# Sparse Multiuser Receiver Design in Large Scale Array System
## 3 Analog Matrix Designing Scheme  
For massive arrays, the sharp increase in the array elements results in extraordinarily narrow and high-gain beams, which can accurately aim at the target user and improve the SNR of the received signal. However, with the beams of highly high directivity, the power of signals transmitted by users beyond the beams will deteriorate to a great extent. This characteristic plays an active role in inter-users interference suppression for the traditional multi-user system. However, for a scenario with a large number of users, a few narrow beams can only cover some users, which may bring about a loss of information.  
Therefore, based on the traditional DFT matrix and random phase matrix, this paper develops an analog matrix applicable for multi-user scenarios [15]. Regarded as an MMF problem, the beam synthesizer designing problem can be written as follows:  
\[\begin{split}&\max_{G_{A}}\min_{k}\frac{\left|G_{A}h_{k}\right|^{ 2}}{\left|G_{A}w\right|^{2}},\\ & s.t.\left|G_{A}(n,m)\right|=1.\end{split} \tag{3}\]  
Given that the \(M\) is far greater than \(N\), and \(w\) is statistically independent and obeys circular Gaussian distribution, \(\left|G_{A}w\right|^{2}\) in the denominator can be given by  
\[\left|G_{A}w\right|^{2}=\ MN\sigma_{n}^{2}. \tag{4}\]  
Figure 1: Large-scale array system based on a hybrid structure serving multiple users simultaneously (the BS adopts \(N\) radio frequency links to serve \(K\) users).  
Hence, the original problem can be converted into  
\[\begin{split}\underset{G_{A}}{\text{max}}\underset{k}{\text{min}}|G_{ A}h_{k}|^{2},\\ s.t.\ |G_{A}(n,m)|=1.\end{split} \tag{5}\]  
Based on the vectorization method, we simplify (5) into  
\[\begin{split}|G_{A}h_{k}|^{2}=\left|\left(h_{k}^{T}\otimes I \right)g\right|^{2}=g^{H}\bigg{(}\bigg{(}h_{k}^{T}\otimes I\bigg{)}^{H}\Big{(} h_{k}^{T}\otimes I\bigg{)}\bigg{)}g=g^{H}\Sigma_{k}g,\end{split} \tag{6}\]  
where \(g=vec(G_{A})\) represents the vectorized form of matrix \(G_{A}\). In addition, \(\Sigma_{k}=\left(h_{k}^{T}\otimes I\right)^{H}\Big{(}h_{k}^{T}\otimes I\Big{)} =\left(\Big{(}h_{k}h_{k}^{H}\Big{)}\otimes I\right)\) represents the channel covariance matrix. Then, we can build the simplified problem as  
\[\begin{split}\underset{g}{\text{max}}\underset{k}{\text{min}}g^{ H}\Sigma_{k}g\\ s.t.\ |g_{i}|=1.\end{split} \tag{7}\]  
Due to the constant modulus constraints in (7), it is a non-convex problem that is difficult to address by general convex optimization methods.  
Thus, we construct a new objective function as follows:  
\[F_{p}(g)=-\frac{1}{p}\text{log}\Bigg{(}\sum_{k}\text{exp}\Big{(}-g^{H}\Sigma_{ k}g\Big{)}\Bigg{)}. \tag{8}\]  
The original objective function can be further transformed into the following convex function:  
\[\begin{split}\underset{g}{\text{max}}-\frac{1}{P}\text{log} \Bigg{(}\sum_{k}\text{exp}\Big{(}-g^{H}\Sigma_{k}g\Big{)}\Bigg{)}\\ s.t.\ |g_{i}|=1.\end{split} \tag{9}\]  
Formula (9) can be handled by the gradient descent method. Specifically, the gradient is expressed as  
\[\
abla_{g}F_{p}(g_{n})=2\sum\frac{\text{exp}\big{(}-g_{n}^{H}\Sigma_{k}g_{n} \big{)}}{\sum\text{exp}(-g^{H}\Sigma_{k}g)}\Sigma_{k}g_{n}. \tag{10}\]  
The gradient update rule is given by  
\[\widetilde{g}_{n+1}=g_{n}+\frac{1}{L}\
abla_{g}F_{p}(g_{n}). \tag{11}\]  
Since the shifter in this system has constant modulus characteristics, it is necessary to normalize the updated shifter coefficients as  
\[g_{n+1}=\frac{\widetilde{g}_{n+1}}{\left|\widetilde{g}_{n+1}\right|}. \tag{12}\]