## 2 Background Principles  
### Eloran Signal System  
The eLoran system is an internationally standardized ground-based radio system [27; 28], an advanced iteration of the Loran-C system incorporating Eurofix data-link techniques [29]. Operating within the frequency band of 90 to 110 kHz with a central carrier frequency of 100 kHz, the eLoran system ensures stable and reliable signal transmission. The most significant improvement in the system was achieved by modulating information onto the third to eighth pulses of each pulse group, thereby achieving efficient data communication. The standard current waveform at the base of the eLoran signal transmitting antenna is defined by Formula (1) [30].  
\[s(t)=\left\{\begin{array}{cc}A(t-\tau)^{2}\exp\Bigl{(}-2\frac{(t-\tau)}{65} \Bigr{)}\sin(2\pi f_{c}t+P_{c}),&\tau\ll t\ll 65+\tau\\ 0,&t<\tau\end{array}\right. \tag{1}\]  
where \(A\) is a constant related to the peak amplitude; \(t\) is the time in \(\mu\)s; \(\tau\) is the envelope-to-cycle difference in \(\mu\)s; \(f_{c}\) = 100 kHz is the carrier frequency, and \(P_{c}\) = 0 or \(\pi\) is the phase code parameter, in radians. According to the above definition, Figure 1 shows the standard Loran C pulse waveform and the normalized amplitude spectrum.  
The signal-to-noise ratio (SNR) of the Loran-C signal has its specific definition. It refers to the ratio of the signal received level to the root mean square level value of the atmospheric noise, and can be expressed as a dimensionless number or in decibels. Among these, the signal level is defined as the effective value level of the continuous wave at 25 \(\mu\)s after the starting point of the pulse envelope [31]. The SNR in the subsequent simulations of this paper is all based on the above definition as the standard.  
Each station chain in the eLoran system consists of a master station and several substations, distinguished by a unique group repetition interval (GRI) that facilitates rapid identification and precise timing [32; 33]. Both the master station and the sub-station have two pulse group phase coding formats of A and B law (A\({}_{\text{master}}\), B\({}_{\text{master}}\) and A\({}_{\text{sub}}\), B\({}_{\text{sub}}\)), alternated in a cyclical pattern for broadcasting, respectively [34; 35].  
By leveraging Eurofix data-link technology, the eLoran system introduces three-state pulse position modulation (PPM) with time modulation quantities of 0 or \(\pm\)1 \(\mu\)s, as shown in Figure 2[36; 37]. This modulation takes advantage of the eLoran signal's pulse characteristics to form frame information.  
Figure 1: Standard Loran-C signal: (**a**) normalized signal waveform; (**b**) normalized amplitude spectrum.  
The frame construction is meticulously designed to carry information through a specific modulation method based on the 128-character ASCII code [38]. Each frame consists of 30 pulse groups with 180 modulated pulses. In each pulse group, the modulation status of six pulses (third to eighth) corresponds to an ASCII character pattern and is converted into a 7-bit binary code, enabling each frame to convey 210 bits of information [27]. It also lays the groundwork for the dissemination of time code information and the potential sharing of additional data, such as differential correction values, in the future.  
The construction of frame information is designed to be resilient against common transmission errors and interference. By integrating Reed-Solomon (RS) codes and Cyclical Redundancy Check (CRC) codes with powerful error correction capabilities, the system's robustness is remarkably enhanced, providing a firm foundation for the demodulation process [39; 40; 41; 42].