*3.3. Calculation of Modulated Clutter Measurement Density*

The calculating process of the modulated clutter measurement density is a crucial part of the LM approach, which can significantly reduce the amount of computation of JIPDA, which evaluates the probabilities of all the feasible joint events that can occur in multiple target tracking for each scan.

The modulated clutter measurement density of measurement **z***k*,*<sup>i</sup>* can be obtained by adding the influence of other tracks to **z***k*,*<sup>i</sup>* by utilizing the probability that measurement **z***k*,*<sup>i</sup>* is generated from other targets to the pure clutter measurement density *ρk*,*<sup>i</sup>* at the position of measurement **z***k*,*i*. Let *ρ*˜*<sup>τ</sup> k*,*i* denote the modulated clutter measurement density, then:

$$\rho\_{k,i}^{\sf T} = \rho\_{k,i} + \sum\_{\substack{\sigma \in \sf T\_k \\ \sigma \ne \tau}} \frac{P\_{k,i}^{\sigma}}{1 - P\_{k,i}^{\sigma}} p\_{k,i}^{\sigma} \tag{16}$$

where *P<sup>σ</sup> <sup>k</sup>*,*<sup>i</sup>* is a measure of the influence of track *σ* on **z***k*,*i*, and it is represented by the prior probability that **z***k*,*<sup>i</sup>* is generated from target *σ* such as:

$$P\_{k,i}^{\sigma} = P\_D P\_G P \left\{ \chi\_k^{\pi} |\mathbf{Z}^{k-1}| \right\} \frac{p\_{k,i}^{\sigma}}{\rho\_{k,i}} / \sum\_{l=1}^{m\_k^{\sigma}} \frac{p\_{k,l}^{\sigma}}{\rho\_{k,l}} \tag{17}$$

where *p<sup>σ</sup> <sup>k</sup>*,*<sup>i</sup>* is a likelihood function of measurement **z***k*,*<sup>i</sup>* with respect to track *σ* such that:

$$p\_{k,i}^{\sigma} = \frac{1}{P\_G} \mathcal{N}(\mathbf{z}\_{k,i}; \mathbf{H} \mathfrak{K}\_{k|k-1}^{\sigma}, \mathbf{S}\_{k|k-1}^{\sigma}),\tag{18}$$

where *PG* is the gate probability [2].

The modulated clutter measurement density of measurement *ρ*˜*k*,*<sup>i</sup>* is used for calculating the data association probabilities in the update step of track trajectory state, as well as the update step of the target existence probability.
