*3.1. Multi-Target Bayesian Theory*

Assume that the state of the sources at time *<sup>k</sup>* is *<sup>x</sup><sup>k</sup>* <sup>=</sup> θ*k*, . θ*k T* , where θ*<sup>k</sup>* is the DOA and moves at a speed of . θ*<sup>k</sup>* rad/s. The state and number of sources are changing at time *k* + 1, which can be described by RFS. From [20], the sources state set in multiple sources tracking can be regarded as an RFS, namely

$$\mathbf{X}\_k = \left\{ \mathbf{x}\_{k,1}, \dots, \mathbf{x}\_{k,P(k)} \right\} \tag{6}$$

where *X<sup>k</sup>* represents a set of sources at time *k*, and the element of the set may be one or more or null. *Z<sup>k</sup>* denotes the measurement set generated by all sources received time *<sup>k</sup>*, and the element is only one.

Single-target Bayesian filtering can be extended to multi-target tracking by modeling the above source states and measured values. The single target posterior probability density function (pdf) *pk*|*k*(*xk*|*Z*1:*k*) is replaced by the joint multi-target posterior *pk*|*k*(*Xk*|*Z*1:*k*). The Bayes joint filter recursion includes two stages: prediction and update. The prediction and update at time *k* in [24] are

$$p\_{k|k-1}(\mathbf{X}\_k|\mathbf{Z}\_{1:k-1}) = \int f\_{k|k-1}(\mathbf{X}\_k|\mathbf{X}\_{k-1}) p\_{k-1|k-1}(\mathbf{X}\_{k-1}|\mathbf{Z}\_{1:k-1}) \delta \mathbf{X}\_{k-1} \tag{7}$$

and

$$p\_{k|k}(\mathbf{X}\_k|\mathbf{Z}\_{1:k}) = \frac{\mathcal{g}(\mathbf{Z}\_k|\mathbf{X}\_k)p\_{k|k-1}(\mathbf{X}\_k|\mathbf{Z}\_{1:k-1})}{\int \mathcal{g}(\mathbf{Z}\_k|\mathbf{X}\_k)p\_{k|k-1}(\mathbf{X}\_k|\mathbf{Z}\_{1:k-1})\delta \mathbf{X}\_k} \tag{8}$$

where <sup>δ</sup> is the set integral and *<sup>Z</sup>*1:*k*−<sup>1</sup> represents all the measurement sets up to time *<sup>k</sup>* <sup>−</sup> 1. *<sup>g</sup>*(*Zk*|*Xk*) is a multi-target joint likelihood function and *fk*|*k*−1(*Xk*|*Xk*−1) is a multi-target state transition probability density function. *pk*|*k*−1(*Xk*|*Z*1:*k*−1) represents the multi-target joint prediction probability density and *pk*|*k*(*Xk*|*Z*1:*k*) is the multi-target joint posterior probability density function.
