac{\pi}{16},\frac{\pi}{16},\frac{\pi}{16}, \frac{\pi}{16},-\frac{15\pi}{4},\frac{\pi}{16},0,-\frac{15\pi}{16},\frac{\pi}{4},-\frac{7\pi}{16},\frac{\pi}{16},\frac{\pi}{4},\frac{where  
\[\beta=\tan^{-1}\frac{\text{Im}\{R^{T}S^{*}\}}{\text{Re}\{R^{T}S^{*}\}} \tag{6}\]  
As \(\theta\) is assumed to be uniformly distributed, then the probability density function of \(R\) given the transmitted symbol sequence \(S\) can be given as  
\[ \begin{split} p(R|S)&=\int_{-\pi}^{\pi}p(R|S, \theta)\,p(\theta)d\theta\\ &=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^{16N}}\exp\left[-\frac{ 1}{2\sigma^{2}}\sum_{k=1}^{N}\sum_{i=1}^{16}\left[\left|r_{kj}\right|^{2}+\rho^ {2}\right|s_{yj}\right|^{2}\right]\right]l_{0}\left(\frac{\rho}{\sigma^{2}} \left|\sum_{k=1}^{N}\frac{16}{r_{k,i}s_{y,i}^{*}}\right|\right)\\ &=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^{16N}}\exp\left[-\frac{ \left\langle R,R^{*}\right\rangle}{2\sigma^{2}}\right]\exp\left[-\frac{\left \langle S,S^{*}\right\rangle}{2\sigma^{2}}\rho^{2}\right]l_{0}\left(\frac{ \left|\left\langle R,S^{*}\right\rangle\right|}{\sigma^{2}}\rho\right)\\ &\sim\,l_{0}\left(\frac{\left|\left\langle R,S^{*}\right\rangle \right|}{\sigma^{2}}\rho\right)\\ &\sim\left|\left\langle R,S^{*}\right\rangle\right|\end{split}  \tag{7}\]  
where \(l_{0}(\cdot)\) is the zeroth order modified Bessel function of the first kind, \(*\) represents the conjugation operation, and \(\left\langle\cdot,\cdot\right\rangle\) denotes the correlation operation. As shown in (7), a decision metric in conjugate correlation form is achieved when no coding is considered. The amplitude information for the fading channel is not required at the receiver, although we assume that perfect acquisition of this instantaneous CSI is available. Furthermore, the CSI for the AWGN channel can also be avoided. This follows from the fact that no contribution from the fade amplitude \(\rho\) and the variance \(\sigma^{2}\) is observed in the decision metric given in (7). Next, we will focus on the LLR calculation when coding is involved.