## 2 Background Principles
### Traditional EPD Algorithm  
In the eLoran system, data acquisition hinges on demodulating the transmitted modulation elements to reconstruct the frame information. For eLoran signals with a carrier cycle of 10 \(\upmu\)s, phase shifts of 0\({}^{\circ}\) or \(\pm\)36\({}^{\circ}\) are generated by emissions of 1 \(\upmu\)s that are on-time, early, or late. The EPD algorithm functions by assessing the phase differences at the time-domain sampling points between the modulated signal and the reference signal (the first pulse of group). It then converts these into corresponding modulation code words, and proceeds with decoding to obtain the final time code information. This is a fundamental logic used in eLoran receivers. First of all, after the modulated signal passes through pre-processing modules (such as BPF and anti-interference) and the phase tracking module, the signal is divided into I and Q channels. Then, the I and Q branches interact with local sine and cosine carriers and undergo low-pass filtering, thereby preserving the sine and cosine components of the modulation phase and phase offset. The resulting quadrature sampled data are denoted as \((a_{i},b_{i})\), with the reference signal's phase offset captured as \((a_{i}^{ref},b_{i}^{ref})\). Using these data, the phase difference (\(\upalpha\)-\(\beta\)) is calculated through Formulas (2)-(4). \[\left\{ \begin{aligned} & tan_{i}=sin\alpha_{i}/cos\alpha_{i}=a_{i}/b_{i} \\ & tan\beta_{i}=sin\beta_{i}/cos\beta_{i}=a_{i}^{ref}/b_{i}^{ref} \end{aligned} \right. \tag{2}\]  
\[tan(\alpha_{i}-\beta_{i})=\frac{tan\alpha_{i}-tan\beta_{i}}{1+tan\alpha_{i}tan \beta_{i}}=\frac{a_{i}b_{i}^{ref}-b_{i}a_{i}^{ref}}{a_{i}a_{i}^{ref}+b_{i}b_{i}^ {ref}} \tag{3}\]  
\[\alpha_{i}-\beta_{i}=arctan\left(\frac{a_{i}b_{i}^{ref}-b_{i}a_{i}^{ref}}{a_{i} a_{i}^{ref}+b_{i}b_{i}^{ref}}\right) \tag{4}\]  
where \(\alpha_{i}\) and \(\beta_{i}\) represent the phase of the modulated signal and the reference signal at the \(i\)-th sampling point, respectively, and \(\alpha_{i}-\beta_{i}\) is the corresponding phase difference. Figure 2: Three-state PPM diagram of eLoran signal. Following this, the algorithm uses multi-threshold processing, set at \(\pm\)18\({}^{\circ}\), to ascertain the polarity of the phase difference. Phase differences within the \(-\)18\({}^{\circ}\)-18\({}^{\circ}\) range are deemed \"0\" modulation; those surpassing 18\({}^{\circ}\) are labeled as \"+\" modulation; and those falling below \(-\)18\({}^{\circ}\) are recognized as \"-\" modulation. Ultimately, based on the statistical outcomes from N sampling points, the modulation state is determined by a majority decision, with error correction applied according to the principle of balanced modulation when an equal number of modulation states occur [18; 43]. Despite its wide usage, the EPD algorithm faces inherent limitations. As a time-domain method, it requires high signal quality. However, eLoran signals are prone to electromagnetic interference such as noise and CWIs, which causes severe waveform distortion and phase detection errors, consequently affecting the decoding success rate. Also, as an AM signal, the eLoran signal is not sampled on a perfect 100 kHz carrier, introducing a certain bias in the 36\({}^{\circ}\) phase detection itself. The example provided in Figures 3 and 4 illustrates the impact of noise and CWI on signal waveform and the resulting phase difference curve after EPD demodulation. In this pulse group, the correct modulation is \"\(-\) 0 + 0 0 0\". However, due to the phase difference often being within the wrong threshold range, the fourth, sixth, and seventh pulses are all wrongly demodulated as \"+\" modulation. As the power of CWI and noise increases, these fluctuations become more pronounced. This highlights the limitations of the EPD algorithm and the potential errors in practical applications. Having evaluated the limitations of the EPD algorithm in the eLoran system, it is clear that advanced demodulation techniques are urgently needed.