## 2 Effect of Spoofing Signals on Acquisition
### Acquisition Performance Analysis  
Receiver's acquisition search for a Global Navigation Satellite System (GNSS) signal is a signal search performed in a two-dimensional space consisting of frequency and time. According to the signal detection theory [24], in the presence and absence of satellite signals, the non-coherent integral amplitude \(V\) is \(\chi^{2}\) distribution and Rayleigh distribution, respectively. When a satellite signal exists and the number of non-coherent integrations \(N_{nc}\) is 1, the number of non-coherent integration means that the receiving channel generates a pair of coherent integration results every other coherent integration time, and the number of incoherent integration results in each search unit is the number of incoherent integrals. Figure 1: Flow chart for the acquisition and tracking loop structure. Figure 2: Relationship between total power of spoofing signal and estimation of receiver noise floor. The detection amount \(V\) of the satellite signal obeys the non-central \(\chi^{2}\) distribution, and the detection probability \(P_{d}\) is:  
\[P_{d}(V)=\int_{D_{t}}^{\infty}\frac{V}{\sigma_{n}^{2}}e^{-\frac{V^{2}+P_{l}}{2 \sigma_{n}^{2}}}I_{0}\left(\frac{VP_{l}}{\sigma_{n}^{2}}\right)dV\,\ (V>0) \tag{7}\]  
Here, \(I_{0}(\cdot)\) is a zero-order first-order modified Bessel function, \(2\sigma_{n}^{2}\) is the noise power, and \(P_{l}\) is the power after the coherent integration of the \(l\)th signal. \(D_{t}\) is the detection threshold, and the above formula can be expressed as follows by pre check signal-to-noise _SNR_ (where _SNR_ is the dimensionless ratio) [25]:  
\[P_{d}=\int\limits_{D_{t}}^{\infty}\frac{V}{\sigma_{n}^{2}}e^{-(\frac{V^{2}}{2 \sigma_{n}^{2}}+SNR)}I_{0}\left(\frac{V\sqrt{2\cdot SNR}}{\sigma_{n}}\right)dV \,\ (V>0) \tag{8}\]  
In fact, Equations (7) and (8) are equivalent, but the calculation methods are different. When satellite signal does not exist, the detection metric \(V\) presents Rayleigh distribution, and the false alarm rate \(P_{fa}\) corresponding to the threshold \(V_{t}\) is expressed as:  
\[P_{fa}=\int\limits_{V_{t}}^{\infty}\frac{v}{\sigma_{n}^{2}}e^{-(\frac{-V^{2}} {2\sigma_{n}^{2}})}\,dv \tag{9}\]  
For multi-channel spoofing signals and authentic signals, in order to detect correctly, all detection units should not appear as a false alarm. Therefore, considering that detection units are independent of each other, the total false alarm probability \(P_{fa-total}\) is defined as:  
\[P_{fa-total}=1-(1-P_{fa})^{N_{c}} \tag{10}\]  
\(N_{c}\) is the number of search units contained in the two-dimensional search range. The number of search units means that the area surrounded by the frequency indeterminate interval and the code phase indeterminate interval constitutes a two-dimensional search range for a received signal. The intersection of each code band and each frequency band is called a search unit. The number of search units means the number of search units within the above search range. Then the detection threshold \(V_{t}\) is calculated from the total false alarm rate \(P_{fa-total}\) of signal acquisition as follows [25]:  
\[V_{t}^{2}=-2\sigma^{2}\ln\Bigl{[}1-(1-P_{fa-total})^{\frac{1}{N_{c}}}\Bigr{]} \tag{11}\]  
The total power of spoofing signal and authentic signal are defined as \(TSP=10\lg(\sum\limits_{m=l}P_{m}^{s})\) and \(TAP=10\lg(\sum\limits_{h=h}P_{h}^{a})\), respectively. The signal-to-noise ratio of power \(P_{m}^{s}\) and \(P_{h}^{a}\) of each spoofing signal and noise variance estimation \(\hat{\sigma}^{2}\) are expressed as \(SNR_{i}^{s}\) and \(SNR_{i}^{a}\), respectively [10]:  
\[SNR_{i}^{s}=\frac{P_{m}^{s}}{2\hat{\sigma}^{2}}\,\ SNR_{i}^{a}=\frac{P_{h}^{a}}{2 \hat{\sigma}^{2}} \tag{12}\]  
In the above formula, let \(\sigma_{n}=1\) (normalized).