*3.2. Multi-Target Multi-Bernoulli Filter*

A Bernoulli set *X* has a probability 1 − *r* of being a null set, and has a probability *r* of containing a single element *x* that is distributed via a pdf *s*(·). The probability of a Bernoulli RFS can be expressed in [21] as

$$\pi(\mathbf{X}) = \begin{cases} 1 - r, \, \mathbf{X} = \bigotimes \\ r\text{s}(\mathbf{X}), \, \mathbf{X} = \{\mathbf{x}\} \\ 0, \, \text{other} \end{cases} \tag{9}$$

A Multi-Bernoulli RFS *X* can be considered as union of a fixed number of independent Bernoulli sets that have existence probability *<sup>r</sup>*(*j*) <sup>∈</sup> (0, 1), *<sup>j</sup>* = 1, ... *<sup>J</sup>* and the pdf *<sup>s</sup>*(*j*), such that

$$\mathbf{X} = \bigcup\_{j=1}^{l} \mathbf{X}^{(j)} \tag{10}$$

where the *j* th Bernoulli set is described by its two parameters: the existence probability *r*(*j*) and the pdf *s* , *X*(*j*) - . So a Multi-Bernoulli RFS can be characterized by a posterior parameter set %*<sup>r</sup>* (*j*) *<sup>k</sup>*|*<sup>k</sup>* ,*sk*|*<sup>k</sup>* % *X*(*j*) *k* && *Jk j*=1 , where *Jk*|*<sup>k</sup>* indicates the number of sources. *<sup>Z</sup><sup>k</sup>* <sup>=</sup> *z*1,*k*, *z*2,*k*, ... , *zM*,*<sup>k</sup> <sup>T</sup>* denotes the sensor measurement data and *<sup>Z</sup><sup>k</sup>* <sup>∈</sup> <sup>Z</sup>, in which <sup>Z</sup> is the measurement space of the sensor. Target birth and survival are determined by birth probabilities *pb*,*k*(*Xk*) and survival probabilities *ps*,*k*(*Xk*), respectively. The source motion model is represented by the transition probability density *fk*|*k*−1(*Xk*|*Xk*−1), and the prior probability of Multi-Bernoulli is described as

$$p\left(\mathbf{X}\_{k-1}|\mathbf{Z}\_{1:k-1}\right) \approx \left\{ r\_{k-1\mid k-1}^{(j)} s\_{k-1\mid k-1} \left(\mathbf{X}\_{k-1}^{(j)}\right) \right\}\_{j=1}^{l\_{k-1}} \tag{11}$$

According to Equation (7), the prediction part can be described as

$$\begin{split} p(\mathbf{X}\_{k}|\mathbf{Z}\_{1:k-1}) &\approx \left\{ \hat{r}\_{k|k-1}^{(j)}, \hat{s}\_{k|k-1} \Big(\mathbf{X}\_{k}^{(j)}\Big) \right\}\_{j=1}^{l|k-1} \\ &= \left\{ r\_{P,k|k-1}^{(j)}, s\_{P,k|k-1} \Big(\mathbf{X}\_{k|k-1}^{(j)}\Big) \right\}\_{j=1}^{l|P,k|k-1} \cup \left\{ r\_{B,k'}^{(j)}, s\_{B,k} \Big(\mathbf{X}\_{k}^{(j)}\Big) \right\}\_{j=1}^{l|B,k} \end{split} \tag{12}$$

where

$$\begin{array}{l} \mathbf{r}\_{k|k-1}^{(j)} = \left(1 - r\_{k-1|k-1}^{(j)}\right) \cdot \int p\_{b,k} \Big(\mathbf{X}\_{k}^{(j)}\Big) \mathbf{s}\_{k-1|k-1} \Big(\mathbf{X}\_{k-1}^{(j)}\Big) \mathbf{d}\mathbf{X}\_{k-1}^{(j)}\\ + r\_{k-1|k-1}^{(j)} \cdot \int p\_{s,k} \Big(\mathbf{X}\_{k-1}^{(j)}\Big) \mathbf{s}\_{k-1|k-1} \Big(\mathbf{X}\_{k-1}^{(j)}\Big) \mathbf{d}\mathbf{X}\_{k-1}^{(j)} \end{array} \tag{13}$$

