# AOA-Based Three-Dimensional Multi-Target Localization in Industrial WSNs for LOS Conditions
## 4 Multi-Target Detection and Localization
### MT-SIMO Signal Transmission  
We utilize the CDMA technology to realize the multi-target detection (separating the signals from different MSNs). Each MSN is assigned a pre-defined spread spectrum sequence, i.e., a PN-sequence with the length of \(N\) chips. The spreading sequence assigned to \(\mathcal{M}_{k}\) is denoted by \(u_{k}(\tau)\) and expressed as  
\[u_{k}(\tau)=\sum_{n=1}^{N}c_{k,n}\mathbf{rect}\left(\tau-n\Delta_{c}\right), \quad c_{k,n}\in\left\{+1,-1\right\}, \tag{2}\]  
where \(\mathbf{rect}\left(\tau\right)\) denotes the rectangular pulse of a chip that has the duration of \(\Delta_{c}\) and power of \(P_{c}=\int_{0}^{\Delta_{c}}\mathbf{rect}^{2}(\tau)d\tau\). The duration of the spreading sequence is \(T_{u}=N\Delta_{c}\). The sequences of the \(K\) MSNs are orthogonal and the auto-correlation functions are pulse functions. Hence, the cross-correlation functions of the spreading sequences are [27]  
\[\langle u_{k_{1}}(\tau),\ u_{k_{2}}(\tau)\rangle=u_{k_{1}}(\tau)*u_{k_{2}}^{*} (-\tau)= \begin{cases}\varepsilon_{k_{1},k_{2}},&k_{1}\
eq k_{2},\\ NP_{c}\delta(\tau),&k_{1}=k_{2},\end{cases}  \tag{3}\]  
where \(<\cdot\cdot>\) denotes the sliding inner-production, \(\varepsilon_{k_{1},k_{2}}\) is a very small number \(\varepsilon_{k_{1},k_{2}}\to 0\) and \(\delta(\tau)\) is the delta function. The peak amplitude of the delta function is \(\int_{0}^{T_{u}}u_{k}^{2}(\tau)d\tau=NP_{c}\). An MSN modulates (e.g., via binary phase shift keying, BPSK) the carrier with its spreading sequence circularly and continuously and uses an omnidirectional antenna to transmit the modulated signal. The MSNs can transmit asynchronously, and each AN captures the signals using an \(M\) element antenna array, as mentioned in Section 3. An AN keeps receiving signals for the duration of more than \(2T_{u}\) such that it can capture at least one complete spreading sequence from each MSN. To reduce the hardware complexity and cost of the ANs, the signal reception can be realized in a time-division-multiplexing (TDM) manner. An AN uses only a single radio and baseband processing chain. The antennas of the array are connected to an \(M\)-input-1-output microwave switch. The switch connects one receiving antenna for more than \(2T_{u}\) to capture the signals in the air, and then changes to the next antenna. Consequently, the antennas receive the signals one-by-one controlled by the microwave switch. After the time of \(2 MT_{u}\), all the antennas receive at least one complete spreading sequence from every MSN. As long as the antenna switching is fast enough, the movement of the MSNs during \(2 MT_{u}\) is ignorable and the signals on the receiving antennas can be regarded as being captured simultaneously. The MT-SIMO links between an MSN and the ANs and the signal flow are illustrated in Figure 2. Please note that synchronization and coordination among the MSNs are not required. As mentioned in Section 3, there are \(L_{i,k}\) propagation paths from \(\mathcal{M}_{k}\) to \(\mathcal{A}_{i}\). Hence, \(L_{i,k}\) MPCs are received on each antenna. In this system, the pattern of one antenna element in an array refers to the complex radiation gain with respect to the reference element (e.g., the first antenna) for a given azimuth and elevation direction. The pattern of the \(m\)-th (\(m=1,2,\cdots,M\)) antenna of \(\mathcal{A}_{i}\) is denoted by \(s_{i,m}(\phi,\theta)\) where \((\phi,\theta)\) is the 2D radiation/incident direction. The amplitude of \(s_{i,m}(\phi,\theta)\) is the scale of the signal magnitude by the antenna gain. The phase of \(s_{i,m}(\phi,\theta)\) is the phase difference between the signals impinging at the \(m\)-th antenna and the reference antenna. When the \(l\)-th path from \(\mathcal{M}_{k}\) arrives at \(\mathcal{A}_{i}\) at the 2D incident angle of \((\phi_{i,k,l},\theta_{i,k,l})\), the pattern of the \(m\)-th antenna will be \(s_{i,m}(\phi_{i,k,l},\theta_{i,k,l})\).