Algorithm 1. Processing Steps of the SMC-UCR-MBerF-AI

Initialization: Let ${\pi }_0={\left\{\left(r_0^{(i)},p_{u,0}^{(i)}\right)\right\}}_{i=1}^{M_0}$ and $p_{u,0}^{(i)}(\tilde{x},A):={\left\{w_{u,0}^{(i,j)},\left({\tilde{x}}_{u,0}^{(i,j)},A_{u,0}^{(i,j)}\right)\right\}}_{j=1}^{L_{u,0}^{(i)}}$ represent the initial state. Input: Given ${\pi }_{k-1}={\left\{\left(r_{k-1}^{(i)},p_{u,k-1}^{(i)}\right)\right\}}_{i=1}^{M_{k-1}}$, $p_{u,k-1}^{(i)}(\tilde{x},A):={\left\{w_{u,k-1}^{(i,j)},\left({\tilde{x}}_{u,k-1}^{(i,j)},A_{u,k-1}^{(i,j)}\right)\right\}}_{j=1}^{L_{u,k-1}^{(i)}}$, the estimated multi-target states ${\widehat{X}}_{k-1}$ and the current measurement set $Z_k$, Prediction: 2. Compute the surviving Bernoulli components ${\left\{\left(r_{k|k-1,\beta =1}^{(i)},p_{u,k|k-1,\beta =1}^{(i)}\right)\right\}}_{i=1}^{M_{k|k-1,\beta =1}}$. (a) Compute the existence probability $r_{k|k-1,\beta =1}^{(i)}$ using Equation (8), for $i=1,2,\cdots ,M_{k-1}$. (b) Compute the weight ${\tilde{w}}_{u,k|k-1,\beta =1}^{(i,j)}$ using Equation (10), for $j=1,2,\cdots ,L_{k-1}^{(i)}$, $i=1,2,\cdots ,M_{k-1}$ and $u=0,1$. (c) Draw the particle $\left({\tilde{x}}_{u,k|k-1,\beta =1}^{(i,j)},A_{u,k|k-1,\beta =1}^{(i,j)}\right)\sim f_{u,k|k-1}\left(·|{\tilde{x}}_{u,k-1}^{\left(i,j\right)},A_{u,k-1}^{\left(i,j\right)}\right)$, for $j=1,2,\cdots ,L_{k-1}^{(i)}$, for $i=1,2,\cdots ,M_{k-1}$ and $u=0,1$. 3. Compute the new-born Bernoulli components ${\left\{\left(r_{k|k-1,\beta =0}^{(i)},p_{u,k|k-1,\beta =0}^{(i)}\right)\right\}}_{i=1}^{M_{k|k-1,\beta =0}}$. (a) Remove measurements near the predicted states ${\widehat{X}}_{k|k-1}$ of the estimated multi-target states ${\widehat{X}}_{k-1}$ and obtain the rest of the measurements $Z_{\mathrm{\Gamma },k}=\left\{z_k^{\left(1\right)},z_k^{\left(2\right)},\cdots ,z_k^{\left({\mathrm{\Gamma }}_k\right)}\right\}$, $M_{k|k-1,\beta =0}={\mathrm{\Gamma }}_k$. (b) Compute the existence probability $r_{k|k-1,\beta =0}^{(i)}={\displaystyle \sum _{u=0,1}{r}_{u,k,\beta =0}^{(i)}}$, for $i=1,2,\cdots ,{\mathrm{\Gamma }}_k$, and $u=0,1$. (c) Compute the weight ${\tilde{w}}_{u,k|k-1,\beta =0}^{(i,j)}=\frac{1}{2N_b^{\left(i\right)}}$ for $j=1,2,\cdots ,N_b^{\left(i\right)}$, for $i=1,2,\cdots ,{\mathrm{\Gamma }}_k$ and $u=0,1$. (d) Draw the particle $\left({\tilde{x}}_{u,k|k-1,\beta =0}^{(i,j)},A_{u,k|k-1,\beta =0}^{(i,j)}\right)\sim b_{u,k}\left(·|Z_{\mathrm{\Gamma },k}\right)$, for $j=1,2,\cdots ,N_b$, for $i=1,2,\cdots ,{\mathrm{\Gamma }}_k$ and $u=0,1$. 4.Using the union of the Bernoulli components, obtain the Pr-MTD as ${\pi }_{k|k-1}={\displaystyle \underset{\beta =0,1}{\cup }{\left\{\left(r_{k|k-1,\beta }^{(i)},p_{u,k|k-1,\beta }^{(i)}\right)\right\}}_{i=1}^{M_{k|k-1,\beta }}}$ Update: 5. Compute the legacy Bernoulli components ${\left\{\left(r_{L,k}^{(i)},p_{L,u,k}^{(i)}\right)\right\}}_{i=1}^{M_k^L}$ (a) Compute the existence probability $r_{L,k}^{\left(i\right)}={\displaystyle \sum _{u=0,1}{r}_{L,u,k}^{\left(i\right)}}$ via Equation (14), for $i=1,2,\cdots ,M_k^L$. (b) Compute the weight ${\tilde{w}}_{L,u,k|k}^{(i,j)}$ via Equation (19), for $j=1,2,\cdots ,L_{u,k|k-1,\beta =1}^{(i)}$, for $i=1,2,\cdots ,M_k^L$ and $u=0,1$. (c) Obtain the particle $\left({\tilde{x}}_{L,u,k|k}^{(i,j)},A_{L,u,k|k}^{(i,j)}\right)=\left({\tilde{x}}_{u,k|k-1,\beta =1}^{(i,j)},A_{u,k|k-1,\beta =1}^{(i,j)}\right)$, for $j=1,2,\cdots ,L_{u,k|k-1,\beta =1}^{(i)}$, for $i=1,2,\cdots ,M_k^L$ and $u=0,1$. 