**3. Analysis of ET-GIW-PHD**

In ET-GIW-PHD, the calculation of *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* is important. If the measurements in Wth cell are generated by clutter, *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* is expected to be smaller than the pruning threshold, then the corresponding component will be eliminated and the clutter will be eliminated.

In Equation (5), *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* contains two parts, one is the weight of the pth partition, denoted by *ωp*, the other is the weight of Wth cell in partition. Without loss of generality, only one partition is considered for clarity, therefore *ω<sup>p</sup>* = 1. Substituting Equation (6) into Equation (5), we arrive at

$$w\_{k|k}^{(j, \mathcal{W})} = \frac{e^{-\gamma^{(j)}} \left(\frac{\gamma^{(j)}}{\lambda\_{\epsilon}c\_k}\right)^{|\mathcal{W}|} p\_D^{(j)} \Lambda\_k^{(j, \mathcal{W})} w\_{k|k-1}^{(j)}}{\delta\_{|\mathcal{W}|,1} + \sum\_{l=1}^{J\_{k|k-1}} e^{-\gamma(l)} \left(\frac{\gamma^{(l)}}{\lambda\_{\epsilon}c\_k}\right)^{|\mathcal{W}|} p\_D^{(l)} \Lambda\_k^{(l, \mathcal{W})} w\_{k|k-1}^{(l)}}. \tag{9}$$

From Equation (9), the numerator is a part of denominator, the measurements of Wth cell is used to correct each GIW component, then <sup>Λ</sup>(*j*,*W*) *<sup>k</sup>* can be obtained, *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* can be given based on some prior parameters, such as *pD*, *γ*, *λ<sup>c</sup>* and *ck* (for brevity, the subscript and superscript are omitted here).

If the measurements in the cell are generated by clutter, the likelihood <sup>Λ</sup>(*j*,*W*) *<sup>k</sup>* of each GIW component will be very small since clutter does not obey the kinematic and extent model of target. If the number of clutter measurements in the cell is equal to one, then |*W*| = 1, *<sup>δ</sup>*|*W*|,1 = 1, *Jk*|*k*−<sup>1</sup> ∑ *l*=1 *e*−*γ*(*l*) *γ*(*l*) *λ<sup>c</sup> ck* <sup>|</sup>*W*<sup>|</sup> *p* (*l*) *<sup>D</sup>* <sup>Λ</sup>(*l*,*W*) *<sup>k</sup> <sup>w</sup>*(*l*) *<sup>k</sup>*|*k*−<sup>1</sup> will be much smaller than 1 because the likelihood <sup>Λ</sup>(*j*,*W*) *<sup>k</sup>* achieves a small value mentioned above and other parameters can be considered as constants, the value of *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* will be close to 0 and is smaller than the pruning threshold, then the corresponding component will be eliminated and the clutter is eliminated. However, if the number of clutter measurements in the cell is more than one, then |*W*| = 1, *<sup>δ</sup>*|*W*|,1 = 0, Equation (9) is the normalization process. Although Λ(*j*,*W*) *<sup>k</sup>* is close to zero, *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* can still take a large value. In this case, ghost targets will emerge and the number of targets will be overestimated. Further details on numerical implementation can be found in Section 5.

According to the analysis above, if the measurement in the cell is clutter, *Jk*|*k*−<sup>1</sup> ∑ *l*=1 *e*−*γ*(*l*) *γ*(*l*) *λ<sup>c</sup> ck* <sup>|</sup>*W*<sup>|</sup> *p* (*l*) *<sup>D</sup>* <sup>Λ</sup>(*l*,*W*) *<sup>k</sup> <sup>w</sup>*(*l*) *<sup>k</sup>*|*k*−<sup>1</sup> (denoted by *Jk*|*k*−<sup>1</sup> ∑ *l*=1 *ψl*,*W*) in Equation (9) should be added by 1. Otherwise, it should be added by 0 and the clutter can be eliminated. However, from Equation (9), if the cell contains only one measurement, *Jk*|*k*−<sup>1</sup> ∑ *l*=1 *ψl*,*<sup>W</sup>* is added by 1, it means that the cell contains only one measurement is considered as clutter in ET-GIW-PHD. Otherwise, it is considered as a target if the

cell contains more than one measurement. In fact, this assumption can be violated under strong clutter. The criterion whether measurements in the cell are generated by clutter based on only the number of measurements can be erroneous. A simple numerical calculation is shown below to illustrate this point.

