### 2.1 Multi-static Doppler radar system

We assume a multi-static Doppler radar system composed of one transmitter and several receivers, as illustrated in Fig. 1. The multi-static Doppler radar system is assumed to be cooperative so that all information of the transmitter and receivers are available.

In the multi-static Doppler system, a transmitter located at \(t=[t_{x},t_{y}]^{\rm T}\) illuminates the target \(x_{k}\) with position \(p_{k}=[p_{x,k},p_{x,k}]^{\rm T}\) and velocity \({\dot{p}}_{k}=[{\dot{p}}_{x,k},{\dot{p}}_{x,k}]^{\rm T}\). If the signal is scattered by the target, receiver *j* with position \(r^{(j)}=[r_{x}^{(j)},r_{y}^{(j)}]^{\rm T}\) will receive it and report a Doppler measurement as

$$\begin{aligned} z_{k}^{(j)}=-{\dot{p}}_{k}^{\mathrm {T}}\left[ \frac{p_{k}-r^{(j)}}{\left\| p_{k}-r^{(j)}\right\| }+\frac{p_{k}-t}{\left\| p_{k}-t\right\| }\right] \frac{f_{c}}{c}+\varepsilon _{k}^{(j)} \end{aligned}$$

(1)

where *c* is the speed of light, \(f_{c}\) is the carrier frequency, and \(\varepsilon _{k}^{(j)}\) is the zero mean white Gaussian measurement noise with standard deviation \(\sigma _{\varepsilon }\). The Doppler shift can be positive or negative. The interval of the Doppler measurements is given as an interval \([-f_{0},+f_{o}]\), where \(f_{o}\) is the maximal possible value of the Doppler measurement. The distribution of false detections (clutter) is time invariant and independent of the target state. It is assumed that the false detections are distributed uniformly over the measurement space, and the number of false detections is Poisson distributed with the constant mean value \(\uplambda ^{(j)}\) for receiver *j*.

### 2.2 Bayes multi-target filter

The Bayes multi-target filter [19, 20] is an extension of the Bayes single-target filter using the RFS to describe the multi-target state. An RFS is a finite-set-valued random variable that the points are random and unordered and that the number of points is random. Assuming that the target states take values from a state space \({\mathbb {X}}\), the multi-target state space is the space of all finite subsets of \({\mathbb {X}}\) and denoted as \({\mathcal {F}}(\mathbb {X\text {)}}\). To distinguish between different targets, a mark \(\ell \in {\mathbb {L}}\) is augmented to the state of each target in the labeled RFS model. In this way, the multi-target state is considered as a finite set on \({\mathbb {X}}\times {\mathbb {L}}\). For convention, single target states are denoted by small letters (e.g., *x*, \({\varvec{x}}\)) and multi-target states are denoted by capital letters (e.g., *X*, \({\varvec{X}}\)). To distinguish labeled states and their distributions from the unlabeled, the labeled ones are denoted by bold face letters (e.g., \({\varvec{x}}\), \({\varvec{X}}\), \(\varvec{\pi }\)).

At time *k*, it is assumed that there are \(N_k\) target states \({\varvec{x}}_{k,1},\ldots ,{\varvec{x}}_{k,N_k}\) taking values in the labeled state space \({\mathbb {X}}\times {\mathbb {L}}\), and \(M_k\) measurements \(z_{k,1},\ldots ,z_{k,M_k}\) taking values in an observation space \({\mathbb {Z}}\). The set of targets is denoted as the multi-target state \({\varvec{X}}_{k}=\{{\varvec{x}}_{k,1},\ldots ,{\varvec{x}}_{k,N_k}\}\in {\mathcal {F}}(\mathbb {X\times {\mathbb {L}}\text {)}}\). The set of observations is treated as the multi-target observation \(Z_{k}=\{z{}_{k,1},\ldots ,z_{k,M(k)}\}\in {\mathcal {F}}(\mathbb {Z\text {)}}\). Let \(\varvec{\mathbf {\pi }}_{k}({\varvec{X}}_{k}|Z_{1:k})\) denote the multi-target filtering density at time *k* and \(\varvec{\mathbf {\pi }}_{k|k-1}({\varvec{X}}_{k}|Z_{1:k-1})\) denote the multi-target prediction density to time *k*. The multi-target Bayes filter propagates \(\varvec{\mathbf {\pi }}_{k}\) in time according to the following update and prediction

$$\begin{aligned} \varvec{\mathbf {\pi }}_{k}({\varvec{X}}_{k}|Z_{1:k})&=\frac{g_{k}(Z_{k}|{\varvec{X}}_{k})\varvec{\mathbf {\pi }}_{k|k-1}({\varvec{X}}_{k}|Z_{1:k-1})}{\int g_{k}(Z_{k}|{\varvec{X}})\varvec{\mathbf {\pi }}_{k|k-1}({\varvec{X}}|Z_{1:k-1})\delta {\varvec{X}}}, \end{aligned}$$

