# Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation
## 3 Bussgang Decomposition-Based MPA
[21] achieved at convergence, SNR\({}^{*}\), which is in turn a function of \(\psi_{p(h)}\) (the PDF of the channel gain) [20]:  
\[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]1  
From the expression for the message error derived in (9), the BER of the proposed Bussgang detector, \(\text{BER}_{\text{Bussgang}}\), is approximately expressed as:  
\[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]2  
where the \(\mathbb{E}\Big{[}\Delta m_{jk}\Big{]}\) is derived in (9). The following insights are drawn from the above analytical result:  
* Notably, (12) quantifies the gap between the BER of the proposed approach and that of a universally optimal MPA (the RFF-based MPA in [12]). As mentioned before, this quantification helps when trading off computational complexity with BER performance subject to achieving a given BER-based level of QoS. * It is further noted that the above deviation is independent of the fading distribution. In this context, it is indeed worth mentioning that the ideal BER, \(\psi_{p(h)}(\text{SNR}^{*})\), is mostly an integral of a Q-function over the concerned PDF \(p(h)\)[2]. However, when \(\psi_{p(h)}(\text{SNR}^{*})\) (and hence its derivative \(\psi_{p(h)}^{\prime}\)) are known, the optimality gap is found to be independent of the underlying distribution. * It is possible to further improve the error approximation in (12) as follows: \[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]3 (13) where \(\psi_{p(h)}^{(l)}(\cdot)\) represents the \(l^{th}\) derivative of \(\psi_{p(h)}(\cdot)\). To simplify, we note from (7) that \(\mathcal{P},\mathcal{Q}\sim\text{Exp}\big{[}\sigma_{n}^{2}(\sigma_{n}^{2}+ \sigma_{n}^{2})\big{]}\) are even powers of normal random variables with average energy \(\sigma_{n}^{2}\big{(}\sigma_{n}^{2}+\sigma_{n}^{2}\big{)}\). Therefore, we obtain the following for \(\mathbb{E}\Big{[}\Delta m_{jk}^{l}\Big{]}\): \[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]4 (14) From ([22] p. 546), this is simplified as: \[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]5 (15) which yields the final expression: \[p(y[k]|\mathbf{x})=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]- \alpha h[k]\sum\limits_{\forall j\in B_{k}}x[k]\right|^{2}}{\sigma_{n}^{2}+ \sigma_{v}^{2}}\right]. \tag{4}\]6 (16) A summary of the proposed Bussgang-based MPA is provided in Algorithm 1. ```
1:Initialization: \(I_{kj}=p\big{(}\mathbf{x}_{j}\big{)}\) according to a uniform distribution. 2:Initialization: \(I_{jk}:=\frac{1}{2\pi\sigma_{n}^{2}}\exp\left[-\frac{\left|y[k]-ah[k]\sum\limits _{y\in I_{jk}}x[k]\right|^{2}}{\sigma_{n}^{2}+\sigma_{n}^{2}}\right]\) \(\alpha=\frac{\mathbb{E}\big{[}\mathbf{y}^{T}\text{diag}(\mathbf{h})\mathbf{x} \big{]}}{\mathbb{E}\big{[}\big{\|}\text{diag}(\mathbf{h})\mathbf{x}\big{\|}^ {2}},\) \(\sigma_{n}^{2}=(1-\alpha)^{2}\mathbb{E}\Big{[}\big{\|}\text{diag}(\mathbf{h}) \mathbf{x}\big{\|}^{2}\Big{]}.\)
3:Initialize the maximum number of iterations, ITER. 4:while\(c<\) ITER do \(I_{jk}:=\log\big{(}p\big{(}\mathbf{x}_{j}\big{)}\big{)}+\sum\limits_{j\in \mathcal{B}_{k}}I_{kj}\).