Step 5: Tracking Update

From the Reference [30], the prior density of the target state is updated as follows:

$$\begin{array}{lcl}p\left(\mathbf{x}\_{p^{\prime}}|\boldsymbol{\chi}\_{\mathcal{P}},\mathbf{Z}^{p}\right) \\ \stackrel{\scriptstyle \mathcal{D}\_{k}}{=} \sum\_{i=0}^{m\_{k}} \sum\_{c=1}^{\mathcal{C}} p\Big(\boldsymbol{\xi}\_{p^{\prime}}(c) \big| \boldsymbol{\chi}\_{\mathcal{P}^{\prime}},\mathbf{Z}^{p-1}\big| \boldsymbol{\beta}\_{p,i} \frac{\boldsymbol{\Lambda}\_{p,i}(\boldsymbol{\xi}\_{p^{\prime}}(c))}{\boldsymbol{\Lambda}\_{p,i}} p\Big(\mathbf{x}\_{p^{\prime}}|\boldsymbol{\xi}\_{p^{\prime}}(c), \boldsymbol{\chi}\_{\mathcal{P}^{\prime}}, \mathbf{z}\_{p,i}, \mathbf{Z}^{p}\big) \\ = \sum\_{c=1}^{\mathcal{C}} p\Big(\boldsymbol{\xi}\_{p+1}(c) \big| \boldsymbol{\chi}\_{\mathcal{P}^{\prime}},\mathbf{Z}^{p}\big) p\Big(\mathbf{x}\_{p^{\prime}}|\boldsymbol{\xi}\_{p+1}(c), \boldsymbol{\chi}\_{\mathcal{P}^{\prime}},\mathbf{Z}^{p}\big) \end{array} \tag{15}$$

The term *p* , **x***p*, - - ξ*p*(*c*), χ*p*, **z***p*,*i*,**Z***<sup>p</sup>* in (15) is the conditional posterior density, which can be denoted as

$$p\left(\mathbf{x}\_{p}|\boldsymbol{\xi}\_{p}(\boldsymbol{c}),\boldsymbol{\chi}\_{p},\mathbf{z}\_{p},\mathbf{z}^{p}\right) = \frac{p\left(\mathbf{y}\_{p,i}^{c}|\mathbf{x}\_{p}\right)p\left(\mathbf{y}\_{p,i}^{d}|\mathbf{x}\_{p}\right)p\left(\mathbf{x}\_{p}|\boldsymbol{\xi}\_{p}(\boldsymbol{c}),\boldsymbol{\chi}\_{p},\mathbf{Z}^{p-1}\right)}{p\left(\mathbf{y}\_{p,i}^{c}|\boldsymbol{\xi}\_{p}(\boldsymbol{c}),\boldsymbol{\chi}\_{p},\mathbf{Z}^{p-1}\right)p\left(\mathbf{y}\_{p,i}^{d}|\boldsymbol{\xi}\_{p}(\boldsymbol{c}),\boldsymbol{\chi}\_{p},\mathbf{Z}^{p-1}\right)}.\tag{16}$$

Because of the nonlinear relationship between target state and target Doppler *h* , **x***k p* - , we have to employ a nonlinear filter for solving Equation (16). In our work, in order to simplify the discussion, we choose not to use the Doppler measurements track state updates. The term *p* % *yd p*,*i* - - - **x***p* & can be approximated as *p* % *yd p*,*i* - - - ξ*p*(*c*), χ*p*,**Z***p*−<sup>1</sup> & . Therefore, the conditional posterior density can be rewritten as

$$p\left(\mathbf{x}\_{p^{\prime}}|\boldsymbol{\xi}\_{p}(\boldsymbol{c}),\boldsymbol{\chi}\_{p^{\prime}},\mathbf{z}\_{p^{j}},\mathbf{Z}^{p}\right) = \mathbf{N}\Big(\mathbf{x}\_{p}(\boldsymbol{\xi}\_{p}(\boldsymbol{c}));\mathbf{x}\_{p^{\prime}p^{\prime}}^{c,j},\mathbf{P}^{c,j}\_{p\|p}\Big).\tag{17}$$

It is worth noting that the conditional posterior density becomes the standard Kalman filter form. That is to say, *p* , **x***p*, - - ξ*p*(*c*), χ*p*, **z***p*,*i*,**Z***<sup>p</sup>* represents the output of a standard Kalman filter with predicted target state **x**ˆ*p*|*p*−<sup>1</sup> , ξ*p*(*c*) and covariance *Pp*|*p*−<sup>1</sup> , ξ*p*(*c*) as stated in step 1.

Considering the problem of multi-target tracking, the LM scheme views all observations from other targets as clutter. Therefore, the density of clutter can be modulated by contributions from other targets. During the *p*-th scan, the LM scheme estimates a revised density of clutter for all tracking gates, which is employed to compute the data association factor, the target existence probability and measurement association probabilities for all *K* targets. As a result, the probability *P<sup>k</sup> <sup>i</sup>* that the *i*-th measurement related to the *k*-th target can be expressed as

*Pk <sup>i</sup>* = *P* % θ*k p*,*i* , χ*<sup>k</sup> p* - - - -**Z***p*−<sup>1</sup> & = *Pk d Pk gP* , χ*k p* - - -**Z***p*−<sup>1</sup> - <sup>Λ</sup>*<sup>k</sup> p*,*i m* \$*p i*=1 Λ*k p*,*i* . (18)

In Equation (18), θ*<sup>k</sup> <sup>p</sup>*,*<sup>i</sup>* denotes the event that the *i*-th measurement is resulted from the *k*-th track at time *p*. Λ*<sup>k</sup> <sup>p</sup>*,*<sup>i</sup>* is presented in Equation (11). *P* , χ*k p* - - -**Z***p*−<sup>1</sup> describes the predicted prior probability of the *k*-th target. Owing to the presence of multi-target, the modified clutter density Ω*<sup>k</sup> <sup>p</sup>*,*<sup>i</sup>* in the gate of the *k*-th track can be expressed as:

$$
\Omega\_{p,i}^k = \rho\_{p,i} + \sum\_{\delta=1,\delta \neq k}^K \Lambda\_{p,i}^\delta \frac{P\_i^\delta}{1 - P\_i^\delta}. \tag{19}
$$

Hence, relying on a linear multi-target scheme, we can obtain the data association factor of the *k*-th target at time *p* as follows:

$$\delta\_p^k = P\_d^k P\_\mathcal{S}^k \left( 1 - \sum\_{i=1}^{m\_p^k} \frac{\Lambda\_{p,i}^k}{\Omega\_{p,i}^k} \right) . \tag{20}$$

The measurement association probability of the *k*-th target at time *p* can be expressed as

$$\beta\_{p,i}^{k} = \frac{1}{1 - \delta\_p^k} \begin{cases} 1 - P\_d^k P\_{\mathcal{S}'}^k & i = 0 \\ P\_d^k P\_{\mathcal{S}}^k \frac{\Lambda\_{p,i}^k}{\Omega\_{p,i}^k} & i > 0 \end{cases} . \tag{21}$$

Therefore, LM-IPDA scheme with DDA can be acquired. The DDA method can be used in the joint IPDA algorithm for target tracking in clutter. The joint IPDA algorithm recursively updates both the probability of target existence and target state estimate. The probability of target existence is used as a track quality measure for false track discrimination. It is worth noting that the contribution of multi-target Doppler information is also embodied in the factor of data association Equation (20), the probability of target existence Equation (18), and the probabilities of data association Equation (21).
