## 2 Signal Transmission and Reception in MIMO-WCDMA Networks  
### Signal Model and Receiver Operations  
In all examined MIMO orientations, it is assumed that downlink transmission takes place and each transmitter/receiver is equipped with \(M_{t}/M_{r}\) antennas, respectively. The transmitted signal for the \(k\)th user (\(1\leq k\leq K\)) can be expressed as a product of individual terms [1]:  
\[\mathbf{s}_{k}(t)=\sqrt{p_{t,k}}c_{k}(t)\mathbf{t}_{k}\mathbf{b}_{k}(t) \tag{1}\]  
where \(p_{t,k}\) is the total power allocated to the \(k\)th user (\(1\leq k\leq K\)), \(\mathbf{b}_{k}\) is the \(dim\times 1\)symbol matrix, \(c_{k}(t)\) is the spreading code and \(\mathbf{t}_{k}\) the \(dim\times dim\) transmission matrix, where \(dim=\min(M_{t},M_{r})\) represents the dimension of the MIMO orientation (i.e., the number of operating transmit or receive antennas for a square channel matrix). It is assumed that the transmitted signal undergoes multiple reflections from \(L\) dominant clusters [17]. Therefore, received signal can be expressed as:  
\[\mathbf{z}_{k}(t)=\sum_{l=1}^{L}\mathbf{H}_{k,l,T_{k}}\mathbf{s}_{k}(t-\tau_{ l})+\sum_{ \begin{subarray}{c}k^{\prime}\
eq k\\ l^{\prime}=-1,\end{subarray} }^{K}\sum_{l=1}^{L}\mathbf{H}_{k,l,T_{k^{\prime} }}\mathbf{s}_{k^{\prime}}(t-\tau_{l})+\mathbf{n}_{k} \tag{2}\]  
where \(\tau_{l}\) and \(\mathbf{H}_{k,l,T_{k}}\) are the delay and channel coefficients, respectively, of the \(l\)th multipath component and \(\mathbf{n}_{k}\) is the corresponding Gaussian \(M_{r}\times 1\) noise. Each element of \(\mathbf{H}_{k,l,T_{k}}\) is assumed to be normally distributed with zero mean value and unity variance. The received signal is processed by a 2-D RAKE receiver, where the time of arrival of each multipath component along with the corresponding channel coefficient are estimated. Hence, the Maximal Ratio Combing (MRC) multiplying vectors per RAKE finger [18] can be formulated as:  
\[\mathbf{r}_{k,l}=\left(\mathbf{H}_{k,l,T_{k}}\mathbf{t}_{k}\right)^{\text{H}} \tag{3}\]  
where the conjugate transpose of a matrix \(\mathbf{A}\) is denoted by \(\mathbf{A}^{\text{H}}\). The output \(dim\times 1\) signal of the \(l\)th RAKE finger of the \(k\)th user after correlation with the desired spreading sequence is given by:  
\[\mathbf{u}_{k,l}=\mathbf{r}_{k,l}\left( \begin{array}{c}\mathbf{H}_{k,l,T_{k }}\mathbf{t}_{k}\mathbf{b}_{k,0}\sqrt{p_{t,k}}+\\ \sum\limits_{\begin{subarray}{c}l^{\prime}=1\\ l^{\prime}=l\\ l^{\prime}\
eq l\end{subarray}}^{L}\mathbf{H}_{k,l^{\prime},T_{k^{\prime}}} \mathbf{t}_{k}\Big{(}\rho_{k,k,l|-l^{\prime}|}\mathbf{b}_{k,-1}+\overline{ \rho}_{k,k,l|-l^{\prime}|}\mathbf{b}_{k,0}\Big{)}\sqrt{p_{t,k}}\\ \sum\limits_{\begin{subarray}{c}k^{\prime}=1\\ k^{\prime}\
eq k\end{subarray}}^{K}\sum\limits_{l^{\prime}=1}^{L}\mathbf{H}_{k, l^{\prime},T_{k^{\prime}}}\mathbf{t}_{k^{\prime}}\Big{(}\rho_{k,k^{\prime},l|-l^{ \prime}|}\mathbf{b}_{k^{\prime},-1}+\overline{\rho}_{k,k^{\prime},l|-l^{ \prime}|}\mathbf{b}_{k^{\prime},0}\Big{)}\sqrt{p_{t,k^{\prime}}}+\mathbf{n}_{k }\end{array} \right) \tag{4}\]  
where \(\rho_{k,k,l|-l^{\prime}|}\) and \(\overline{\rho}_{k,k,|l-l^{\prime}|}\) are the partial cross correlations [19], \(\mathbf{b}_{k,0}\) is the symbol matrix at the current transmission period and \(\mathbf{b}_{k,-1}\) is the corresponding transmission matrix at the previous symbol period. The partial cross correlations are given by:  
\[\rho_{k,k^{\prime},l}=\int\limits_{0}^{IT_{c}}\overline{c_{k}(t-IT_{c})}c_{k^ {\prime}}(t)dt \tag{5}\]  
\[\overline{\rho}_{k,k^{\prime},l}=\int\limits_{T_{c}}^{T}\overline{c_{k}(t-IT_{ c})}c_{k^{\prime}}(t)dt \tag{6}\]  
where \(\overline{\text{x}}\) is the conjugate of x.