*3.1. SCM-Based JTC Method: Single-Target Case*

The joint target state can be modeled as ξ*k*−<sup>1</sup> - (*xk*−1, *<sup>c</sup>*), where *<sup>x</sup>k*−<sup>1</sup> is the kinematic state and *<sup>c</sup>* is the class label that can be taken from the set of the target classes *<sup>C</sup>* <sup>=</sup> {*c*1, *<sup>c</sup>*2, ··· , *<sup>c</sup>nc* }. *nc* and *<sup>c</sup><sup>m</sup>* represent the total number of the target class and the *m*th target class, respectively. In the SCM-based JTC method, the available measurement at time *k* consists of kinematic (position) measurement *z* p *<sup>k</sup>* = [*r*, <sup>θ</sup>] <sup>T</sup> and signature (HRRP) measurement *z*<sup>c</sup> *<sup>k</sup>* <sup>=</sup> *<sup>d</sup>*, and the joint measurement is denoted as '*z<sup>k</sup>* - (*z* p *<sup>k</sup>* , *<sup>z</sup>*<sup>c</sup> *k* ). The measurement set up to time *k* is represented by '*Z k* = '*z*τ *k* <sup>τ</sup><sup>=</sup>0.

The purpose of Bayesian JTC is to estimate the target state and class simultaneously at time *k*, under the condition that the distribution *<sup>p</sup>*(*xk*−1, *<sup>c</sup>*<sup>|</sup> '*Z k*−1 ) at time *<sup>k</sup>* <sup>−</sup> 1 and the measurement '*z<sup>k</sup>* at time *<sup>k</sup>* are available. That is, to obtain the posterior probability-mass distribution

$$p(\mathbf{x}\_k, \mathbf{c} | \overleftarrow{\mathbf{Z}}^k) = p(\mathbf{x}\_k | \mathbf{c}, \overleftarrow{\mathbf{Z}}^k) p(\mathbf{c} | \overleftarrow{\mathbf{Z}}^k) \tag{21}$$

For target tracking, the class-dependent probability density function (PDF) for a specific target class *c<sup>m</sup>* can be represented as

$$p(\mathbf{x}\_k|\mathbf{c}^m, \overline{\mathbf{Z}}^k) = \frac{p(\widetilde{\mathbf{Z}}\_k|\mathbf{x}\_{k'}c^m)p(\mathbf{x}\_k|\mathbf{c}^m, \overline{\mathbf{Z}}^{k-1})}{p(\widetilde{\mathbf{z}}\_k|\mathbf{c}^m, \overline{\mathbf{Z}}^{k-1})} \tag{22}$$

where *<sup>p</sup>*('*zk*|*cm*, '*<sup>Z</sup> k*−1 ) = # *<sup>p</sup>*('*zk*|*xk*, *cm*)*p*(*xk*|*cm*, '*<sup>Z</sup> k*−1 )*dx<sup>k</sup>* is the normalized factor. Accordingly, for target classification, the probability function can be obtained by

$$\mathbb{P}\_k \mu\_k^m \triangleq p(c^m | \overline{\mathbf{Z}}^k) = \frac{p(\overline{\mathbf{z}}\_k | c^m, \overline{\mathbf{Z}}^{k-1}) p(c^m | \overline{\mathbf{Z}}^{k-1})}{p(\overline{\mathbf{z}}\_k | \overline{\mathbf{Z}}^{k-1})} \tag{23}$$

where *<sup>p</sup>*('*zk*<sup>|</sup> '*Z k*−1 ) = \$*nc <sup>m</sup>*=<sup>1</sup> *<sup>p</sup>*('*zk*|*cm*, '*<sup>Z</sup> k*−1 )*p*(*cm*| '*Z k*−1 ) is the normalized factor.

To obtain the recursive equations of the SCM-based JTC method, two assumptions should be followed.

**Assumption 1.** *All the targets have the same motion model, i.e., the single state transition function fk*|*k*−1(*xk*, *<sup>c</sup><sup>j</sup>* <sup>|</sup>*xk*−1, *<sup>c</sup>m*) *is*

$$f\_{k|k-1}(\mathbf{x}\_k, \mathbf{c}^j | \mathbf{x}\_{k-1}, \mathbf{c}^m) = f\_{k|k-1}^k(\mathbf{x}\_k | \mathbf{x}\_{k-1}) f\_{k|k-1}^c(\mathbf{c}^j | \mathbf{c}^m) \tag{24}$$

*where f* <sup>k</sup> *<sup>k</sup>*|*k*−1(*xk*|*xk*−1)*is the kinematic state transition function and is decided by the system model and <sup>f</sup>* <sup>c</sup> *<sup>k</sup>*|*k*−1(*c<sup>j</sup>* |*cm*) *is the class state transition function and can be represented by the Dirac function* δ(·) *as*

$$f\_{k|k-1}^{\mathbb{C}}(c^j|c^m) = \delta\_m(j) = \begin{cases} \text{1, if } j = m \\ \text{0, if } j \neq m \end{cases} \tag{25}$$