(2)

$$\begin{aligned} \varvec{\mathbf {\pi }}_{k|k-1}({\varvec{X}}_{k}|Z_{1:k-1})&=\int {\varvec{f}}_{k|k-1}({\varvec{X}}_{k}|{\varvec{X}})\varvec{\mathbf {\pi }}_{k-1}({\varvec{X}}|Z_{1:k-1})\delta {\varvec{X}}, \end{aligned}$$

(3)

where \({\varvec{f}}_{k|k-1}(\cdot |\mathbf {\cdot })\) is the multi-target transition density, \(g_{k}(\cdot |\mathbf {\cdot })\) is the multi-target likelihood, and \(Z_{1:k}=(Z_{1},\ldots ,Z_{k})\) contains all the measurements accumulated up to time *k*. The integrals in Eqs. (2)–(3) are set integrals but not ordinary integrals. The set integral for a function \(f:{\mathcal {F}}({\mathbb {X}}\mathbb {\times L})\rightarrow {\mathbb {R}}\) is given by

$$\begin{aligned} \int {\varvec{f}}({\varvec{X}})\delta {\varvec{X}}=\sum _{i=0}^{\infty }\frac{1}{i!}\sum _{(\ell _{1},\ldots ,\ell _{i})\in {\mathbb {L}}^{i}}\int {\varvec{f}}(\{(x_{1},\ell _{1}),\ldots ,(x_{i},\ell _{i})\})d(x_{1},\ldots ,x_{i}). \end{aligned}$$

(4)

### 2.3 Labeled multi-Bernoulli filter

The multi-Bernoulli filter was proposed in [23] as an approximation of the Bayes multi-target filter by approximating the posterior as a multi-Bernoulli RFS. Compared to the multi-Bernoulli filter, the LMB filter does not exhibit a cardinality bias and can output target tracks. The LMB distribution is described by \(\varvec{\pi }=\{(r^{(\ell )},p^{(\ell )}(\cdot ))\}_{\ell \in {\mathbb {L}}}\) in which \(r^{(\ell )}\) indicates the existence probability of a target with label \(\ell \in {\mathbb {L}}\), and \(p^{(\ell )}(x)\) is the the spatial distribution of the target’s state \(x\in {\mathbb {X}}\) when it exists [27]. The LMB RFS density is given by

$$\begin{aligned} \varvec{\pi }({\varvec{X}})=\Delta ({\varvec{X}})w({\mathcal {L}}({\varvec{X}}))[p]^{{\varvec{X}}}, \end{aligned}$$

(5)

where \({\mathcal {L}}({\varvec{X}})\) denotes the set of all possible labels obtained from \({\varvec{X}}\), and

$$\begin{aligned} w(L)&=\prod _{i\in {\mathbb {L}}}(1-r^{(i)})\prod _{i\in {\mathbb {L}}}\frac{1_{{\mathbb {L}}}r^{(\ell )}}{(1-r^{(\ell )})}, \end{aligned}$$

(6)

$$\begin{aligned}[p]^{{\varvec{X}}}&=\prod _{(x,\ell )\in {\varvec{X}}}p{}^{(\ell )}(x). \end{aligned}$$

(7)

If the prior distribution is an LMB distribution denoted as \(\{(r^{(\ell )},p^{(\ell )}(\cdot ))\}_{\ell \in {\mathbb {L}}}\), the predicted LMB distribution under the Bayes filtering framework evolves the survival LMB components and the birth LMB components, as follows:

$$\begin{aligned} \varvec{\pi }_{+}=\{(r_{+,S}^{(\ell )},p_{+,S}^{(\ell )}(\cdot ))\}_{\ell \in {\mathbb {L}}}\cup \{(r_{B}^{(\ell )},p_{B}^{(\ell )}(\cdot ))\}_{\ell \in {\mathbb {B}}}, \end{aligned}$$

