# 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  
In this letter, we consider that the transmission data of the \(k\)-th user is encoded using a linear convolutional code with a rate of \(R_{c}=\frac{k_{c}}{n_{c}}\). Supposing that the interleaver is ideal, then the correlated bit metrics are sufficiently separated in the decoder. Through the appropriate subscript combinations, we let \(\mathbf{x}_{k}=[x_{k,1},\cdots,x_{k,d}]^{T}\) and \(\hat{\mathbf{x}}_{k}=[\hat{x}_{k,1},\cdots,\hat{x}_{k,d}]^{T}\), where \(d\) is Hamming distance of the code and \(\hat{x}_{k,l}\) is the nearest constellation point to \(x_{k,l}\). According to [25], the upper-bound for convolutionally encoded massive MIMO-MMSE systems can be evaluated using  
\[P_{b}\leq\frac{1}{k_{c}}\sum_{d=d_{\text{free}}}^{\infty}A_{d}f(d,\mu,\mathcal{ A}), \tag{23}\]  
where \(A_{d}\) is the total information weight of all error events at Hamming distance \(d\) and \(d_{\text{free}}\) is the free Hamming distance of the code. \(f(d,\mu,\mathcal{A})\) is the Average codeword Pairwise Error Probability (APEP) that is generated by error events at Hamming distance \(d\). Similar to the analysis in [25], the pairwise error probability of two codewords can be computed as follows:  
\[ \begin{split} P\{\mathbf{x}_{k}\rightarrow\hat{\mathbf{x}}_{k}\}& =\mathbb{E}_{\xi_{k,1},\cdots,\hat{\xi}_{k,d}}\left[Q\left(\sqrt{ \frac{\sum_{l=1}^{d}|\xi_{k,l}(x_{k,l}-\hat{x}_{k,l})|^{2}}{2\mathbb{E}[|\eta_{ k}|^{2}]}}\right)\right]\\ &=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\prod_{l=1}^{d}\underbrace {\mathbb{E}_{\xi_{k,l}}\left[\exp\left(-\frac{|\xi_{k,l}(x_{k,l}-\hat{x}_{k,l})| ^{2}}{4\mathbb{E}[|\eta_{k}|^{2}\sin^{2}\theta}\right)\right]}_{\Delta_{\hat{x} _{l}}(\theta)}\right]}_{\Delta_{\hat{x}_{l}}(\theta)}\mathrm{d}\theta,\end{split}  \tag{24}\]  
where the linear expectation is over all pairs of signal symbols \(\mathbf{x}_{k}\) and \(\hat{\mathbf{x}}_{k}\) whose labels differ in only one bit. Consequently, the result of the \(f(d,\mu,\mathcal{A})\) is given by  
\[ \begin{split} f(d,\mu,\mathcal{A})&=\mathbb{E}_{ \mathbf{x}_{k},\hat{\mathbf{x}}_{k}}\left[P\{\mathbf{x}_{k}\rightarrow\hat{ \mathbf{x}}_{k}\}\right]=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\mathbb{E}_{ \mathbf{x}_{k},\hat{\mathbf{x}}_{k}}\left[\prod_{l=1}^{d}\Delta_{k,l}(\theta )\right]\mathrm{d}\theta\\ &\leq\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\left\{\mathbb{E}_{ \lambda_{U},\hat{\lambda}_{U}}\left[\Delta_{k,l}(\theta)\right]\right\}^{d} \mathrm{d}\theta\\ &\overset{(a)}{=}\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\left\{ \frac{1}{|\mathcal{A}|}\sum_{\hat{x}_{k}}\sum_{\hat{x}_{k}}\Delta_{k}(\theta) \right\}^{d}\mathrm{d}\theta.\end{split}  \tag{25}\]In (a) of Equation (25), the subscript \(l\) is omitted. The \(\Delta_{k}(\theta)\), as shown in Equation (24), can be written as  
\[\Delta_{k}(\theta)=\mathbb{E}_{\gamma_{k}}\left[\exp\left(-\frac{\gamma_{k}|x_{k }-\hat{x}_{k}|^{2}}{4\mathbb{E}[|x_{k}|^{2}]\sin^{2}\theta}\right)\right]. \tag{26}\]  
For convenience, let the normalized distance between symbols be \(c_{k}=\frac{|x_{k}-\hat{x}_{k}|^{2}}{4\mathbb{E}[|x_{k}|^{2}]\sin^{2}\theta}\). Then, using (20), the form of the expectation operation in Equation (26) be changed as  
\[ \begin{split}\mathbb{E}_{\gamma_{k}}\left[e^{-c_{k}\gamma_{k}} \right]&=\int_{0}^{\infty}e^{-c_{k}x}f_{\gamma_{k}}(x)\mathrm{d}x \\ &=A\sum_{m=1}^{2}\sum_{l=1}^{l_{m}}\frac{B_{m,l,k,\delta}}{(t_{m}- l)!}\int_{0}^{\infty}x^{t_{m}-l}e^{-(\delta_{m}+c_{k})x}\mathrm{d}x\\ &=A\sum_{m=1}^{2}\sum_{l=1}^{l_{m}}\frac{B_{m,l,k,\delta}}{( \frac{1}{\delta_{m}}+c_{k})^{t_{m}-l+1}}.\end{split}