6. Compute the updated Bernoulli components ${\left\{\left(r_{U,k}\left(z\right),p_{U,u,k}\left(·;z\right)\right)\right\}}_{z\in Z_k}$ (a) Compute the existence probability $r_{U,k}\left(z\right)={\displaystyle \sum _{u=0,1}{r}_{U,u,k}\left(z\right)}$ via Equation (17), for $i=1,2,\cdots ,\left|Z_k\right|$. (b) Compute the weight ${\tilde{w}}_{U,u,k|k,\beta }^{(i,j)}$ via Equations (20) and (21), for $j=1,2,\cdots ,L_{u,k|k-1,\beta }^{(i)}$, for $i=1,2,\cdots ,M_{k|k-1,\beta }$, $\beta =0,1$ and $u=0,1$. Obtain the weight ${\tilde{w}}_{U,u,k|k,}^{(i,j)}={\displaystyle \underset{\beta =0,1}{\cup }{\tilde{w}}_{U,u,k|k,\beta }^{(i,j)}}$. (c) Obtain the particle $\left({\tilde{x}}_{U,u,k|k}^{(i,j)},A_{U,u,k|k}^{(i,j)}\right)={\displaystyle \underset{\beta =0,1}{\cup }\left({\tilde{x}}_{u,k|k-1,\beta }^{(i,j)},A_{u,k|k-1,\beta }^{(i,j)}\right)}$, for $j=1,2,\cdots ,L_{u,k|k-1,\beta }^{(i)}$, for $i=1,2,\cdots ,M_{k|k-1,\beta }$ and $u=0,1$. 7.Using the union of the Bernoulli components, obtain the Po-MTD as ${\pi }_k={\left\{\left(r_{L,k}^{(i)},p_{L,u,k}^{(i)}\right)\right\}}_{i=1}^{M_k^L}{{\displaystyle \cup \left\{\left(r_{U,k}\left(z\right),p_{U,u,k}\left(·;z\right)\right)\right\}}}_{z\in Z_k}$ Resample: 8.Discard the Bernoulli components with existence probability below a threshold $G$ and obtain ${\pi }_k={\left\{\left(r_k^{\left(i\right)},p_{u,k}^{\left(i\right)}\right)\right\}}_{i=1}^{M_k}$, and $p_{u,k}^{(i)}(\tilde{x},A):={\left\{{\tilde{w}}_{u,k|k}^{(i,j)},\left({\tilde{x}}_{u,k|k}^{(i,j)},A_{u,k|k}^{(i,j)}\right)\right\}}_{j=1}^{L_{u,k|k}^{\left(i\right)}}$. 9.Resample $L_k^{(i)}=\max \left(r_k^{(i)}{L}_{\max },L_{\min }\right)$ times from ${\left\{{\tilde{w}}_{u,k|k}^{(i,j)},\left({\tilde{x}}_{u,k|k}^{(i,j)},A_{u,k|k}^{(i,j)}\right)\right\}}_{j=1}^{L_{u,k|k}^{\left(i\right)}}$ to obtain ${\left\{w_{u,k}^{(i,j)},\left({\tilde{x}}_{u,k}^{(i,j)},A_{u,k}^{(i,j)}\right)\right\}}_{j=1}^{L_{u,k}^{\left(i\right)}}$ with weights $w_{u,k}^{(i,j)}=1/L_k^{\left(i\right)}$ and $L_{u,k}^{(i)}=L_k^{(i)}$. Multi-target state and clutter rate estimation: 10.Estimate number of actual targets ${\widehat{N}}_k={\displaystyle {\sum }_{i=1}^{M_k}{r}_{1,k}^{\left(i\right)}}$ 11.Estimate actual targets’ state ${\widehat{X}}_k=\left\{{\widehat{x}}_k^{\left(1\right)},{\widehat{x}}_k^{\left(2\right)},\cdots ,{\widehat{x}}_k^{\left({\widehat{N}}_k\right)}\right\}$ with ${\widehat{x}}_k^{\left(i\right)}={\displaystyle {\sum }_{j=1}^{L_{1,k}^{\left(i\right)}}{w}_{1,k}^{\left(i,j\right)}}{\tilde{x}}_{1,k}^{\left(i,j\right)}$. 12.Estimate clutter rate ${\widehat{\lambda }}_{c,k}={\displaystyle {\sum }_{i=1}^{M_k}{r}_{0,k}^{\left(i\right)}}{\displaystyle {\sum }_{j=1}^{L_{0,k}^{\left(i\right)}}{w}_{0,k}^{\left(i,j\right)}}{p}_{D,0,k}^{\left(i,j\right)}$. Output: ${\pi }_k={\left\{\left(r_k^{\left(i\right)},p_{u,k}^{\left(i\right)}\right)\right\}}_{i=1}^{M_k}$, $p_{u,k}^{(i)}(\tilde{x},A):={\left\{w_{u,k}^{(i,j)},\left({\tilde{x}}_{u,k}^{(i,j)},A_{u,k}^{(i,j)}\right)\right\}}_{j=1}^{L_{u,k}^{(i)}}$, ${\widehat{X}}_k$, ${\widehat{\lambda }}_{c,k}$