**Assumption 2.** *The kinematic measurement and HRRP measurement are independent of each other, and the kinematic measurement is independent of the target class, so the measurement likelihood can be written as*

$$p(\overleftarrow{\bf z}\_{k}|\mathsf{x}\_{k},\mathsf{c}^{m},\overrightarrow{\bf Z}^{k-1}) = p(\mathsf{z}\_{k}^{\mathrm{P}}|\mathsf{x}\_{k})p(\mathsf{z}\_{k}^{\mathrm{c}}|\mathsf{x}\_{k},\mathsf{c}^{m})\tag{26}$$

*where p*(*z* p *<sup>k</sup>* <sup>|</sup>*xk*) *g*<sup>k</sup> *<sup>k</sup>* (*xk*) = <sup>N</sup>(*zk*; *<sup>h</sup>*(*xk*), *<sup>R</sup>*) *is the likelihood function of kinematic measurement. <sup>p</sup>*(*z*<sup>c</sup> *k* <sup>|</sup>*xk*, *<sup>c</sup>m*) - *g*c *k* (*xk*, *<sup>c</sup>m*) = ( *dk*, *g*(φ*k*, *Sc<sup>m</sup>* ) ) /(*dk*·*g*(φ*k*, *<sup>S</sup>cm* )) *is the likelihood function of HRRP measurement, and is defined as normalized correlation coe*ffi*cient of observed HRRP with model-predicted HRRP. Scm is the SCM corresponding to target class cm.*

Therefore, the SCM-based JTC method can be constructed through the following two steps. The prediction steps of the target state and class are given by

$$p(\mathbf{x}\_{k}|\boldsymbol{\varepsilon}^{m},\boldsymbol{\overline{Z}}^{k-1}) = \int f\_{k|k-1}^{k}(\mathbf{x}\_{k}|\mathbf{x}\_{k-1}) p(\mathbf{x}\_{k-1}|\boldsymbol{\varepsilon}^{m},\boldsymbol{\overline{Z}}^{k-1}) d\mathbf{x}\_{k-1} \tag{27}$$

$$
\mu\_{k|k-1}^{m} = \mu\_{k-1}^{m} \tag{28}
$$

Similarly, the update steps of target state and class are

$$p(\mathbf{x}\_k|\mathbf{c}^m, \overleftarrow{\mathbf{Z}}^k) = \frac{g\_k^k(\mathbf{x}\_k)g\_k^c(\mathbf{x}\_{k'}\mathbf{c}^m)p(\mathbf{x}\_k|\mathbf{c}^m, \overleftarrow{\mathbf{Z}}^{k-1})}{p(\overleftarrow{\mathbf{z}}\_k|\mathbf{c}^m, \overleftarrow{\mathbf{Z}}^{k-1})}\tag{29}$$

$$\mu\_k^m = p(c^m | \overline{\mathbf{Z}}^k) = \frac{p(\overline{\mathbf{z}}\_k | c^m, \overline{\mathbf{Z}}^{k-1}) p(c^m | \overline{\mathbf{Z}}^{k-1})}{p(\overline{\mathbf{z}}\_k | \overline{\mathbf{Z}}^{k-1})} \tag{30}$$

with

$$p(\overline{\mathbf{z}}\_k | \mathbf{c}^m, \mathbf{Z}^{k-1}) = \int \mathcal{G}\_k^k(\mathbf{x}\_k) \mathcal{G}\_k^c(\mathbf{x}\_k, \mathbf{c}^m) p(\mathbf{x}\_k | \mathbf{c}^m, \mathbf{Z}^{k-1}) d\mathbf{x}\_k \tag{31}$$

$$p(\widetilde{\boldsymbol{z}}\_k | \widetilde{\boldsymbol{Z}}^{k-1}) = \sum\_{m=1}^{n\_c} p(\widetilde{\boldsymbol{z}}\_k | \boldsymbol{c}^m, \widetilde{\boldsymbol{Z}}^{k-1}) p(\boldsymbol{c}^m | \widetilde{\boldsymbol{Z}}^{k-1}) \tag{32}$$

