## 3 BER Calculation in Frequency Selective Fading MIMO Orientations  
Each diagonal element of \(\mathbf{P}_{s,k}\) can be equivalently written using the Equation below. \[\gamma_{k,d}^{2}=\left(\sum\limits_{l=1}^{L}\sum\limits_{q=1}^{N_{r}}\zeta_{k, d,l,q}\right)^{2}P_{k,d} \tag{8}\]  
where  
\[\zeta_{k,d,l,q}=\left|\sum\limits_{p=1}^{N_{t}}H_{k,l,q,p}\mathbf{t}_{k,p,d} \right|^{2} \tag{9}\]  
In Equation (9), \(P_{k,d}\) is the power allocated to the \(d^{\mathrm{th}}\) transmission mode. In the remainder of this article, the \(i^{\mathrm{th}}\) entry of matrix \(\mathbf{X}\) will be denoted as \(X_{i}\). In this context, in Equations (8) and (9), \(H_{k,l,q,p}\) represents the channel coefficient from the \(v^{\mathrm{th}}\) transmit antenna (\(1\leq v\leq N_{t}\)) to the \(q^{\mathrm{th}}\) receiver antenna (\(1\leq q\leq M_{r}\)) for the \(l^{\mathrm{th}}\) multipath and \(t_{k,p,d}\) the (\(v\),\(d\)) element of the precoding matrix \(\mathbf{t}_{k}\). The summation of a large number of random variables (RVs) is normally distributed, where mean and standard deviation values are given by the equation below. \[ \begin{array}{c}\mu_{o,d}=\sum\limits_{l=1}^{L}\sum\limits_{q=1}^{N_{t}}E \left[\zeta_{k,d,l,q}\right]\\ \sigma_{o,d}=\sqrt{\sum\limits_{l=1}^{L}\sum\limits_{q=1}^{N_{r}}\sum\limits_{ l^{\prime}=1}^{L}\sum\limits_{q^{\prime}=1}^{N_{r}}E\left(\left(\zeta_{k,d,l,q}- \left\langle\zeta_{k,d,l,q}\right\rangle\right)\left(\zeta_{k,d,l^{\prime},q^{ \prime}}-\left\langle\zeta_{k,d,l^{\prime},q^{\prime}}\right\rangle\right) \right)}\end{array}  \tag{10}\]  
according to the Central Limit Theorem (CLT) [19]. In Equation (10), \(\left\langle\zeta_{k,d,l,q}\right\rangle\) represents the mean value of \(\zeta_{k,d,l,q}\). At this point, it is reasonable to assume that all \(\zeta_{k,d,l,q}\) variables have similar probability density function (pdf) curves with the same mean (\(\mu_{d}\)) and standard deviation (\(\sigma_{d}\)). Under this assumption, Equation (10) can be written by using the formula below. \[\mu_{o,d}=LN_{r}\mu_{d},\sigma_{o,d}=\sigma_{d}\left\{\sqrt{LN_{r}\left(1+ \rho_{o,d}(LN_{r}-1)\right)}\right\} \tag{11}\]  
where \(\rho_{o,d}\) is the correlation coefficient for an arbitrary pair \(\left(\zeta_{k,d,l,q},\zeta_{k,d,l^{\prime},q^{\prime}}\right)\). Moreover, MAI from the \(k^{\prime\mathrm{th}}\) user can be expressed by Equation (12) below. \[ \begin{array}{c}MAI_{k^{\prime},d}=\sum\limits_{l=1}^{L}\sum\limits_{q=1}^{N_{ t}}\left[\sum\limits_{v=1}^{N_{t}}h_{k,l,q,p}t_{k,p,d}\right]\sum\limits_{p^{ \prime}=1}^{L}\sum\limits_{q^{\prime}=1}^{dim}\left[\sum\limits_{v=1}^{N_{t}} h_{k,l^{\prime},q,v}h_{v^{\prime},p,d^{\prime}}\right]b_{o,k^{\prime},q^{\prime}} \sqrt{P_{k^{\prime},d}}\\ b_{o,k,d^{\prime}}=\rho_{k,k,l,q^{\prime}}\left[b_{o,k^{\prime},-1}+\overline{ \rho}_{k,k,l,-p}\left|b_{k,d^{\prime},0}\right.\right.\end{array}  \tag{12}\]where, as before the real and complex part of each, the \(MAI_{k^{\prime},d}\) variable is normally distributed, according to CLT. The equivalent SINR for the \(k^{\text{th}}\) user and \(d^{\text{th}}\) transmission mode will be given by Equation (13) below (\(Im(\text{x})\) is the imaginary part of x). \[ \begin{split} SINR_{k,d}\approx\frac{\left(\sum\limits_{k=1}^{L} \sum\limits_{q=1}^{N_{L}}\left|\sum\limits_{q=1}^{N_{L}}H_{l,q,v}\mathbf{k}_{v,\rho,d}\right|^{2}\right)^{2}P_{k,d}}{Re\left(\sum\limits_{k^{\prime}=1,k^{ \prime}\
eq k}^{K}MAI_{k^{\prime},d}\right)^{2}+Im\left\{\sum\limits_{k^{ \prime}=1,k^{\prime}\
eq k}^{K}MAI_{k^{\prime},d}\right\}^{2}}\end{split}  \tag{13}\]  
where ISI and noise power can be neglected for increased MAI scenarios. The denominator of \(SINR_{k,d}\) is Chi-squared distributed [19], since it represents sum of squares of normal RVs. However, an alternate representation based on Gamma distribution will be considered, in order to include the special case of correlated sums.