# A High Dynamic Weak Spread Spectrum Signal Acquisition Strategy Based on Iterative Local Search
## 2 Space Station Ranging Signal Model  
The China Space Station is in low Earth orbit at an altitude of 400 km to 450 km. It has a high flight speed, and the center frequency of the microwave link ranging signal is in the Ka band with a high frequency [1]. Owing to the high speed of the space station, the spread spectrum signal not only has a large frequency deviation, but also includes a Doppler rate of change and a high-order rate of change. According to the number of orbital elements of the space station released on the website of the China Manned Space Engineering Office and using Beijing in WGS-84 coordinates coordinates (\(-\)2,172,495 m, 4,389,278 m, 4,072,343 m) as the ground station coordinates, the variation of the distance and radial velocity between the space station and the ground station in the over the top arc segment is shown in Figure 1. Taking the 26 GHz frequency point in the Ka band as the carrier center frequency, the dynamic situation of the spread spectrum signal is analyzed according to the above flight situation of the space station, according to the equation:  
\[f_{d}=\frac{v}{c}f_{IF} \tag{1}\]  
where \(f_{d}\) is the Doppler frequency of the signal; \(f_{IF}\) is the center carrier frequency; \(c\) is the speed of light (\(c\) = 299,792,458 m/s). Doppler changes can be obtained through higher\(-\)order fitting, as shown in Figure 2.  
According to Figure 2, under the background of the satellite-ground microwave link of the space station, the Doppler change rate above the third order is low, which has little influence on the acquisition and can be neglected. Given that the high-order dynamics over the cubic term of the time variable are negligible, the signal model can be given by:  
\[s(t)=\sqrt{2 PC}(t)\times\cos\left[\left(2\pi(f_{IF}+f_{d})t+\pi f_{g}t^{2}+ \frac{\pi}{3}f_{d}t^{3}+\varphi_{0}\right)\right]+n(t) \tag{2}\]  
where \(P\) is the power of the signal; \(C(t)\) is the spreading code; \(\varphi_{0}\) is the initial carrier phase, which is assumed to be 0; \(f_{g}\) is the Doppler change rate; \(f_{a}\) is the Doppler second-order rate of change; \(n(t)\) is the noise. The coherent integral gain is given by:  
Figure 1: Satellite-ground distance and radial velocity variation of the space station obtained by STK simulation. Figure 2: Carrier dynamics of the spread spectrum signal, including Doppler frequency, Doppler change rate, Doppler second-order change rate, and Doppler third-order change rate. \[S=\sqrt{\frac{P}{2}}\times R(\Delta\tau)\times T_{c}\times\sin c\bigg{[}\pi\times \bigg{(}f_{e}+\frac{f_{ve}}{2}\times T_{c}+\frac{f_{ae}}{6}\times T_{c}^{2} \bigg{)}\times T_{c}\bigg{]}+\tilde{N} \tag{3}\]  
where \(R\) is the code auto-correlation function; \(\Delta\tau\) is the code phase delay error; \(T_{c}\) is the coherent integration time; \(f_{e}\) is the Doppler frequency error; \(f_{ve}\) is the Doppler change rate error; \(f_{ae}\) is the Doppler second-order rate of the change error; \(\tilde{N}\) is the noise after correlation. When the coherence integration time is short, the influence of the Doppler change rate and the second-order change rate can be ignored. When the coherence integration time is long, the influence of the Doppler change rate and second-order change rate must be considered. In one spreading code period, the code auto-correlation function is given by [2]:  
\[R(\Delta\tau)=\left\{ \begin{array}{cc}1-\frac{|\Delta\tau|}{T_{chip}},&| \Delta\tau|<T_{chip}\\ 0,&\text{otherwise}\end{array} \right. \tag{4}\]  
where \(T_{chip}\) is the duration of a chip. When \(\Delta\tau=0\), the correlation value is at most 1; when \(|\Delta\tau|\geq T_{chip}\), the correlation approaches the lowest value of 0. In addition, high dynamics will cause not only the carrier Doppler, but also the code Doppler:  
\[f_{co}=\frac{f_{c}}{f_{IF}}\bigg{(}f_{d}+f_{o}t+\frac{1}{2}f_{a}t^{2}\bigg{)} \tag{5}\]  
where \(f_{c}\) is the spreading code rate. Given the high dynamic characteristics of the signal, the code phase slip caused by the code Doppler cannot be ignored.