# Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation
## 3 Bussgang Decomposition-Based MPA  
In this section, we elaborate on the Bussgang decomposition-based MPA detector. The MPA detector iteratively exchanges the \(\log(\cdot)\) of the conditional likelihood as messages across the function nodes, indexed as \(j=1,2,\cdots,V\), and the variable nodes, indexed as \(k=1,2,\cdots,J\). Also, for the resulting Tanner graph of the function nodes and variable nodes ([2] Section 12.1.1.3), the graph neighborhood of node \(k\) is denoted as \(\mathcal{B}_{k}\). In this regard, we invoke the Bussgang theorem [10], and re-express (1) as:  
\[\mathbf{y}=\alpha\text{diag}(\mathbf{h})\mathbf{x}+\mathbf{v}+\mathbf{n}, \tag{3}\]  
where \(\alpha\) denotes a correlation-coefficient and \(\mathbf{v}\) denotes an independent distortion term with variance \(\sigma_{v}^{2}\). Using this equivalent form, we obtain the following expression for the conditional PDF, \(p(y[k]|\mathbf{x})\):  
\[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]  
Generally, classical MPA-based detection propagates the \(\log(\cdot)\) of the conditional PDF across the function nodes, \(j\), and variable nodes, \(k\) ([2] p. 377). For AWGN channels, the conditional PDF of \(y[k]\) given \(\mathbf{x}\) is provided below:  
\[\log p(y[k]|\mathbf{x})=-\log\Bigl{(}2\pi\sigma_{n}^{2}\Bigr{)}-\frac{\left|y[ k]-\alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{v}^{2}+ \sigma_{n}^{2}}. \tag{5}\]  
The parameters \(\alpha\) and \(\sigma_{v}^{2}\) are estimated using the available pilots and the channel estimates \(\mathbf{h}\) from (1) as follows:  
\[\alpha=\frac{\mathbb{E}\left[\mathbf{y}^{T}\text{diag}(\mathbf{h}) \mathbf{x}\right]}{\mathbb{E}\left[\left\|\text{diag}(\mathbf{h})\mathbf{x} \right\|^{2}\right]}, \tag{6}\] \[\sigma_{v}^{2}=(1-\alpha)^{2}\mathbb{E}\left[\left\|\text{diag}( \mathbf{h})\mathbf{x}\right\|^{2}\right].\]  
Figure 3: Depiction of the Tanner Graph. For the log-max MPA approaches over AWGN channels, the messages, \(m_{jk}\), are essentially given by the log likelihood \(\log[p(y[k]|\mathbf{x})]\). Considering the Bussgang representation of (1) in (3), \(m_{jk}\) is explicitly written as:  
\[m_{jk}=\frac{-\left|y[k]-\alpha h[k]\sum\limits_{\forall j\in\mathcal{B}_{k}}x[ k]\right|^{2}}{\sigma_{n}^{2}+\sigma_{v}^{2}}. \tag{7}\]  
The difference between the value of this message and its corresponding ideal value is expressed as follows:  
\[\Delta m_{jk}=\frac{\left|y[k]-h[k]\sum\limits_{\forall j\in\mathcal{B}_{k}}x[ k]\right|^{2}\left(\sigma_{n}^{2}+\sigma_{v}^{2}\right)-\left|y[k]-\alpha h[k] \sum\limits_{\forall j\in\mathcal{B}_{k}}x[k]\right|^{2}\sigma_{n}^{2}}{ \sigma_{n}^{2}(\sigma_{n}^{2}+\sigma_{v}^{2})}. \tag{8}\]  
If the appropriate expression for the Kullback-Leibler divergence between Gaussian PDFs having zero mean and variances \(\sigma_{n}^{2}\) and \(\sigma_{n}^{2}+\sigma_{d}^{2}\) is invoked, the difference between \(m_{jk}\) and its corresponding ideal value, \(\mathbb{E}\left[\Delta m_{jk}\right]\) (with \(\alpha=1\) and \(\sigma_{v}^{2}=0\)), is given by [19]:  
\[\mathbb{E}\Big{[}\Delta m_{jk}\Big{]}=\frac{1}{2}\log\Bigg{[}\frac{\sigma_{n} ^{2}+(1-\alpha)^{2}\sigma_{h}^{2}\sigma_{x}^{2}+\sigma_{n}^{2}}{\sigma_{n}^{ 2}}\Bigg{]}+\frac{\sigma_{n}^{2}}{2\Big{(}\sigma_{n}^{2}+(1-\alpha)^{2}\sigma _{h}^{2}\sigma_{x}^{2}\Big{)}}-\frac{1}{2}, \tag{9}\]  
where  
\[\sigma_{h}^{2}=\mathbb{E}\Big{[}h^{2}[k]\Big{]}, \tag{10}\] \[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]0  
Next, we directly link the converged log likelihood ratio for the ideal linear channel to the generalized signal-to-noise ratio (GSNR) [20], ref.