*Sensors* **2020**, *20*, 1679

Because of the complexity and high nonlinearity of the observation model, there is no analytic form to obtain the recursive estimation of target state and class, so we have to resort to the SMC technique (also known as particle filter, PF). Given the particle set *wj k*−1 , *xj k*−1 , *l j nk*−1,*<sup>p</sup> <sup>j</sup>*=<sup>1</sup> , where the superscript *<sup>j</sup>* denotes the index of the particles, *l <sup>j</sup>* <sup>∈</sup> *<sup>C</sup>* represents the class label corresponding to the *<sup>j</sup>*th particle and *nk*−1,*<sup>p</sup>* is the number of particles at time *k* − 1. The posterior target state and associated class probability can be represented by

$$p(\mathbf{x}\_{k-1}|\mathbf{c}^{m}, \overleftarrow{\mathbf{Z}}^{k-1}) = \sum\_{j=1}^{n\_{k-1,p}} w\_{k-1}^{m,j} \delta\_{\mathbf{x}\_{k-1}^{m,j}}(\mathbf{x}\_{k-1}) \tag{33}$$

$$p(\boldsymbol{c}^{\text{m}}|\mathcal{Z}^{k-1}) = \sum\_{j=1}^{n\_{k-1,p}} w\_{k-1}^j \delta\_{l^j}(\boldsymbol{c}^{\text{m}}) / \sum\_{j=1}^{m\_{k-1,p}} w\_{k-1}^j \tag{34}$$

with

$$w\_{k-1}^{m,j} = w\_{k-1}^j \delta\_{l^j}(\mathbf{c^m}) / \sum\_{j=1}^{n\_{k-1,p}} w\_{k-1}^j \delta\_{l^j}(\mathbf{c^m}) \tag{35}$$

⎞ ⎟⎟⎟⎟⎟⎠

$$\mathbf{x}\_{k-1}^{m,j} = \mathbf{x}\_{k-1}^{j} \delta\_{l^j}(c^m) \tag{36}$$

Then, a complete recursive procedure from time *k* − 1 to *k* can be summarized as Algorithm 1.

**Algorithm 1** Single-time step recursion of the scattering center model (SCM)-based joint tracking and classification (JTC) method

Step 1. Model prediction


Step 2. Likelihood evaluation

1) Kinematic observation likelihood: *g*<sup>k</sup> *k* (*xj k* ) = <sup>N</sup>(*zk*; *<sup>h</sup>*(*x<sup>j</sup> k* ), *R*)

$$\text{(2)}\quad \text{HRRP correlation coefficient: } \mathcal{g}\_k^c(\mathbf{x}\_{k'}^{j'}l^{j'}) = \left<\mathbf{d}\_k, \hat{\mathbf{d}}\_{k'}^{j}\right> / \left(\left\|\mathbf{d}\_k\right\| \cdot \left\|\hat{\mathbf{d}}\_{k'}^{j}\right\|\right)$$

Step 3. Particle weight evaluation


The posterior target state estimation *<sup>x</sup>*ˆ*<sup>k</sup>* and class probabilities *<sup>p</sup>*(*cm*<sup>|</sup> '*Z k* ) at time *k* can be obtained by

$$\hat{\mathfrak{x}}\_{k} = \sum\_{m=1}^{n\_{c}} p(c^{m}|\overrightarrow{\mathbf{Z}}^{k}) \hat{\mathfrak{x}}\_{k}^{m} \tag{37}$$

$$p(\boldsymbol{c}^{\mathfrak{m}}|\boldsymbol{\widetilde{\boldsymbol{Z}}}^{k}) = \sum\_{j=1}^{n\_{k-1,p}} \boldsymbol{\widetilde{w}}\_{k}^{j} \delta\_{l^{j}}(\boldsymbol{c}^{\mathfrak{m}}) / \sum\_{j=1}^{n\_{k-1,p}} \boldsymbol{\widetilde{w}}\_{k}^{j} \tag{38}$$

with

$$\mathfrak{x}\_k^m = \sum\_{j=1}^{n\_{k-1}} \overleftarrow{w}\_k^{m,j} \delta\_{\mathfrak{x}\_k^{m,j}}(\mathfrak{x}\_k) \tag{39}$$

$$
\langle \overline{w}\_k^{m,j} \rangle = \overline{w}\_k^j \delta\_{lj}(\mathfrak{c}^m) / \sum\_{j=1}^{m\_{k-1,p}} \overline{w}\_k^j \delta\_{lj}(\mathfrak{c}^m) \tag{40}
$$

$$\mathbf{x}\_{k}^{m,j} = \mathbf{x}\_{k}^{j} \delta\_{l^{j}}(\mathbf{c}^{m}) \tag{41}$$

To reduce the effect of particle degeneracy, the resampling operation [33] should be considered in method implementation. Specifically, the class-dependent resampling strategy is adopted to avoid particle degeneracy caused by an incorrect target classification. In this strategy, the maximum number of particles for the SCM-based JTC method is set as *L*max, while the minimum number of particles for each target class is set as *L*min. In this paper, the standard resampling operation is used to resample each class, that is, for the particle with *l <sup>j</sup>* = *cm*, *xj <sup>k</sup>* , 1/*nk*,*p*,*m*, *l j nk*,*p*,*<sup>m</sup> j* =1 = resample *xj k* , *w*'*<sup>j</sup> k* , *l j lj* <sup>=</sup>*cm* , *nk*,*p*,*<sup>m</sup>* max(*L*min, *nk*−1,*<sup>p</sup>* · *<sup>p</sup>*(*cm*<sup>|</sup> '*Z k* )).
