*3.2. Multiple GPS Signals Separation and Reconstruction*

The GPS signal uses C/A code *C*(*t*) and P code *P*(*t*) to spread the data code *D*(*t*). This paper only considers the GPS satellite signal modulated by C/A code [25,26]. The signal received by the reference channel can be expressed as

$$\mathbf{x}\_r(t) = \sum\_{k=1}^{M} \sqrt{P\_k} \cdot \mathbb{C}\_k \left(t - \tau\_k\right) \cdot D\_k \left(t - \tau\_k\right) \cdot e^{j2\cdot\pi kt} + n(t),\tag{4}$$

where *Pk* is the power of the transmitted signal of the kth GPS satellite, *Ck*(*t*) represents the C/A code of the kth GPS satellite, *Dk*(*t*) represents the navigation data of the kth GPS satellite, *fk* is the carrier frequency of the received signal, and *τ<sup>k</sup>* is the delay of the received signal. By using the CDMA principle of the GPS system and the characteristics of the C/A code disclosure, a successive interference canceller (SIC) can be used to effectively separate and reconstruct multiple GPS signals *xr*(*t*) by the reference channel.

SIC is implemented in multiple steps, where each step requires signal acquisition to reconstruct the signal; then, the interference signal is removed from the received signal, and the "detection-reconstruction-cancel" step is repeated until all GPS signals are recovered. The steps of the method are as follows:

Step 1: Using a GPS acquisition algorithm to detect the GPS signal in the received signal of the reference channel, and obtain its corresponding spreading code information *Cl*(*t*), amplitude estimation value *Pl* , phase offset value *τl*, and frequency offset value *fl*;

Step 2: Demodulate and reconstruct the signal *xl*(*t*); the received signal is down-converted by using the frequency offset information *fl*, and then the phase offset *τ<sup>l</sup>* is obtained by the lth local C/A code *Cl*(*t*) to obtain *Cl* (*t* − *τl*), and, according to the orthogonality of the C/A codes of different satellites, the information *D l* (*t*) of the satellite is de-spreaded. In order to correctly recover the navigation data, the de-spreaded data are processed by the envelope averaging method, and finally the navigation data *Dl*(*t*) are determined. The process is shown in Figure 8.

**Figure 8.** Reference signal demodulation process.

The recovered navigation data *Dl*(*t*) is modulated using the phase-synchronized local C/A code, and the reconstructed reference signal *xl*(*t*) is obtained by using the amplitude *Pl* of the signal and then up-converting, which is expressed as

$$
\hat{\mathbf{x}}\_{l}(t) = \sqrt{\overline{P}\_{l}} \cdot \mathbb{C}\_{l} \left( t - \tau\_{l} \right) D\_{l} \left( t - \tau\_{l} \right) e^{j2\pi f\_{l}t}.\tag{5}
$$

The correlation coefficient between the original signal and the reconstructed signal obtained by simulation calculation is 0.99. As shown in Figure 9, it shows that the original signal is well reconstructed.

**Figure 9.** Compare the reconstructed signal with the original signal.

Step 3: Subtract the signal recovered in step 2 from the received signal of the reference channel to reduce the interference when reconstructing the next signal. In this paper, the adaptive filtering method is used to eliminate the interference signal, that is, the input end of the reference signal in Figure <sup>3</sup> is replaced by the reconstructed signal *<sup>x</sup>l*(*t*), and the input end of the monitoring channel signal is replaced by *xr*(*t*), so that the signal after strong interference cancellation can be obtained

$$\mathbf{x}\_r^{(1)}(n) = \boldsymbol{\varepsilon}(n) = \mathbf{x}\_l(n) - w^T(n-1)\mathbf{\hat{x}}\_l(n). \tag{6}$$

Step 4: Repeat the "capture-reconstruction-cancellation" process from step one to step three for the signal *x* (1) *<sup>r</sup>* (*t*) of the output signal of step three until the GPS signal is not detected in the reference channel. Finally, a plurality of reconstructed GPS reference signals can be obtained. At this time, the reference signals *x*ˆ1(*t*), *x*ˆ2(*t*)... *x*ˆ*M*(*t*) have been separated and there is no noise, so there is no other GPS signals and noise interfering with the reference signals. The process not only eliminates the noise in the reference signal, but also separates multiple GPS signals in the reference channel, providing a good reference signal for the DPI and MPI suppression processes of multiple GPS satellites in the monitoring channel. At the same time, it also provides a useful reference signal for the joint detection of multiple GPS weak echoes. In addition, the GPS signal that interferes with the original reference channel is converted into a reference signal that is advantageous to the system.
