## 2 MIMO Spatial Multiplexing Transceiver Model  
Considering the downlink of an arbitrary MIMO-CDMA orientation with \(N_{t}\) antennas at the transmitter and \(N_{r}\) antennas at the receiver (\(N_{t}\times N_{r}\)) then the transmitted signal for the \(k^{\text{th}}\) (\(1\leq k\leq K\)) user will be given by the equation below [17]. \[\mathbf{x}_{k}(t)=\mathbf{t}_{k}\mathbf{P}_{k}^{1/2}\mathbf{b}_{k}(t)c_{k}(t) \tag{1}\]By defining \(dim=\min\) (\(N_{t}\), \(N_{r}\)) as the dimension of the MIMO channel, then \(\mathbf{x}_{k}(t)\) is the \(dim\times 1\) transmission signal matrix, \(\mathbf{P}_{k}\) is a diagonal \(dim\times dim\) matrix indicating the transmission power per data stream, \(\mathbf{b}_{k}(t)\) is the \(dim\times 1\) symbol matrix, \(c_{k}\) is the Gold coding sequence [17], and \(\mathbf{t}_{k}\) is the \(dim\times dim\) precoding matrix. The various data streams in the \(\mathbf{b}_{k}\) matrix will also be referred to as modes throughout the rest of this article. The received signal per active user can be described as a superposition of the \(L\) multipath components from all active users. \[\mathbf{y}_{k}(t)=\sum\limits_{l=1}^{L}\mathbf{H}_{k,l}\mathbf{x}_{k}(t-\tau_ {l})+\sum\limits_{l=1}^{L}\sum\limits_{k^{\prime}=1,k^{\prime}\
eq k}^{K} \mathbf{H}_{k,l}\mathbf{x}_{k^{\prime}}(t-\tau_{l})+\mathbf{n}_{k} \tag{2}\]  
where \(\tau_{l}/\mathbf{H}_{k,l}\) are the delay and channel coefficient of the \(l^{\text{th}}\) multipath component (\(1\leq l\leq L\)) and \(\mathbf{n}_{k}\) is the corresponding Gaussian \(dim\times 1\) noise. The real and complex parts of each element of the channel matrix are assumed to be normally distributed with zero mean value and a variance equal to 0.5. This signal is processed by a bank of \(L\) 2-D RAKE receivers, where the time of arrival of each multipath component along with the corresponding channel coefficient are estimated [18]. The output \(dim\times 1\) signal of the \(l^{\text{th}}\) RAKE finger after correlation with the desired spreading sequence and MRC will be given by the Equation below. \[\mathbf{z}_{k,l}=\mathbf{r}_{k,l}\left( \begin{array}{c}\mathbf{H}_{k,l} \mathbf{k}_{k}^{\text{P}_{k}/2}\mathbf{b}_{k,0}+\\ \sum\limits_{l^{\prime}=1,l^{\prime}\
eq l}^{L}\mathbf{H}_{k,l^{\prime}} \mathbf{t}_{k}\mathbf{P}_{k}^{1/2}\Big{(}\rho_{k,k,l|-l^{\prime}|}\mathbf{b}_ {k,-1}+\overline{\rho}_{k,k,l|-l^{\prime}|}\mathbf{b}_{k,0}\Big{)}+\\ \sum\limits_{k^{\prime}=1,k^{\prime}\
eq k}^{K}\sum\limits_{l^{\prime}=1}^{L} \mathbf{H}_{k,l^{\prime}}\mathbf{t}_{k^{\prime}}\mathbf{P}_{k^{\prime}}^{1/2 }\Big{(}\rho_{k,k^{\prime},l|-l^{\prime}|}\mathbf{b}_{k^{\prime},-1}+\overline {\rho}_{k,k^{\prime},l|-l^{\prime}|}\mathbf{b}_{k^{\prime},0}\Big{)}+\\ \mathbf{n}_{k}\end{array} \right) \tag{3}\]  
where \(\rho_{k,k,l|-l^{\prime}|}\) and the corresponding conjugate \(\overline{\rho}_{k,k,l|-l^{\prime}|}\) are the partial cross correlations [4], \(\mathbf{r}_{k,l}=\left(\mathbf{H}_{k,l}\mathbf{t}_{k}\right)^{\text{H}}\) is the MRC multiplying matrix (\(\mathbf{A}^{\text{H}}\) is the conjugate transpose of matrix \(\mathbf{A}\)), \(\mathbf{b}_{k,0}\) is the symbol matrix at the current symbol period, and \(\mathbf{b}_{k,-1}\) is the corresponding matrix at the previous symbol period. Assuming independent symbol matrices among the various users of the MIMO, CDMA network, the signal power of the desired user, the Intersymbol Interference (ISI), MAI power, and the total noise power will be given by Equations (4)-(7), respectively. \[\mathbf{P}_{s,k}=\mathbf{P}_{k}\mathbf{k}^{\text{H}}_{k}\left(\left(\sum \limits_{l=1}^{L}\mathbf{r}_{k,l}\mathbf{H}_{k,l}\right)^{\text{H}}\left(\sum \limits_{l=1}^{L}\mathbf{r}_{k,l}\mathbf{H}_{k,l}\right)\right)\mathbf{t}_{k} \tag{4}\]  
(5)\[\mathbf{P}_{MAI,k^{\prime}}=\sum\limits_{ \begin{array}{c}k^{\prime}=1\\ k^{\prime}\
eq k\end{array}