# Accurate Performance Analysis of Coded Large-Scale Multiuser MIMO Systems with MMSE Receivers
## 4 Performance Analysis of Coded Multiuser MIMO System with an MMSE Detector
### BER Upper Bound in Both Small- and Large-Scale Fading
 \tag{27}\]  
By substituting \(c_{k}\) with \(\frac{|x_{k}-\hat{x}_{k}|^{2}}{4\mathbb{E}[|x_{k}|^{2}]\sin^{2}\theta}\) inside Equation (27), and combining Equation (23), it holds that  
\[P_{b}\leq\frac{1}{\pi k_{c}}\sum_{d=d_{\text{true}}}^{\infty}A_{d}\int_{0}^{ \frac{\pi}{2}}\left[\frac{A}{|\mathcal{A}|}\sum_{x}\sum_{\hat{x}}\sum_{m=1}^{2 }\sum_{l=1}^{l_{m}}\frac{B_{m,l,k,\delta}}{(\frac{1}{\delta_{m}}+\frac{|x_{k} -\hat{x}_{k}|^{2}}{4\mathbb{E}[|x_{k}|^{2}]\sin^{2}\theta})^{t_{m}-l+1}}\right] ^{d}\mathrm{d}\theta. \tag{28}\]  
It is obvious that Equation (25) is a function in the form of an integral. We note that a simple approximated expression can be obtained for \(Q(x)\) as [18]  
\[Q(x)\simeq\frac{1}{12}e^{-\frac{x^{2}}{2}}+\frac{1}{6}e^{-\frac{2x^{2}}{3}}. \tag{29}\]  
By applying (29) into (24), Equation (24) can be simplified as  
\[ \begin{split}& P\{\mathbf{x}_{k}\rightarrow\hat{\mathbf{x}}_{k}\} \\ &=\mathbb{E}_{\xi_{k,1},\cdots,\xi_{k,l}}\Big{[}\frac{1}{12}\exp \big{(}-\frac{\sum_{l=1}^{d}|\xi_{k,l}(x_{k,l}-\hat{x}_{k,l})|^{2}}{4\mathbb{E} [|\eta_{k}|^{2}]}\big{)}+\frac{1}{6}\exp\big{(}-\frac{\sum_{l=1}^{d}|\xi_{k,l} (x_{k,l}-\hat{x}_{k,l})|^{2}}{3\mathbb{E}[|\eta_{k}|^{2}]}\big{)}\Big{]}\\ &=\frac{1}{12}\prod_{l=1}^{d}\mathbb{E}_{\xi_{k,l}}\exp\big{(}- \frac{|\xi_{k,l}(x_{k,l}-\hat{x}_{k,l})|^{2}}{4\mathbb{E}[|\eta_{k}|^{2}]} \big{)}+\frac{1}{6}\prod_{l=1}^{d}\mathbb{E}_{\xi_{k,l}}\exp\big{(}-\frac{| \xi_{k,l}(x_{k,l}-\hat{x}_{k,l})|^{2}}{3\mathbb{E}[|\eta_{k}|^{2}]}\big{)}\\ &\leq\frac{1}{12}\bigg{[}\mathbb{E}_{\gamma_{k}}\exp\big{(}- \frac{\gamma_{k}|(x_{k}-\hat{x}_{k})|^{2}}{4\mathbb{E}[|x_{k}|^{2}]}\big{)} \bigg{]}^{d}+\frac{1}{6}\bigg{[}\mathbb{E}_{\gamma_{k}}\exp\big{(}-\frac{ \gamma_{k}|(x_{k}-\hat{x}_{k})|^{2}}{3\mathbb{E}[|x_{k}|^{2}]}\big{)}\bigg{]} ^{d}.\end{split}  \tag{30}\]  
For the convenience of calculation, we define \(\frac{|(x_{k}-\hat{x}_{k})|^{2}}{4\mathbb{E}[|x_{k}|^{2}]}\) as \(b_{1}\), and \(\frac{|(x_{k}-\hat{x}_{k})|^{2}}{3\mathbb{E}[|x_{k}|^{2}]}\) as \(b_{2}\). Similar to that in Equation (25), we have  
\[ \begin{split} f(d,\mu,\mathcal{A})&=\mathbb{E}_{ \mathbf{x}_{k},\hat{\mathbf{x}}_{k}}\left[P\{\mathbf{x}_{k}\rightarrow\hat{ \mathbf{x}}_{k}\}\right]\\ &=\frac{1}{12}\bigg{[}\frac{1}{|\mathcal{A}|}\sum_{x_{k}}\sum_{ \hat{x}_{k}}\mathbb{E}_{\gamma_{k}}\left[e^{-\gamma b_{1}}\right]\bigg{]}^{d} +\frac{1}{6}\bigg{[}\frac{1}{|\mathcal{A}|}\sum_{x_{k}}\sum_{\hat{x}_{k}} \mathbb{E}_{\gamma_{k}}\big{[}e^{-\gamma b_{2}}\big{]}\bigg{]}^{d}\\ &=\frac{1}{12}\bigg{[}\frac{A}{|\mathcal{A}|}\sum_{x_{k}}\sum_{ \hat{x}_{k}}\sum_{m=1}^{2}\sum_{l=1}^{l_{m}}\frac{B_{m,l,k,\delta}}{(\frac{1} {\delta_{m}}+b_{1})^{t_{m}-l+1}}\bigg{]}^{d}\\ &+\frac{1}{6}\bigg{[}\frac{A}{|\mathcal{A}|}\sum_{x_{k}}\sum_{ \hat{x}_{k}}\sum_{\hat{x}_{k}}\sum_{m=1}^{2}\sum_{l=1}^{l_{m}}\frac{B_{m,l,k, \delta}}{(\frac{1}{\delta_{m}}+b_{2})^{t_{m}-l+1}}\bigg{]}^{d}.\end{split}  \tag{31}\]Now, by replacing \(b_{1}\) and \(b_{2}\) with \(\frac{|(x_{k}-\delta_{k})|^{2}}{4\mathbb{E}[|x_{k}|^{2}]}\) and \(\frac{|(x_{k}-\delta_{k})|^{2}}{3\mathbb{E}[|x_{k}|^{2}]}\) respectively, and combining Equations (31) and (23), the BER upper bound for the coded systems is given by  
\[ \begin{split} P_{b}\leq\frac{1}{\pi k_{c}}\sum_{d=d_{\text{free}}}^ {\infty}A_{d}&\bigg{\{}\frac{1}{12}\bigg{[}\frac{A}{|\mathcal{A }|}\sum_{x_{k}}\sum_{\delta_{k}}\sum_{m=1}^{2}\sum_{l=1}^{t_{m}}\frac{B_{m,l, \mathcal{L},\mathcal{E}}}{(\frac{1}{\delta_{m}}+\frac{|(x_{k}-\delta_{k})|^{2} }{4\mathbb{E}[|x_{k}|^{2}]})^{t_{m}-l+1}}\bigg{]}^{d}\\ &+\frac{1}{6}\bigg{[}\frac{A}{|\mathcal{A}|}\sum_{x_{k}}\sum_{ \delta_{k}}\sum_{m=1}^{2}\sum_{l=1}^{t_{m}}\frac{B_{m,l,\mathcal{E}}}{(\frac{1 }{\delta_{m}}+\frac{|(x_{k}-\delta_{k})|^{2}}{3\mathbb{E}[|x_{k}|^{2}]})^{t_{m }-l+1}}\bigg{]}^{d}\bigg{\}}.\end{split}  \tag{32}\]