# SMC-CPHD Filter with Adaptive Survival Probability for Multiple Frequency Tracking
## 2 Processing Steps of the SMC-CPHD Filter
The covariance of the fused cardinality and gains in (35) are set as \(\alpha=P_{C_{f}}\times\left(\left(P_{C_{1}}\right)^{-1}-\gamma(\gamma P_{C_{1} }+(1-\gamma)P_{C_{2}}\right)^{-1}\right)\) and \(\beta=P_{C_{f}}\times\left(\left(P_{C_{1}}\right)^{-1}-(1-\gamma)(\gamma P_{ C_{1}}+(1-\gamma)P_{C_{2}})^{-1}\right)\). \(C_{1}\) is the estimated cardinality using the CPHD filter, and \(P_{C_{1}}\) is the variance of \(C_{1}\). \(C_{2}\) is the additionally generated cardinality by iterative clustering to compensate for the original cardinality estimate, \(C_{1}\). The iterative clustering algorithm [39] repeatedly performs fuzzy c-means (FCM) clustering using the resampled particles obtained from the CPHD filter. The clustering number that maximizes the clustering evaluation index among the candidates of the clustering number is selected as cardinality information, \(C_{2}\). \(P_{C_{2}}\) defines the quality information and represents the variance of \(C_{2}\). Finally, the covariance of the fused cardinality can be calculated by:

\[\left(P_{C_{f}}\right)^{-1}=\left(P_{C_{1}}\right)^{-1}+\left(P_{C_{2}}\right)^ {-1}-\left(\gamma P_{C_{1}}+(1-\gamma)P_{C_{2}}\right)^{-1} \tag{36}\]

The compensation method using the ICI method does not always guarantee good performance before fusing. A certain condition is required to continuously obtain the well-fused results and is analyzed by an inequality equation. Because the cardinality is a scalar value, Equation (36) can be expressed as:

\[\frac{1}{P_{C_{f}}}-\frac{1}{P_{C_{1}}}=\frac{1}{P_{C_{2}}}-\frac{1}{\gamma P_ {C_{1}}+(1-\gamma)P_{C_{2}}} \tag{37}\]

If the difference in Equation (37) is positive, the fused result using the ICI method always has better estimated results compared with those of the CPHD filter. Therefore, to always make the above equation positive, the \(\gamma P_{C_{1}}+(1-\gamma)P_{C_{2}}\) term guarantees the positive condition or is equal to zero. The term is multiplied in both sides of (37) and is then rewritten as follows:

\[ \begin{split}&\left(\frac{1}{P_{C_{f}}}-\frac{1}{P_{C_{1}}} \right)\times\left(\gamma P_{C_{1}}+(1-\gamma)P_{C_{2}}\right)\\ &=\left(\frac{1}{P_{C_{2}}}-\frac{1}{\gamma P_{C_{1}}+(1-\gamma) P_{C_{2}}}\right)\times\left(\gamma P_{C_{1}}+(1-\gamma)P_{C_{2}}\right)=\gamma \left(\frac{P_{C_{1}}}{P_{C_{2}}}-1\right)\end{split}  \tag{38}\]

where \(\gamma\in[0,1]\). In addition, if \(P_{C_{1}}\geq P_{C_{2}}\), \(\gamma\left(\frac{P_{C_{1}}}{P_{C_{2}}}-1\right)\geq 0\). This means that the fused result with the ICI method has a lower estimated error compared with the original CPHD filter. Finally, the estimated cardinality is only used in the \(P_{C_{1}}\geq P_{C_{2}}\) case when performing the ICI-based cardinality compensation process. ## 5 Simulations

Simulations were performed to verify the performance of the proposed algorithm. This paper aims to track multiple frequencies using the CPHD filter to remove GNSS interference when GNSS signals are received with multiple GNSS interference signals. In each simulation, GNSS data were generated by a SatGen simulator. The frequencies of received GNSS interference signals were designed as the combination of a constant frequency and a linear changed frequency in the time-frequency domain. We simulated two cases, and the true values of trajectory including the cardinality of each case are shown in Figures 2 and 3. In Figures 2a and 3a, the solid red line indicates the true trajectory, and the gray crosses show the frequency measurement, including the clutter of the CPHD filter. In both cases, the clutter conditions were set differently, as shown in Figure 2c, to confirm how resistant the proposed method is to the clutter. The jamming to signal ratio was set to 30 dB, the power of the received GNSS signal was set to \(-130\) dBm, and the overall noise floor was set to \(-114\) dB/MHz. In addition, the chirp rate was set to \(3.65\times 10^{12}\)Hz/s [40,41]. The initial digital frequency was set to 0.25 in case 1 and was respectively set to 0 and 0.5 in case 2.