## 3 Method Description
### Phase Spectrum Smoothing Demodulation  
Phase detection demodulation technology is widely used in eLoran receivers. As introduced in Section 2.2, the EPD algorithm relies on time-domain phase detection and typically employs a BPF to improve signal quality. However, it has great limitations when dealing with in-band noise and CWIs. To overcome these challenges, this paper proposes a novel phase spectrum smoothing demodulation (PSSD) algorithm based on phase spectrum filtering. This algorithm aims to more effectively suppress noise and interference through frequency domain processing, thereby enhancing the demodulation performance of eLoran signals. The following sections will detail its principles and steps. 1. Fast Fourier Transform (FFT) processing. For a given eLoran pulse signal \(x(t)\), the FFT is performed to obtain its frequency domain representation \(X(f)\), which includes the signal's amplitude and phase spectra. This information will be used for the next step of phase spectrum smoothing. The FFT operation can be mathematically represented as: \[X(f)=FFT(x(t))=\int_{-\infty}^{+\infty}x(t)e^{-j2\pi ft}\] (5) Notably, in the FFT process for eLoran signals, the choice and application of the truncation window are crucial. It is designed to pick out the effective portion of the signal and reduces the noise energy. This enables a more accurate frequency components analysis and lays the groundwork for subsequent phase smoothing processing. 2. Phase spectrum filtering. For the eLoran signal's specific bandwidth, the key frequency range components (denoted as \(L_{freq}\) in the frequency domain \(X(f)\)) are subjected to filtering. This process aims to maintain the integrity of the signal while enhancing its phase stability and mitigating the impact of noise and interference on the phase spectrum. Let \(H(f)\) represent the transfer function of the smoothing filter. The phase spectrum filtering can be mathematically expressed as: \[X_{filt}(f)=X(f)\cdot H(f)\] (6) where \(X_{filt}(f)\) represents the frequency domain signal after smoothing. The choice of the smoothing filter's length and shape is critical, as it dictates the degree of smoothing applied. The appropriate Gaussian window filter is selected through experimentation, which effectively smooths the phase spectrum, reducing noise levels and controlling phase jumps. This is crucial for the subsequent calculation of phase differences, ensuring that the signal information used is both stable and reliable. 3. Phase difference vector acquisition. The purpose of this step is to calculate the phase difference between the test signal and the reference signal within the frequency domain. By comparing the complex spectra within the range of \(L_{freq}\), a phase difference vector that reflects the changes in phase shift between the two signals can be obtained. Moreover, it is necessary to address the discontinuity of the phase vector caused by \(2\pi\) jumps. For \(X_{filt,test}(f)\) and \(X_{filt,ref}(f)\) in the frequency domain, calculate the phase vector according to the real and imaginary parts of each frequency point as follows: \[\varphi_{test}(f)=phase\Big{(}X_{filt,test}(f)\Big{)}\] (7)\[\varphi_{ref}(f)=phase\Big{(}X_{filt,ref}(f)\Big{)} \tag{8}\]  
Given the phase \(\varphi(f)\) at frequency point \(f\), the unwrapping process is mathematically defined to correct for discontinuities due to \(2\pi\) jumps. Specifically, for each frequency point \(f\), the phase is adjusted as follows:  
* If \(\varphi(f)-\varphi(f-1)>\pi\) in the positive direction: \[\varphi(f)\leftarrow\varphi(f)-2\pi\] (9)
* Alternatively if \(\varphi(f)-\varphi(f-1)<-\pi\) in the negative direction: \[\varphi(f)\leftarrow\varphi(f)+2\pi\] (10)  
The unwrapping operation is applied to both \(\varphi_{test}(f)\) and \(\varphi_{ref}(f)\), yielding the unwrapped phases \(\varphi_{test,unwarapped}(f)\) and \(\varphi_{ref,unwarapped}(f)\), respectively. This process involves adjusting the phase at each frequency point to ensure continuity.