(8)

where

$$\begin{aligned} r_{+,S}^{(\ell )}&=\eta _{S}(\ell )r^{(\ell )}, \end{aligned}$$

(9)

$$\begin{aligned} p_{+,s}^{(\ell )}(\cdot )&=\frac{\left\langle p_{S}(\cdot ,\ell )f(x|\cdot ,\ell ),p(\cdot ,\ell )\right\rangle }{\eta _{S}(\ell )}, \end{aligned}$$

(10)

\(f(x|x',\ell )\) is the transition density for track \(\ell\), \(p_{S}(\cdot ,\ell )\) is the state dependent survival probability, \(\eta _{S}(\ell )=\left\langle p_{S}(\cdot ,\ell ),p(\cdot ,\ell )\right\rangle\) is the survival probability of track \(\ell\), and the standard inner product \(\left\langle f,g\right\rangle \triangleq \int f(x)g(x)dx\).

Let the predicted LMB distribution denote as \(\varvec{\pi }_{+}=\{(r_{+}^{(\ell )},p_{+}^{(\ell )}(\cdot ))\}_{\ell \in {\mathbb {L}}_{+}}\), the posterior multi-target density is approximated as follows [27]:

$$\begin{aligned} \varvec{\pi }(\cdot |Z)=\{(r^{(\ell )},p^{(\ell )}(\cdot ))\}_{\ell \in {\mathbb {L}}_{+}}, \end{aligned}$$

(11)

where

$$\begin{aligned} r^{(\ell )}&=\sum _{(I_{+},\theta )\in {\mathcal {F}}({\mathbb {L}}_{+})\times \Theta _{I_{+}}}w^{(I_{+},\theta )}(Z)1_{I_{+}}(\ell ), \end{aligned}$$

(12)

$$\begin{aligned} p^{(\ell )}(x)&=\frac{1}{r^{(\ell )}}\sum _{(I_{+},\theta )\in {\mathcal {F}}({\mathbb {L}}_{+})\times \Theta _{I_{+}}}w^{(I_{+},\theta )}(Z)\times 1_{I_{+}}(\ell )p^{(\theta )}(x,\ell ), \end{aligned}$$

(13)

$$\begin{aligned} w^{(I_{+},\theta )}(Z)&\propto w_{+}(I_{+})[\eta _{Z}^{(\theta )}(\ell )]^{I_{+}}, \end{aligned}$$

(14)

$$\begin{aligned} p^{(\theta )}(x,\ell |Z)&=\frac{p_{+}(x,\ell )\psi _{Z}(x,\ell ;\theta )}{\eta _{Z}^{(\theta )}(\ell )}, \end{aligned}$$

(15)

$$\begin{aligned} \eta _{Z}^{(\theta )}(\ell )&=\left\langle p_{+}(\cdot ,\ell ),\psi _{Z}(\cdot ,\ell ;\theta )\right\rangle , \end{aligned}$$

(16)

$$\begin{aligned} \psi _{Z}(x,\ell ;\theta )&={\left\{ \begin{array}{ll} \frac{p_{D}(x,\ell )g(z_{\theta (\ell )}|x;\ell )}{\kappa (z_{\theta (\ell )})}, &{} \text {if }\theta (\ell )>0,\\ 1-p_{D}(x,\ell ), &{} \text {if }\theta (\ell )=0, \end{array}\right. } \end{aligned}$$

(17)

and \(g(z|x;\ell )\) is the single target likelihood, \(\kappa (\cdot )\) is the clutter intensity, \(\Theta _{I_{+}}\) is the space of mappings \(\theta :I_{+}\rightarrow \{0,1,\ldots ,|Z|\}\), and the inclusion function \(1_{S}(X)=1, \mathrm {if}~X\subseteq S\), otherwise, \(1_{S}(X)=0\).

There are two implementations of the LMB recursion: one is using the sequential Monte Carlo (SMC) method and the other is using Gaussian mixtures (GM). The GM implementation is popular because it provides a closed form analytic solution to the recursions under linear Gaussian target dynamics and measurement models. Alternatively, SMC implementation has the natural ability of handling nonlinear target dynamics and measurement models. In this paper, the SMC implementation is adopted to handle the nonlinear dynamic and measurement models.