# Accurate Performance Analysis of Coded Large-Scale Multiuser MIMO Systems with MMSE Receivers
## 3 Distribution of the SINR
Since all the non-zero eigenvalues have the same edge distribution, \(\mathcal{G}\) is equivalent to the mean of \(\frac{1}{\lambda+\rho}\), where the subscripts of the eigenvalues is emitted. Therefore, we can also use the statistics of \(\lambda\) to obtain \(\mathcal{G}\). In theory, for matrix \(\mathbf{H}_{k}\mathbf{D}_{kk}\mathbf{H}_{k}^{H}\) with arbitrary dimensions, we can use edge distribution (9) of the eigenvalues or statistics to obtain \(\mathcal{G}\). However, because the edge distribution function is relatively complicated, the parameter \(\mathcal{G}\) becomes more and more difficult to obtain when the size of the matrix becomes large. In order to solve this problem, we first introduce an important concept from [23], that is, for an \(n\times n\) Hermitian matrix \(\mathbf{A}\) with empirical spectral/eigenvalue distribution \(\mathbf{F}_{n}\), which \(\eta\)-transform is defined as  
\[L=\prod_{l=1}^{K-1}\frac{1}{\xi_{l}^{R}(R-l)!}\prod_{k<l}^{K-1}\frac{\xi_{k}\xi_ {l}}{\xi_{k}-\xi_{l}}. \tag{8}\]1  
where \(\varphi\) is a nonnegative real number. In addition, for the high-dimensional diagonal matrix \(\mathbf{D}_{kk}\), we can consider replacing the diagonal elements with the first moment of the empirical spectral distribution [20] (\[L=\prod_{l=1}^{K-1}\frac{1}{\xi_{l}^{R}(R-l)!}\prod_{k<l}^{K-1}\frac{\xi_{k}\xi_ {l}}{\xi_{k}-\xi_{l}}. \tag{8}\]23.2) of \(\mathbf{D}_{kk}\) to reduce the computational complexity. Obviously, \(\mathbf{H}_{k}\) and \(\mathbf{D}_{kk}\) are independent. In contrast to _Theorem_ 2.39 in [23], when \(K-1\) satisfies \((K-1)/R\to c>0\) as \(R\rightarrow\infty\), then, almost surely, the empirical spectral distribution of \(\mathbf{H}_{k}\mathbf{D}_{kk}\mathbf{H}_{k}^{H}\) converges to a distribution whose \(\eta\)-transform can be written as  
\[L=\prod_{l=1}^{K-1}\frac{1}{\xi_{l}^{R}(R-l)!}\prod_{k<l}^{K-1}\frac{\xi_{k}\xi_ {l}}{\xi_{k}-\xi_{l}}. \tag{8}\]3  
where  
\[L=\prod_{l=1}^{K-1}\frac{1}{\xi_{l}^{R}(R-l)!}\prod_{k<l}^{K-1}\frac{\xi_{k}\xi_ {l}}{\xi_{k}-\xi_{l}}. \tag{8}\]4  
and \(\bar{\xi}=\sum_{i=1,j\
eq k}^{K}\zeta_{i}\). Replacing the parameter \(\varphi\) in Equation (14) with \(1/\rho\), we can easily get  
\[L=\prod_{l=1}^{K-1}\frac{1}{\xi_{l}^{R}(R-l)!}\prod_{k<l}^{K-1}\frac{\xi_{k}\xi_ {l}}{\xi_{k}-\xi_{l}}. \tag{8}\]5  
Since the meaning of the parameter is the same, for the convenience of future use, here, we still let \(\mathcal{G}\) represent the expectation values in Equation (11). Up to now, we have derived parameters \(\mathcal{G}\) that are suitable for traditional multiuser MIMO systems and large-scale multiuser MIMO systems, respectively. Let \(\mathbf{\beta}=[\beta_{1},\cdots,\beta_{R}]^{T}\), where \(\beta_{r}\) is the coefficient of \(|\bar{k}_{r}|^{2}\) in Equation (11). Note that two of the \(R\) entries are distinct in \(\mathbf{\beta}\). Let's divide the elements \(\beta_{1},\cdots,\beta_{R}\) into two groups; each group is the collection of the entries equal to the same value, respectively. Moreover, let \(\mathbf{t}=[t_{1},t_{2}]^{T}\), whose entries are the number of elements in each group, i.e., \(t_{1}=K-1,t_{2}=R-K+1\). Similar to that in [24], we define the following parameters:  
\[L=\prod_{l=1}^{K-1}\frac{1}{\xi_{l}^{R}(R-l)!}\prod_{k<l}^{K-1}\frac{\xi_{k}\xi_ {l}}{\xi_{k}-\xi_{l}}. \tag{8}\]6  
\[B_{m,l,r,\delta}=(-1)^{l+1}\sum_{\mathbf{i}\in\Omega_{m,l}}\prod_{j=1,j\
eq m }^{2}\left( \begin{array}{c}i_{j}+t_{j}-1\\ i_{j}\end{array} \right)\left(\frac{1}{\delta_{j}}-\frac{1}{\delta_{m}}\right) ^{-(t_{j}+i_{j})}, \tag{18}\]  
where the values of \(\delta_{1}\) and \(\delta_{2}\) are equal to \(\zeta_{k}\mathcal{G}\) and \(\zeta_{k}/\rho\), respectively. The vector \(\mathbf{i}=[i_{1},i_{2}]^{T}\) is extracted from the set \(\Omega_{m,l}\) of all nonnegative integer partitions of \(l-1\) (with \(i_{m}=0\)).