# A Nonbinary LDPC-Coded Probabilistic Shaping Scheme for a Rayleigh Fading Channel
## 2 Methods
\begin{array}{c}X_{I}\\ X_{Q}\end{array} \right]=\left[ \begin{array}{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array} \right]\left[ \begin{array}{c}S_{I}\\ S_{Q}\end{array} \right], \tag{4}\]  
where \(\theta\) is the rotated angle. \(X_{I}\) and \(X_{Q}\) denote the in-phase signal of \(X\) and the quadrature signal of \(X\), respectively. In order to improve the reliability performance in fading channels, a component interleaver is used, which can make in-phase and quadrature signals go through different fading. Then, the \(X_{int}\) signals can be obtained after the component interleaver. Assume that the receiver can get the perfect channel state information; the received signals can be expressed as  
\[Y_{int}=H_{int}X_{int}+N, \tag{5}\]  
where \(N\) is the complex Gaussian noise with zero mean and variance \(\sigma^{2}\). \(H_{int}\) denotes the channel coefficient with zero mean and unit variance. In the receiver, after passing through the phase equalizer, the received signal is converted into  
\[\frac{H_{\text{int}}^{\dagger}}{\|H_{\text{int}}\|}Y_{\text{int}}=\|H_{\text{ int}}\|X_{\text{int}}+\frac{H_{\text{int}}^{\dagger}}{\|H_{\text{int}}\|}N. \tag{6}\]  
After the component deinterleaver, the signal \(Y=[Y_{I},Y_{Q}]^{T}\) can be obtained. After maximum a posteriori (MAP) detection, we can get NB LDPC symbol-level log-likelihood ratios. Assume that the label \(B=B_{1}B_{2}\cdots B_{p}=\text{label}(x_{i})\), which means that one \(GF(2^{p})\) LDPC code corresponds to one QAM symbol. The received noisy symbol \(y_{i}\) is demodulated by calculating \(l_{i,j}\),  
\[l_{i,j}=\log\frac{P_{B_{j}}(0)}{P_{B_{j}}(1)}+\log\frac{q_{i,j}(y_{i}|0)}{q_{i, j}(y_{i}|1)}, \tag{7}\]  
Figure 1: The system model of the proposed NB LDPC-coded probabilistic shaping scheme for Rayleigh fading channel. \[P_{B_{j}}(b)=\sum_{a\in\{0,1\}^{p^{\prime}}:a_{j}=b}P_{B}(a), \tag{8}\]  
\[q_{i,j}(y|b)=\sum_{a\in\{0,1\}^{p^{\prime}}:a_{j}=b}q_{c}(y_{i}|x_{a})\frac{P_{B} (a)}{P_{B_{j}}(b)}, \tag{9}\]  
where \(i=1,2,\cdots,n_{c},j=1,2,\cdots,p\). \(q_{c}(\cdot|\cdot)\) is a conditional probability density function. \(x_{a}=\left\{x_{a_{1}},x_{a_{Q}}\right\}\), \(x_{a_{1}}\) and \(x_{a_{Q}}\) denote the real part of \(x_{a}\) and the image part of \(x_{a}\), respectively. \[q_{c}(y|x_{a})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\text{exp}\,\frac{-(y_{l}-|h_{l} |\cdot x_{a_{l}})^{2}-(y_{Q}-|h_{Q}|\cdot x_{a_{Q}})^{2}}{2\sigma^{2}}. \tag{10}\]  
The demodulated bit probability is  
\[P(y_{i,j}=0)=\tfrac{1}{1+\exp(-l_{ij})},P(y_{i,j}=1)=\tfrac{\exp(-l_{ij})}{1+ \exp(-l_{ij})}, \tag{11}\]  
for \(i=1,2\cdots n_{c},j=1,2,\cdots p\). The GF\((2^{p})\) LDPC decoder input is calculated as  
\[P_{i}(c)=\frac{\tilde{P_{i}}(c)}{\sum_{c^{\prime}\in\text{GF}(2^{p})}\tilde{P_ {i}}(c^{\prime})},\tilde{P_{i}}(c)=\prod_{j=1}^{p}\tilde{P_{i,j}}, \tag{12}\]  
for \(i=1,2\cdots n_{c},j=1,2,\cdots p\), where  
\[\tilde{P}_{i,j}=\left\{ \begin{array}{ll}P(y_{i,j}=0),&if[\phi_{\text{GF}(2^ {p})}^{-1}(c)]_{j}=0\\ P(y_{i,j}=1),&if[\phi_{\text{GF}(2^{p})}^{-1}(c)]_{j}=1\end{array} \right., \tag{13}\]  
\([\phi_{\text{GF}(2^{p})}^{-1}(c)]_{j}\) is the \(j\)-th binary mapping bit of \(c\) for \(c\in\)GF\((2^{p})\). According to [15], the decoding complexity comparison of per bit per iteration for the Log-FFT-BP decoding algorithm of NB LDPC codes and the Log-BP algorithm of binary LDPC codes are shown in Table 1, where \(d_{v}\) denotes the average column weight. The regular GF(256) LDPC codes with \(d_{v}=2\) and regular GF(2) LDPC codes with \(d_{v}=3\) are compared, which shows that the decoding complexity comparison of per bit per iteration for the GF(256) LDPC codes is about 144.7 times as much as that of GF(2) LDPC codes. The NB LDPC decoding symbols sequence \(\hat{u}^{A}\), \(\hat{u}^{5}\) are then mapped via P/S converter. After \(\chi_{A}^{-1}(\cdot)\) amplitude inverse labeling function and inverse DM operation, we can finally get the estimated data bits \(\{\hat{u}_{\hat{b}\hat{b}\hat{b}\hat{b}}^{A}\}\).