## 2 Materials and Methods
### Multichannel Orthogonal Correlation Detection Method
#### 2.2.1 Signal Digital Filtering  
Before signal detection, the received signal must be denoised, and three frequency signals must be extracted. Digital signal filtering entails two steps. First, a digital bandpass  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Station** & \multicolumn{6}{c}{**The Transmission Sequence of a Frame Signal**} & \multicolumn{2}{c}{**Frequency**} \\ \hline Master & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & 0 & 0 & 0 \\ \hline Western secondary & \(f_{3}\) & 0 & \(f_{1}\) & \(f_{2}\) & 0 & 0 \\ \hline Eastern secondary & 0 & \(f_{3}\) & \(f_{2}\) & \(f_{1}\) & 0 & 0 \\ \hline Northern secondary & \(f_{2}\) & 0 & 0 & 0 & \(f_{1}\) & \(f_{3}\) \\ \hline Southern secondary & 0 & \(f_{1}\) & 0 & 0 & \(f_{3}\) & \(f_{2}\) \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c c} \hline \hline
**Station** & \multicolumn{6}{c}{**The Transmission Sequence of a Frame Signal**} & \multicolumn{2}{c}{**Frequency**} \\ \hline Master & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & 0 & 0 & 0 \\ \hline Western secondary & \(f_{3}\) & 0 & \(f_{1}\) & \(f_{2}\) & 0 & 0 \\ \hline Eastern secondary & 0 & \(f_{3}\) & \(f_{2}\) & \(f_{1}\) & 0 & 0 \\ \hline Northern secondary & \(f_{2}\) & 0 & 0 & 0 & \(f_{1}\) & \(f_{3}\) \\ \hline Southern secondary & 0 & \(f_{1}\) & 0 & 0 & \(f_{3}\) & \(f_{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Alpha navigation system broadcast signals and sequences.  
Figure 2: Implementation process of detection method.  
filter is used to attenuate interference and the out-of-band noise to improve the signal detection probability. The bandpass filter uses an FIR filter. The bandwidth of the filter was 10-15 kHz, and the order was 256-order Hamming. The out-of-band signal suppression ability of the filter exceeded 40 dB. Subsequently, three frequency signals, denoted \(f_{1}\), \(f_{2}\), and \(f_{3}\), were extracted using three frequency notch filters. The notch filter uses a second-order IIR filter [29; 30]. We used the error signals of the three notch filters as the three extracted frequency signals. The three frequency signals were obtained after processing using three notch filters, which reduced the interaction of the frequency signals between different stations. The transfer function of the second-order IIR notch filter is as follows [31]:  
\[H(z)=\frac{1+\alpha}{2}\frac{1-2\beta z^{-1}+z^{-2}}{1-(1+\alpha)\beta z^{-1}+ \alpha z^{-2}}, \tag{1}\]  
where \(\alpha\) is the bandwidth control parameter of the notch filter; \(\beta\) is the frequency control parameter of the notch filter. The two parameters are designed as  
\[\beta=cos(\omega), \tag{2}\]  
\[\alpha=\frac{\sqrt{r_{SNR}{}^{-1}}-\sqrt{1-r_{SNR}{}^{-1}}\ tan(\Delta\omega/2 )}{\sqrt{r_{SNR}{}^{-1}}+\sqrt{1-r_{SNR}{}^{-1}}\ tan(\Delta\omega/2)}, \tag{3}\]  
where \(\omega\) signifies the angular frequency of the notch signal; \(\Delta\omega\) denotes the bandwidth of the trap; and \(r_{SNR}\) indicates the SNR of the notch signal. Equations (2) and (3) show that the corresponding notch filter can be designed based on the frequency of the three signals of the Alpha navigation signal.  
The signal processed by the notch filter is expressed as  
\[s_{i}(t)=A_{i}sin(2\pi f_{i}t)+n_{i}(t), \tag{4}\]  
where \(i\) represents the three frequency signals of \(f_{1}\), \(f_{2}\), and \(f_{3}\); \(f_{i}\) denotes the signal frequency; \(A_{i}\) is the signal amplitude; and \(n_{i}\) indicates the noise in the signal after the notch filter.