## 3 The Signal Adaptive Iterative Optimization Resampling (SAIOR) Algorithm
### Optimization Resampling  
For the PFM interference \(j_{m}(t)\), let \(n\) be any positive integer; then, from Equation (5), we can obtain:  
\[\begin{array}{l}j_{m}(t+nT_{m})=A_{m}\exp\left[2\pi f_{0m}(t+nT_{m})+\frac{ \Delta f_{m}T_{m}}{2}\cdot\cos(\frac{2\pi}{T_{m}}(t+nT_{m}))+\theta_{m}\right] \\ =A_{m}\exp\left[2\pi f_{0m}t+2\pi f_{0m}nT_{m}+\frac{\Delta f_{m}T_{m}}{2} \cdot\cos(\frac{2\pi}{T_{m}}t)+\theta_{m}\right]\\ =A_{m}\exp\left[2\pi f_{0m}t+\frac{\Delta f_{m}T_{m}}{2}\cdot\cos(\frac{2\pi}{ T_{m}}t)+\theta_{m}\right]\exp(2\pi f_{0m}nT_{m})\\ =j_{m}(t)\exp(2\pi f_{0m}nT_{m})\end{array} \tag{6}\]  
From the above equation, it can be seen that, when a single-component PFM signal is resampled with an interval of \(nT_{m}\), the consecutive resampled data points differ only by a scaling factor.  
Next, we analyze different interference scenarios separately:  
* When the interference scenario is a single-component PFM signal, it can be seen from Equation (6) that, after optimization resampling with an interval of \(nT_{m}\), the interference signal is concentrated at a single frequency point in the resampled data. This means that optimization resampling can transform the PFM interference into a single-frequency interference, while the energy distribution of the LEO satellite signal \(\sum\limits_{i}^{N}s_{i}(t)\) and noise \(n(t)\) is essentially unaffected.
* When the interference scenario is multi-component PFM signals with the same interference carrier frequency, let the starting point be \(j_{m}(\theta_{m})\), and the periods of the \(m\) PFM interference signals be denoted as \(T_{1},T_{2},\ldots,T_{m}\). Then, they must have a least common multiple \(T_{M}\), that is: \[T_{M}=n_{1}T_{1}=n_{2}T_{2}=\cdots n_{m}T_{m}\] (7) where \(n_{1},n_{2},\ldots,n_{m}\) are positive integers. When performing optimization resampling with an interval of \(nT_{M}\) (where \(n\) is a positive integer) on the \(m\) PFM interference signals, then: \[\begin{array}{l}\sum\limits_{m=1}^{M}j_{m}=\sum\limits_{m=1}^{M}A_{m}\exp(2 \pi\;\text{f}_{0m}MnT_{M}+\theta_{m})\\ =\exp(2\pi\;\text{f}_{0m}MnT_{M})\sum\limits_{m=1}^{M}A_{m}\exp(\theta_{m}) \end{array}\] (8)  
Equation (8) shows that, by performing optimization resampling with an interval of \(nT_{M}\), multiple-component PFM interference signals with the same carrier frequency can still be transformed into single-frequency interference.
* When the interference scenario involves multiple-component PFM signals with inconsistent carrier frequencies, similarly, let the periods of \(k\) PFM interference signals be denoted as \(T_{1},T_{2},\ldots,T_{k}\), and their least common multiple as \(T_{K}\). Let the carrier frequencies of the PFM interference signals be denoted as \(f_{01},f_{02},\ldots,f_{0k}\). By performing optimization resampling with an interval of \(nT_{K}\) (where n is a positive integer) on the \(k\) PFM interference signals, the results from Equations (6) and (8) can be obtained: \[\sum\limits_{k=1}^{K}j_{k}=\sum\limits_{k=1}^{K}A_{k}\exp(2\pi f_{0k}KnT_{k}+ \theta_{m})\] (9)  
Equation (9) indicates that, through optimization resampling with an interval of \(nT_{K}\), multiple-component PFM interference signals with different carrier frequencies can still be concentrated into several single-frequency interferences in the newly sampled data, andthe specific number of single-frequency points is determined by the carrier frequencies of the PFM interference.  
In summary, for PFM signals, the carrier frequency \(f_{0m}\) and modulation period \(T_{m}\) are the most critical parameters affecting optimization resampling performance. When subjected to PFM interference, regardless of whether the interference pattern is single/multiple components or whether the carrier frequencies of interferences are consistent, appropriate optimization resampling can transform the interference into one or several single-frequency interferences. Meanwhile, the energy distribution of LEO satellite signals and noise is essentially unaffected.