*2.1. NOTATION*


$$\delta\_Y(X) = \begin{cases} 1, ifX = \mathcal{Y} \\ 0, otherwise, \end{cases}$$

where inclusion function is denoted as:

$$1\_Y(X) = \begin{cases} 1\_"ifX \subseteq \mathcal{Y} \\ 0\_"otherwise. \end{cases}$$


In the multi-target environment based on RFS framework in time k, the states of multiple targets can be denoted by a set as *Xk* = x*k*,1, ··· , x*k*,*N*(*k*) ∈ F (X ) [13], where F (X ) is the all of finite subsets of state space and *N* (*k*) is the number of surviving targets. In the similar way, the observation about TDOAs and AOAs of the *<sup>q</sup>*th sensor pair can be described as *<sup>Z</sup>*[*q*] *<sup>k</sup>* = z [*q*] *<sup>k</sup>*,1, ··· , z[*q*] *k*,*Mq*(*k*) ∈ F (Z), where F (Z) is the space of finite subsets of observation space Z and *Mq* (*k*) is the number of observed measurement.

In the location tracking area, there are many uncertainties in the number and measurement of the detection process, such as birth, death, derivation, false alarm and missed detection. Consequently, The multiple targets state in time *k* can be defined as [59,60]:

$$X\_k = \left[ \bigcup\_{\mathbf{x} \in X\_{k-1}} S\_{k|k-1} \left( \mathbf{x} \right) \right] \cup \left[ \bigcup\_{\mathbf{x} \in X\_{k-1}} B\_{k|k-1} \left( \mathbf{x} \right) \right] \bigcup \Gamma\_{k'} \tag{1}$$

where *Sk*|*k*−<sup>1</sup> (*x*), *Bk*|*k*−<sup>1</sup> (*x*) and <sup>Γ</sup>*<sup>k</sup>* are the RFS of survival target at time *<sup>k</sup>* − 1, the RFS of the target spawn at time *k* from the survival target at time *k* − 1 and the RFS of target new-born, respectively. There are clutter or false alarms in the tracking area, which can be expressed as:

$$Z\_k^{[q]} = \left[ \bigcup\_{\mathbf{x} \in X\_k} \Theta\_k^{[q]}(\mathbf{x}) \right] \bigcup \mathcal{K}\_k^{[q]},\tag{2}$$

where Θ[*q*] *<sup>k</sup>* (*x*) is the measurements with the RFS which produced by targets in the tracking area:

$$\Theta\_k^{[q]}\left(\mathbf{x}\_k\right) = \begin{cases} \boldsymbol{\phi}, & \boldsymbol{H}\_{\text{miss}}\\ \left\{\mathbf{z}\_k^{[q]}\right\}, \boldsymbol{H}\_{\text{miss}} \end{cases} \tag{3}$$

here, the *H*¯ *miss* and *Hmiss* are the hypotheses of detection and miss detection, more generally, whether the sensor has received the signal generated by targets. Moreover, <sup>K</sup>[*q*] *<sup>k</sup>* is the measurement set of alarms or clutter false which follows a poisson distribution with a uniform density U (*z*) on the observation area and is given by:

$$
\mathcal{K}\_k \left( z\_k \right) = \frac{\lambda\_\varepsilon}{\int \mathcal{U} \left( z \right) dz} \mathcal{U} \left( z \right) \,. \tag{4}
$$

In RFS framework, the probability density function that the state of multi-target makes a transition from state *Xk*−<sup>1</sup> to *Xk* can be described as:

$$f\_{k|k-1}\left(\mathbf{X}\_k | \mathbf{X}\_{k-1}\right) = \sum\_{\mathcal{W} \in \mathcal{X}\_k} \pi\_{T,k|k-1}\left(\mathcal{W} | \mathbf{X}\_{k-1}\right) \times \pi\_{\Gamma,k}\left(\mathbf{X}\_k - \mathcal{W}\right),\tag{5}$$

where *<sup>π</sup>T*,*k*|*k*−<sup>1</sup> (· |·) is the probability density of spontaneous target birth and *<sup>π</sup>*Γ,*<sup>k</sup>* (·) is the probability density of target new-born.
