## 3 Theoretical Analysis
### Polar Codes for VLC Channels  
#### 3.1.1 Encoding  
The fundamental principle of channel polarization centers on optimally selecting information bits. While binary erasure channels (BECs) offer relatively straightforward analysis, evaluating capacity for complex non-BECs remains a significant research challenge. VLC channels exemplify this complexity, presenting substantial obstacles in accurately assessing polarization reliability. Our research introduces an innovative approach to evaluating the reliability of the polarization channel, along with a thorough analysis of the relevant polarization phenomenon. In a VLC IM/DD system, the On-Off Keying (OOK) modulation technique enables a probabilistic characterization of channel output. The conditional probability of a channel producing an output \(y\) when the transmitted bit is either 0 or 1 is described by a normal distribution, also known as a Gaussian distribution, and is expressed as follows:  
\[P(y\mid x=0)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\biggl{(}-\frac{y^{2}}{2\sigma ^{2}}\biggr{)}, \tag{4}\]  
\[P(y\mid x=1)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\biggl{(}-\frac{(y-h)^{2}}{2 \sigma^{2}}\biggr{)}. \tag{5}\]  
The Bhattacharyya parameter is a key metric used in the design and analysis of polar codes to evaluate the reliability of a channel. It provides a way to measure how distinguishable the transmitted signal is from the noise and is closely related to the channel error probability [6]. For a binary-input channel \(W(y|x)\) with input \(x\in\{0,1\}\) and output \(y\), the Bhattacharyya parameter \(Z(W)\) is defined as \[Z(W)=\sum_{y\in\mathcal{Y}}\sqrt{W(y|0)W(y|1)} \tag{6}\]  
where \(\mathcal{Y}\) is the set of possible outputs of the channel. The parameter \(Z(W)\) lies in the range \([0,1]\). If \(Z(W)=0\), the channel is perfectly reliable, as the outputs for \(x=0\) and \(x=1\) are completely distinguishable. If \(Z(W)=1\), the channel is completely unreliable, as the outputs for \(x=0\) and \(x=1\) are indistinguishable. By the definition of the Bhattacharyya parameter, its mathematical expression in a VLC channel is given by  
\[ \begin{split} Z(W)&=\int_{-\infty}^{\infty}\sqrt{P (y\mid 0)P(y\mid 1)}dy\\ &=\int_{-\infty}^{\infty}\sqrt{\frac{1}{\sqrt{2\pi\sigma^{2}}} \mathrm{e}^{-\frac{y^{2}}{2\sigma^{2}}}\frac{1}{\sqrt{2\pi\sigma^{2}}} \mathrm{e}^{-\frac{(y-k)^{2}}{2\sigma^{2}}}}dy\\ &=\mathrm{e}^{-\frac{k^{2}}{8\sigma^{2}}}.\end{split}  \tag{7}\]  
In the context of VLC channel modeling, as shown in Equation (1), the log-likelihood ratio (LLR) for signal outputs demonstrates a near-Gaussian probabilistic behavior. The signal transformations of sub-channel \(W_{N}^{(i)}\) can be characterized through recursive computational approaches, where each channel's logarithmic likelihood representation \(L_{N}^{(i)}\) exhibits a distinctive mean value \(m_{N}^{(i)}\). Based on a Gaussian approximation principle, these mean values emerge from the following recursive mathematical formulations [26]:  
\[m_{N}^{(2i)}=2m_{N/2}^{(i)}, \tag{8}\]  
\[m_{N}^{(2i-1)}=\phi^{-1}\bigg{(}1-\Big{(}1-\phi\Big{(}m_{N/2}^{(i)}\Big{)} \Big{)}^{2}\bigg{)}. \tag{9}\]  
The computational complexity of the function \(\phi(x)\) is prohibitively high, making it challenging to solve directly. To address this, researchers have proposed an approximate calculation formula, referred to as Approximate GA [27]. The function \(\phi(x)\) in this context is expressed as follows:  
\[\phi(x)=\left\{ \begin{array}{cc}\exp\big{(}-0.4527x^{0.86}+0.0218\big{)},&0< x<10\\ \\ \sqrt{\frac{\pi}{x}}\Big{(}1-\frac{10}{7x}\Big{)}\exp\big{(}-\frac{\pi}{4} \big{)},&10\leq x.\end{array} \right. \tag{10}\]  
The initial value \(m_{1}^{(1)}\) is determined using Equation (7), while the remaining mean values \(m_{N}^{(i)}\) are computed iteratively based on Equations (8) and (9).