## 2 Materials and Methods
### Multichannel Orthogonal Correlation Detection Method
#### 2.2.2 Multichannel Orthogonal Correlation  
After obtaining the three frequency signals via notching, an orthogonal correlation method was used to obtain the correlation peaks of the three frequency signals. The correlation method not only obtains high-gain correlation peaks but also further suppresses noise and other frequency signals left by the notch. We performed a sliding correlation operation of length \(T\) between \(s_{i}(t)\) and the orthogonal carrier signals at frequency \(f_{i}\). The two correlation results can be expressed as follows [32; 33]:  
\[R_{i,I}(t)=\frac{1}{T}\int\limits_{t}^{t+T}s(\tau)cos(2\pi f_{i}\tau+2\pi f_{i }t)d\tau=\frac{1}{T}\int\limits_{t}^{t+T}[A_{i}sin(2\pi f_{i}t)+n_{i}(t)]cos( 2\pi f_{i}\tau+2\pi f_{i}t)d\tau, \tag{5}\]  
\[R_{i,Q}(t)=\frac{1}{T}\int\limits_{t}^{t+T}s(\tau)sin(2\pi f_{i}\tau+2\pi f_{i }t)d\tau=\frac{1}{T}\int\limits_{t}^{t+T}[A_{i}sin(2\pi f_{i}t)+n_{i}(t)]sin( 2\pi f_{i}\tau+2\pi f_{i}t)d\tau, \tag{6}\]  
where \(T\) denotes the correlation length, and \(2\pi f_{i}t\) indicates the phase difference between the Alpha navigation signal and the local orthogonal carrier signal of the same frequency. Owing to the strong correlation between the orthogonal carrier signal and the received signal as well as the weak correlation between the orthogonal carrier signal and the noise, the received signal can be enhanced. However, the noise is not completely independent of the orthogonal carrier signal. Therefore, residual noise exists after correlation. Equations (5) and (6) can be expressed as follows:\[R_{i,l}(t)=\frac{A_{i}}{2T}\int\limits_{t}^{t+T}[sin(4\pi f_{i}\tau+2\pi f _{i}t)-sin(2\pi ft)]d\tau+\frac{1}{T}\int\limits_{t}^{t+T}n_{i}(t)cos(2\pi f_{i }\tau+2\pi f_{i}t)d\tau \tag{7}\] \[\qquad\qquad\qquad=-\frac{A_{i}sin(2\pi f_{i}t)}{2T}+\frac{A_{i}}{ 2T}\int\limits_{t}^{t+T}sin(4\pi f_{i}\tau+2\pi f_{i}t)d\tau+n_{i,l}(t),\] \[R_{i,Q}(t)=\frac{A_{i}}{2T}\int\limits_{t}^{t+T}[sin(2\pi f_{i}t )-cos(4\pi f_{i}\tau+2\pi f_{i}t)]d\tau+\frac{1}{T}\int\limits_{t}^{t+T}n_{i}( t)sin(2\pi f_{i}\tau+2\pi f_{i}t)d\tau\] (8) \[\qquad\qquad\qquad=-\frac{A_{i}cos(2\pi f_{i}t)}{2T}-\frac{A_{i}} {2T}\int\limits_{t}^{t+T}cos(4\pi f_{i}\tau+2\pi f_{i}t)d\tau+n_{i,Q}(t),\]  
where \(n_{i,l}(t)\) and \(n_{i,Q}(t)\) represent the residual noises after the correlation. The result of the second term in the formula is so small that it can be disregarded [34]. Thus, Equations (7) and (8) can be expressed as follows:  
\[R_{i,l}(t)=-\frac{A_{i}sin(2\pi f_{i}t)}{T}+n_{i,l}(t), \tag{9}\]  
\[R_{i,Q}(t)=\frac{A_{i}cos(2\pi f_{i}t)}{T}+n_{i,Q}(t). \tag{10}\]  
To eliminate the phase difference \(2\pi f_{i}t\), the operations in Equation (11) are performed on Equations (9) and (10). Thus, we can obtain the orthogonal correlation results:  
\[R_{i}(t)=\sqrt{\left[R_{i,l}(t)\right]^{2}+\left[R_{i,Q}(t)\right]^{2}=\frac {A_{i}}{2}+N_{i}(t)}. \tag{11}\]  
where \(N_{i}(t)\) represents the residual noise after orthogonal correlation. As observed in the orthogonal correlation process, the correlation length \(T\) is a critical parameter that can be used to obtain an ideal correlation peak. We obtained six correlation peaks through simulation with \(T\) values of 50, 100, 150, 200, 250, and 300 \(\upmu\)s. Figure 3 presents the correlation results. As illustrated in Figure 3, with an increase in the correlation length \(T\), the amplitude of the correlation peaks not only increases but also becomes sharper, which can improve the sensitivity of detection. However, a longer correlation length \(T\) leads to a wider correlation peak, resulting in a correlation peak overlap between the two stations with the same frequency signal and the detection of the signal not being used. When the correlation length \(T\) is too short, the correlation gain decreases. Therefore, we comprehensively consider selecting a correlation length of \(T=150\) ms to ensure the gain of the correlation peaks and prevent overlap between them. This is beneficial to improve the detection sensitivity of the Alpha navigation signal.