## 2 MIMO-WCDMA Orientation  
A building orientation with \(M\) floors is considered, with \(M_{i}\) sensor nodes per floor (\(1\leq i\leq M\)). It is assumed that all nodes can communicate wirelessly with the central BMS. This is a rather realistic assumption, since we consider a dynamic environment where all nodes can be placed ad hoc. In order to reduce overall transmission power, each node is equipped with \(M_{t}\) antennas, the central BS with \(M_{t}\) antennas, while the WCDMA physical layer protocol has been adopted for internode communications [17]. In WCDMA, which is already used by mobile terminals in 3G communications, all nodes can transmit simultaneously occupying the whole transmission bandwidth. This is achieved with the use of orthogonal spreading sequences, which multiply the information signal. These sequences have low cross correlation values, thus minimizing unwanted effects from multipath propagation. The transmitted WCDMA signal for diversity combining transmission mode (i.e., the same signal is send and received from all antennas) is given by [17]  
\[\mathbf{x}_{n}(t)=\sqrt{p_{n}}b_{n}(t)c_{n}(t)\mathbf{w}_{n} \tag{1}\]  
where \(p_{n}\) is the power of the \(n\)th node (\(1\leq n\leq N\)), \(c_{n}(t)\) the coding sequence, and \(b_{n}(t)\) the information signal. For \(M_{t}\) antennas at the transmitter, \(\mathbf{w}_{n}\) represents the \(M_{t}\times 1\) transmission vector that corresponds to the complex power distribution per antenna (i.e., \(\|\mathbf{w}_{n}\|_{F}^{2}=1\), where \(\|\mathbf{x}\|_{F}\) is the [PERSON] norm of vector matrix \(\mathbf{x}\)). It is assumed that \(b_{n}(t)\) has a symbol duration equal to \(T\), while the corresponding duration for the coding sequence is \(T_{c}\). The ratio \(T/T_{c}\) is also called a spreading factor (\(SF\)). In a  
Figure 1: Typical building management system (BMS) infrastructure. multipath environment, the received signal from the \(M_{r}\) antennas of the central BS (considering uplink transmission) will be given by  
\[\mathbf{y}_{n}(t)=\sqrt{\frac{p_{n}}{TL_{n}}}\sum\limits_{l=1}^{L}\mathbf{H}_{n,l}\mathbf{x}_{n}(t-\tau_{n,l})+\sqrt{\frac{p_{n^{\prime}}}{TL_{n^{\prime}}}} \sum\limits_{ \begin{array}{c}n^{\prime}=1\\ n^{\prime}\
eq n\end{array} }^{N}\sum\limits_{l=1}^{L}\mathbf{H}_{n^{\prime},l }\mathbf{x}_{n^{\prime}}\big{(}t-\tau_{n^{\prime},l}\big{)}+\mathbf{n}_{n} \tag{2}\]  
where \(\mathbf{y}_{n}(t)\) is the \(M_{r}\times 1\) received signal vector matrix, \(L\) is the number of multipath components and \(\tau_{l}\) the corresponding delay of the \(l\)th multipath (\(1\leq l\leq L\)). Moreover, the term \(TL\) denotes the total losses (i.e., due to pathloss, shadowing, antenna radiation patterns, etc.) from the transmitter to the receiver. Finally, \(\mathbf{H}_{n,l}\) is the \(M_{r}\times M_{t}\) channel matrix, and \(\mathbf{n}_{n}\) the \(M_{r}\times 1\) additive white Gaussian noise. In a rich scattering environment such as the inner of a building, each element of \(\mathbf{H}_{n,l}\) is assumed to be a zero mean complex Gaussian random variable with standard deviation equal to one [18]. In order to exploit multipath propagation, it is assumed that each node is equipped with \(L\) rake receivers, also called fingers, which estimate the corresponding delay and coherently combine the individual components. The output from each finger will be given by [18]  
\[z_{n,l}=\mathbf{r}_{n,l}\left( \begin{array}{c}\sqrt{\frac{p_{n}}{TL_{n}}} \mathbf{H}_{n,l}\mathbf{w}_{n}b_{n,0}+\\ \sqrt{\frac{p_{n}}{TL_{n}}}\sum\limits_{\begin{array}{c}L\\ l^{\prime}=1\end{array}}^{L}\mathbf{H}_{n,l^{\prime}}\mathbf{w}_{n}\Big{(} \rho_{n,n,|l-l^{\prime}|}b_{n,-1}+\bar{\rho}_{n,n,|l-l^{\prime}|}b_{n,0}\Big{)} +\\ \hskip 14.226378 ptl^{\prime}\
eq l\end{array} \right) \tag{3}\]  
In Equation (3), \(\mathbf{b}_{n,0}\) is the symbol at the current time offset, while \(b_{n,-1}\) the corresponding symbol at the previous time offset.