# Sparse Multiuser Receiver Design in Large Scale Array System
## 4 BPSK Demodulator Based on GAMP Method  
For traditional hybrid structure systems, each RF chain corresponds to one user. Once the number of users exceeds the number of RF chains, the extra users will be unable to communicate with the BS. Section 3 presents an analog matrix designing method aiming to support simultaneous service for multiple users by limited RF chains. In this section, we propose a technique that enables the BS equipped with limited RF chains to receive BPSK data transmitted by multiple users. The received signal is expressed as  
\[y=G_{A}Hx+G_{A}w=Bz+n=q+n, \tag{13}\]  
where \(B=G_{A}H\), \(z=x\sim N(0,\mathbf{\Sigma}_{z})\), \(n=G_{A}w\sim N(0,\mathbf{\Sigma}_{N})\).  
Therefore, for the traditional Bayesian estimation method, the optimal estimation of \(z\), namely, \(\hat{z}\), can be obtained as follows:  
\[\hat{z}=Ty=T(Bz+n), \tag{14}\]  
where \(T\) represents the estimation matrix. Further, the optimization problem is given by  
\[\text{min}_{T}f(T)=E\Big{[}|T(Bz+n)-z|_{F}^{2}\Big{]}. \tag{15}\]  
Subsequently, we take the derivative of the objective function, and then set the derivative to 0, as shown in (16).  
\[\frac{\partial f(T)}{\partial T}=0. \tag{16}\]  
It can be obtained that the optimal form of the estimation matrix \(T\) is  
\[T=\Big{(}\mathbf{\Sigma}_{z}H^{H}G_{A}^{H}\Big{)}\Big{(}G_{A}H\mathbf{\Sigma}_{Z}H^{H} G_{A}^{H}+G_{A}G_{A}^{H}\Big{)}^{-1}. \tag{17}\]  
Thus, we have  
\[\hat{z}=Ty. \tag{18}\]  
\(G_{A}H\) is not of full column rank, thus \(\hat{z}\) cannot be obtained through the optimal Bayesian estimation. Further, we shift to use the prior modulation information to solve this problem. Since BPSK modulation is adopted, the distribution of \(z\) can be written as follows:  
\[p(z)=\lambda\times\delta(z+1)+(1-\lambda)\times\delta(z-1). \tag{19}\]  
In the GAMP method, to simplify the calculation, we use the central limit theorem to approximate the Gaussian distribution of the intermediate variables \(z\) and \(q\) on the premise of large signal dimensions [16; 17; 18]. Thus, the conditional posterior distribution of \(z\) and \(q\) is represented by  
\[\begin{split} P_{Z|S}(z|s;\hat{r},\mu^{r})&=\frac{P_ {Z}(z)\times N(z;\hat{r},\mu^{r})}{\int_{z^{\prime}}P_{Z}\big{(}z^{\prime} \big{)}\times N\big{(}z^{\prime};\hat{r},\mu^{r}\big{)}}\\ &=\frac{[\lambda\delta(z+1)\times N(-1;\hat{r},\mu^{r})+(1- \lambda)\delta(z-1)N(1;\hat{r},\mu^{r})]}{\lambda N(-1;\hat{r},\mu^{r})+(1- \lambda)N(1;\hat{r},\mu^{r})}.\end{split} \tag{20}\]  
\[\begin{split} P_{Q|S}(q|s;\hat{q},\mu^{q})&=\frac{P_ {S|Q}(s|q)\times N(q;\hat{q},\mu^{q})}{\int_{q^{\prime}}P_{S|Q}\big{(}s|q^{ \prime}\big{)}\times N\big{(}q^{\prime};\hat{q},\mu^{q}\big{)}}=\frac{1}{C}N \Big{(}s;q,\sigma_{n}^{2}\Big{)}\times N(q;\hat{q},\mu^{q})\\ &=N\bigg{(}q,\frac{s}{\sigma_{n}^{2}}+\frac{\hat{q}}{\mu^{q}}, \frac{\mu^{q}\sigma_{n}^{2}}{\mu^{q}+\sigma_{n}^{2}}\bigg{)}.\end{split} \tag{21}\]  
Because of the Gaussian approximation, we acquire the expectation and variance of \(z\) subject to the conditional posterior distribution, as shown in (22) and (23).  
\[\begin{split} E[Z|s;\hat{r},\mu^{r}]&=\int_{z}zP_ {Z|Y}(z|s;\hat{r},\mu^{r})\\ &=\frac{[-\lambda\times N(-1;\hat{r},\mu^{r})+(1-\lambda)N(1; \hat{r},\mu^{r})]}{\lambda N(-1;\hat{r},\mu^{r})+(1-\lambda)N(1;\hat{r},\mu^{ r})}.\end{split} \tag{22}\]  
\[\hat{z}_{n}(t+1)=g_{in}(\hat{r}_{n}(t),\mu_{n}^{r}(t)). \tag{40}\]  
For \(z\) to be estimated, the GAMP algorithm will produce a series of estimated results \(\hat{z}_{n}(t)\) along with their variance \(\mu_{n}^{z}(t)\). The final \(\hat{z}_{n}(t)\) is the modulation data of the corresponding user.