In ET-GIW-PHD, the probability of the measurements of the cell generated by clutter is obtained based on the Bayesian theorem, see Equation (10). Note that, only the number of measurement is considered in this calculation.

$$\begin{split}P(\mathbf{Z}\_W \subset \mathbb{C} | n\_W = 1) &= 1 - P(\mathbf{Z}\_W \subset \mathbb{T} | n\_W = 1) \\ &= \frac{P(n\_W = 1 | \mathbf{Z}\_W \subset \mathbb{C}) P(\mathbf{Z}\_W \subset \mathbb{C})}{P(n\_W = 1 | \mathbf{Z}\_W \subset \mathbb{T}) P(\mathbf{Z}\_W \subset \mathbb{T}) + P(n\_W = 1 | \mathbf{Z}\_W \subset \mathbb{C}) P(\mathbf{Z}\_W \subset \mathbb{C})} \end{split} \tag{10}$$

where **Z***<sup>W</sup>* presents the measurements in cell, *nW* is the number of measurements in cell, C and T mean clutter and target respectively, *P*(**Z***<sup>W</sup>* ⊂ C) and *P*(**Z***<sup>W</sup>* ⊂ T) are the prior information.

The number of measurements generated by the target is subject to Poisson distribution with Poisson rate *γ*, the detection probability is *pd*, then

$$\begin{split} P(n\_W = 1 | \mathbf{Z}\_w \subset \mathbf{T}) &= \sum\_{i=1}^{\infty} p\_d (1 - p\_d)^{i-1} \mathbb{C}\_i^1 \frac{\gamma^i}{i!} e^{-\gamma} \\ &= \sum\_{j=0}^{\infty} p\_d (1 - p\_d)^j (j+1) \frac{\gamma^{j+1}}{(j+1)!} e^{-\gamma} \\ &= p\_d \gamma e^{-p\_d \gamma} \sum\_{j=0}^{\infty} \frac{((1 - p\_d)\gamma)^j}{j!} e^{-(1 - p\_d)\gamma} \\ &= p\_d \gamma e^{-p\_d \gamma} .\end{split} \tag{11}$$

where *C<sup>n</sup> <sup>m</sup>* = *<sup>n</sup>*! *<sup>m</sup>*!(*m*−*n*)! denotes the combinatorial number of the events that *<sup>m</sup>* out of *<sup>n</sup>*.

Remark: *pd* is not equal to *pD* in Equation (6). *pd* is the probability that one measurement generated by target or clutter is detected, while *pD* is the probability that an extended target will generate a measurement set [15]. *pD* can be derived if *pd* is already known.

The clutter measurements are assumed to be uniformly distributed over the surveillance area, and the number of clutter is subject to Poisson distribution with Poisson rate *λc*. So we have

$$\begin{split} P(n\_W = 1 | \mathbf{Z}\_w \subset \mathbb{C}) &= \sum\_{i=1}^{\infty} p\_d (1 - p\_d)^{i-1} \mathbb{C}\_i^1 \frac{\lambda\_c^{\frac{i}{d}}}{i!} e^{-\lambda\_c} \\ &= p\_d \lambda\_c e^{-p\_d \lambda\_c} .\end{split} \tag{12}$$

When the number of measurements is 1, the probability of the measurement in the cell generated by clutter is shown in Table 1 with different *γ* and *λc*. In this simulation, the prior information is set to 0.5, then *P*(**Z***<sup>W</sup>* ⊂ C) = *P*(**Z***<sup>W</sup>* ⊂ T) = 0.5.

**Table 1.** The probability of the measurement generated by clutter.


From Table 1 we can see that when *γ* = 10, *λ<sup>c</sup>* = 35 and *pd* = 0.9, *P*(**Z***<sup>W</sup>* ⊂ C|*nW* = 1) = 0.013. Although the cell contains only one measurement, the probability of the measurement in the cell generated by clutter is close to 0. Consequently, the criterion of ET-GIW-PHD does not work well in this case. when *γ* = 10 and *λ<sup>c</sup>* = 5, *P*(**Z***<sup>W</sup>* ⊂ C|*nW* = 1) = 1. In this case, the clutter is distinguished correctly based on the criterion of ET-GIW-PHD. In summary, the determination whether measurements are generated by clutter based on only the number of measurements can be erroneous.
