# Accurate Performance Analysis of Coded Large-Scale Multiuser MIMO Systems with MMSE Receivers
## 2 System Model  
Considering the uplink of a multiuser massive MIMO system, the system consisting of one BS and \(K\) single-antenna users. The BS are equipped with \(R\) antennas, i.e., \(R>K\). We assume that single antenna users are independent of each other. The \(K\) transmit symbols of the users can be organized into a \(K\times 1\) vector \(\mathbf{x}=[x_{1},\cdots,x_{K}]\), which is normalized such that \(\mathbb{E}[\mathbf{x}\mathbf{x}^{H}]=\mathbf{I}\). For a given time instant, the received \(R\times 1\) signal vector at BS reads  
\[\mathbf{y}=\sqrt{\frac{E_{s}}{K}}\mathbf{G}\mathbf{x}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{G}=\mathbf{H}\mathbf{D}^{\frac{1}{2}}\) represents the \(R\times K\) propagation matrix from the BS to the \(K\) users. The elements \(h_{r,k}\sim\mathcal{CN}(0,1)\), \(r=1,2,\cdots,R\) and \(k=1,2,\cdots,K\), of the \(R\times K\) channel matrix \(\mathbf{H}\) denotes uncorrelated ideal fast fading coefficients from the BS to the \(K\) users. \(K\)-dimensional diagonal matrix \(\mathbf{D}^{\frac{1}{2}}=\text{diag}\{\sqrt{\zeta_{1}},\sqrt{\zeta_{2}}, \cdots,\sqrt{\zeta_{K}}\}\) models the geometric attenuation and shadow fading. Furthermore, \(\sqrt{\zeta_{k}}\) is independent of parameter \(r\) and changes slowly over time, which can be assumed to be a constant [10, 17]. Moreover, the average transmit power per user is \(\frac{E_{s}}{K}\), and \(\mathbf{n}\sim\mathcal{CN}(\mathbf{0},N_{0}\mathbf{I})\), is the \(R\times 1\) additive white Gaussian noise. Then, the average received SNR is given by  
\[\mathrm{SNR}=\frac{E_{s}}{K}\frac{\mathbb{E}[||\mathbf{H}\mathbf{D}^{\frac{1}{ 2}}\mathbf{x}||]^{2}}{\mathbb{E}[||\mathbf{n}||^{2}]}=\frac{E_{s}\mathrm{tr}( \mathbf{P})}{KN_{0}}. \tag{2}\]  
The choice of MMSE filtering matrix is given by the following well-known formula  
\[\mathbf{W}=\left(\mathbf{G}^{H}\mathbf{G}+\rho\mathbf{I}\right)^{-1}\mathbf{G }^{H}, \tag{3}\]  
where \(\rho=\frac{KN_{0}}{E_{s}}\).  
Analogous to [18] (Equation (10)), with the MMSE detector, the SINR for the \(k\)-th data stream is expressed as  
\[\begin{array}{c}\gamma_{k}=\zeta_{k}\mathbf{h}_{k}^{H}(\sum\limits_{r=1,r \
eq k}^{K}\zeta_{r}\mathbf{h}_{r}\mathbf{h}_{r}^{H}+\rho\mathbf{I}_{R})^{-1} \mathbf{h}_{k}\\ \stackrel{{(a)}}{{=}}\zeta_{k}\mathbf{\tilde{h}}_{k}^{H}\mathbf{ \Lambda\tilde{h}}_{k},\end{array} \tag{4}\]  
where (a) of Equation (4) is obtained by the eigenvalue decomposition of the matrix \(\sum_{r=1,r\
eq k}^{K}\zeta_{r}\mathbf{h}_{r}\mathbf{h}_{r}^{H}\). \(\mathbf{h}_{k}\) is the \(k\)-th column of \(\mathbf{H}\), \(\mathbf{\tilde{h}}_{k}=[\tilde{h}_{1,k},\ldots,\tilde{h}_{R,k}]^{T}\) has the same distribution as \(\mathbf{h}_{k}\), and  
\[\mathbf{\Lambda}=\mathrm{diag}\big{(}\frac{1}{\lambda_{k,1}+\rho},\ldots, \frac{1}{\lambda_{k,K-1}+\rho},\frac{1}{\rho},\ldots,\frac{1}{\rho}\big{)}. \tag{5}\]  
Moreover, \(\lambda_{k,1},\cdots,\lambda_{k,K-1}\) are the \(K-1\) non-zero eigenvalues of the matrix \(\sum_{r=1,r\
eq k}^{K}\zeta_{r}\mathbf{h}_{r}\mathbf{h}_{r}^{H}\). Therefore, \(\mathrm{SINR}_{i}\) can be represented as  
\[\gamma_{k}=\zeta_{k}\sum_{i=1}^{K-1}\frac{|\tilde{h}_{i,k}|^{2}}{\lambda_{k,i }+\rho}+\zeta_{k}\sum_{j=K}^{R}\frac{|\tilde{h}_{j,k}|^{2}}{\rho}. \tag{6}\]