## 4 Multi-Sensor Detection with Particle Swarm Optimization
### Decorrelation Detection with Particle Swarm Optimization at Cooperative Sensor
The position and velocity of \(p\)-th particle at \(i\)th iteration for \(m\)-th symbol vector in \(K\) dimensional space are \(\mathbf{X^{m}_{p,i}}=\left[ \begin{array}{cc}X^{m}_{p,i,s},X^{m}_{p,2j,i},\cdots,X^{m}_{ p,k,i},\cdots,X^{m}_{p,k,i}\end{array} \right]^{T}\) and \(\mathbf{V^{m}_{p,i}}=\left[ \begin{array}{cc}V^{m}_{p,i,s},V^{m}_{p,2j,i},\cdots,V^{ m}_{p,k,i},\cdots,V^{m}_{p,k,i}\end{array} \right]^{T}\) respectively. The cooperative sensor \(P_{l}\) with D-PSO algorithm sets the output \(\mathbf{\hat{d}_{D}^{m}_{S,\mathbf{\eta}_{l}}}\) of D-MSD given by Equation (15) as the initial position of its first particle at the first iteration, _i.e._, \(\mathbf{X^{m}_{l,t}}=\mathbf{\hat{d}_{D}^{m}_{S,\mathbf{\eta}_{l}}}\). The objective function for \(m\)-th symbol vector is applied by cooperative sensor \(P_{l}\) to evaluate the position of \(p\)-th particle at \(i\)-th iteration:  
\[\mathrm{F}\left(\mathbf{X^{m}_{p,i}}\right)=\mathrm{Re}\left[ \begin{array}{cc} \mathbf{2(X^{m}_{p,i})^{T}}\mathbf{A_{S}(Q^{m}_{S,\mathbf{\eta}_{l}})^{T}}\mathbf{y^{m}_{S,\mathbf{ \eta}_{l}}}-\left(\mathbf{X^{m}_{p,i}}\right)^{T}\mathbf{A_{S}(Q^{m}_{S,\mathbf{\eta}_{l}} )^{T}}\mathbf{A_{S}Q^{m}_{S,\mathbf{\eta}_{l}}}\mathbf{X^{m}_{p,i}}\end{array} \right] \tag{17}\]  
where \(\mathbf{A_{S}}\) is the average amplitude matrix of the transmitting signals of \(K\) source sensors, \(\mathbf{A_{S}}=\mathrm{diag}(\sqrt{E_{S_{1}}},\cdots,\sqrt{E_{S_{k}}},\cdots, \sqrt{E_{S_{k}}})\); \(\mathrm{Re}\left(\bullet\right)\) represents the real part. The higher the value of objective function is, the better the position of particle is. The position and velocity of \(p\)-th particle at \(i\)-th iteration in \(k\)-th dimension for \(m\)-th symbol vector at cooperative sensor \(P_{l}\) are updated as follows:  
\[V^{m}_{p,i+1}=w\times V^{m}_{p,i}+c_{1}\times r_{1}\times(pbest^{m}_{p,k,i}-X^ {m}_{p,i})+c_{2}\times r_{2}\times(gbest^{m}_{k,i}-X^{m}_{p,i}) \tag{18}\]  
\[X^{m}_{p,k,i+1}=\left\{ \begin{aligned} & 1,&\frac{1}{1+\exp(-V^{m}_{p,i+1})} \geq rand\\ & -1,&\frac{1}{1+\exp(-V^{m}_{p,i+1})}<rand\end{aligned} \right. \tag{19}\]  
where \(w\) is inertia weight coefficient which is used to control the update speed of particles; \(r_{l}\) and \(r_{2}\) are random numbers between [0, 1]. \(c_{l}\) and \(c_{2}\) are learning factors which are applied to accelerate the particle to the best position of particle swarm; \(V_{max}\leq V^{m}_{p,k,i}\leq-V_{max}\), where \(V_{max}\) is the maximum velocity of the particles; \(\mathbf{best^{m}_{p,i}}\) is the local best position, that is the best position of \(p\)-th particle at \(i\)-th iteration for \(m\)-th symbol vector; \(\mathbf{gbest^{m}_{i}}\) is the global best position, which denotes the optimal detection result at \(i\)-th iteration for \(m\)-th symbol vector. \(\mathbf{gbest^{m}_{p,i}}\) and \(\mathbf{gbest^{m}_{i}}\) are determined by:  
\[\mathbf{gbest^{m}_{p,i}}=\max\left\{\mathrm{F}(\mathbf{X^{m}_{p,l}}), \mathrm{F}(\mathbf{X^{m}_{p,2}}),\cdots,\mathrm{F}(\mathbf{X^{m}_{p,i}})\right\} \tag{20}\]  
\[\mathbf{gbest^{m}_{i}}=\max\left\{\mathrm{F}(\mathbf{gbest^{m}_{1,i}}), \mathrm{F}(\mathbf{gbest^{m}_{2,i}}),\cdots,\mathrm{F}(\mathbf{gbest^{m}_{N_{p,i}}})\right\} \tag{21}\]  
The detection result of \(m\)-th symbol vector is obtained by D-PSO algorithm after the iteration number arrives at \(Max\_dt\), which is given by:  
\[\mathbf{\hat{d}_{\mathbf{\mathit{PSO}}}}^{m}_{s,\mathbf{\eta}}=[\hat{d_{1}}\left(m\right), \cdots,\hat{d_{k}}\left(m\right),\cdots,\hat{d_{k}}\left(m\right)]^{T}=\mathbf{gbest ^{m}_{Max\_dt}} \tag{22}\]  
Therefore, the cooperative sensor \(P_{l}\) can get the estimated \(K\times M\) matrix for the \(M\) bits of time-frequency coded frames of \(K\) source sensors in phase 1, which is given by:\[