## 4 Multi-Sensor Detection with Particle Swarm Optimization
### Decorrelation Detection with Particle Swarm Optimization at Cooperative Sensor  
As shown in Figure 3, the cooperative sensors \(P_{i}\) (\(l=1,\cdots,K\)) receive signals from \(K\) source sensors in phase 1. In order to assist its source sensor \(S_{i}\) (\(l=1,\cdots,K\)), cooperative sensor \(P_{i}\) detects the original data stream of source sensor \(S_{i}\) with multi-sensor detection (MSD). As depicted in Figure 2b, the signal \(\mathbf{r_{s,p}}\) received at cooperative sensor \(P_{i}\) is serial-to-parallel converted into a \(N\)-path parallel signals. Then, the \(N_{c}\)-path parallel signals are turned into frequency-domain signals \(\boldsymbol{y}_{S,\boldsymbol{\eta}_{t}}\) by Fast Fourier Transform (FFT), which is given by:  
\[ \begin{split}\boldsymbol{y}_{S,\boldsymbol{\eta}_{t}}=& [y_{S,\boldsymbol{\eta}_{t}}(1),y_{S,\boldsymbol{\eta}_{t}}(2),\cdots,y_{S, \boldsymbol{\eta}_{t}}(N_{c})]^{T}\\ =&\left[\begin{array}{c}\sum_{k=1}^{K}\sqrt{E_{S_{ k}}}c_{k}(1)d_{k}^{1}(1)h_{S_{k},\boldsymbol{\eta}_{t}}(1)+n_{k}e^{-j2\pi|_{t}}\\ \sum_{k=1}^{K}\sqrt{E_{S_{k}}}c_{k}(2)d_{k}^{1}(1)h_{S_{k},\boldsymbol{\eta}_{t }}(2)+n_{2}e^{-j2\pi|_{t}}\\ \vdots\\ \sum_{k=1}^{K}\sqrt{E_{S_{k}}}c_{k}(G)d_{k}^{1}(M)h_{S_{k}, \boldsymbol{\eta}_{t}}(Nc)+n_{N_{c}}e^{-j2\pi|_{N_{c}}}\end{array}\right]\end{split}  \tag{13}\]  
Assume that the \(m\)-th symbols of all time-frequency coding frames of \(K\) source sensors form the symbol vector \(\boldsymbol{d}_{S,\boldsymbol{\eta}_{t}}^{m}\) which can be expressed as \(\boldsymbol{d}_{S,\boldsymbol{\eta}_{t}}^{m}=[d_{1}^{1}(m),\cdots,d_{k}^{1}(m ),\cdots,d_{K}^{1}(m)]^{T}\). The detection vector of \(\boldsymbol{d}_{S,\boldsymbol{\eta}_{t}}^{m}\) is given by:  
\[ \begin{split}\boldsymbol{y}_{S,\boldsymbol{\eta}_{t}}^{m}=& [y_{S,\boldsymbol{\eta}_{t}}^{m}(1),y_{S,\boldsymbol{\eta}_{t}}^{m}(2), \cdots,y_{S,\boldsymbol{\eta}_{t}}^{m}(G)]^{T}\\ =&[y_{S,\boldsymbol{\eta}_{t}}((m-1)G+1),y_{S, \boldsymbol{\eta}_{t}}((m-1)G+2),\cdots,y_{S,\boldsymbol{\eta}_{t}}(mG)]^{T} \\ =&\left[\begin{array}{c}\sum_{k=1}^{K}\sqrt{E_{S_{ k}}}c_{k}(1)d_{k}^{1}(m)h_{S_{k},\boldsymbol{\eta}_{t}}((m-1)G+1)+n_{(m-1)G+1}e^{-j2 \pi|_{m-1}(m-1)G+1}\\ \vdots\\ \sum_{k=1}^{K}\sqrt{E_{S_{k}}}c_{k}(G)d_{k}^{1}(m)h_{S_{k}, \boldsymbol{\eta}_{t}}(mG)+n_{mG}e^{-j2\pi|_{m-G}}\end{array}\right]\end{split}  \tag{14}\]  
The cooperative sensor \(P_{t}\) apply D-MSD to estimate the vector of \(\boldsymbol{d}_{S,\boldsymbol{\eta}_{t}}^{m}\), which can be expressed as:  
\[\boldsymbol{\hat{d}}_{\boldsymbol{-}}\boldsymbol{D}_{S,\boldsymbol{\eta}_{t }}^{m}= [\hat{d}_{1}^{1}(m),\cdots,\hat{d}_{k}^{1}(m),\cdots,\hat{d}_{K}^{1}(m )]^{T}=pinv(\boldsymbol{Q}_{S,\boldsymbol{\eta}_{t}}^{m})\times\boldsymbol{y} _{S,\boldsymbol{\eta}}^{m} \tag{15}\]  
where \(pinv()\) represents the pseudo inverse matrix. Matrix \(\boldsymbol{Q}_{S,\boldsymbol{\eta}_{t}}^{m}\) is defined as:  
\[\boldsymbol{Q}_{S,\boldsymbol{\eta}_{t}}^{m}= \left[ \begin{array}{c}c_{1}(1)h_{S_{k},P_{t}}((m-1)G+1),\cdots,c_{K }(K)h_{S_{K},P_{t}}((m-1)G+1)\\ c_{1}(2)h_{S_{k},P_{t}}((m-1)G+2),\cdots,c_{K}(2)h_{S_{K},P_{t}}((m-1)G+2)\\ \vdots\\ c_{1}(G)h_{S_{k},P_{t}}(mG),\cdots,c_{K}(G)h_{S_{K},P_{t}}(mG)\end{array} \right] \tag{16}\]  
When the channel noise is ignored, \(\boldsymbol{Q}_{S,\boldsymbol{\eta}_{t}}^{m}\) meets the equation: \(\boldsymbol{y}_{S,\boldsymbol{\eta}_{t}}^{m}=\boldsymbol{Q}_{S,\boldsymbol{\eta }_{t}}^{m}\boldsymbol{d}_{S,\boldsymbol{\eta}_{t}}^{m}\)[31]. Particle swarm optimization (PSO) is a kind of heuristic, stochastic and iterative optimization search algorithm, which has the advantages of good convergence performance and simple implementation, _etc._ In phase 1, cooperative sensors use the D-PSO algorithm to iteratively update the output of D-MSD with particle swarm optimization [18,32]. Consequently, the \(K\times M\) estimated matrix of time-frequency coded frames of \(K\) source sensors is obtained at each cooperative sensor. Assume that the iteration number is \(Max\_dt\) and the particle swarm scale is \(N_{p}\) for each detection.