$$\begin{split} \mathfrak{S}\_{k|k-1} \left( \mathbf{X}\_{k|k-1}^{(j)} \right) &= \frac{\mathbb{1}\_{\times k} \Big( \mathbf{X}\_{k-1}^{(j)} \Big) r\_{k-1|k-1}^{(j)} \cdot \int f\_{k|k-1} \Big( \mathbf{X}\_{k-1}^{(j)} \Big) \mathbb{S}\_{k-1|k-1} \Big( \mathbf{X}\_{k-1}^{(j)} \Big) d\mathbf{X}\_{k-1}^{(j)}}{r\_{k|k-1}^{(j)}} \\ &+ \frac{\mathbb{1}\_{\times k} \Big( \mathbf{X}\_{k}^{(j)} \Big) \Big( 1 - r\_{k-1|k-1}^{(j)} \right) \cdot b\_{k|k-1} \Big( \mathbf{X}\_{k}^{(j)} \Big)}{r\_{k|k-1}^{(j)}} \end{split} \tag{14}$$

where *Jk*|*k*−<sup>1</sup> = *JP*,*k*|*k*−<sup>1</sup> + *JB*,*k*, *JP*,*k*|*k*−<sup>1</sup> = *Jk*−1. The number of Multi-Bernoulli parameter sets for survival sources and newborn sources are represented by *JP*,*k*|*k*−<sup>1</sup> and *JB*,*k*, respectively. According to Equation (8), if the predicted Multi-Bernoulli parameter set can be expressed as *r*ˆ (*j*) *k*|*k*−1 ,*s*ˆ*k*|*k*−<sup>1</sup> % *X*(*j*) *k* & *Jk*|*k*−<sup>1</sup> *j*=1 , then the update process can be expressed as

$$p(\mathbf{X}\_k | \mathbf{Z}\_{1:k}) \approx \left\{ r\_{k \mid k'}^{(j)} s\_{k \mid k} \left( \mathbf{X}\_{k \mid k-1}^{(j)} \right) \right\}\_{j=1}^{f\_k} \tag{15}$$

where

$$r\_{k|k}^{(j)} = \frac{\hat{r}\_{k|k-1}^{(j)} \int \lg\left(\mathbf{Z}\_k \middle| \mathbf{X}\_k^{(j)}\right) \mathfrak{s}\_{k|k-1}\left(\mathbf{X}\_k^{(j)}\right) \mathrm{d}\mathbf{X}\_k^{(j)}}{1 - \hat{r}\_{k|k-1}^{(j)} + \hat{r}\_{k|k-1}^{(j)} \int \lg\left(\mathbf{Z}\_k \middle| \mathbf{X}\_k^{(j)}\right) \hat{\mathfrak{s}}\_{k|k-1}\left(\mathbf{X}\_k^{(j)}\right) \mathrm{d}\mathbf{X}\_k^{(j)}} \tag{16}$$

$$\mathbf{s}\_{k|k}\left(\mathbf{X}\_{k}^{(j)}\right) = \frac{\operatorname{g}\left(\mathbf{Z}\_{k}\middle|\mathbf{X}\_{k}^{(j)}\right)\mathbf{s}\_{k|k-1}\left(\mathbf{X}\_{k}^{(j)}\right)}{\int \operatorname{g}\left(\mathbf{Z}\_{k}\middle|\mathbf{X}\_{k}^{(j)}\right)\mathbf{s}\_{k|k-1}\left(\mathbf{X}\_{k}^{(j)}\right)\mathbf{d}\mathbf{X}\_{k}^{(j)}}\tag{17}$$

where *<sup>g</sup>*(*Zk*|*Xk*) denotes the likelihood function. If the covariance of the general sensor array at time *<sup>k</sup>* in Gaussian noise environment is *Rk*, the likelihood function can be expressed as

$$\log(\mathbf{Z}\_k|\mathbf{X}\_k) = \frac{1}{\pi^M \text{det}(\mathbf{R}\_k)} \exp\left(-(\mathbf{Z}\_k - A(\mathbf{X}\_k)\mathbf{S}\_k)^H \mathbf{R}\_k^{-1} (\mathbf{Z}\_k - A(\mathbf{X}\_k)\mathbf{S}\_k)\right) \tag{18}$$

The frame of Formula (18) is not held in impulse noise, so we propose to replace the likelihood function with a spatial spectrum method.
