## 3 Method Description
### Phase Spectrum Smoothing Demodulation
Subsequently, the phase difference vector \(\Delta\varphi(f)\) is calculated as:  
\[\Delta\varphi(f)=\varphi_{test,unwarapped}(f)-\varphi_{ref,unwarapped}(f) \tag{11}\]  
This approach ensures a continuous and consistently oriented representation of the phase difference, which is essential for reliable frequency domain analysis. * Weighted phase difference calculation. Considering the PPM characteristics of eLoran AM signals, the phase difference curves obtained from different modulated signals exhibit a predominantly linear trend with different slopes in the frequency band, and are symmetrically distributed near the 100 kHz frequency point. Based on this feature, a suitable normalized weighting window is selected. Through testing multiple window functions, the Gaussian window is chosen for its optimal performance and is configured with suitable parameters to perform a weighted summation on the phase difference vector \(\Delta\varphi(f)\), resulting in a single weighted phase difference. \[\Delta\Phi_{weighted}=\sum_{f}\Delta\varphi(f)\mathrm{e}^{-\frac{1}{2}(\frac {f-f_{c}}{\varphi})^{2}}\] (12)  
Subsequently, the weighted phase difference \(\Delta\Phi_{weighted}\) is unwrapped in the range \(-\pi/2\) to \(\pi/2\) to obtain \(\Delta\Phi\), ensuring an effective demodulation range. * Modulation decision. Utilize the obtained \(\Delta\Phi\) to determine the modulation state of the eLoran signal through the modulation decision process. The modulation decision criteria are consistent with the EPD algorithm. Based on the value of \(\Delta\Phi\), apply the following decision criteria:
* If \(\Delta\Phi\geq\pi/10\), then the modulation decision is \"+\". * If \(-\pi/10<\Delta\Phi<\pi/10\), then the modulation decision is \"0\". * If \(\Delta\Phi\leq-\pi/10\), then the modulation decision is \"\(-\)\". A flowchart based on the above algorithm steps is shown in Figure 5. It is evident that the FFT signal length, FFT resolution, signal bandwidth selection, and Gaussian window \(\sigma\) are key parameters affecting the algorithm's performance in this study. They collectively determine the accuracy, stability, and computational efficiency of signal processing. We have selected the following parameter values as relatively optimal configurations based on an in-depth understanding of eLoran signal characteristics, previous design research experience with eLoran receivers, and meticulous experimental analysis of parameter sensitivity. 1. **FFT Signal Length**: The width of the truncation window U is set at 170 \(\upmu\)s (specifically, from \(-\)42.5-127.5 \(\upmu\)s centered on the highest point of the standard signal). This selection considers the pulse characteristics, effective duration, and SNR of the eLoran signal. 2. **FFT Resolution**: Extensive experiments showed that increasing the FFT resolution (e.g., by 100 Hz) does not significantly improve demodulation performance. Therefore, a resolution of 1 kHz is set for the FFT, which balances the frequency resolution with a sampling rate of \(f_{s}\) = 2 MHz and computational efficiency. 3. **Signal Frequency Bandwidth**: The phase spectrum processing is focused on the \(L_{freq}\) = 92-108 kHz band, which covers the main frequencies and high-power components of the eLoran signal. 4. **Gaussian Window**: Based on experiments, considering the spectral characteristics and central frequency selectivity of the signal, we have adopted two sets of different Gaussian window parameters: * Phase Spectrum Smoothing: The window width is set to 5 (corresponding to 5 kHz), and \(\sigma\) is set to 1.5. This parameter selection aims to effectively smooth the signal's phase spectrum while maintaining its spectral characteristics. * Phase Difference Weighted Summation: The window width is set to 17 (corresponding to 17 kHz), and \(\sigma\) is set to 2. This parameter selection helps to improve the accuracy of the weighted phase difference in summation process.