## Appendix A  
In the Gaussian approach for the theoretical calculation of the BER in the spatial multiplexing mode, the \(n\)th user is considered as the desired user, and MAI from the remaining \(N\)-1 users as well as ISI are expressed in terms of the desired user signal. With respect to Equation (9) [23],  
\[E\Bigg{\{}\sum_{l=1}^{L}\|\mathbf{r}_{n,l}\mathbf{H}_{n,l}\mathbf{w}_{n^{ \prime}}\|_{F}^{2},d\Bigg{\}}\rightarrow\sum_{l=1}^{L}\Big{\{}\|\mathbf{r}_{ n,l}\|_{F}^{2},d\Big{\}}p_{n^{\prime},d},\ 1\leq d\leq dim \tag{1}\]  
The term {**A**,x} indicates the x\({}^{\text{th}}\) line of Matrix **A**. Moreover,  
\[E\Bigg{\{}\sum_{l=1}^{L}\|\mathbf{r}_{n,l}\mathbf{H}_{n,l}\mathbf{w}_{n}\|_{F} ^{2},d\Bigg{\}}\rightarrow\sum_{l=1}^{L}\Big{\{}\|\mathbf{r}_{n,l}\|_{F}^{2},d\Big{\}}^{2},1\leq d\leq dim \tag{2}\]  
The SINR for the \(n\)th user and \(d\)th spatial multiplexing mode will be given by [23]  
\[SINR_{n,d}\approx\frac{\sum_{l=1}^{L}\{\|\mathbf{r}_{n,l}\|_{F}^{2},d\}p_{n,d }}{(N-1)\Big{(}\frac{1}{6\mathcal{P}\mathcal{C}}\Big{)}+\Big{(}\frac{L-1}{L} \Big{)}\Big{(}\frac{1}{4\mathcal{P}\mathcal{C}}\Big{)}+N_{0}},1\leq d\leq dim \tag{3}\]  
In Equation (3), \(p_{n,d}\) is the transmission power per active user and mode, subject to  
\[\sum_{d=1}^{dim}p_{n,d}=P_{n} \tag{4}\]  
The desired user signal will be given by  
\[\gamma_{n,d}=\sum_{l=1}^{L}\Big{\{}\|\mathbf{r}_{n,l}\|_{F}^{2},d\Big{\}}p_{n,d} \tag{5}\]  
For BPSK modulation, BER will be given by [18]  
\[P_{e}=\frac{1}{dim}\sum_{d=1}^{dim}\int_{0}^{\infty}\mathcal{Q}\bigg{(}\sqrt{ SINR_{n,d}(\gamma_{n,d})}\bigg{)}p_{\gamma_{n,d}}(\gamma_{n,d}) \tag{6}\]where \(P_{e}\) is the error probability, \(p_{\gamma_{n,d}}\left(\gamma_{n,d}\right)\) is the probability density function (pdf) of the desired user signal of Equation (A5), and the Gaussian function is given by  
\[Q(\mathrm{x})=\frac{1}{\sqrt{2\pi}}\int_{\mathrm{x}}^{\infty}\exp\left(-\frac{ \mu^{2}}{2}\right)du\] (A7)  
In order to numerically evaluate the pdf of the desired signal per transmission mode, samples are taken from \(10^{7}\) different channel realizations, where \(\gamma_{n,d}\) is calculated for each transmission mode. For a 4 \(\times\) 2 MIMO network with six multipath components, the pdf functions of the two transmission modes follow the Gamma distribution [24]:  
\[f_{1}(\mathrm{x})=\mathrm{x}^{27}\frac{e^{-\mathrm{x}/0.11}}{0.11^{28}\Gamma( 28)}\] (A8)  
\[f_{2}(\mathrm{x})=\mathrm{x}^{18}\frac{e^{-\mathrm{x}/0.09}}{0.09^{19}\Gamma( 19)}\] (A9)  
where \(\Gamma(\mathrm{x})\) = \((\mathrm{x}-1)!\), as shown in Figure 11. Simulation results are presented in Figures 12-14. The mean BER is presented on a logarithmic scale versus throughput considering three different MIMO orientations: 4 \(\times\) 2, 4 \(\times\) 3, and 8 \(\times\) 4, respectively, with two and six resolvable multipath components. An SNR equal to 10 dB is considered, as well as a uniform power delay profile regarding the individual powers of the multipath components. The number of active users may vary from 1 to 40 with a step of 5, so the total network throughput will vary from 30 \(\times\) min(\(M_{t},M_{r}\)) Kbps to 1200 \(\times\) min(\(M_{t},M_{r}\)) Kbps, i.e., from 60 to 2400 Kbps. Note that the basic transmission rate of 30 Kbps corresponds to a \(PG\) equal to 128. As can be observed in Figures 12-14, the Gaussian approximation for the error probability and consequently the mean BER can be quite accurate for the performance evaluation of a MIMO-WCDMA system operating in spatial multiplexing mode. Moreover, the selection of all multipath components (i.e., 6) leads to an improved BER compared to the case of two multipath components. As expected, this gain reduces for high values of overall throughput, since for an increased number of active users, MAI can be significantly increased.  
Figure A2: Simulation and theoretical BER for a 4 \(\times\) 2 MIMO—WCDMA network.